is an emerging technology , utilizing both fundamental electronic charge and spin .spin is the intrinsic angular momentum of the electron .the spin - transfer - torque magnetoresistive random access memory ( stt - mram ) , which is based on spintronic effects , has revolutionized the magnetic storage industry - . in the past few years , extensive research on modeling this device has been carried out - .the developed models of stt - mram enable estimation of the performance of stt - mrams together with its cmos circuits , further accelerating the development of stt - mram technology .meanwhile , the spin torque oscillator ( sto ) , which is another interesting spintronics - based device , has recently received a rapidly increased attention .the sto provides a widely tunable voltage oscillation at microwave frequencies , greatly extending the possible application range of spintronics .possible applications of stos include frequency detection , , magnetic field sensing , , microwave sources , , and microwave communications , .particularly , the magnetic tunnel junction ( mtj ) sto , which provides comparatively large output power , show great potential to be used in different applications .this motivates the focus of this work on the mtj sto .however , unlike for the case of stt - mram , very little progress has been achieved in modeling the stos for circuit- and system - level design , impeding the development of sto - based applications .the only existing mtj sto models , are limited by several factors . for instance , they offer a limited applicable range , inaccurate dc operating point , and utilize expressions that are not fully validated by experiments or theory . a new analytical mtj sto model , which can overcome these issues , has been proposed in parti . to further allow being used by a circuit simulator , such as cadence spectre - rf , which can analyze the analog and rf performances of sto - based circuits and systems ,the analytical mtj sto model should be implemented in a hardware description language .verilog - a is a hardware description language , which uses mathematical expressions to model the behaviors of arbitrary types of devices and components , while allowing circuit- and system - level design and analyses .therefore , verilog - a is suitable and will be used for modeling mtj stos . in the existing mtj sto models , , however , the information of full verilog - a implementation is absent . in the sto verilog - a model , which has been proposed by the same research group as , and has not been validated by mtj stos , it is not possible to change the bias magnetic field in the verilog - a model since the calcuation of the effective magnetic field is not included .instead , the effective magnetic field is calculated in advance in matlab and manually imported to circuit simulation platforms by the user for every change of field bias condition .therefore , this model does not allow tuning the applied field in the circuit simulator , and it can not be considered to be fully implemented in verilog - a , which brings difficulties in designing or optimizing the dedicated circuits for stos . besides , to generate the frequency or phase fluctuation of the sto , the model in employs an approach that can cause signal discontinuity . as a result ,this existing sto verilog - a model is not ready to be used .  here , in part ii , we present a full verilog - a implemention of the anlytical mtj sto model proposed in part i .this mtj sto verilog - a model uses a new approach to reproduce the phase fluctuation of the mtj sto and avoid convergence issues , enabling a reliable mtj sto model. moreover , efficient simulations of both the stand - alone mtj sto model and the mtj sto model combined with cmos circuits , are demonstrated .as detailed in part i , the characteristics of an mtj sto include the dc operating point , operating frequency , output peak power , and linewidth ( the full width at half - maximum ) .these characteristics vary greatly as the biasing condition changes .the biasing condition for the mtj sto is typically composed of the amplitude and the in - plane angle of the external magnetic field , as well as the applied dc current .the complete verilog - a code of the comprehensive and compact mtj sto model is available from .   to fully implement the mtj sto model in verilog - a , solving eq.(4a , 4b ) in for the angle and magnitude of the effective fieldis realized solely by using verilog - a .the effective field angle ( see eq.(4a ) in ) is solved numerically using the fixed - point iteration method , based on which the effective field magnitude can be simply obtained using eq.(4b ) in . the equation solver , as well as the the large amount of calculations involved in obtaining the nonlinear coefficients and characteristics of the mtj sto , make the transient simulation time - consuming . for most of the analyses , none of the parameters in the effective field change for a single run , so that all the heavy calculations can be executed in the _ initial step _ event in order to improve the efficiency of the simulation .the _ initial step _event is pre - defined in verilog - a and called on the first point of a simulation .therefore , in these cases , all the coefficients , parameters and characteristics are computed only once in the entire simulation .   in order to generate the time domain signal , which includes all the characteristics of the mtj stos , the output of the model should comprise of both rf and dc terms , implemented in verilog - a as +  [ cols="<",options="header " , ]   +  + where is the amplitude of the resistance oscillation , whose value depends on the biasing condition .the function returns the absolute simulation time .since mtj stos have considerable phase noise amplitude , is used to represent the phase fluctuation ( phase noise ) , reflecting the linewidth of the proposed model . for the timescales of circuit - level simulations ( generally on the order of ) , white frequency noise extending over fluctuation frequencies wider than 1 mhz 100 mhz will be the noise type dominating the linewidth .hence , for this application , it is natural and adequate to approximate the frequency noise by white frequency noise alone .the amplitude noise of mtj stos is coupled to the phase noise due to the nonlinearity of the governing magneto - dynamic equation ( eq.(1 ) in ) .such a coupling greatly enhances the phase noise , making the impact of the amplitude noise less significant for applications . for this reason ,the amplitude noise is not explicitly addressed in this work , but its effect is manifested in the linewidth of the proposed mtj sto model .nevertheless , gaussian amplitude noise could be added to the proposed model by further specifying its standard deviation .  an accurate verilog - a model of oscillators should memorize its phase along the simulation time . however , a solution to keep the phase information between adjacent simulation time steps has not been found . in one common verilog - a phase noise generation method ,the step - specific perturbed frequency + is integrated all the way from = 0 up to , resulting in a discontinuous phase jump whenever is updated .the method used in the existing verilog - a sto model instead implements a phase noise that is discontinuously updated every couple of nanoseconds . in both methods , the comparatively large and discontinuous phase changes result in significant signal discontinuities for sinusoidal signals .this type of discontinuity is not a characteristic of mtj stos and , moreover , may cause convergence issues during simulations .the solution for achieving an accurate and reliable mtj sto model is to instead implement a , which bears a continuous , linear phase change in between the fixed points of randomized phase fluctuation .the flow chart of the divergence - free implementation of is illustrated in fig .1 . the first step is to create a vector  ] follows a normal distribution with a mean of 0 and a standard deviation of , resulting in a phase variance that is growing linearly with time in a rate consistent with the specific level of white frequency noise for the specific linewidth  .the virtual time step is related to the upper cut - off frequency for white noise in the frequency noise spectrum , and should be set to a value smaller than one order of magnitude lower than the inverse cut - off frequency in order to produce white frequency noise all the way up to the cut - off . for cut - off frequencies of 100 mhz or 1 ghz, this means that the virtual time step should be set to lower than 1 ns or 100 ps respectively .failing to set a short enough will result in a too narrow white band in the frequency noise spectrum , resulting in a decreased spectral linewidth . in this work , = 100 ps is employed .the length of  ] , which is used to store the phase deviation accumulated from the reference time ( =0 ) to each virtual time step .the relationship between  ] is illustrated in fig .2 . in the first period between =0 and the first virtual time step ( =1 ) ,  ] to  ] . a large  ] up to the last virtual time step , and the phase deviation produced from the last virtual time step to the absolute time .the parameter is employed to count the past number of virtual time steps , and it is updated every by using the function in verilog - a . it should be noted that the initial value of is set to -1 since the function is called at the beginning of each time period .the absolute time is fetched by calling the function .the relationship between ,  ] is presented in fig .specifically , the slope of is the instantaneous frequency deviation  ] ) . by summing  ] ) , the required is realized .the proposed phase noise generation approach successfully overcomes the phase discontinuity issue identified in the existing verilog - a models for oscillators .however , the proposed noise generation approach makes the proposed mtj sto model less suitable for simulations which involve momentary variations in the operating conditions , such as the modulation of current or field . nevertheless , at the early stage of evaluating stos in various applications and designing sto - based building blocks towards applications , these simulations involving momentary variations in the operating conditions are not yet critical .transient simulations of the aforementioned analytical model implemented in verilog - a , are carried out using the cadence spectrerf circuit simulator .   to validate the proposed function for generating the frequency fluctuation of the mtj sto , transient simulations of the stand - alone ( unloaded ) mtj sto model using the parameters from different mtj stos - performed .the time domain signals of the transient simulation using the parameters from and = 1.5 ma , while sweeping , are depicted in fig .3 as an example .they are compared with the measured time domain signal from , since time domain measurements are not included in - .the general nature of these simulated time domain signals agree with the measured one .especially , the phase fluctuation generated by the proposed function , has continuous changes and is very similar to that of the measured time domain signal . aside from the discrepancies between the modeled ( theoretical ) and measured linewidth, this comparison demonstrates that the proposed mtj sto model can reproduce the phase fluctuation generated by mtj stos , so as to achieve a reliable mtj sto model .  the time domain signals in fig. 3 . are further analyzed to validate that the model implemented in verilog - a is equivalent to the analytical model given in .  for different , the time domain signals shown in fig .3 present different characteristics including the operating frequency , phase noise ( linewidth ) , amplitude , and dc biasing points .time domain signals with large output power ( voltage amplitude ) appear at the in - plane external field angles between and ( corresponding in - plane effective field angles between and ) . among these ,the signal generated at shows less random fluctuations in the phase than that at .the estimation of output power and linewidth based on fig .3 are in agreement with the analytical results given in and fig .4(a ) of respectively . as it can also be seen in fig .3 , the phase noise at large in - plane external field angles degrades significantly , which is in accord with the theoretical linewidth in fig .4(a ) of . in such cases , where the phase noise is considerable, the simulation does not suffer from any signal discontinuity or convergence issues , thanks to the proposed method used to generate the phase noise . additionally , noticed from fig .3 , the dc voltage across the mtj sto increases as a function of , which is in agreement with eq.(5 ) in .  to quantify the characteristics of the mtj sto verilog - a model as a function of ,the time domain signals obtained from the transient simulations ( 1 ) are converted ( in cadence ) to the frequency domain using fft so as to obtain power spectral densities ( psds ) of the signals , which are depicted in . to perform the fft ,a hamming window of the full waveform length 1 ( 16384 samples ) is employed .this results in a resolution bandwidth of ( 1 ) = .4(a ) shows the psds , which contain the information of the operating frequency and linewidth of the mtj sto s signals as a function of .as is increased , the operating frequency initially decreases , reaches a minimum and thereafter increases , agreeing with the measured data given in fig .2(a ) of . for different , the operating frequency and linewidth of the proposed mtj sto model implemented in verilog - a are in accordance with the one obtained from the theoretical analysis , as given in fig .2(a ) and fig .4(a ) of . particularly , as it can be noticed from fig .4(a ) , the linewidth at , is much narrower than that at , indicating less random frequency fluctuations and confirming the estimation based on fig .moreover , as it can be observed from fig .4(a ) , the signal with comparatively large output power can be found between and , which is also in agreement with the analytical and measured results in fig .3(a ) of .the dependence of on the characteristics of the mtj sto is also examined by performing transient simulations of the proposed verilog - a model with different .the psds of the transient simulation results as a function of at is depicted in fig .the rise in causes a decline in the operating frequency and an increase in the output power , which matches the theoretical results and experiments .effort has been made in this work to achieve the compact mtj sto model and improve the simulation speed .therefore , the time required for transient simulations of the proposed model is of interest . to benchmark the simulation speed , a 1 simulation with a maximum time step of 5 psis performed . by averaging the runtime of 10 simulations containing approximately 470900 transient steps, a simulation takes only 52.6 s. the simulations are performed on a server with 2 opteron 6172 , and occupy one core . regarding the micromagnetics - based model , which is most efficiently simulated on graphic processing units ( gpus ) or supercomputer clusters , the typical runtime is roughly 24 hours for a similar device . compared with the micromagnetics - based model discussed in part i , the proposed model offers rapid simulations with a similar degree of accuracy .hybrid simulation of the mtj sto model with cmos circuits is of great importance since it provides validation of the proposed model at system- and circuit - level throughout its range of operation .furthermore , it can fully cross - verify the model and the designed cmos circuits . to perform the hybrid simulation ,a system including the proposed mtj sto model as well as cmos circuits is considered . in this system, the mtj sto model employs the parameters from since this mtj sto has a resistance close to 50 , which eases analyses .firstly , proper biasing circuits for driving the mtj sto are investigated and analyzed .for instance , the current mirror , which has been employed in , to provide the current biasing for the mtj sto , is simulated with the proposed mtj sto model and examined .the simulation results , however , suggest that using the current mirror to bias the mtj sto is not suitable .the reason is that the resistance ( biasing voltage ) of the mtj sto changes when is varied , as illustrated in fig .3 . therefore, different resistances at different make it impossible for the current mirror to accurately copy the current from a current source to the mtj sto under all circumstances .thereafter , a traditional rc bias - t , as shown in fig .5 , is simulated with the proposed mtj sto model .  according to the simulation results , this rc bias - tcan more successfully be utilized by mtj stos since the performance of the bias - t is not influenced by the variable resistance ( biasing voltage ) of mtj sto .consequently , the rc bias - t is used in this work to build the system .the selection of the biasing circuits for the mtj sto demonstrates that the proposed mtj sto model is very useful for the device and circuit community to identify the suitable circuit topologies and to design dedicated circuits for mtj stos , owing to the complete implementation of the proposed model in verilog - a .  to complete a system , which provides low - noise amplification to mtj sto signals andcan be used in either applications or measurements , a wideband balun - low noise amplifier ( lna ) is employed .an output buffer and an ac coupling capacitor are added to present a system ( fig .5 ) that is able to drive 50 load .the buffered wideband balun - lna is fully - esd protected , and it is implemented in cmos 65 nm process with a 1.2 v power supply .  before performing the transient simulation of the system ,the unloaded mtj sto is simulated at and , where a comparatively high voltage generated by the mtj sto can be obtained .this voltage will be used as the reference voltage to compare with the voltage signals obtained from the mtj sto together with the balun - lna .the transient simulation of the system is then conducted at the same biasing condition .the obtained time domain signals , including the voltage generated by the unloaded mtj sto , the voltage that can be delivered from the mtj sto to the balun - lna , and the amplified voltage at the differential output of the balun - lna , are plotted in figthe output signals of the mtj sto before and after being connected by the balun - lna show that the dc voltage is sustained due to employed bias - t , and the dc resistance ( ) is close to 50 .in addition , approximately 2/3 of the ac voltage generated by the mtj sto is delivered to the balun - lna , which is due to the fact that an ac voltage divider is formed between and the input impedance of the balun - lna ( ) . to quantify the power delivery and amplificationso as to evaluate the hybrid simulation , psds of these signals are plotted in fig .figure 7 shows that the power delivered from the mtj sto to the balun - lna is about -3 db less than the power generated by the unloaded mtj sto .based on this loss ( /( ) = 10 ) , it can be calculated that the return loss s11=()/( ) at the operating frequency of the mtj sto ( under the applied biasing condition ) is .since is close to 50 , the calculated s11 should be similar to the s11 of the balun - lna that is characterized with a 50 termination .the s11 of the balun - lna reported in is about , which verifies the s11 calculated based on the hybrid simulation . as it can be also approximated from fig .7 , the difference between the simulated psd signals at the input and output of the balun - lna is approximately , which corresponds to the gain of the balun - lna reported in . in summary , the behavior of the proposed mtj sto model has been validated at circuit- and system- level .the evaluated hybrid simulation demonstrates that the performance of an mtj sto - based system can be easily , reliably and accurately predicted by the circuit simulator using the proposed mtj sto verilog - a model .the analytical model of the mtj sto proposed in of this paper has been fully implemented in verilog - a , enabling its direct use in sto - based systems . during the implementation , an approach to replicate the phase noise ,hence the generated signal of the mtj stos , has been developed .this approach makes a reliable mtj sto model possible , and allows different performance analyses so as to extensively explore possible applications .the simulation results of the stand - alone mtj sto model and the mtj sto - based system show that the implemented model gives identical characteristics as those obtained from the proposed analytical model .additionally , the results demonstrate that the proposed mtj sto model is useful for estimating as well as improving overall performance of the mtj sto - based circuits and systems .consequently , the proposed mtj sto has the potential to accelerate the development of mtj sto technology towards its future applications .s. a. wolf , d. d. awschalom , r. a. buhrman , j. m. daughton , s. von molnar , m. l. roukes , a. y. chtchelkanova , and d. m. treger , `` spintronics : a spin - based electronics vision for the future , '' _ science _ , vol . 294 , no . 5546 , pp . 14881495 , 2001 .k. c. chun , h. zhao , j. d. harms , t .- h .wang , and c. h. kim , `` a scaling roadmap and performance evaluation of in - plane and perpendicular mtj based stt - mrams for high - density cache memory , '' _ ieee j. solid - state circuits _ ,2 , pp . 598610 , 2013 .   c. augustine , n. n. mojumder , x. fong , s. h. choday , s. p. park , and k. roy , `` spin - transfer torque mrams for low power memories : perspective and prospective , '' _ ieee sens . j. _ , vol .4 , pp . 756766 , 2012 .g. d. panagopoulos , c. augustine , and k. roy , `` physics - based spice - 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prejbeanu , j. a. katine , d. mauri , a. zeltser , m. prigent , j. -c .nallatamby and r. sommet , `` amplitude and phase noise of magnetic tunnel junction oscillators , '' _ appl .18 , p. 182507, 2010 .z. m. zeng , p. upadhyaya , p. k. amiri , k. h. cheung , j. a. katine , j. langer , k. l. wang , and h. w. jiang , `` enhancement of microwave emission in magnetic tunnel junction oscillators through in - plane field orientation , '' _ appl ._ , vol . 99 .3 , p. 032503, 2011 .p. gangmei , a. neudert , m. k. marcham , r. j. hicken , m. a. gubbins , x. cao , r. r. lamberton , and a. b. johnston , `` thermal and spin - transfer - torque excitation of precessional modes in magnetic tunnel junction nanopillars with symmetric interfaces and a thick free layer , '' _ phys . rev .b _ , vol . 88 , no .13 , p. 134415, 2013 .p. k. muduli , y. pogoryelov , s. bonetti , g. consolo , f. mancoff , and j. kerman , `` nonlinear frequency and amplitude modulation of a nanocontact - based spin - torque oscillator , '' _ phys .b _ , vol . 81 , no .14 , p. 140408, 2010 .tingsu chen ( s11 ) received the b.sc .degree in communication engineering from the nanjing university of information science and technology , china , and the m.sc .degree in system - on - chip design from the kth royal institute of technology , sweden , in 2009 and 2011 , respectively .she is currently working toward the ph.d .degree at kth with the research area of high frequency circuit design for spin torque oscillator technology .anders eklund ( s13 ) received the m.sc .degree in engineering physics from kth royal institute of technology , sweden , in 2011 .he is currently working towards a ph.d .degree in physics at kth , experimentally investigating the frequency stability of spin torque oscillators by means of electrical characterization and synchrotron x - ray measurements .ezio iacocca ( s08 ) received the b.sc .degree in electronic engineering from the simn bolvar university , caracas , venezuela ( 08 ) , the m.sc . in nanotechnology from the royal institute of technology , stockholm , sweden ( 10 ) , and the ph.d . in physics from the university of gothenburg , gothenburg , sweden ( 14 ) .his research focuses on the magnetodynamical modes of spin transfer torque driven nano oscillators and their applications in communication and storage technology .saul rodriguez ( m06 ) received the b.sc .degree in electrical engineering from the army polytechnic school ( espe ) , quito , ecuador , and the m.sc .degree in system - on - chip design and the ph.d .degree in electronic and computer systems from the kth royal institute of technology , stockholm , sweden . in 2001 , 2005 , and 2009 , respectively .his current research interests include rf cmos circuit design for wideband frond - ends , ultralow - power circuits for medical applications and graphene - based rf , and ams circuits .b. gunnar malm ( m98 - sm10 ) was born in stockholm , sweden , in 1972 .he received the m.s . from uppsala university , sweden , in 1997 , the phd in solid - state electronics 2002 , from royal institute of technology ( kth ) , stockholm .he is an associate professor at the school of ict , kth since 2011 .his recent work includes silicon photonics , silicon - carbide technology for extreme environments and spintronics .he also serves on the kth sustainability council .johan ( m06 ) has an ing .dipl . degree ( 94 ) from epfl , switzerland , a m.sc . in physics ( 96 ) from lth , sweden , and a ph.d . in materials physics ( 00 ) from kthroyal institute of technology , stockholm . in 2008he was appointed full professor at university of gothenburg and is a guest professor at kth royal institute of technology .he is also the founder of nanosc ab and nanosc instruments ab .ana rusu ( m92 ) received the m.sc .degree in electronics and telecommunications and ph.d .degree in electronics in 1983 and 1998 , respectively .she has been with kth royal institute of technology , stockholm , sweden , since 2001 , where she is professor in electronic circuits for integrated systems .her research interests include low / ultralow power high performance cmos circuits and systems , sto - based systems , rf graphene and high temperature sic circuits .

Abstract: the rapid development of the magnetic tunnel junction ( mtj ) spin torque oscillator ( sto ) technology demands an analytical model to enable building mtj sto - based circuits and systems so as to evaluate and utilize mtj stos in various applications . in part i of this paper , an analytical model based on the macrospin approximation , has been introduced and verified by comparing it with the measurements of three different mtj stos . in part ii , the full verilog - a implementation of the proposed model is presented . to achieve a reliable model , an approach to reproduce the phase noise generated by the mtj sto has been proposed and successfully employed . the implemented model yields a time domain signal , which retains the characteristics of operating frequency , linewidth , oscillation amplitude and dc operating point , with respect to the magnetic field and applied dc current . the verilog - a implementation is verified against the analytical model , providing equivalent device characteristics for the full range of biasing conditions . furthermore , a system that includes an mtj sto and cmos rf circuits is simulated to validate the proposed model for system- and circuit - level designs . the simulation results demonstrate that the proposed model opens the possibility to explore sto technology in a wide range of applications .   = 1  spin torque oscillator , magnetic tunnel junction , macrospin , analytical model .


we consider the _ thinnest path _ problem in wireless ad hoc networks . for a given source and a destination , the thinnest path problem asks for a path from the source to the destination that results in the minimum number of nodes hearing the message .such a path is achieved by carefully choosing a sequence of relaying nodes and their corresponding transmission power .  at the first glance, one may wonder whether the thinnest path problem is simply a shortest path problem with the weight of each hop given by the number of nodes that hear the message in this hop . realizing that a node may be within the transmission range of multiple relaying nodes andshould not be counted multiple times in the total weight ( referred to as the width ) of the resulting path , we see that the thinnest path problem does not have a simple cost function that is summable over edges .but rather , the width of a path is given by the cardinality of the _ union _ of all receiving nodes in each hop , which is a highly nonlinear function of the weight of each hop .one may then wonder whether we can redefine the weight of each hop as the number of nodes that hear the message for the first time .such a definition of edge weight indeed leads to a summable cost function .unfortunately , in this case , the edge weight can not be predetermined until the thinnest path from the source to the edge in question has already been established .a more fundamental difference between the thinnest path and the shortest path problems is that the thinnest path from a single source to all other nodes in the network do not form a tree . in other words ,the thinnest path to a node does not necessarily go through the thinnest path to any of its neighbors .the loss of the tree structure is one of the main reasons that the thinnest path problem is much more complex than the shortest path problem . indeed , as shown in this paper , the thinnest path problem is np - complete , which is in sharp contrast with the polynomial nature of the shortest path problem .another aspect that complicates the problem is the choice of the transmission power at each node ( within a maximum value that may vary across nodes ) . in this case, the network can not be modeled as a simple graph in which the neighbors of each node are prefixed . in this paper, we adopt the _ directed hypergraph _ model which easily captures the choice of different neighbor sets at each node . while a graph is given by a vertex set and an edge set consisting of cardinality-2 subsets of , a hypergraph  is free of the constraint on the cardinality of an edgespecifically , any non - empty subset of can be an element ( referred to as a hyperedge ) of the edge set .hypergraphs can thus capture group behaviors and higher - dimensional relationships in complex networks that are more than a simple union of pairwise relationships . in a directed hypergraph , each hyperedgeis directed , going from a source vertex to a non - empty set of destination vertices .an example is given in fig .[ fig : hypergraph]-(a ) where we have directed hyperedges rooted at a source node with each hyperedge modeling a neighbor set of under a specific power .the directed hypergraph model of the thinnest path problem is thus readily seen : rooted at each node are multiple directed hyperedges , each corresponding to a distinct neighbor set feasible under the maximum transmission power of this node .the problem is then to find a minimum - width hyperpath from the source to the destination where the width of a hyperpath is given by the cardinality of the union of the hyperedges on this hyperpath .based on the directed hypergraph formulation , we show that the thinnest path problem in -d networks is np - complete even under a simple disk propagation model .this result is established through a reduction from the minimum dominating set problem in graphs , a classic np - complete problem .the most challenging part of this reduction is to show the reduced problem is realizable under a 2-d disk model .we further establish that even with a fixed transmission power at each node ( in this case , the resulting hypergraph degenerates to a standard graph ) , the thinnest path problem is np - complete .we then propose two polynomial - time approximation algorithms that offer and approximation ratios for general directed hypergraphs ( which can model non - isomorphic signal propagation in space ) and constant approximation ratios for ring hypergraphs ( which result from isomorphic signal propagation ) .we also establish the polynomial nature of the problem in 1-d and 1.5-d networks , where a 1.5-d network is a 1-d network embedded in a 2-d field of eavesdroppers with arbitrary unknown locations .we propose an algorithm based on a nested backward induction ( nbi ) starting at the destination .we show that this nbi algorithm has time complexity . since the size of the input date is , the proposed algorithm is order optimal .it solves the thinnest path problem in both the 1-d and 1.5-d networks .in particular , no algorithm , even with the complete location information of the eavesdroppers , can obtain a thinner path than the nbi algorithm which does not require the knowledge of eavesdropper locations .there is a large body of literature on security issues in wireless ad hoc networks ( see , for example , ) .however , the thinnest path problem has not been studied in the literature except in .chechik studied the thinnest path ( referred to as the secluded path in ) and the thinnest steiner tree in graphs .they showed that the problem in a general graph is np - complete and strongly inapproximable .they proposed an approximation algorithm with an approximation ratio of where is the maximum degree of the graph .they further studied the problem in several special graph models including graphs with bounded degree , hereditary graphs , and planar graphs .however , their study focuses on the problem in topological graphs , whereas we focus on hypergraphs and geometric graphs .the results obtained in  do not apply to special hypergraphs satisfying certain geometric properties that result naturally from the communication problem studied in this paper .this paper also includes several new results on the thinnest path problem under the graph model . specifically , we established the np - completeness of the problem in -d disk graphs and -d unit disk graphs . the results in  andthis work thus complement each other to provide a more complete picture of the thinnest path problem under different ( hyper)graph models .  in the general context of algorithmic studies in hypergraphs ,ausiello  tackled the problem of finding the -optimal hyperpath where is a general measure on hyperpaths that satisfies a certain monotone property .they established the np - completeness of this problem for general measures .the thinnest path problem can be seen as an -optimal traversal problem with the measure given by the number of vertices covered by the path .since it is a special measure , their np - completeness result developed under general measures does not apply .furthermore , in many applications , the resulting hypergraphs have certain topological and/or geometrical properties , and the computational complexities under these special models require separate analysis .another realated problem is the shortest path problem in hypergraphs .the shortest path problem in hypergraphs remains a polynomial - time problem as its counterpart under the graph model .the static shortest hyperpath problem was considered by knuth  and gallo  in which dijkstra s algorithm for graph was extended to obtain the shortest hyperpaths .ausiello proposed a dynamic shortest hyperpath algorithm for directed hypergraphs , considering only the incremental problem ( network changes contain only edge insertion and weight decrease ) with the weights of all hyperedges limited to a finite set of numbers  . in, gao developed the first fully dynamic shortest path algorithms for general hypergraphs .as discussed in [ sec : intro - thinnest ] , the thinnest path problem is fundamentally different and significantly more complex than the shortest path problem .the widest path problem has been well studied under the graph model , and the existing results can be easily extended to hypergraphs .the widest path problem asks for a path whose minimum edge weight along the path is maximized . in other words ,the width of a path is given by the minimum edge weight on that path , which is different from the definition of path width in the thinnest path problem studied in this paper . as a consequence ,the widest path problem is not the complement of the thinnest path problem .since the tree structure is preserved in the widest path problem ( the widest path to a node must go through the widest path to one of its neighbors ) , it remains a polynomial time problem .the thinnest path problem , however , is np - complete in general .a _ directed hypergraph _ contains a set of vertices and a set of directed hyperedges .each directed hyperedge has a single source vertex and a non - empty set of destination vertices .[ c](a ) [ c](b ) [ c](c )  a _ disk hypergraph _ is a special directed hypergraph whose topology is determined by a set of points located in a -dimensional euclidean space and a maximum range associated with each vertex .there exists a hyperedge from source to destination set if and only if consists of vertices located within the -dimensional sphere centered at with a radius $ ] . a _ unit disk hypergraph _ ( udh ) is a disk hypergraph with unit maximum range ( ) for all vertices .[ fig : hypergraph ] shows examples of a directed hypergraph , a disk hypergraph , and a unit disk hypergraph .a _ ring hypergraph _ is a generalized disk hypergraph where associated with each vertex is a minimum range as well as a maximum range .hyperedges rooted at are formed by spheres centered at with radiuses satisfying .it is easy to see that a disk hypergraph is a ring hypergraph with , a disk _ graph _ is a ring hypergraph with for all , and a unit disk graph ( udg ) is a ring hypergraph with for all .consider a wireless ad - hoc network with nodes located in a -dimension euclidean space .each node can choose the power , within a maximum value , for the transmission of each message .the chosen power , along with the signal propagation model , determines the set of neighbors that can hear the message .the maximum transmission power is in general different across nodes .the objective is to find a path between a given source - destination pair that involves the minimum number of nodes hearing the message .  as discussed in sec .[ sec : intro - thinnest ] , we formulate the problem using a directed hypergraph . each node is a vertex .the directed hyperedges rooted at a node are given by distinct neighbor sets of this node feasible under its maximum transmission power and the signal propagation model . under a general nonisomorphic propagation model, we end up with a general hypergraph .the only property the resulting hypergraph has is the monotonicity of the hyperedge set .specifically , the hyperedges rooted at each node can be ordered in such a way ( say , ) that and .this is due to the nature of wireless broadcasting where nodes reachable under transmission power can also be reached under any power greater than . under an isomorphic propagation model , we end up with a disk hypergraph .if all nodes have the same maximum range , we have a unit disk hypergraph .this hypergraph model also applies to networks with eavesdroppers .each eavesdropper can be seen as a node with zero transmission range .it is thus a vertex with no outgoing hyperedges .given a source - destination pair , a hyperpath from to is defined as a sequence of hyperedges such that for , , and .define the cover of to be the set of vertices in , the width is then given by the thinnest path problem asks for a hyperpath from to with the minimum width .note that choosing a hyperedge simultaneously chooses the relaying node and its transmission power ( determined by ) .in this section , we show that the thinnest path ( tp ) problem is np - complete in several special geometric hypergraphs and graphs . this implies the np - completeness of the problem in general directed hypergraphs .   in this subsection , we prove the np - completeness of the thinnest path problem in -d disk hypergraphs .while a stronger result is shown in the next subsection , the proof of this result provides the main building block for the proof of the next result .the result is established through a reduction from the maximum dominating set ( mds ) problem .the mds problem asks for the minimum subset of vertices in a given graph such that every vertex in the graph is either in the subset or a direct neighbor of one vertex in the subset .the following theorem formally establishes the polynomial reduction ( denoted by ) from mds to tp in -d disk hypergraphs .since the thinnest path problem is clearly in the np space , this theorem establishes the np - completeness of tp in 2-d disk hypergraphs .mds tp in -d disk hypergrpahs .[ thm : np-2ddh ]  to prove theorem [ thm : np-2ddh ] , consider an mds problem in an arbitrary graph .we first construct a _ general _ directed hypergraph based on such that a thinnest path in leads to an mds in .the main challenge in the proof is to show that is realizable under a 2-d disk model .there are two main difficulties .first , line crossing is inevitable when we draw on a 2-d plane .the implementation of hyperedges that cross each other needs special care to avoid unwanted overhearing that may render the reduction invalid .second , the geometric structure of 2-d disk hypergraphs dictates that there are at most 5 vertices ( even with arbitrary ranges ) that can reach a common sixth vertex but not each other .it is thus challenging to implement a vertex with up to incoming hyperedges in while preserving the reduction .our main approach to the above difficulties is to allow _ directed _ overhearing .specifically , messages transmitted along one hyperedge may be heard by vertices implementing another hyperedge in , but not vise verse . by carefully choosing the directions of the introduced overhearing ,we ensure that the resulting 2-d disk hypergraph , while having a different topological structure from , preserves the reduction from mds in  .another challenge in constructing is to ensure the polynomial nature of the reduction .the number of additional vertices added in needs to be in a polynomial order with the size of .this often limits using reduced transmission ranges as a way to avoid unwanted overhearing : exponentially small transmission ranges may require exponentially many vertices to connect two fixed points .  a detailed proof is given in [ app : np_2ddh ] .we now establish the np - completeness of tp in 2-d _ unit _ disk hypergraphs ( udh ) .the proof builds upon the proof of theorem [ thm : np-2ddh ] .the only difference is that when implementing the general directed hypergraph , we no longer have the freedom of choosing the maximum transmission range of each vertex .this presents a non - trivial challenge . as stated in sec .[ sec : npc-2ddh ] , our approach to circumventing the constraints imposed by the geometrical structures of 2-d disk hypergraphs is to allow directed overhearing , which is achieved by carefully choosing different maximum transmission ranges of various vertices .to implement a 2-d udh for the reduction , however , all vertices must have the same maximum transmission range .  to address this issue, we introduce a special type of disk hypergraphs , called _ exposed disk hypergraphs _ , and show that tp in -d exposed disk hypergraphs can be reduced to tp in -d udh for any .we then show that the -d disk hypergraph constructed in the proof of theorem [ thm : np-2ddh ] can be modified to an exposed hypergraph while preserving the reduction , .we thus arrive at the np - completeness of tp in 2-d udh based on the transitivity of polynomial time reduction .  in a disk hypergraph ,let denote the closest non - neighbor if it is outside the maximum range of .] of .define can be change to an arbitrary positive value smaller than . ] where is the distance between and ( is set to when does not have non - neighbors ) .an _ exposed area _ of is defined as where denotes the closed ball centered at with radius .a disk hypergraph is _ exposed _ if every vertex has a non - empty exposed area .[ c] [ c] [ c]  [ lma : expose ] tp in -d exposed disk hypergraphs tp in -d udh .the basic idea is to place super vertices at specific locations in exposed areas to force vertices on a thinnest path to use transmission ranges smaller than the maximum value .the problem is thus transformed to the case with disk hypergraphs where vertices may have different maximum transmission ranges . a detailed proof is given in [ app : expose_base ] .  with lemma [ lma :expose ] providing a bridge between disk and unit disk hypergraphs , all we need to show is that mds can be reduced to tp in -d _ exposed _ disk hypergraphs .mds tp in -d exposed disk hypergrpahs .[ lma : np-2dedh ]  see [ app : expose ] .  based on lemma [ lma : expose ] and lemma [ lma : np-2dedh ] , we arrive at the following theorem .mds tp in -d udh . [ thm : np_2dudh ]   in this subsection , we consider the thinnest path problem in disk graphs and unit disk graphs ( udg ) .recall that disk and unit disk graphs are special ring hypergraphs with and , respectively .in other words , they can be seen as hypergraphs where each vertex has only one outgoing hyperedge directing to its prefixed neighbor set ( determined by its fixed transmission power ) .this also shows that disk hypergraphs and disk graphs are not special cases of each other .given the same set of vertices and their associated maximum ranges , a disk hypergraph has a different topology from a disk graph : each vertex in general has more than one outgoing hyperedges due to the freedom of using smaller transmission ranges . the same holds for udh and udg . as a consequence, the complexity of tp in disk and unit disk graphs can not be inferred from theorems [ thm : np-2ddh ] and [ thm : np_2dudh ] , and needs to be studied separately .mds tp in -d disk graphs .[ thm : np_2ddg ]  in the proof of theorem [ thm : np-2ddh ] , the vertices along the thinnest path in the constructed 2-d disk hypergraph all use their maximum ranges .thus , mds in can be reduced to tp in a disk graph constructed based on by including only those hyperedges associated with the maximum range of each vertex .next we consider tp in udg .unfortunately , the approach through exposed disk hypergraphs used in showing the np - completeness of tp in udh does not apply , since it hinges on vertices being able to use any transmission ranges smaller than a maximum value .the difficulty , however , can be circumvented for 3-d udg as shown in the following theorem .mds in degree- graphs tp in -d udg .[ thm : np_3dudg ]  the proof is similar to that of theorem [ thm : np-2ddh ] with two main differences .first , line crosses are implemented by using the third dimension to `` go around '' , rather than using different transmission ranges ( a luxury absent in udg ) to create directed crosses .second , reduction from mds in graphs with a maximum degree of ensures that there are at most incoming edges to each super vertex in the reduced udg .this makes the geometric constraint on the number ( at most in a -d euclidean space ) of vertices that can reach a common vertex but not each other inconsequential ( see [ app : np_3dudg ] ) . ] . a detailed proof is given in [ app : np_3dudg ] .note that using a reduction from mds in graphs with a constant maximum degree rather than mds in general graphs leads to a weaker statement .while mds in both cases are np - complete , the former is approximable with a constant ratio , and the latter a ratio of .theorems [ thm : np-2ddh]-[thm : np_2ddg ] thus give a order lower bound on the approximation ratio of those problems whereas theorem [ thm : np_3dudg ] a constant lower bound .in this section , we consider the thinnest path problem in 1-d networks . we show that the problem is polynomial time by constructing an algorithm with time complexity of .since the input data has size , the proposed algorithm is order - optimal .we then consider the -d problem and show that the algorithm developed for 1-d networks directly applies to the -d problem .consider a network under a general propagation model with nodes located on a straight line .each vertex is associated with a coordinate on the line ( the vertex index and its location are often used interchangeably ) . without loss of generality, we assume that .[ c] [ c] [ c] [ c] [ c] [ c] [ c] [ c] [ c]  it is clear that every node located between the source and the destination ( see fig [ fig:1d ] ) will hear the message no matter which path is chosen and all nodes to the right of can be excluded from the thinnest path .therefore , finding the thinnest path is to minimize the number of vertices to the left of that can overhear the message .the problem is nontrivial . due to the arbitrariness of the node locations and propagation range ,a forward path ( every hop moves the message to the right toward ) from to may not exist and nodes to the left of may need to act as relays .the question is thus how to efficiently find out whether a forward path exists and if not , which set of nodes to the left of need to relay the message .we propose an algorithm based on nested backward induction ( nbi ) . for each vertex , we define its predecessor to be the rightmost vertex on the left side of that can reach : thus , in order to reach , its predecessor or a vertex to the left of has to transmit .in other words , those vertices between and can not directly reach .equivalently , any vertex to the right of can only hear a message from through a relay by or a vertex to the left of .the nbi algorithm is then carried out in two steps . in the first step ,the predecessors of certain vertices are obtained one by one starting from moving toward . specifically , the predecessor of , denoted by ,is first obtained .if , then the first step terminates .otherwise , the predecessor of , denoted by , is obtained and its location compared with .the same procedure continues until the currently obtained predecessor is to the left of or is itself .the first step thus produces a sequence of vertices with , , and .let , which is a valid path from to .if , the algorithm terminates , and the thinnest path from to is given by .otherwise , we carry out step 2 of the algorithm where we find a path from to .specifically , let denote the set of vertices located between and including but not .let denote the set of all hyperedges whose source and destination vertices are in . as shown in [ app : f ] on the correctness of the algorithm , any hyperpath from to in the sub - hypergraph concatenated with gives a thinnest path from to .finding such an can be easily done by a breadth - first search ( bfs ) in .however , the resulting time complexity is .next we propose a special bfs procedure to reduce the time complexity to .the trick here is to set up two pointers , and , to the locations of the leftmost and the rightmost vertices in that have been discovered . due to the geometric structure of the 1-d network , each time we only need to search vertices to the left of and vertices to the right of .the detailed algorithm is given below .* enqueue , set and to the index of .* repeat until the queue is empty or is found : * * dequeue a vertex and examine it * * if , go to step 4 . ** otherwise , * * * while can reach * * * * enqueue and * * * * set the parent of to * * * while can reach * * * * enqueue and * * * * set the parent of to * if the queue is empty , return `` no path from to '' . *trace back to and return .the following theorem establishes the correctness of the proposed nbi algorithm .furthermore , it reveals a strong property of the path obtained by nbi under a disk propagation model .specifically , under a disk propagation model , we define the _ covered area _ of a hyperpath as where is the minimum transmission range that induces hyperedge , theorem [ thm : nbi_property ] shows that the covered area of the path obtained by nbi is a subset of the covered area of any feasible path from to .nbi algorithm finds the thinnest path .furthermore , under a disk propagation model , given any valid path from to , we have .[ thm : nbi_property ]  see [ app : f ] .the time complexity of the nbi algorithm is .[ thm : nbi_complexity ]  the complexity of the first step of nbi is readily seen .in the second step , the time complexity is dominated by updating the queue at each iteration .let denote the number of iterations in step 2 .note that we only check vertices at iteration , where is the new vertices that have been enqueued at this iteration and .also is bounded by .hence the total time complexity of this step is bounded by .we thus arrive at the theorem .we now consider the -d problem where in - network nodes are located on a line and eavesdroppers are located in a -dimensional space that contains the line network .we focus on the disk propagation model .a unit cost is incurred for each in - network node that hears the message and a non - negative cost is incurred for each eavesdropper that hears the message .the objective is to find a path from to with the minimum total cost : where is the cost for vertex , and is the covered area of path as defined in .based on theorem [ thm : nbi_property ] , it is easy to see that nbi provides the optimal solution to the 1.5-d thinnest path problem without the knowledge of the eavesdroppers locations .more specifically , no algorithm , even with the complete knowledge of the locations of the eavesdroppers , can obtain a thinner path than nbi which does not require the location knowledge of the eavesdroppers .in this section , we introduce two approximation algorithms for the thinnest path problem and analyse their performance in different types of hypergraphs .   given a general directed hypergraph with source vertex and destination vertex , we set the weight of a hyperedge to be the number of destination vertices in this hyperedge : the shortest hyperpath algorithm from to is then obtained under this weight definition as an approximation of the thinnest path .the following theorem gives performance of this shortest path based algorithm ( spba ) .the spba algorithm provides a -approximation for tp in general directed hypergraphs , a -approximation for ring hypergraphs with . additionally , the ratio of the spba algorithms is asymptotically tight even in -d disk hypergraphs .[ thm : approx ]  see [ app : spba ] .approximation occurs in two places in spba .first , the width of a path is approximated by the sum of the width of the hyperedges on that path .second , the thinnest path to a vertex is assumed to go through the thinnest path to one of its incoming neighbors .the first approximation can be avoided while maintaining the polynomial nature of the approximation algorithm . in particular, we can ensure that the width of a path is correctly obtained by using the set union operation instead of summation .the assumption on the tree structure of the thinnest paths still allows us to use dijkstra s algorithm with some modifications .specifically , for each vertex , we need to store the current thinnest path from to this vertex rather than only the width of this path and the parent of this vertex on this path .this allows us to take the set union operation when we update the neighbors of this vertex .given below is the performance of this tree structure based algorithm ( tsba ) .[ thm : dpapprox ] the tsba algorithm provides a -approximation for general directed hypergraphs , -approximation for ring hypergraphs with . additionally , the ratio of the tsba algorithm is tight in general directed hypergraphs and asymptotically tight in disk hypergraphs in the worst case .see [ app : tsba ] .while the approximation ratio of tsba is better than that of spba , these are worst - case performances and do not imply that tsba outperforms spba in every case as shown in fig [ fig : spdp ] .[ c] [ c]s [ c]t  [ c ]  fig .[ fig : simu_dh_1_5 ] shows the average performance of these two algorithms .we see that both algorithms have relatively small approximation ratios growing sub - linearly with the number of vertices . in general, tsba outperforms spba on average , as also demonstrated in a number of other simulation results ( omitted due to the space limit ) .we studied the complexity and developed optimal and approximation algorithms for the thinnest path problem for secure communications in wireless ad hoc networks . in establishing the np - completeness of the problem, our techniques of using directed crosses and exposed disk hypergraphs may spark new tools for complexity studies in geometrical hypergraphs and graphs .the bounding techniques and the use of sphere packing results in analyzing the performance of the two approximation algorithms may also find other applications in algorithmic analysis .consider an mds problem in a graph with vertices .we construct a directed hypergraph based on as follows .the vertex set of includes the vertices of augmented by a destination vertex and super vertices .a super vertex corresponds to the normal vertex and is a set of normal vertices .the hyperedges in are all rooted at the normal vertices .specifically , rooted at ( ) are directed hyperedges , where is the degree of in .each hyperedge rooted at has two destinations : and a super vertex whose corresponding normal vertex dominates in the original graph .[ fig : rsh ] is an example illustrating the construction of from .  from the construction of , we see that any path from to must traverse through all normal vertices one by one .there are multiple hyperedges leading from to , each involving a super vertex that corresponds to a dominating node of in .thus , choosing a hyperedge going from to is equivalent to choosing a dominating node of in .since every path from to includes all the normal vertices , the thinnest path is given by the one with the minimum number of super vertices , thus leading to the mds in .at this point , the size of a super vertex can be any positive integer . as will become clear later , to implement under a 2-d disk model , additional normal vertices need to be added . as a consequence , paths from  may include different number of normal vertices . to preserve the reduction, we need to make sure that the width of a path is dominated by the number of super vertices it covers .this can be achieved by choosing an sufficiently large ( see [ subsec : appa - mp ] )  [ c] [ c] [ c] [ c] [ c] [ c] [ c] [ c] [ c] [ c] [ c] [ c] [ c] [ c] [ c] [ c] [ c](a ) [ c](b )  the following lemma formally establishes the correctness of the reduction .[ lma : gelh ] there is a dominating set with size in if and only if there is a path from to in with width .first , assume that has a dominating set with size .by the definition of dominating set , for each vertex in , there is a vertex that dominates . from the construction of ,there exists a hyperedge ( ) in directing from to vertex and super vertex corresponding to the dominating node in .thus , the hyperpath is a path from to with width .the width comes from the fact that all vertices in is on the path along with super vertices , each consisting of normal vertices .conversely , assume a path from to in with width .based on the construction of , every path from to consists of hyperedges rooted at each of the normal vertices .thus , a path with width must contain super vertices . from the construction of the hyperedges , we conclude that the vertices in that correspond to those super vertices along the given path form a dominating set with size .the directed hypergraph obtained above does not satisfy the geometric properties of 2-d disk hypergraphs . to prove theorem [ thm : np-2ddh ] , we need to modify to a 2-d disk hypergraph while preserving the reduction from mds in .our approach is to realizing the topological structure of each hyperedge in by adding additional vertices with carefully chosen locations and maximum ranges to lead from the source vertex to the destination vertices of this hyperedge .the number of additional vertices , however , should be kept at a polynomial order with the problem size to ensure the polynomial nature of the reduction .this can be achieved by adding vertices on a 2-d grid with a constant grid spacing , which allows a constant maximum range thus polynomially many additional vertices .the detailed implementation of under a 2-d disk model is given in the next subsection .as a preparation step , we show in this subsection that the hyperedges in can be represented by line segments of a 2-d grid with a constant grid spacing .we first embed the normal vertices of evenly in a horizontal line in a 2-d space ( see fig . [fig : rsg_r ] for an illustration ) . below this line is a unit grid .there are vertical lines between and ( ) that are partitioned into three zones ( , , ) of , , and vertical lines , respectively .the super vertices are embedded evenly on a horizontal line below the grid .the horizontal position of super vertex is between and .next , we specify how a hyperedge traverses on the grid from its source vertex to its destination vertices . recall that every hyperedge in directs from a normal vertex to a super vertex and the next normal vertex . to preserve the reduction , we need to ensure that each hyperedge can only reach its normal vertex destination after reaching its super vertex destination . to facilitate the implementation around the super vertices ( see [ subsubsec : super ] ) , we designate the middle zone between and for traveling down to super vertex and then up to the corresponding normal vertex destination ( see region in fig .[ fig : rsg_r ] ) .each hyperedge involving has two designated vertical lines in ( one for going down to , the other going up from the super vertex ) . to connect the designated vertical lines in zone with the source vertex and then to the normal destination vertex , we designate two horizontal lines for each hyperedge .the traverse of the hyperedge completes by designating one vertical line in and one in to connect the normal vertices with the corresponding designated horizontal lines .since there are at most hyperedges , the designed grid size is sufficient to ensure that each hyperedge traverses through a distinct set of line segments in the grid . .[ c] [ c] [ c] [ c] [ c] [ c] [ c] [ c] [ c] [ c] [ c] [ c] [ c] [ c]   based on the 2-d grid representation of , we can construct a 2-d disk hypergraph that preserves the reduction .specifically , we place a sequence of evenly spaced normal vertices with a constant maximum range along the line segments in the grid that form each hyperedge of .the distance of two adjacent vertices is set to their maximum range .the constant maximum range can be set sufficiently small ( say , ) to avoid overhearing across vertices on different hyperedges that may render the reduction invalid .there are two issues remain to be addressed : the implementation of crosses and that around super vertices .the line crossing in the grid representation of make overhearing across hyperedges inevitable .however , by exploiting the freedom of choosing the maximum range for each vertex , we can implement _ directed _ crosses that allow us to preserve the reduction . specifically ,when two line segments in the grid representation cross , we can choose the maximum ranges of the vertices along these two lines in such a way that messages transmitted over one line can be heard by vertices on the other but not vise verse .a specific implementation is given in fig .[ fig : cross_1 ] .next , we show how carefully choosing the direction of each cross allows us to preserve the reduction .the cross directions are defined by assigning a level index to each line segment in the grid representation .specifically , for a hyperedge rooted at in , its line segments before and after reaching the super vertex destination have levels and , respectively .then , each cross has a direction pointing from the higher level segment to the lower one ( messages transmitted on the higher level segment can be heard by the vertices along the lower level segment but not vise verse ) .if the two segments have the save level , the direction of the cross can be arbitrary . to see that this directed implementation of crosses preserves the reduction, we only need to notice that any path from to still needs to go through all the normal vertices one by one and must reach one super vertex before reaching the next normal vertex .recall that a super vertex in is a set of normal vertices that have no outgoing hyperedges .it can be implemented by points with zero maximum range and located sufficiently close to each other ( so that any path from to in includes either all of them or none of them ) .consider first the implementation of one incoming hyperedge to a super vertex . recall that in the 2-d grid representation of , a hyperedge approaches and leaves through two vertical lines in zone ( see fig .[ fig : rsg_r ] ) .one implementation of this u - turn around is to add normal vertices with specific maximum ranges and locations . as shown in fig .[ fig : super_2 ] , these vertices include three anchor vertices , , and with maximum range , two interface vertices and that connect with the grid , and a bridging vertex , all with maximum range . the value of and the connection with the grid will be specified later .[ c] [ c] [ c] [ c] [ c] [ c] [ c]  the challenge remains in the implementation of up to incoming hyperedges to the same super vertex .note that under a 2-d disk model , one can at most have vertices ( even with arbitrary ranges ) that reach a common sixth vertex but not each other .the key to circumvent this difficulty is to allow directed overhearing , similar to the idea behind the implementation of the crosses .specifically , the reduction is preserved as long as a hyperedge rooted at can not overhear a message transmitted over a hyperedge rooted at for any .the detailed implementation is as follows .the fist step is to designate the vertical lines in zone to the incoming hyperedges of based on the indexes of their source vertices .specifically , the incoming hyperedge with the smallest source vertex index takes the two most center lines in , and so on .consider first the implementation of the two incoming hyperedges ( say , and ) with the smallest source vertex indexes .as shown in fig .[ fig : super_2_2 ] , we first implement as described above ( see fig . [ fig : super_2 ] ) .the structure of the implementation of is similar except that the maximum range of the anchor vertices , , and is set to to present unwanted overhearing . as a consequence ,more bridging vertices ( with maximum range , , and , respectively ) are needed to connect the interface vertex to the anchor vertex .note that no vertices along ( the centers of the blue circles in fig .[ fig : super_2_2 ] ) are in the range of any vertices along ( the green circles ) .the correct direction of overhearing is thus ensured .the same procedure continues for any additional incoming hyperedges to , in the ascending order of their source vertex indexes in .note that the range of the anchor vertices in the hyperedge is , growing exponentially with .the maximum ranges ( specifically , ) of the bridging vertices are chosen to preserve the polynomial nature of the reduction . in this way , the number of additional vertices for implementing the hyperedge is , and the total number of additional vertices around one super vertex is at most .[ c] [ c] [ c] [ c] [ c] [ c] [ c] [ c] [ c]  next we consider the value of which should be set sufficiently small to avoid overhearing across hyperedges leading to different super vertices .note that the width of the area covered by the additional vertices around a super vertex is times the largest maximum range of the anchor vertices .we thus set , considering the distance between two adjacent super vertices being .the last issue is to connect the interface vertices with the grid .each interface vertex needs to be connected with a designated vertical line in .while the vertical lines in are evenly spaced , the horizontal positions of the interface vertices have an exponential structure due to the exponentially growing range of the anchor vertices .furthermore , the vertices realizing the vertical lines in the grid have a constant range , whereas the interface vertices have an exponentially smaller range of . if we connect them using a sequence of vertices with a constant range , unwanted overhearing will occur near the interface vertices . on the other hand , connecting them using vertices with range results in an exponential number of additional vertices . to preserve the correctness and the polynomial nature of the reduction, we propose the scheme detailed in fig .[ fig : gridinterfacegeo ] .[ c] [ c] [ c] [ c] [ c] [ c] [ c] [ c] [ c]  since the generated sequence of circles are within the boundary given by lines and and the boundary lines corresponding to different interface vertices do not cross ( see fig .[ fig : super_3 ] ) , the above scheme does not introduce overhearing , thus preserving the reduction .the polynomial nature of the reduction can be shown based on the following lemma .[ c] [ c]  [ lma : super ] consider the geometrical scheme described in fig .[ fig : gridinterfacegeo ] .assume .the number of circles , denoted by , satisfies when , and when , where , denote the radiuses of circles and , and the distance between lines and .assume first .without loss of generality , assume . since line and are parallel , the three lines , and intersect at one point , denoted by in fig . [ fig : gridinterfacegeo ] .let , , and denote the angles , and , respectively .it can be shown that all the circles tangent with the same boundary line . without loss of generality , assume that they tangent with line , and . based on simple geometry , the lengths of the line segments of forms an equal ratio sequence : with .we thus have based on the stopping condition of the procedure , the number of circles is given by the minimum index such that .we thus have   since and are similar triangles , the ratio equals to the ratio .also because for , ( [ equ : super_1 ] ) can be written as because is a triangle and , the value of can be lower bounded as the following :   furthermore , since , the length of has a upper bound : . hence ( [ equ : super_2 ] ) leads to : substituting ( [ equ : super_4 ] ) into ( [ equ : super_3 ] ) , we have   consider next .the sequence of circles have the same radius . since , the bound holds .  to satisfy the assumption of in lemma [ lma : super ], we set the distance between the last horizontal line of the grid and the horizontal line of super vertices to .this ensures the angle .note that in the downward part from the grid to a left interface vertex , is a constant and . hence the bound on given in lemma [ lma : super ] can be written as : which is in the order of .similar argument can be made in the upward part where and is a constant .the same holds for .hence the total number of additional vertices to connect the grid to the interface vertices of a super vertex is in the order of .with constructed , we now establish the correctness of the reduction from the mds in to the tp from to in .[ lma : dh ] let where is the total number of normal vertices in .there is a dominating set with size in if and only if there is a path from to in with width between and .the chosen value of ensures that the width of a path from to is dominated by the number of super vertices that it covers .the correctness of the reduction thus follows from the same arguments in the proof of lemma [ lma : gelh ] based on the construction of  .the polynomial nature of the reduction is clear from the construction of .we thus arrive at theorem [ thm : np-2ddh ] .consider a tp problem from to in a -d exposed disk hypergraphs .we construct a -d udh as follows .first , the normal vertex set of is given by , except that the ranges of any equals to .next , for each vertex , we place a super vertex in ( i.e. , the exposed area of the corresponding vertex in ) that contains normal vertices located sufficiently normal vertices are sufficiently close such that any transmission from one of these vertices to a vertex outside this super vertex reaches all the normal vertices in this super vertex . ] close to each other .the super vertices have the same range as the normal vertices in , ensuring is a udh .the reduction can thus be seen by noticing that while the enlarged ranges introduce additional hyperedges in , these hyperedges can not be on a thinnest path due to the fact that they all contain at least one super vertex .in this proof , we modify the -d disk hypergraph in the proof of theorem [ thm : np-2ddh ] to a -d exposed disk hypergraph while preserving the polynomial reduction . based on the definition , a sufficient condition for a -d disk hypergraph to be exposed is that none of the maximum range disks are completely inside any other .the vertices in for realizing the line segments of the grid satisfy this condition .we only need to modify the implementations of the crosses and around the super vertices .[ c](a ) [ c](b ) [ c](c ) [ c](d ) [ c] [ c] [ c] [ c] [ c] [ c]  in the implementation of directed crosses in ( see fig .[ fig : cross_1 ] ) , some vertices on the line with a lower level index may have an empty exposed area ( see the red disks in fig .[ fig : cross_1 ] that are completely covered by blue ones ) . to implement a direct cross in a -d exposed disk hypergraph , the maximum ranges of vertices on the line with a lower level index need to be small enough to preserve the direction of the cross but also large enough to make the vertices exposed .we propose the scheme described in fig .[ fig : expose_cross ] .   in the previous implementation around a super vertex ,all the vertices are exposed except the anchor vertices side and the bridging vertices .however , we notice that these vertices would all be exposed if there were no interface vertices .our solution is thus to move all the interface vertices away from its original position with a constant distance and adding a constant number of vertices to connect each new interface vertex to the bridging vertex or the anchor vertex on the right side .a detailed implementation is shown in fig .[ fig : expose_super ] .consider an mds problem in a graph with a maximum degree of .we first follow the first two steps in the proof of theorem [ thm : np-2ddh ] to build the grid representation of hypergraph .note that due to the unit range of all vertices , we set the size of the grid to a constant greater than ( say , ) to avoid unwanted overhearing .next , we implement this representation in a -d udg while preserving the reduction .any line segment of hyperedges in is replaced by a sequence of unit disks , one tangent to another .any cross between two line segments can be easily implemented by using the third dimension , as shown in fig .[ fig:3udg_cross ] . in this implementation, there is no overhearing between vertices on these two line segments at all .since has a maximum degree of , there are at most hyperedges passing through a super vertex .it can be easily implemented without any unwanted overhearings ( see fig . [ fig:3udg_super ] ) . to prevent the super vertices from relaying messages, we place a _ mega _vertex besides each super vertex .this mega vertex is only within the range of this super vertex and contains more normal vertices than the total number of normal vertices in the reduced graph ( including the normal vertices contained in all the super vertices but not those in other mega vertices ) . in this way , a path via any super vertex covers at least one mega vertex , thus can not be the thinnest path .[ fig:3udg_super ] illustrates the implementation around a super vertex . in this case, there are at most incoming hyperedges . along with the mega vertex, they can be packed around a super vertex without overhearing . ] .the correctness of the reduction follows from the same arguments as in the proof of lemma [ lma : gelh ] .[ c]mega vertexwe first show that as long as there exists a path from to , there exists a path from to that traverses only in the sub - hypergraph .this can be shown by noticing that must hear the message from before and any vertex to the right of .this is due to the monotonicity of wireless broadcast and the definition of predecessor .consequently , there must exist a path from to in .since is covered by the hyperedge leading from to in , the concatenation of with any path to in covers the same set of vertices .specifically , the cover of the path returned by nbi is the set of vertices located between ( and including ) and . since any path from to covers this set of vertices , the correctness of the algorithm is established .next , we prove the property of under the disk propagation model . we first state the following lemma that follows directly from triangle inequality .[ lma : cir ] let and denote two closed balls in with radiuses and , respectively .let denote the distance between the centers of and .if , then .  based on lemma [ lma : cir ] , for any vertex between and , we have . therefore ( let ) .next , consider an arbitrary path from to .we show that for any ( ) , . specifically , since must first hear the message from or a vertex to the left of , is a subset of the covered area of this hop in based on lemma [ lma : cir ] .this completes the proof .let denote the path from to provided by spba and the thinnest path .if multiple thinnest path exists , let be the one with the minimum number of hyperedges .let denote the length ( the sum of hyperedge weights ) of .since each vertex covered in ( except the source ) contributes to the weight of at least one hyperedge in , the width is no larger than the length of this path plus one .also because is the shortest path , its length is no larger than the length of .we thus have we then obtain the approximation ratio by deriving an upper bound of as a function of .note that the destination set of hyperedge on can not contain vertices : its own source vertex and vertices in .the later holds because otherwise is not the the thinnest path with minimum number of hyperedges .we thus have where ( [ equ : approx_spba_1 ] ) comes from . substituting ( [ equ : approx_spba_1 ] ) into ( [ equ : approx_spba_basic ] ), we have where ( [ equ : approx_spba_2 ] ) holds since .  based on ( [ equ : approx_spba_2 ] ) , if , then .otherwise , we have . in summary, spba provides a approximation .since a ring hypergraph is a special directed hypergraph , all the analysis in the previous subsection applies . specifically , inequality ( [ equ : approx_spba_basic ] ) holds .the problem then remains on obtaining a tighter upper bound of based on the geometrical properties of ring hypergraphs .first , note that the length of a hyperpath equals to the sum of the number of times each vertex is reached .let denote the set of hyperedges on that include in their destination sets , now we construct a subset of by iteratively removing one from any pair of hyperedges whose positions in are adjacent until no such pair exists . because at most half of the hyperedges are removed from , the size of is at least half of the size of , in another word .let and denote the largest maximum range and the smallest minimum range among all vertices in the given ring hypergraph , respectively .let be the larger one between and the smallest distance between any two vertices in .based on the construction of , the set of source vertices of hyperedges in satisfies two properties .first , based on the definition of ring hypergraphs , the distance between any source vertex in the set and is no larger than the maximum range of this vertex and hence no larger than .second , the distances between any two source vertices in the set are larger than and hence .otherwise the two hyperedges rooted at these two vertices can reach the source vertex of each other and hence they are adjacent in ( recall that can not reach any vertex in ) .  given these two properties ,the size of thus is upper bounded by the maximum number of points in the euclidean space that are at most away from and at least apart from each other .this is equivalent to a sphere packing problem of arranging the maximum number of small spheres with radius inside a large sphere with radius .an upper bound of this packing problem is the ratio between the volumes of the large and small spheres .we thus have : .recall that .note that the destination can only be reached by the last hyperedge and hence .we thus have substituting ( [ equ : approx_spba_ring_2 ] ) into ( [ equ : approx_spba_basic ] ) .we have spba provides a -approximation for tp in -d ring hypergrpahs .we now prove that -ratio is asymptotically tight even for -d disk hypergrpahs .the proof has two steps .first , we construct a directed hypergraph for which the worst case ratio is asymptotically reached .next , we show a 2-d disk implementation of .  consider the the following hypergraph illustrated in fig .[ fig : example_spba ] with red vertices and blue vertices along with the source and the destination .each red vertex has one outgoing hyperedge with ( let denote ) .each blue has one outgoing hyperedge  with ( let denote ) . finally , we add two hyperedges that connect source to and respectively .[ c] [ c] [ c] [ c] [ c] [ c] [ c] [ c] [ c] [ c] .since the shortest path traverses through the blue hyperedges while the thinnest path through the red ones , the approximation ratio is given by : note that the total number of vertices is when is large , and .next , we implement the above hypergraph under a 2-d disk model as illustrated in fig .[ fig : spdhexample ] .the red vertices are located on a straight line with for .the source vertex is located on the line to the left of , and both its maximum range and its distance to equals to .the terminal vertex has a maximum range of and is located to the right of with a distance of .the maximum range of a blue vertex is where is a small positive value to prevent from overhearing messages transmitted by . andthe blue vertices are located on a route from to that contains two vertical line segments of length and a horizontal one of length , as demonstrated by the blue dash lines in fig .[ fig : spdhexample ] .the positive parameter is used to prevent a blue vertex from overhearing the last red vertex . in the asymptotic regime with large , can be set sufficiently large so that the blue vertices can be implemented along the depicted route from to .[ c] [ c] [ c] [ c] denote the path in hypergraph from to given by the tsba algortihm and the thinnest path .let and denote the paths from to a vertex given by spba and tsba , respectively .the following lemma establishes a property of .let denote the thinnest path .for the ease of presentation , let the sequence of source vertices and the final destination be denoted as .let . based on lemma [ lma : dp_property ], we have , for all , where ( [ equ : approx_tsba_1 ] ) holds since does not contain vertices in . summing ( [ equ : approx_tsba_1 ] ) over , and noticing and , we have : next , since we can upper bound by for any . thus the right side of this inequality is a quadratic function of with the maximum at .we thus have       let be the shortest path from to .let denote the source vertex of and .we prove , through induction , the following inequality for all :    now assume that ( [ equ : lma_ratio2 ] ) holds for , . based on lemma [ lma : dp_property ] and this induction assumption , we have : this completes the induction . considering , we have     we first construct a directed hypergraph as illustrated in fig .[ fig : dpexample1 ] .the vertex set of consists of two types of vertices : normal vertices and , and super vertices , each containing normal vertices .rooted at each normal vertex are two hyperedges and .hyperedge has destination vertices and hyperedge has destination vertices .it is easy to see that the thinnest path from to is with width .however , tsba returns the path with width in the worst case can update through both and with the same width .since the order of hyperedges used in the update is arbitrary , could be used to update for all in the worst case . ] .the approximation ratio is   [ c] [ c] [ c] [ c] [ c] [ c] [ c] [ c] [ c] [ c] [ c] [ c] [ c] [ c] [ c] [ c] [ c]  given the similarity between and the hypergraph constructed in the proof of theorem [ thm : np-2ddh ] , we can follow the same approach given in [ app : np_2ddh ] to implement under a 2-d disk model .however , this implementation requires adding additional vertices ( referred to as auxiliary vertices ) that may render our previous approximation analysis invalid . to maintain the ratio , each original vertex ( including the vertices in a super vertex ) in replaced with vertices ( clustered together ) in its 2-d disk implementation , where is the number of auxiliary vertices introduced by the implementation . in this case, tsba returns a path that covers along with a set of auxiliary vertices .the thinnest path covers and another set of auxiliary vertices .the approximation ratio in this -d disk hypergraph is given by where and denote the number of auxiliary vertices covered by the path returned by tsba and the thinnest path . since and , when is large , we have , the approximation ratio is asymptotically tight in -d disk hypergraphs .

Abstract: we formulate and study the thinnest path problem in wireless ad hoc networks . the objective is to find a path from a source to its destination that results in the minimum number of nodes overhearing the message by a judicious choice of relaying nodes and their corresponding transmission power . we adopt a directed hypergraph model of the problem and establish the np - completeness of the problem in 2-d networks . we then develop two polynomial - time approximation algorithms that offer and approximation ratios for general directed hypergraphs ( which can model non - isomorphic signal propagation in space ) and constant approximation ratios for ring hypergraphs ( which result from isomorphic signal propagation ) . we also consider the thinnest path problem in 1-d networks and 1-d networks embedded in 2-d field of eavesdroppers with arbitrary unknown locations ( the so - called 1.5-d networks ) . we propose a linear - complexity algorithm based on nested backward induction that obtains the optimal solution to both 1-d and 1.5-d networks . this algorithm does not require the knowledge of eavesdropper locations and achieves the best performance offered by any algorithm that assumes complete location information of the eavesdroppers .