We initiate a thorough study of -- producing algorithms for the approximation problems of property testing in the CONGEST model. In particular, for the so-called testing model we emulate sequential tests for nearly all graph properties having @math -sided tests, while in the and models we obtain faster tests for triangle-freeness and bipartiteness respectively. In most cases, aided by parallelism, the distributed algorithms have a much shorter running time as compared to their counterparts from the sequential querying model of traditional property testing. The simplest property testing algorithms allow a relatively smooth transitioning to the distributed model. For the more complex tasks we develop new machinery that is of independent interest. This includes a method for distributed maintenance of multiple random walks.
Related to having information being sent to, or received by, a central authority, is the concept of proof-labelling schemes, introduced by @cite_9 (for extensions see, e.g., @cite_21 ). In this setting, each vertex is given some external label, and by exchanging labels the vertices need to decide whether a given property of the graph holds. This is different from our setting in which no information other than vertex IDs is available. Another setting that is related to proof-labelling schemes, but differs from our model, is the prover-verifier model of @cite_35 .
@cite_35 @cite_9 @cite_21
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Given a graph G, a function f:V(G)->Z, and an initial 0 1-vertex-labelling c"1:V(G)-> 0,1 , we study an iterative 0 1-vertex-labelling process on G where in each round every vertex v never changes its label from 1 to 0, and changes its label from 0 to 1 if at least f(v) neighbours have label 1. Such processes model opinion disease spreading or fault propagation and have been studied under names such as irreversible threshold majority processes in a large variety of contexts. Our contributions concern computational aspects related to the minimum cardinality irr"f(G) of sets of vertices with initial label 1 such that during the process on G all vertices eventually change their label to 1. Such sets are known as irreversible conversion sets, dynamic irreversible monopolies, or catastrophic fault patterns. Answering a question posed by Dreyer and Roberts [P.A. Dreyer Jr., F.S. Roberts, Irreversible k-threshold processes: graph-theoretical threshold models of the spread of disease and of opinion, Discrete Appl. Math. 157 (2009) 1615-1627], we prove a hardness result for irr"f(G) where f(v)=2 for every [email protected]?V(G). Furthermore, we describe a general reduction principle for irr"f(G), which leads to efficient algorithms for graphs with simply structured blocks such as trees and chordal graphs. This paper addresses the problem of locally verifying global properties. Several natural questions are studied, such as “how expensive is local verification?” and more specifically, “how expensive is local verification compared to computation?” A suitable model is introduced in which these questions are studied in terms of the number of bits a vertex needs to communicate. The model includes the definition of a proof labeling scheme (a pair of algorithms- one to assign the labels, and one to use them to verify that the global property holds). In addition, approaches are presented for the efficient construction of schemes, and upper and lower bounds are established on the bit complexity of schemes for multiple basic problems. The paper also studies the role and cost of unique identities in terms of impossibility and complexity, in the context of proof labeling schemes. Previous studies on related questions deal with distributed algorithms that simultaneously compute a configuration and verify that this configuration has a certain desired property. It turns out that this combined approach enables the verification to be less costly sometimes, since the configuration is typically generated so as to be easily verifiable. In contrast, our approach separates the configuration design from the verification. That is, it first generates the desired configuration without bothering with the need to verify it, and then handles the task of constructing a suitable verification scheme. Our approach thus allows for a more modular design of algorithms, and has the potential to aid in verifying properties even when the original design of the structures for maintaining them was done without verification in mind. We study the problem of predicting the labelling of a graph. The graph is given and a trial sequence of (vertex,label) pairs is then incrementally revealed to the learner. On each trial a vertex is queried and the learner predicts a boolean label. The true label is then returned. The learner’s goal is to minimise mistaken predictions. We propose minimum p-seminorm interpolation to solve this problem. To this end we give a p-seminorm on the space of graph labellings. Thus on every trial we predict using the labelling which minimises the p-seminorm and is also consistent with the revealed (vertex, label) pairs. When p = 2 this is the harmonic energy minimisation procedure of [22], also called (Laplacian) interpolated regularisation in [1]. In the limit as p → 1 this is equivalent to predicting with a label-consistent mincut. We give mistake bounds relative to a label-consistent mincut and a resistive cover of the graph. We say an edge is cut with respect to a labelling if the connected vertices have disagreeing labels. We find that minimising the p-seminorm with p = 1 + where → 0 as the graph diameter D → ∞ gives a bound of O(Φ logD) versus a bound of O(ΦD) when p = 2 where Φ is the number of cut edges. We consider vertex-labeled graphs, where each vertex v is attached with a label from a set of labels. The vertex-to-label distance query desires the length of the shortest path from the given vertex to the set of vertices with the given label. We show how to construct an oracle for a vertex-labeled planar graph, such that (O( 1 n n) ) storing space is needed, and any vertex-to-label query can be answered in (O( 1 n ) ) time with stretch 1 + e. Here, Δ is the hop-diameter of the given graph. For the case that Δ = O(logn), we construct a distance oracle that achieves (O( 1 n) ) query time, without changing space usage.