Sampling and output estimation in distributed algorithms and LCAs

We consider the distributed message-passing model and the Local Computational Algorithms (LCA) model. In both models a network is represented by an n-vertex graph G=(V,E). We focus on labeling problems, such as vertex-coloring, edge-coloring, maximal independent set (MIS) and maximal matching. In the distributed model the vertices of v perform computations in parallel in order to compute their own solution for solving the problem for G. In contrast, in the LCA model probes are performed on certain vertices in order to compute their labels in a solution to a given problem. In this work we study the possibility of estimating a solution produced by an algorithm, much before the algorithm terminates. This estimation not only allows for size approximation of a solution, but also for early detection of failure in randomized algorithms. We do this such that a correcting procedure can be executed. To this end, we propose a sampling technique, in which the labels in the sampling are distributed proportionally to the distribution in the algorithm's output. However, the sampling running time is significantly smaller than that of the algorithm in hand. We achieve the following results, in terms of the maximum degree Δ and the arboricity a of the input graph. The running time of our procedures is O(log⁡a+log⁡log⁡n), for sampling vertex-coloring, edge-coloring, maximal matching and MIS. This significantly improves upon previous sampling techniques, which incur additional dependency on the maximum degree Δ that can be much higher than the arboricity, as well as more significant dependency on n. Not only that, we also show that our technique extends naturally for the power graph G^r for any constant integer r>1 for the problems of MIS and coloring. Our techniques for sampling in the distributed model provide a powerful and general tool for estimation in the LCA model. In this setting the goal is estimating the size of a solution to a given problem, by making as few vertex probes as possible. For the above-mentioned problems, we achieve estimations with probe complexity d^{O(log⁡a+log⁡log⁡n)}, where d=min(Δ,a⋅poly(log⁡(n)). Our results extend as well to power graphs for the coloring and MIS problems.