Silent MST approximation for tiny memory

Abstract

In network distributed computing, minimum spanning tree (MST) is one of the key problems, and silent self-stabilization one of the most demanding fault-tolerance properties. For this problem and this model, a polynomial-time algorithm with O(log 2 n) memory is known for the state model. This is memory optimal for weights in the classic [1, poly(n)] range (where n is the size of the network). In this paper, we go below this O(log 2 n) memory, using approximation and parametrized complexity.More specifically, our contributions are two-fold. We introduce a second parameter s, which is the space needed to encode a weight, and we design a silent polynomial-time self-stabilizing algorithm, with space O(log n • s). In turn, this allows us to get an approximation algorithm for the problem, with a trade-off between the approximation ratio of the solution and the space used. For polynomial weights, this trade-off goes smoothly from memory O(log n) for an n-approximation, to memory O(log 2 n) for exact solutions, with for example memory O(log n log log n) for a 2-approximation.

Note on distributed certification of minimum spanning trees

“…This can be problematic if we want to build the certificates in a distributed manner without exceeding some space limit. A recent preprint [2] tackles the problem of how to construct the labels to a get an optimal space self-stabilizing algorithm, and solve this problem by allowing the nodes to check the size of their labels.…”

“…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.…”

“…This note originates from a careful reading of [7], while working on [2]. Comments are most welcome.…”

Introduction to local certification

“…A general open problem is to determine which results from distributed decision can be used for self-stabilization. A more concrete one is the following: in [8] the self-stabilizing algorithm for MST first computes the solution, and then computes the certificates, and this is unusual: algorithms usually compute the solution and its certification at the same time. The paper [8] provides some elements showing that this form might be necessary, but no complete proof is known.…”

“…Second, many lower bounds work only in a ring, which tells us little about denser topologies. Third, the only result we have with other parameters is the Θ(log n log W ) bound for minimum spanning tree [32], where W is the maximum edge weight; and this parametrization led to improvements of self-stabilizing algorithms [8].…”