leader election is one of the fundamental problems in distributed computing , first stated in .each entity in some set has a boolean variable initialized to 0 and , after the election , exactly one of these entities , called the _ leader _ , should change this value to 1 .all other entities should know which one becomes the leader .in this paper we consider the problem of leader election among mobile agents that operate in a network .we assume that neither nodes of the network nor agents have labels that can be used for leader election .this assumption is motivated by scenarios where nodes and/or agents may refrain from revealing their identities , e.g. , for security or privacy reasons .hence it is desirable to have leader election algorithms that do not rely on identities but exploit asymmetries in the initial configuration of agents due to its topology and to port labelings . with unlabeled nodes and agents ,leader election is impossible for symmetric initial configurations , e.g. , in a ring in which ports at each node are 0,1 , in the clockwise direction and agents are initially situated at every node .our goal is to answer the following question :  _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ for which initial configurations of agents is leader election possible and how to do it when it is possible ? _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  a network is modeled as an undirected connected graph with unlabeled nodes .it is important to note that the agents have to be able to _ locally _ distinguish ports at a node : otherwise , the adversary could prevent an agent from choosing a particular edge , thus making navigation in the network impossible even in the simple case of trees .this justifies a common assumption made in the literature : ports at a node of degree have arbitrary fixed labelings  . throughout the paper, we will use the term `` graph '' to mean a graph with the above properties .we do not assume any coherence between port labelings at various nodes .agents can read the port numbers when entering and leaving nodes .  at the beginning ,identical agents are situated in some nodes of the graph , at most one agent at each node .the graph with bicolored nodes ( black if the node is occupied , white if it is not ) is called an initial configuration .agents do not have labels and have unlimited memory : they are modeled as identical turing machines .they execute the same deterministic algorithm .agents navigate in the graph in an asynchronous way which is formalized by an adversarial model used in and described below .two important notions used to specify movements of agents are the _ route _ of the agent and its _walk_. intuitively , the agent chooses the route _ where _ it moves and the adversary describes the walk on this route , deciding _ how _ the agent moves .more precisely , these notions are defined as follows .the adversary initially places an agent at some node of the graph .the route is chosen by the agent and is defined as follows .the agent chooses one of the available ports at the current node .after getting to the other end of the corresponding edge , the agent learns the port number by which it enters and the degree of the entered node .then it chooses one of the available ports at this node or decides to stay at this node .the resulting route of the agent is the corresponding sequence of edges , which is a ( not necessarily simple ) path in the graph .we now describe the walk of an agent on its route .let be the route of an agent .let .let , where , be an increasing sequence of reals , chosen by the adversary , that represent points in time .let \rightarrow [ v_i , v_{i+1}] ] , we define .the interpretation of the walk is as follows : at time the agent is at the point of its route .this general definition of the walk and the fact that ( as opposed to the route ) it is designed by the adversary , are a way to formalize the asynchronous characteristics of the process .the movement of the agent can be at arbitrary speed , the adversary may sometimes stop the agent or move it back and forth , as long as the walk in each edge of the route is continuous and covers all of it .this definition makes the adversary very powerful , and consequently agents have little control on how they move .this , for example , makes meetings between agents hard to achieve .note that agents can meet either at nodes or inside edges of the graph .agents with routes and and with walks and meet at time , if points and are identical .a meeting is guaranteed for routes and , if the agents using these routes meet at some time , regardless of the walks chosen by the adversary .when agents meet , they notice this fact and can exchange all previously acquired information . however , if the meeting is inside an edge , they continue the walk prescribed by the adversary until reaching the other end of the current edge . new knowledge acquired at the meetingcan then influence the choice of the subsequent part of the routes constructed by each of the agents .since agents do not know a priori the topology of the graph and have identical memories at the beginning , the only way to elect a leader among agents is by learning the asymmetries in their initial positions in the graph . hence agents must gain this knowledge by navigating in the network and share it with other agents to accomplish leader election . sharing the knowledge can be done only as a result of meetings of agents , which is difficult because of the asynchronous way in which they move .it is not hard to see ( cf . proposition [ no - bound ] ) that in the absence of a known upper bound on the size of the graph , leader election is impossible even for asymmetric configurations .hence we assume that all agents know a priori a common upper bound on the size of the graph .this is the only information about the environment available to the agents when they start the task of leader election .having described our model , we can now make the initial problem more precise .call an initial configuration _ eligible _ if , starting from this configuration , leader election can be accomplished regardless of the actions of the adversary . thus in order that a configuration be eligible , it is enough to have some leader election algorithm starting from it , even one dedicated to this specific configuration .now our problem can be reformulated as follows ._ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ which initial configurations are eligible ? find a universal leader election algorithm that elects a leader regardless of the actions of the adversary , for all eligible configurations in graphs of size at most , where is known to the agents ._ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _   assuming an upper bound on the size of the graph , known a priori to all agents , we characterize all eligible initial configurations and construct an algorithm that accomplishes leader election for all of them .more precisely , we formulate a combinatorial condition on the initial configuration , which has the following properties . on the one hand , if this condition does not hold , then the adversary can prevent leader election starting from the given initial configuration . on the other hand, we construct an algorithm that elects a leader , regardless of the adversary , for all initial configurations satisfying the condition , in graphs of size at most equal to the given bound .intuitively , leader election is possible when the initial configuration is asymmetric and when agents can learn this , regardless of the actions of the adversary .both these requirements are contained in the necessary and sufficient condition on eligibility , which we formulate in section [ sec : feasibility ] .in fact , the process of learning the asymmetries by the agents is the main conceptual and technical challenge in the design and analysis of our algorithm .agents acquire and share this knowledge as a result of meetings .the difficulty is to design the algorithm in such a way that all asymmetries be finally learned by all agents and that all agents be aware of this fact and thus capable to correctly elect the leader .leader election in networks was mostly studied assuming that all nodes have distinct labels and election has to be performed among nodes .this task was first studied for rings . a synchronous algorithm , based on comparisons of labels , and using messages was given in .it was proved in that this complexity is optimal for comparison - based algorithms .on the other hand , the authors showed an algorithm using a linear number of messages but requiring very large running time .an asynchronous algorithm using messages was given , e.g. , in and the optimality of this message complexity was shown in .deterministic leader election in radio networks has been studied , e.g. , in and randomized leader election , e.g. , in . in leader election problem is approached in a model based on mobile agents for networks with labeled nodes .many authors studied various computing problems in anonymous networks .in particular , characterize message passing networks in which leader election can be achieved when nodes are anonymous . in authors study the problem of leader election in general networks , under the assumption that labels are not unique .they characterize networks in which this can be done and give an algorithm which performs election when it is feasible .they assume that the number of nodes of the network is known to all nodes .in the authors study feasibility and message complexity of sorting and leader election in rings with nonunique labels , while in the authors provide algorithms for the generalized leader election problem in rings with arbitrary labels , unknown ( and arbitrary ) size of the ring and for both synchronous and asynchronous communication .characterizations of feasible instances for leader election and naming problems have been provided in  .memory needed for leader election in unlabeled networks has been studied in .the asynchronous model for mobile agents navigation in unlabeled networks has been previously used in in the context of rendezvous between two agents . in agentshad different labels and in agents were anonymous , as in our present scenario .the synchronous model , in which agents traverse edges in lock - step , has been used , e.g. , in , also in the context of rendezvous .   in section [ sec : memory_states ] we formalize the description of how agents decide .this concerns both navigation decisions ( on what basis the agents construct their routes ) and the final decision who is the leader .we define memory states of the agents that are the basis of all these decisions . in section [ sec : feasibility ] we formulate the combinatorial condition ec concerning initial configurations that is then proved to be equivalent to eligibility , and we formulate our main result . in section [ sec : negative ] we prove two negative results concerning leader election : one saying that condition ec is necessary for eligibility and the other saying that the assumption concerning knowledge of the upper bound can not be removed . in section[ sec : algorithm ] we give our main contribution : we construct a universal algorithm electing a leader for all configurations satisfying condition ec , if agents know an upper bound on the size of the graph .section [ sec : conclusions ] contains conclusions .in this section we describe formally on what basis the agents make decisions concerning navigation in the graph ( i.e. , how they construct their routes ) and on what basis they make the decision concerning leader election .all these decisions depend on the _ memory states _ of the agents . at every time memory state of an agent is a finite sequence of symbols defined as follows . before an agent is woken up by the adversary , its memory state is blank : it is the empty sequence .when an agent is woken up , it perceives the degree of its initial position , i.e. , its memory state becomes the sequence .further on , the memory state of an agent changes when it visits a node .it is caused by the following three types of events : entering a node by the agent , meeting other agents , and leaving a node by the agent .a change of a memory state of an agent is done by appending to its current memory state a sequence of symbols defined as follows .the change due to entering a node of degree by port number , consists of appending the sequence to the current memory state of the agent .the change due to leaving a node by port consists of appending to the current memory state of the agent .the change due to meeting other agents is defined as follows . when entering a node the agent considers all meetings with other agents that occurred since leaving the previous node .suppose that the current memory states of the agents met in this time interval by agent were , in lexicographic order , regardless of the order of meetings in this time interval and disregarding repeated meetings corresponding to the same memory state ( and thus to the same agent ) .agent appends the sequence of symbols ... [\sigma_k]) ] due to a meeting with an agent with current memory state since leaving the previous node , is performed by at most once .since the number of agents is finite , this implies that , by any given moment in time , the memory state of an agent has changed only a finite number of times , and each time a finite sequence of symbols has been appended .hence memory states are indeed finite sequences of symbols .the decisions of agents are made always when an agent is at a node and they are of three possible types : an agent can decide to stay idle , it can decide to exit the current node by some port , or it can elect a leader and stop forever .all these decisions are based on the memory state of the agent after entering the current node and are prescribed by the algorithm .( recall that agents execute the same deterministic algorithm . )if an agent decides to stay at a given node , then it remains idle at it until another agent enters this node . at this time the memory state of the idle agent changes , and in the new memory state the agent makes a new decision .if an agent decides to leave the current node by a given port , it walks in the edge in the way prescribed by the adversary and makes a new decision after arriving at the other end of the edge .finally , if an agent decides to elect a leader , it either elects itself , or it decides that it is not a leader , in which case it has to give a sequence of port numbers leading from its own initial position to the initial position of the leader : this is the meaning of the requirement that every non - leader has to know which agent is the leader .in this section we express the necessary and sufficient condition on eligibility of an initial configuration and we formulate the main result of this paper .we first introduce some basic terminology .we will use the following notion from .let be a graph and a node of .we first define , for any , the _ truncated view _ at depth , by induction on . is a tree consisting of a single node . if is defined for any node in the graph , then is the port - labeled tree rooted at and defined as follows . for every node , , adjacent to ,there is a child of in such that the port number at corresponding to edge is the same as the port number at corresponding to edge , and the port number at corresponding to edge is the same as the port number at corresponding to edge .now node , for becomes the root of the truncated view .the _ view _ from is the infinite rooted tree with labeled ports , such that is its truncation to level , for each . for an initial configurationin which node is the initial position of an agent , the view is called the view of this agent .  we will also use a notion similar to that of the view but reflecting the positions of agents in an initial configuration . consider a graph and an initial configuration of agents in this graph .let be a node occupied by an agent .a function that assigns either or to each node of is called a _ binary mapping _ for .a pair , where is a binary mapping for , such that if and only if corresponds to an initial position of an agent , is called the _ enhanced view _ from .thus , the enhanced view of an agent additionally marks in its view the nodes corresponding to initial positions of other agents in the initial configuration .  for any route such that ,we denote and , and we say that _ leads _ from to in . since nodes of are unlabeled , agents traveling on a route are aware only of the port numbers of the edges they traverse .hence , it will be usually more convenient to refer to these sequences of port numbers rather than to the edges of the route .any finite sequence of non - negative integers will be called a _trail_.  we define an operator , that provides the trail corresponding to a given route .more formally , if is a route in , then define to be the trail such that and are the port numbers of at and , respectively , for .we say that a trail is _ feasible from _ in , if there exists a route in such that and , and in such a case the route is denoted by .  for a sequence denote by the sequence . for two sequences and we write to refer to the sequence .  for any agent ,let denote its initial position .consider two agents and .consider any route leading from to and let .if , then we say that the route is a palindrome . for a given initial configuration ,a palindrome is called _ uniform _ , if for any route such that , whenever is occupied by an agent , then is also occupied by an agent .we are now ready to formulate our condition on an initial configuration , that will be called ec ( for eligibility condition ) in the sequel :  enhanced views of all agents are different _ and _ + ( there exist agents with different views _ or _ there exists a non - uniform palindrome )  we now formulate our main result whose proof is the objective of the rest of the paper .[ main ] assume that all agents are provided with an upper bound on the size of the graph .then an initial configuration is eligible if and only if condition _ ce _ holds for this configuration .moreover , there exists an algorithm electing a leader for all eligible configurations , regardless of the actions of the adversary .in this section we prove two negative results concerning the feasibility of leader election .the first result shows that condition ce is necessary to carry out leader election , even if the graph ( and hence its size ) is known to the agents .[ pro : negative1 ] suppose that the condition _ ce _ does not hold for the initial configuration .then there exists an adversary , such that leader election can not be accomplished for this configuration , even if the graph is known to the agents .fix an initial configuration . condition ce can be abbreviated as , where is `` enhanced views of all agents are different '' , is `` there exist agents with different views '' , and is `` there exists a non - uniform palindrome '' .suppose that ec does not hold .first consider the case when is false .this means that there exist agents and with the same enhanced view .this in turn implies that for every agent there exists an agent that has the same enhanced view as .indeed , if is the trail corresponding to a route that leads from to , agent is the agent whose initial position is at the end of the route corresponding to and starting at . for every agent we will call the agent its twin . consider a hypothetical leader election algorithm and the `` perfectly synchronous '' adversary that starts the execution of the algorithm simultaneously for all agents and moves all of them with the same constant speed. such an adversary induces rounds which are units of time in which all agents traverse an edge .the beginning of a round coincides with the end of the previous round .hence at the beginning and at the end of each round every agent is at a node .if agents meet inside an edge , they must meet exactly in the middle of a round in which they traverse an edge in opposite directions .we will show that the memory state of twins is identical at the end of each round .this implies that leader election is impossible , as an agent elects a leader when it is at a node , and consequently if some agent elects itself as a leader , its twin would elect itself as well , violating the uniqueness of the leader .the invariant that the memory state of twins is identical at the end of each round is proved by induction on the round number .it holds at the beginning , due to the same degree of initial positions of twins .suppose that after some round the memory states of twins are identical .consider twins and . in round they exit by the same port number and enter the next node by the same port number . if in round they do nt meet any agent in the middle of the edge , at the end of the round must meet agents with the same memory states as those met by ( if any ) , and hence memory states of and at the end of round are identical .if in round agent meets some agents in the middle of the edge , then agent must meet exactly the twins of these agents in the middle of the edge . by the inductive hypothesis, these twins have the same memory states as agents met by and hence again , at the end of the round the memory states of and are identical .this concludes the proof if is false .next consider the case when is false .this means that views of all agents are identical and every palindrome for the initial configuration ( if any ) is uniform . for any trail that yields a uniform palindrome , this gives a partition of all agents into pairs of agents at the ends of routes that correspond to this trail .again we consider the `` perfectly synchronous '' adversary described above .there are two subcases .if there is no palindrome in the initial configuration , then we prove the following invariant , holding at the beginning of each round , by induction on the round number : the memory state of all agents is the same and there is no palindrome between agents .the invariant holds at the beginning by assumption .suppose it holds after round . in round agents choose the same port number and enter the next node by the same port number .there are no meetings in round .indeed , the only meeting could be in the middle of an edge but this would mean that agents were joined by a one - edge palindrome at the beginning of round . if a pair of agents were joined by a palindrome after round , they would have to be joined by a palindrome longer or shorter by two edges at the beginning of the round , contradicting the inductive assumption .hence the invariant holds by induction .the second subcase is when there is a palindrome in the initial configuration ( and hence all such palindromes are uniform ) .now we prove the following invariant holding in the beginning of each round : the memory state of all agents is the same and every agent is at the end of a palindrome corresponding to the same trail .the invariant holds at the beginning by the assumption .suppose the invariant holds after round . in round agents choose the same port number and enter the next node by the same port number .if after round no pair of agents were at the ends of an edge with both ports equal , or they were but agents did not choose this port in round , then no meeting occurred and the invariant carries on after round .if after round every pair of agents were at the ends of an edge with both ports and the agents chose this port in round , then meetings of agents with identical memory states occurred in pairs in the middle of each joining edge . since meeting agents had identical memory state during the meeting , this holds also after round and agents are again in pairs at the ends of edges with both ports the invariant holds at the end of round .hence at the beginning of each round the memory state of all agents is the same , both when and when is violated .this implies that with the `` perfectly synchronous '' adversary leader election is impossible whenever condition ec is violated for the initial configuration .notice that the argument holds even when agents know the graph in which they operate .our second negative result shows that the assumption about the knowledge of an upper bound on the size of the graph can not be removed from theorem [ main ] .[ no - bound ] there is no algorithm that accomplishes leader election regardless of the adversary for all initial configurations satisfying condition _ce_.  suppose for a contradiction that such a universal algorithm exists .consider an `` almost ''oriented ring of size : ports 0,1 are in the clockwise direction at each node except one , where they are counterclockwise .this node is called _special_. the initial configuration on this ring consists of two agents : one at the special node , and one at the neighbor clockwise from it .call this configuration ( cf . fig .[ fig : negative ] ( a ) ) .this configuration satisfies condition ce : agents have different views .hence algorithm must elect a leader for this configuration , regardless of the adversary .consider a `` perfectly synchronous '' adversary that starts the execution of the algorithm simultaneously for all agents and moves all of them with the same constant speed .it induces rounds corresponding to edge traversals by all agents .suppose that a leader is elected for this adversary after rounds .now consider a ring of size in which there are special nodes at distances : at these nodes ports 0,1 are in the counterclockwise direction , and in all other nodes they are in the clockwise direction .the initial configuration consists of agents .there is an agent at every special node and at every clockwise neighbor of a special node .additionally there is an agent at the counterclockwise neighbor of one special node . call this configuration ( cf .[ fig : negative ] ( b ) ) .this configuration satisfies condition ce .indeed , due to the single group of three consecutive agents , all agents have distinct enhanced views .on the other hand , agents at special nodes have a different view from agents at clockwise neighbors of special nodes .hence algorithm must elect a leader for this configuration as well , regardless of the adversary .consider the same `` perfectly synchronous '' adversary as before .consider the agent in the configuration that is initially situated at the special node antipodal to the special node with both neighbors hosting agents .consider the agent initially situated at the special node that is clockwise from and closest to . in the first rounds of the execution of starting from configuration , memory states of agents and are identical to memory states of the agent initially situated at the special node of configuration .this is due to the large size of the ring in configuration .hence if in configuration agent elects itself , then in configuration agents and elect each of them itself as the leader after rounds .if in configuration agent elects its neighbor , then in configuration agents and elect each of them their neighbor as the leader after rounds . in both cases two different agents are elected , which is a contradiction .in this section we present an algorithm that elects a leader for all initial configurations satisfying condition ec , assuming that an upper bound on the size of the graph is known to all agents . in view of proposition[ no - bound ] , this assumption can not be removed .this upper bound , denoted by , is an input of our algorithm .the section is divided into three subsections . in the first subsection we provide additional terminology and notation , as well as some auxiliary results used in the algorithm and in its analysis . in the second subsectionwe give the intuitive overview of the algorithm and its formal description , and we provide some illustrative examples of its functioning .finally , the third subsection is devoted to the proof that the algorithm is correct .let be a graph and let be any node of . for any integer , we define the _ code _ of as the sequence of integers obtained as follows .perform the complete depth first search traversal of , starting at its root , in such a way that at each node an edge with a smaller port number is selected prior to an edge with a larger port number . then , if is the -th traversed edge , when going from to during the traversal of , then the -th and -th elements of are the port numbers of at and , respectively .the following is a direct consequence of this definition .[ prop : codes_distinguish ] let and be any nodes of and let be an integer . if and only if .  let be an integer .we extend the notion of binary mappings to the truncated views .we say that is a _ binary mapping _ for , if assigns either or to each node of .if is a binary mapping for ( or for for some ) , then ( where , respectively ) refers to restricted to the nodes of .given two binary mappings for , we write if for each node of . if is the enhanced view from and is a binary mapping for such that , then the pair is called a _ partially enhanced view _ from . intuitively , in a partially enhanced view only some nodes corresponding to initial positions of agents are marked .let be the enhanced view from , where is selected so that there exists an agent with .then , is called the _complete identifier _ of .the significance of the notion of a complete identifier is the following .an agent can never get the entire view or the entire enhanced view , as these are infinite objects . however ,the following propositions from show that to differentiate two views or two enhanced views , it is enough to consider their truncations to depth . thus , as stated in corollary  , complete identifiers identify agents with different enhanced views .[ prop : no ] for a -node graph and for all nodes and of , if and only if .   [ prop : no2 ] for a -node graph , for all nodes and of , if and are the enhanced views from and , respectively , then if and only if .  [ cor : no2 ] for a -node graph and for any agents and , the enhanced views from and are equal if and only if the complete identifiers of and are equal .  a sequence of the form is called a _ label _ , if the following conditions hold :  1 . is a node of , and and are integers , 2 . is a binary mapping for for each . is the binary mapping for that assigns only to the root , and for every index .moreover , we say that is the _ length _ of the label , denoted by .let be the set of all labels of length at most .in this section we give a high - level description of the algorithm and its pseudo - code formulation .an important ingredient of the algorithm are meetings between agents during which information is exchanged .the method that guarantees that some meetings between pairs of agents will occur uses the idea of tunnels introduced in .the routes and form a _ tunnel _ if for some and for some .moreover , we say that the route is the _ tunnel core with respect to _ .note that if is the tunnel core with respect to , then is the tunnel core with respect to .[ prop : tunnel ] let be an agent with route , . if and form a tunnel with the tunnel core , then and are guaranteed to have a meeting such that and are the routes of the agents traversed till the meeting , where .  informally speaking , if the routes of two agents form a tunnel , then they are guaranteed to have a meeting with the property that the routes traversed to date by the agents give ( by taking one of the routes and the reversal of the other ) the tunnel core .let be the set of all integer sequences with terms in , whose length is even and equals at most .then , we define and let be the -th triple in , .  in our leader election algorithm we will proceed in phases , and in each phase the label of each agent is fixed .( due to the fact that the model is asynchronous , the adversary may force the agents to be in different phases in a particular point of time . )the total number of phases for each agent is .after the first phase each agent computes its label used in phase .these labels are defined in such a way that there exist two agents and with different labels .the aim of phase is that agents and correctly identify each other s initial positions in their respective views . after phase agent can identify the initial positions of all agents in its view and hence is able to perform leader election .the label of an agent used in phase is denoted by , .label is computed before the start of phase , and is computed at the end of phase , for .label is used to elect the leader at the end of the algorithm .each phase is divided into stages . by denote the route traversed by agent till the end of stage in phase , , .as we prove later , each agent starts and ends each stage at its initial position .let be the route of an agent traversed till the beginning of phase , and hence till the end of phase , whenever .hence , is the route traversed by prior to the beginning of the first phase .now we give an informal description of algorithm .this algorithm is executed by each agent , and in the pseudo - code is used to refer to the executing agent .note that the upper bound on the number of nodes of is given as an input .the pseudo - code of the algorithm and pseudo - codes of its subroutines are in frames . in the informal descriptionwe refer to lines of these pseudo - codes .first , we discuss procedure  that is called at the beginning of algorithm .the agent starts by computing .this can be done knowing , without any exploration of the graph .then , the agent computes by performing a dfs traversal of to the depth ( line  ) .the function is set ( line  ) to be the binary mapping for that assigns to all nodes of except for the root .hence , is a partially enhanced view from .the value of , that will be the label of in the first phase , is set to ( line  ) .now we informally describe the main part of algorithm , refering to the lines of the pseudo - code given below .the -th iteration of the main ` for ' loop in lines - is responsible for the traversal performed by the agent in phase , .an internal ` for ' loop in lines - is executed and its -th iteration determines the behavior of in stage .the stage of each phase ` processes ' the -th element of .if , then the agent does not move in this stage and proceeds to the next one .otherwise let ( we describe only this case as the other one is symmetric ) . the agent checks in line whether a certain trail is feasible from , and if it is not , then the stage ends .as we prove later , the verification of the feasibility of can be done by inspecting .if is feasible from , then follows the route ( line  ) , which guarantees that :  * if the agent is located at at the beginning of stage , then is located at at the end of stage ( see lemma [ lem : closed ] ) , and * if the route leads from to the initial position of another agent and , then the routes and form a tunnel .the agent ends phase by updating its label .this is done by calling function in line  , which produces a binary mapping that is used to update the label in line  . after completing the tree phases ( for ) the agent calls procedure which completes the task of leader election .in order to formally describe the trail mentioned above we need the following notation .let be a label of length , let and let be a node of .we define to be the trail that corresponds to the route performed till the end of stage of phase by an agent whose label equals in phase , , and whose initial position corresponds to the root of the truncated view in .we prove ( see lemma [ lem : history_simulation ] ) that , for any and the trail can be computed on the basis of and .the trail mentioned above is .figure [ fig : tunnel ] illustrates the routes of a pair of agents and that execute one iteration of the internal ` for ' loop in lines - of , for the same values of and , such that .we assume in this example that leads from to in .figure [ fig : tunnel ] gives the prefixes of two routes and traversed by the two agents .the fact that the routes form a tunnel ( as shown in figure [ fig : tunnel ] ) follows from lemma [ lem : guaranteed_t_confirmation ] proven later .  the routes in figure [ fig : tunnel ] are extended to and once goes from to its initial position by following the route in , and follows in order to return to , respectively .note that the routes , and are not edge disjoint .although we use tunnels , we use them differently from .first , we are able to construct much simpler routes that form tunnels .this is due to the fact that deals with the rendez - vous problem in finite graphs of unknown size and in infinite graphs .as argued before , for leader election we have to assume that an upper bound on the size of the graph is known , and we take advantage of knowing to construct ` shorter ' tunnels which simplifies our analysis .second and more importantly , it is not sufficient for our purposes to just generate a meeting for a particular pair of agents the meetings are generated to perform the exchange of information .in particular , as a result of a meeting that occurs in a tunnel an agent should be able to determine the node ( in its own view ) corresponding to the initial position of the other agent .this leads us to the following concept of meetings with ` confirmation ' of a trail .[ def : confirms ] let .suppose that agents and meet .we say that _ confirms _ as a result of this meeting if  1 .[ it : confirms : stage ] is in stage of phase and is in stage of phase , where , or and , 2 .[ it : confirms : label ] , where , 3 .[ it : confirms : trail ] if and are the trails traversed by and , respectively , till the meeting , then where and when , and and otherwise .as we prove in section [ subsec : alg : analysis ] , if an agent confirms as a result of a meeting with , then .figure [ fig : verification ] depicts the equation in part [ it : confirms : trail ] of definition [ def : confirms ] .figure [ fig : verification](a ) presents the trail that is a perfix of the trail corresponding to the route followed by the agent till the end of stage of phase .figure [ fig : verification](b ) presents the trails ( dashed line ) and ( dotted line ) .it remains to describe function and procedure that are called in the algorithm .we start by giving intuition of the first of them .this procedure is crucial for the entire algorithm , as it takes advantage of memory state exchanges between agents that meet and permits every agent to insert initial positions of all agents in its view .this in turn allows the agents to learn asymmetries in the initial configuration and thus correctly perform leader election .function takes as an input the current memory state of an agent and returns a binary mapping for its view , such that is a partially enhanced view for agent .agent considers memory states of all previously met agents at the times of the meetings .the memory state , , of an agent and the memory state of at the time of their meeting permit the agent to verify whether or confirmed as a result of their meeting .if one of the agents confirms , then takes the advantage of this fact to determine the nodes of its view corresponding to initial positions of agents . in particular, is able to locate a node in its own view that corresponds to the initial position of , because there exists a route corresponding to and connecting the initial positions of the two agents . afterwards, if this route is of length at most , then recursively calls function for the memory state ( which is shorter than the current memory state of and thus recursion is correct ) .hence , can compute the binary mappings corresponding to memory states of all previously met agents at the times of the meetings . using trails between initial positions of these agents and , as well as the obtained binary mappings , agent correctly position all partially enhanced views of these agents in its own view .a call to function at the end of phase executed by agent permits to compute .  in the formulation of function we use the following notions .let and be two nodes of .we say that a function assigning to each node of a node of is a _ transition _ from to , if and correspond to the same node of for each node of . for any trail and any node , we say that a node at depth in is at the end of from the root , if the length of is and the sequence of ports corresponding to the simple path from the root of to is  .in the following example we illustrate one iteration of the ` for ' loop in lines - of function  . the graph is given in figure [ fig : meeting](a ) , and let be an upper bound that was initially provided to each agent .  ;( b ) ; a subtree of   the black nodes of are the initial positions of some agents .denote by and the agents whose initial positions are and , respectively .note that the views from any two nodes of are identical in this case .however , the enhanced view from each node of is unique .we focus on the instance of function  executed by during its meeting with . for simplicity , we show only some subtrees of and in figures [ fig : meeting](b ) and [ fig : meeting](c ) , respectively .note that the nodes of , and therefore the nodes of any view , are unlabeled and we provide the labels only for the illustrative purpose . in this examplethe trail , computed in line  of function equals , which determines the node of the truncated view that corresponds to the root of . the black nodes of both views correspond to the initial positions of agents that and determined prior to this meeting .the dotted arrows give the part of the transition that maps the nodes of to the black nodes of .it follows from the definition of the view that , in general , more than one node of can be mapped by to a node of .we finally present procedure that is called by algorithm after the third phase .the leader is selected by an agent on the basis of the label .let be a node at depth at most in and satisfying .let be a subtree of depth of rooted at .we prove later that the pair is the complete identifier of some agent .since the initial positions of all agents have been detected till the end of phase , the agent can determine all complete identifiers and hence elect the leader .this section is devoted to the proof that algorithm correctly elects a leader whenever an initial configuration satisfies condition , regardless of the actions of the adversary .the proof is split into a series of lemmas .the role of the first lemma is to show that knowing is enough to check if a trail of any length is feasible from .[ lem : view_extension ] let be any node of . using ,the truncated view can be computed for any positive integer .if , then is a subtree of , so we may assume that .we extend the view to for each . to this endwe perform the following computation for each node at depth of .let be the node of that corresponds to .note that the subtree of rooted at and containing all descendants of in is equal to .hence , there exists a node in such that corresponds to , in view of the connectedness of .this implies that there exists a node in such that the subtree of consisting of and all its descendants to depth from is equal to .hence , one can find any such node of .let be the node of that corresponds to . due to proposition [ prop : no ] , is equal to .since belongs to we obtain that is a subtree of rooted at , and therefore we can extend the subtree by replacing in the subtree rooted at with .[ cor : other_views_computable ] let be any node of and let be any node of . then , the subtree of to depth and rooted at can be computed for any , using .  [ cor : trail_verification ] let be any trail and let be any node of . using can be verified if is feasible from in .  the next lemma shows that given any label of the agent and the stage number , it is possible to compute the trail which , informally speaking , is the history of the moves of the agent with label till this stage .[ lem : history_simulation ] using a label and an integer , the trail can be computed .suppose that is of length , . by definition , , where is a binary mapping for and is a node of .first note that can be reconstructed from its code .suppose that is an agent whose initial position is and whose label in phase is , .the trail can be computed by simulating the execution of algorithm for the agent . by its formulation ,the agent executed procedure , iterations of the main ` for ' loop in lines - of algorithm , and exactly iterations of the nested ` for ' loop in lines - of algorithm in the -th iteration of the main ` for ' loop .thus results in traversing the route .note that is unknown to , but we will reconstruct the route by simulating edge traversals in .( while is infinite , it can be reconstructed from to any finite depth , using corollary [ cor : trail_verification ] . )we prove the lemma by induction on the total number of stages ` processed ' by an agent .note that if and , then the trail corresponds to the route that is the dfs traversal of to depth and starting at .this trail can be obtained by performing the dfs traversal of that starts and ends at the root .now assume that . in order to simulate the behavior of in any stage , , of phase , , one needs to know . by construction , .the induction hypothesis and corollary [ cor : trail_verification ] imply that the second part of the condition in line  of algorithm can be checked . if , where , then does not move in stage of phase .hence , assume without loss of generality that .this implies that executes the instruction in line  of algorithm . by the induction hypothesis ,the trail in lines and can be computed on the basis of and .we say that a route is _ closed _ if .the following lemma implies that at the end of each stage each agent comes back to its initial position .[ lem : closed ] let be any agent . for every and for every the route is closed .denote by the route that the agent follows in stage , , of phase , .we prove the lemma by induction on the total number of stages processed in all phases by an agent .first note that , i.e. , the route of performed as a result of the execution of line  of procedure is closed .hence , it remains to prove that if is closed for each agent , for some and for some , then is closed as well .note that .hence , and therefore it is enough to argue that is closed .let .if , then according to lines  and  of algorithm , is empty , i.e. does not move in stage of phase . in this casethe proof is completed .otherwise , we obtain that where and ( see lines  and  of algorithm ) .let .hence , is closed if and only if is closed . however , by definition , the latter route equals for an agent such that its initial position is and ( if such an agent exists ) .it follows from the induction hypothesis that is closed , which completes the proof of the lemma .the next lemma shows the importance of confirmation of a trail .it implies that if an agent confirms as a result of a meeting with , then it can correctly situate the initial position of in its view .[ lem : tconfirmation ] let , let and let .let be an agent such that .suppose that the agent meets an agent , when is in stage of phase .if confirms as a result of this meeting , then , where if and if .suppose without loss of generality that .condition [ it : confirms : trail ] in definition [ def : confirms ] implies that the route leads from to in .let .by lemma [ lem : closed ] , both and are closed .this implies that as required .the following lemma shows that processing an appropriate triple by two agents guarantees their meeting confirming .[ lem : guaranteed_t_confirmation ] let .let and be two agents such that for some and . if , then prior to the first moment when one of the agents completes phase , the agents and have a meeting as a result of which either or confirms .suppose without loss of generality that ends the traversal of at the same time or earlier than ends the traversal of .let be the route traversed by in stage of phase .by lemma [ lem : closed ] , .the route is constructed as a result of the execution of lines - of algorithm by . since in line of algorithm , we obtain that and form a tunnel with the tunnel core . by proposition [ prop :tunnel ] , and will have a meeting while is in stage of phase and before ends the traversal of .this implies that [ it : confirms : stage ] of definition [ def : confirms ] holds .moreover , [ it : confirms : label ] of definition [ def : confirms ] is satisfied by assumption .let and be the trails traversed by and , respectively , till the meeting . by proposition [ prop :tunnel ] , , where is the tunnel core .this proves that [ it : confirms : trail ] of definition [ def : confirms ] holds .hence , the agent confirms as a result of the meeting .the role of the next lemma is to show that an agent never marks falsely an initial position of another agent in its view .[ lem : no_false_noting ] let be the label of any agent in phase . if is any node of corresponding to a node of that is not an initial position of an agent , then .suppose for a contradiction that an agent sets to be , and corresponds to a node of that is not an initial position of an agent .suppose that the input memory state of the agent is the shortest that satisfies this property .this assumption implies that if is computed by agent in line  of function for any memory state , , then is a partially enhanced view from , where corresponds to the node at the end of from the root in .hence , if is an initial position of an agent , then , due to the definition of transition , each node from line  of function corresponds to an initial position of an agent .the latter implies that the assignment of to occurs in line  of , as a consequence of a meeting with some agent whose memory state was at the time of the meeting .hence , is at the end of from the root in . due to line of , either or confirms as a result of their meeting . by lemma [ lem : tconfirmation ] , , where is determined in line  of .thus , corresponds to the node at the end of from the root in .the latter implies that corresponds to , contradicting our assumption .the next lemma is a companion result to lemma [ lem : no_false_noting ] .it says that if an agent confirms a trail as a result of a meeting with , then both of them correctly mark their respective initial positions in their views .[ lem : updatef ] let .let and be two agents with labels and . if agent confirms as a result of a meeting with agent in phase , , then  1 . , where is at the end of from the root in , such that and . , where is at the end of from the root in , and .suppose that is in some stage of phase , , when a meeting with occurs as a result of which confirms .hence , one of the memory states in line  of function called at the end of phase is the memory state , , of at the time of the meeting .we consider the -th iteration of the ` for ' loop in lines - of function , i.e. , informally speaking , the iteration in which ` analyzes ' the meeting with .the agent determines in line  of function the fact that either or confirms as a result of the meeting .then , in line  of function , is set to , where is at the end of from the root of . by lemma [ lem : tconfirmation ] and by the choice of in line  of function , corresponds to .function returns .due to line  of algorithm , .  in order to prove the second part of the lemma notice that agent has also access to the memory states of and at the time of the meeting .hence , upon completing phase it performs analogous computations as during its execution of function at the end of phase and sets .let be an agent and let be a node of the truncated view at the end of a trail from the root of .if the agent sets during the execution of function and corresponds to the initial position of some agent , then we say that _ noted on .  a label , where , of an agent is _ complete with respect to _ if for each node of that corresponds to an initial position of .we say that is _ semi - complete with respect to _ if for each node that corresponds to an initial position of and belongs to a level of .  the next lemma explains how agents confirm trails as a result of their meetings .agents with different labels can confirm any trail in between their initial positions , while agents with equal labels are able to confirm only ` palindromes ' .[ lem : label_complete ] let and let and be any two agents .  1 .[ it : different_labels ] if , then is complete with respect to .[ it : identical_labels ] if , then prior to the end of its phase the agent has noted on each trail such that , and .  to prove [ it : different_labels ] let be any trail such that leads from to in .by construction of , for some index , because . by lemma [ lem : guaranteed_t_confirmation ] ,one of the agents or confirms as a result of a meeting that occurs when it is in phase .this is done by checking the conditions [ it : confirms : stage ] , [ it : confirms : label ] and [ it : confirms : trail ] in definition [ def : confirms ] , which can be accomplished by analyzing the memory states of and at the time of the meeting . by lemma [ lem : updatef ] , , where is at the end of in ,regardless of which agent confirmed as a result of the meeting .the proof of [ it : identical_labels ] is analogous .  [lem : phase1 ] there exist two agents and such that .  if there exist two agents and with different views , , then by proposition [ prop : no ] , .proposition [ prop : codes_distinguish ] implies , and consequently .thus , assume in the following that the views from the initial positions of all agents are equal . in view of condition there exists a non - uniform palindrome .hence , there exist two agents and a trail , such that , leads from to a node that is an initial position of an agent , and leads from to a node that is not an initial position of any agent .let and be the nodes of and , respectively , at the end of from the roots .note that corresponds to and corresponds to . by lemma [ lem : label_complete][it : identical_labels ] , noted on in phase and , by lemma [ lem : no_false_noting ] , does not note any agent on during the entire execution of its algorithm .hence , and after the execution of procedure at the end of phase .since the views and are equal , we have and , where .hence , .the role of the next lemma is to explain indirect learning of initial positions of other agents .if two agents and have equal labels , then they mutually situate their initial positions using a third agent with a different label as an intermediary .such an agent exists by lemma [ lem : phase1 ] .agents and can correctly fill their initial positions to depth in their views due to the fact that the intermediary has their initial positions to depth .  [lem : completetosemicomplete ] let and be two agents such that is complete with respect to , . if is any agent such that , then is semi - complete with respect to .  since we obtain , by the definition of , that for some , where is such a trail that the route contains at most edges and .we analyze the execution of function by at the end of phase . by lemma [ lem : guaranteed_t_confirmation ] , or confirms as a result of their meeting prior to the completion of phase by , i.e. , prior to the execution of function we consider .the agent learns this fact in line  of function .then , the condition in line  of function is satisfied by assumption , and verifies this condition by checking whether the length of does not exceed .consequently , finds in line  of function a transition that maps the node at the end of in to the root of , because . since is of length at most , sets ( in the ` for ' loop in lines - of function ) for each node at depth at most in corresponding to .the latter is due to the fact that is complete with respect to .this proves that is semi - complete with respect to .see figure [ fig : complete ] , where the root and some nodes that correspond to the initial position of have been marked on , and we show a transition that allows to determine all nodes in the view of to the depth corresponding to .  in view of the next lemma , upon completion of phase an agenthas correctly situated all nodes corresponding to initial positions of all agents to depth in its view .[ lem : phase3 ] let be any agent . then, is semi - complete with respect to each agent .  if , then by lemma [ lem : label_complete][it : different_labels ] , is complete with respect to , and therefore it is semi - complete with respect to .hence , suppose that , which implies .hence , lemma [ lem : phase1 ] implies that there exists an agent such that . due to lemma[ lem : label_complete][it : different_labels ] , is complete with respect to . note that implies .lemma [ lem : completetosemicomplete ] completes the proof .our final lemma says that any agent can correctly reconstruct complete identifiers of all other agents .this is due to the fact that every agent has filled initial positions of all agents to depth in its view .[ lem : complete_identifiers ] let be the set computed in procedure executed by an .then , for every agent , there exists an index such that is the complete identifier of and leads from to in .let be the label of obtained at the end of phase .initially is set to be empty in line  of procedure .consider the nodes and and the function from line  of procedure . by lemma [ lem : no_false_noting ], is an initial position of an agent , because .thus , in view of lemma [ lem : phase3 ] , is the complete identifier of . due to lines - of procedure , for some trail such that . by lemma [ lem : phase3 ] , for any agent , there is a node in corresponding to .this implies that , for any agent , the set contains ( at the end of the ` for ' loop in lines - of procedure ) a pair , , such that is the complete identifier of and leads from to in .[ thm : le_part2 ] if the condition is satisfied for an initial configuration , then algorithm correctly elects a leader regardless of the actions of the adversary .let be the agent whose complete identifier is lexicographically smallest among the complete identifiers of all agents . by condition ( more precisely by its part saying that all enhanced views are different ) and by corollary [ cor : no2 ] , agent is unique .  each agent computes as a result of the execution of algorithm . by lemma [ lem : complete_identifiers ] , computes ( in lines - of procedure ) , for each agent , the complete identifier of and a trail such that the route of length at most leads from to in . by corollary [ cor : no2 ], the complete identifiers uniquely distinguish the agents . using the lexicographic order on the set of all complete identifiers, the agent finds in line of procedure the complete identifier such that and for each such that for some trail . by definition ,then , decides in line  of procedure that the agent with the initial position is the leader .this is agent .theorem [ thm : le_part2 ] , together with proposition [ pro : negative1 ] implies our main result which is theorem [ main ] from section [ sec : feasibility ] .we characterized all initial configurations of agents for which leader election is possible and we constructed a universal algorithm electing a leader for all such configurations , assuming that agents know an upper bound on the size of the graph .we observed that the latter assumption can not be removed . in this paper we focused on the feasibility of leader election under a very harsh scenario in which the adversary controls the speed and the way in which agents move along their chosen routes .this adversarial scenario captures the totally asynchronous nature of mobile agents .while we gave a complete solution to the problem of feasibility of leader election , we did not try to optimize the efficiency of the algorithm , e.g. , in terms of its cost , i.e. , of the total or of the maximum number of edge traversals performed by the mobile agents .in fact , any kind of such optimization appears to be quite challenging .it is clear that in order to elect a leader agents have to meet .already the much simpler problem of optimizing the cost of meeting of two agents in our asynchronous model is open , both when agents have different labels and when they are anonymous , as in our present scenario . in particular , in the latter paper the authors asked if rendezvous of two agents can be accomplished ( whenever it is feasible ) at a cost polynomial in the size of the graph .e. bampas , j. czyzowicz , l. gasieniec , d. ilcinkas , a. labourel , almost optimal asynchronous rendezvous in infinite multidimensional grids , _ proc .24th international symposium on distributed computing ( disc 2010 ) _ , _ lncs _ 6343 , 297 - 311  p. boldi , s. shammah , s. vigna , b. codenotti , p. gemmell , and j. simon , symmetry breaking in anonymous networks : characterizations .4th israel symposium on theory of computing and systems , ( istcs 1996 _ ) , 16 - 26    j. chalopin , local computations on closed unlabelled edges : the election problem and the naming problem _ proc .31st conference on current trends in theory and practice of computer science ( sofsem 2005 ) _ , 82 - 91               m.a .haddar , a.h .kacem , y. mtivier , m. mosbah , and m. jmaiel , electing a leader in the local computation model using mobile agents .6th acs / ieee international conference on computer systems and applications ( aiccsa 2008 _ ) , 473 - 480

Abstract: we study the problem of leader election among mobile agents operating in an arbitrary network modeled as an undirected graph . nodes of the network are unlabeled and all agents are identical . hence the only way to elect a leader among agents is by exploiting asymmetries in their initial positions in the graph . agents do not know the graph or their positions in it , hence they must gain this knowledge by navigating in the graph and share it with other agents to accomplish leader election . this can be done using meetings of agents , which is difficult because of their asynchronous nature : an adversary has total control over the speed of agents . when can a leader be elected in this adversarial scenario and how to do it ? we give a complete answer to this question by characterizing all initial configurations for which leader election is possible and by constructing an algorithm that accomplishes leader election for all configurations for which this can be done .  * keywords : * leader election , anonymous network , asynchronous mobile agents  [ section ] [ section ] [ section ] [ section ] [ section ] [ section ] [ section ] [ section ]  [ 1][proof ]  * # 1 : *


a significant goal of plasma physics research is the characterization of mass , momentum , and energy transport in a wide variety of complex systems . in particular ,the question of what mechanisms mediate the transfer of energy between turbulent fields and distributions of plasma particles , leading to the eventual damping and dissipation of turbulence , is open .one system which displays turbulent behavior is the solar wind , a hot , diffuse emanation from the sun s surface that fills the heliosphere . while lacking the precise control over conditions afforded in a laboratory setting , the large volume of in situ measurements of the solar wind over the last half century has led to the accrual of observations of turbulence with a wide variety of plasma parameters .such measurements have proven useful to the study of phenomena in magnetized turbulence.  a limitation of these in situ observations is that they are taken at a single - point in space at a given time .this raises at least two significant complications ; one must disentangle the dynamics associated with spatial and temporal variation , and track the evolution of spatially integrated quantities , such as the energy content of a field or distribution of charged particles , given only single - point measurements .the first of these complications is addressed by invoking taylor s hypothesis, the conjecture that for single - point measurements of sufficiently fast flows , the time evolution is essentially frozen and the measurement traces out the spatial structure of the turbulence ; a review of the application of taylor s hypothesis to solar wind observations can be found in klein et al 2014.  to address the second complication of inaccessible spatially integrated quantities , one may consider the dynamics of the phase - space energy density rather than the total energy . a technique has been recently proposed to measure the local - in - phase - space energy transfer between fields and plasma distributions using single - point time series of simple plasma systems. by correlating the product of the electric field and velocity derivative of the particle distribution measured at a single point in space , the velocity structure of the transfer of energy between fields and particles is obtained . by averaging this correlation over a selected time interval , the oscillatory energy transfer between the fields and particles is removed , leaving only the secular energy transfer .the mechanisms responsible for this energy transfer can be identified by the velocity space structure of the field - particle correlation .initial work applied this correlation to systems that damp via the landau resonance. here , we consider the transfer of energy in linearly unstable systems , and show that field - particle correlations are able to identify the presence of both fluid and kinetic instabilities in such systems .the unstable systems under consideration are reviewed in section [ sec : drifts ] . in section [ sec : vp ] , we present the nonlinear numerical code employed in this work , ` vp ` , as well as the three simulations under consideration . in section [ sec : fpc ] , the field - particle correlation is presented and applied to the three simulations . analysis and discussion related to the correlations are found in section [ sec : method ] .application of field - particle correlations to simple , homogeneous , linearly unstable systems enables the construction of signatures of basic energy transfer mechanisms . combined with signatures for all relevant energy transfer mechanisms , such correlations may be usefully employed to diagnose the behavior of more complex systems .the 1d1v electrostatic systems of interest in this work are governed by the vlasov and poisson equations and which evolve the distribution of species , and the electric field .recent work has applied field - particle correlations to electrostatic stable systems as a means of extracting the velocity dependent signature of landau damping from single - point time series. here , we consider electrostatic systems unstable to either fluid or kinetic instabilities , and characterize the signature of the associated growth using field - particle correlations . for fluid instabilities , the behavior of the plasma depends on the bulk parameters of the system and the energy transfer is not organized by characteristic velocities such as the resonant wave phase speed ; energy transfer for kinetic instabilities depends on such characteristic velocities , as the electrostatic field acts to exchange particles of higher and lower kinetic energies near the field s resonant velocity .for both types of instabilities , one or more of the distributions lose energy , while the fields and other distributions gain energy , resulting in an inverse damping .general discussion of plasma instabilities , as well as particular treatments of the electron drift instabilities of interest in this work , can be found in many plasma textbooks.  .electron bulk parameters [ cols="^,^,^,^,^",options="header " , ]   [ tb : params ]  to highlight the distinct velocity space structure of energy transfer in unstable systems , we consider three cases , each with distinct sets of parameters for one population of ions and two populations of electrons .the equilibrium distributions for the three populations take the maxwellian form \label{eqn : f0.fpc}\ ] ] where is the population s drift velocity .the linear normal mode behavior for such equilibria is governed by solutions of the dispersion relation \label{eqn : disp.fpc}\ ] ] where is a function of wavenumber and complex frequency , is the species charge , is the plasma dispersion function with argument , with the sum taken over the three plasma populations .the electron plasma frequency and debye length , both defined using the total electron density , normalize the time and length scales of our system .values for the normalized density of electron population , , and bulk velocity , normalized by the electron thermal velocity , are given in table [ tb : params ] .for all three cases , the electron populations have equal temperatures ; the ions are singly ionized , and initialized with , , and .case 1 is stable to the effects of the counter - streaming electrons , while the increase in for case 2 yields the classic counter - streaming instability .case 3 is unstable to the bump - on - tail instability .cases 2 and 3 serve as examples of fluid and kinetic instabilities respectively .  in fig[ fig : linear ] , solutions to eqn [ eqn : disp.fpc ] for the three cases are presented . in panela , the complex frequency solutions are given for fixed . for case 1 ( black circles )all modes are shown to be damped . for case 2 ( red triangles ), the increase in leaves the frequency and damping rate of the langmuir modes , the modes with , largely unaffected .the pair of least damped acoustic modes from case 1 are now both unstable , having and two distinct growth rates , . the parametric path from stability ( , open triangles ) to instability ( , filled triangles ) for these two modes is illustrated as a function of complex frequency in panel b. for case 3 , the langmuir mode and the least damped acoustic modes are negligibly affected by the bump distribution . by parametric variation of from ( open diamonds in panel c ) to ( filled diamonds ) , it is observed that the growing mode for the bump - on - tail instability arises from an acoustic mode which is strongly damped in the stable regime ( solid line ) , while the langmuir mode ( dashed line ) becomes heavily damped .the dispersion relations for the least damped and/or fastest growing langmuir ( acoustic ) modes as a function of for the three cases are presented in panels d and f ( e and g ) illustrating the wavelengths for which linear instabilities arise for cases 2 and 3 ( solid lines in panel g ) .to evaluate the time evolution of these systems , we have extended the vlasov - poisson solver ` vp` to allow for the inclusion of an arbitrary number of drifting plasma populations .` vp ` solves the nonlinear vlasov - poison system using second - order finite differencing for spatial and velocity derivatives and a third - order adams - bashforth scheme in time . as a test of this extension , we compare numerical solutions of eqn .[ eqn : disp.fpc ] for the three cases described in section [ sec : drifts ] to frequencies and damping rates extracted from time traces of the electrostatic field energy from linear and small - amplitude nonlinear ` vp ` simulations for a range of wavevectors , given as points in fig .[ fig : linear ] .agreement between the dispersion relation and ` vp ` are close as long as the mode of interest is not heavily damped and unstable modes supported by the system do not grow too quickly with respect to the damped modes .    for the evaluation of the field - particle correlation , we perform three nonlinear simulations corresponding to the three cases from section [ sec : drifts ] . for these simulations ,we add a sinusoidal perturbation to the ion s equilibrium distribution of the form with .256 ( 128 ) points in velocity ( cordinate ) space are resolved over the interval  ] .each simulation is run for longer than and the total energy in the system is conserved to better than a few tenths of a percent .changes in the electrostatic energy as well as energy in the three plasma populations , , from their initial values are shown in fig .[ fig : energy ] , with energy normalized by . in run 1 , most of the energy damped from the electric field is equally partitioned between the two electron populations , with little energy transferred to the ions . for run 2 ,an instability is clearly triggered , with significant losses of energy from the electrons and both and growing from their initial values .a more virulent instability is triggered in run 3 , with the bump electron population losing a significant fraction of its energy to the core electron population , with little energy transferred to the ions .while tracking the change in energy is sufficient to identify instabilities in systems with complete knowledge of spatial and velocity structure , we seek the signature of unstable behavior given limited , single - point measurements of a system of the type available to spacecraft in the solar wind .the application of field - particle correlations to such measurements allows for a local observation of secular energy transfer .we define the field - particle correlation for a discrete set of measurements of and with timestep , taken at a single point as the correlation averages the field - particle interaction term in the vlasov equation , the third term in eqn .[ eqn : vlasov.ld ] , over a time interval of length . as the ballistic term , the second in eqn .[ eqn : vlasov.ld ] , does not lead to net energy transfer, the product in represents the energy density transferred at one point in phase space between the electric field and velocity distribution . by averaging over a selected time interval , oscillatory energy transfer between and is removed , leaving only the secular energy transfer . to track the accumulated change of the phase - space energy density , we integrate the correlation over time , defining .  .the resonant velocities for the system are indicated by dashed grey lines.,width=623 ]  unlike eqn . 2 in klein & howes 2016, we include the full distribution function in our definition of , rather then only the perturbed component .previous studies had focused on particular cases where the equalibria were even with respect to , ensuring that they would not contribute to net energy transfer . forthe cases under consideration in this work , the equilibrium electron distributions have odd components and therefore may contribute to a secular transfer of energy between the fields and distributions . to show that the two forms of the correlation obtain similar results when is even , we apply both correlations to single - point field and distribution data from a landau damped counter - propagating langmuir wave simulation , case 1 in klein & howes 2016 , and plot at with in fig .[ fig : compare ] .we see that both correlations produce qualitatively similar structure in the accumulated phase - space energy density , especially in regards to production of a plateau surrounding the resonant velocities of the system , , which serves as the key velocity - space signature of landau damping .    with our correlation defined , we next select an appropriate correlation interval for the three simulations . by averaging over a particular interval ,we remove the transfer of energy between the fields and particles which oscillates with frequency of the order , leaving the secular , or non - oscillatory , component .plotted in fig .[ fig : tau ] are velocity integrated correlations , , between and the three plasma populations for a range of correlation lengths $ ] at a single spatial location .there is significant oscillatory transfer for small correlations for run 1 , panels a - c of fig .[ fig : tau ] .this oscillatory transfer is reduced for longer intervals , with nearly all of the oscillations removed for , a correlation length corresponding to the period of langmuir waves supported by the system with frequency .this leaves the velocity integrated correlations for all three distribution functions nearly monotonic , while slightly longer correlations reintroduce some oscillatory behavior .as the langmuir wave is the least damped , finite frequency linear mode supported by this system , correlating over its period is physically justified . for run 2 ,panels d - f , correlating over the interval , which corresponds to the langmuir wave frequency , removes significant oscillatory energy transfer . averaging over the time scale associated with the least damped modes , rather than the unstable modes ,is motivated by the fact that the unstable modes of this system have zero frequency ; see fig [ fig : linear ] . for run 3, we see that there does not exist a single value of for which the oscillatory energy transfer is completely removed .this due to the fact that the system supports both a weakly damped langmuir wave as well as a finite - frequency unstable acoustic mode .correlating over the langmuir period retains some of the acoustic oscillations , while correlating over the acoustic period retains some of the langmuir oscillations .we choose , corresponding to the period of the growing acoustic mode with and acknowledge that some oscillatory contributions from langmuir waves will persist .with an appropriate interval selected , we calculate the velocity dependent field - particle correlation for the three simulations . by retaining the velocity dependence ,we are able to address the question of where the energy transfer occurs in phase space , and use the structure of this energy transfer to characterize the nature of the underlying instability .    for the stable , counter - streaming electron case , run 1, the phase - space energy transfer obtained from the field - particle correlation is a fairly regular function of velocity . at a given point in cordinate space , for example shown in the first row of fig .[ fig : c_single ] , one of the electron populations gains energy from , while the other population loses energy to , the electric field .the energy transfer to the ions from the electric field is an odd function of velocity , meaning that when the correlation is integrated over , there is no net energy transfer to the ions .there is no evidence for the dependence of the phase - space energy transfer on drift velocities ( black vertical dashed lines ) or the resonant velocities of either the langmuir ( green lines ) or acoustic ( magenta ) modes for any of the three populations .  increasing the speed of the counter - streaming electrons for run 2 , shown in panels e - h of fig .[ fig : c_single ] , the structure of the energy transfer is altered .the sign of the field - particle correlation changes at due to a change in sign of . unlike for case 1, this change in sign occurs for electrons which exchange significant energy with the electric field . for the ions , a small , even component in the field - particle correlation arises , yielding a net transfer of energy from the fields to the ions , as seen in panel h. this transfer of energy to the ions , which does not depend on either the langmuir or acoustic resonant velocities , serves as a phase - space signature for the fluid instability that arises for this system .  for the bump - on - tail instability , case 3 shown in panels i - l of fig .[ fig : c_single ] , the velocity - space structure of the field - particle correlation is significantly different .as we have correlated over the unstable acoustic period , the oscillatory structure from the weakly damped langmuir mode is evident .we also see in the bump electron population that the energy transfer changes sign across the acoustic resonance and at later time across the langmuir resonance .this resonant structure is the signature of the transfer of energy from the bump population to the core electron population as mediated by the electric field .we note that the resonant structure is maintained for other choices of , including an interval corresponding to the langmuir wave period , not shown .the velocity - integrated correlation confirms that the core population has a net gain of energy , and as expected for this instability , we see that the electrons that receive the energy are near the resonant velocities .this explicit dependence of the phase - space energy transfer on resonant coupling between the fields and particles serves as a distinct signature of kinetic instabilities when compared to the fluid instability in run 2 .f6.eps  the field - particle correlations presented in fig .[ fig : c_single ] were calculated at a single point in cordinate space .an obvious question arises as to how the correlation and the associated phase - space energy transfer change as a function of position within the simulation . to assesthis question , we calculate at five other points within the simulation and present the velocity integrated accumulated change of the phase - space energy density , , in fig .[ fig : int_c ] . for the stable case ,run 1 , we see that the energy transfer to or from the two electron populations changes sign and amplitude as a function of the position in the simulation , with the transfer passing through zero at the two nodes of the initial standing wave pattern at . the correlation with the ions maintains an even velocity space structure such that the ions continue to neither lose nor gain net energy from the electric field regardless of spatial position . for the unstable counter - streaming electrons , run 2 , the same pattern of shifting sign and amplitude for the energy transfer to and from the electron holds .the ions gain energy from the electric field regardless of which electron population is gaining or losing energy .the amplitude of the ion energy gain changes with the amplitude of the electron field - particle correlation , going to zero at the nodes and having its largest value at the anti - nodes .this phase space structure serves as a distinct signature of growing fluid instabilities . for the bump - on - tail instability ,run 3 , there is no regular spatial variation in the transfer of energy between the beam and core electron populations ; the core gains energy at the expense of the beam , with the energy density accumulating at nearly the same rate regardless of spatial position , as expected for a kinetic instability .field - particle correlations of the form defined in eqn .[ eqn : fpc.fpc ] , modified from klein & howes 2016 to account for drifting equilibrium distributions , are applied to a set of simulations where fluid and kinetic instabilities are present .the structure of the resulting correlations , which can be interpreted as the secular phase - space energy density transferred between the fields and distributions , can be used to identify the presence of instabilities as well as to characterize the mechanisms driving the unstable growth .the form of the correlation allows for such characterizations to be made from observations at a single point , or a few points , in cordinate space , as opposed to requiring knowledge of spatially integrated quantities typically not accessible to experimental measurements .we consider simplified 1d-1v electrostatic systems in an attempt to characterize field - particle correlations as measurements of secular energy transfer in advance of future work applying such correlations to systems of higher dimensionality , where magnetization , turbulence , and inhomogeneities may complicate the interpretation of the correlation . by determining signatures of basic plasma physics phenomena responsible for energy transfer between fields and distributions , such as landau damping and one - dimensional instabilities, we lay the foundation for the determination of the velocity distribution signatures of more complex interactions , such as cyclotron and transit time damping, heating by large amplitude , stochastic fluctuations, and magnetic reconnection, which have all been proposed to play a role in the dissipation of turbulent fluctuations .future work will also consider the effects of solar wind expansion , electron conduction , and other inhomogeneous mechanisms on the plasma to clearly identify the role of such energy transfer mechanisms in the solar wind . by constructing the correlation to be obtained from single - point measurements , we allow for the identification of such damping mechanisms from in situ observation of the solar wind on current and future missions including _ deep space climate observatory ( dscovr ) _ , _ mms_, and _ solar probe plus_.     notable exceptions to the single - point limitationare the are the _cluster_, _ themis _ , and _ _ magnetospheric multiscale ( mms)__ missions , which are comprised of four or five spacecraft arranged in particular geometric configurations .

Abstract: a recently proposed technique correlating electric fields and particle velocity distributions is applied to single - point time series extracted from linearly unstable , electrostatic numerical simulations . the form of the correlation , which measures the transfer of phase - space energy density between the electric field and plasma distributions and had previously been applied to damped electrostatic systems , is modified to include the effects of drifting equilibrium distributions of the type that drive counter - streaming and bump - on - tail instabilities . by using single - point time series , the correlation is ideal for diagnosing dynamics in systems where access to integrated quantities , such as energy , is observationally infeasible . the velocity - space structure of the field - particle correlation is shown to characterize the underlying physical mechanisms driving unstable systems . the use of this correlation in simple systems will assist in its eventual application to turbulent , magnetized plasmas , with the ultimate goal of characterizing the nature of mechanisms that damp turbulent fluctuations in the solar wind .