Distributedly Testing Cycle-Freeness

Abstract

Abstract. We tackle local distributed testing of graph properties. This framework is well suited to contexts in which data dispersed among the nodes of a network can be collected by some central authority (like in, e.g., sensor networks). In local distributed testing, each node can provide the central authority with just a few information about what it perceives from its neighboring environment, and, based on the collected information, the central authority is aiming at deciding whether or not the network satisfies some property. We analyze in depth the prominent example of checking cycle-freeness, and establish tight bounds on the amount of information to be transferred by each node to the central authority for deciding cycle-freeness. In particular, we show that distributedly testing cycle-freeness requires at least ⌈log d⌉ − 1 bits of information per node in graphs with maximum degree d, even for connected graphs. Our proof is based on a novel version of the seminal result by Naor and Stockmeyer (1995) enabling to reduce the study of certain kinds of algorithms to order-invariant algorithms, and on an appropriate use of the known fact that every free group can be linearly ordered.

Randomized Local Network Computing

“…Observe that, since the language L in the statement of the theorem may not be in LD, this first stage of the proof requires a different approach as the one in [28], because the reduction to orderinvariant algorithms in [28] requires the languages to be in LD. Nevertheless, it was recently shown (see Theorem 1 in [3]) that the LD assumption is not necessary. The result hereafter does not even need L ∈ BPLD.…”

“…In fact, as far as the order-invariant reduction is concerned, the local checkability assumption is actually not much of an issue, as shown in [3], as long as the node identities are not restricted in size. Indeed, if the only requirement is that the identities given to the nodes are pairwise distinct integers, then [3] proved that any constant-time (deterministic) algorithm can be turned to an order-invariant algorithm performing in the same amount of time.…”

“…Indeed, if the only requirement is that the identities given to the nodes are pairwise distinct integers, then [3] proved that any constant-time (deterministic) algorithm can be turned to an order-invariant algorithm performing in the same amount of time.…”

Randomized Proof-Labeling Schemes

“…Proof-labeling schemes, where nodes may communicate at distance greater than 1, i.e., may take their individual decision based on the labels of the nodes in their vicinity at distance t > 1, was recently studied in [20]. Finally, distributed decision and verification processes in which the global interpretation of the collection of individual outputs is not restricted to be the logical conjunction of these outputs has been studied in [5,6].…”

Solving the Induced Subgraph Problem in the Randomized Multiparty Simultaneous Messages Model

“…In [7] the authors prove that it is impossible to decide whether the input graph G has diameter at most 3 or whether G has a triangle unless the messages sent by the nodes are all of size Ω(n), even if randomness is allowed. Deciding whether the input graph G contains a cycle requires at least one node to write a message of length at least log d − 1, where d is the maximum degree of G [4].…”