Note on distributed certification of minimum spanning trees

Abstract

A distributed proof (also known as local certification, or proof-labeling scheme) is a mechanism to certify that the solution to a graph problem is correct. It takes the form of an assignment of labels to the nodes, that can be checked locally. There exists such a proof for the minimum spanning tree problem, using O(log n log W ) bit labels (where n is the number of nodes in the graph, and W is the largest weight of an edge). This is due to Korman and Kutten who describe it in concise and formal manner in [7]. In this note, we propose a more intuitive description of the result, as well as a gentle introduction to the problem.This note originates from a careful reading of [7], while working on [2]. Comments are most welcome. Funding Support by ANR ESTATE IntroductionDistributed checking of minimum spanning tree. The problem is the following: we are given a weighted graph where some edges are selected, and we want to enable the nodes of the graph to collectively check whether the selected edges form a minimum spanning tree (MST) of this weighted graph or not.More precisely, we want the nodes to take a distributed decision. For this, every node takes its own decision whether to accept or to reject, and the collective decision is considered to be an acceptance, if and only if, all nodes accept. The difficulty is that the nodes have a limited knowledge of the graph. Each node has a local view that contains only its adjacent edges and nodes, along with the weight of these adjacent edges and whether they are selected or not. In addition, we assume that the nodes are given unique identifiers.Actually we will need to provide more information to the nodes. Indeed, it is impossible to check whether the selected edges form a minimum spanning tree or not in this restricted setting, as the following example shows. Consider a ring, where the nodes have arbitrary distinct identifiers, and all edges are selected. Now take an arbitrary node. Given its local view, it cannot distinguish whether it is indeed in a ring, or if it is in the middle of a long path, where all edges are selected. If this node chooses to reject, then the distributed decision will automatically be a rejection, and in the long path this would be a wrong decision. Thus it has to accept. But then, in the ring, by symmetry, every node has to accept, and then the distributed decision is an acceptance, although the ring is not a correct instance.Distributed proof. The mechanism used to bypass the impossibility above is called a distributed proof. In such a mechanism, every node will be given a label, and a node can see not only its own label but also the ones of its neighbors. These labels are supposed to certify the correctness of the minimum spanning tree, in the following sense:If the set of selected edges form a minimum spanning tree, then there exists a label assignment such that all nodes accept. If the set of selected edges does not form a minimum spanning tree, then for any label assignment, at least one node will reject.

Introduction to local certification

“…Actually one can prove a more fine-grain result: minimum spanning tree has proofs of size Θ(log n log W ), where W is the maximum weight of an edge. The proof of this statement is in [32], and a more intuitive explanation of the upper bound can be found in [15].…”