Deterministic Partially Dynamic Single Source Shortest Paths for Sparse Graphs

Abstract

In this paper we consider the decremental single-source shortest paths (SSSP) problem, where given a graph G and a source node s the goal is to maintain shortest paths between s and all other nodes in G under a sequence of online adversarial edge deletions. (Our algorithm can also be modified to work in the incremental setting, where the graph is initially empty and subject to a sequence of online adversarial edge insertions.)In their seminal work, Even and Shiloach [JACM 1981] presented an exact solution to the problem with only O(mn) total update time over all edge deletions. Later papers presented conditional lower bounds showing that O(mn) is optimal up to log factors.In SODA 2011, Bernstein and Roditty showed how to bypass these lower bounds and improve upon the Even and Shiloach O(mn) total update time bound by allowing a (1 + ) approximation. This triggered a series of new results, culminating in a recent breakthrough of Henzinger, Krinninger and Nanongkai [FOCS 14], who presented a (1 + )-approximate algorithm whose total update time is near linear: O(m 1+O(1/ √ log n) ). However, every single one of these improvements over the Even-Shiloach algorithm was randomized and assumed a non-adaptive adversary. This additional assumption meant that the algorithms were not suitable for certain settings and could not be used as a black box data structure. Very recently Bernstein and Chechik presented in STOC 2016 the first deterministic improvement over Even and Shiloach, that did not rely on randomization or assumptions about the adversary: in an undirected unweighted graph the algorithm maintains (1+ )-approximate distances and has total update timeÕ(n 2 ). In this paper, we present a new deterministic algorithm for the problem with total update timeÕ(n 1.25 √ m) = O(mn 3/4 ): it returns a (1 + ) approximation, and is limited to undirected unweighted graphs. Although this result is still far from matching the randomized near-linear total update time, it presents important progress towards that direction, because unlike the STOC 2016Õ(n 2 ) algorithm it beats the Even and Shiloach O(mn) bound for all graphs, not just sufficiently dense ones. In particular, theÕ(n 2 ) algorithm relied entirely on a new sparsification technique, and so could not hope to yield an improvement for sparse graphs. We present the first deterministic improvement for sparse graphs

Dynamic Algorithms for Graph Coloring

“…We wish to bound this probability Pr[E τ ], which is evaluated over the past random choices of the algorithm which the adversary fixing the order of edge insertions is oblivious to. 6 We do this by using the principle of deferred decision.…”

“…We believe that this is an important question, since it may help in understanding how to develop deterministic dynamic algorithms in general. It is very challenging and interesting to design deterministic dynamic algorithms with performances similar to the randomized ones for many dynamic graph problems such as maximal matching [5,10,9,11,12,13], connectivity [24,28,32,27], and shortest paths [6,8,7,21,22].…”

Deterministic Algorithms for Decremental Approximate Shortest Paths: Faster and Simpler

“…This, improves the algorithms by Bernstein and Chechik [BC16, BC17, Ber17] over the entire sparsity m = O(n 1.5−o(1) ) and dominates it heavily for the important case when m = O(n) where our algorithm improves the current running time ofÕ(n 1+3/4 ) to just O(n 1.5+o(1) ) total update time. This matches a natural barrier to the problem encountered by all current approaches as pointed out by Bernstein and Chechik [BC17]. We present the algorithm for the more challenging decremental setting but point out that an adaption to the incremental setting is straight-forward.…”

“…To overcome this problem, Bernstein and Chechik [Che + 16] presented the first deterministic algorithm running inÕ(n 2 ) total time. They also presented an algorithm for sparse graphs [BC17] with update timeÕ(n 1.25 √ m) =Õ(mn 3/4 ). Bernstein [Ber17] later extended the algorithm for dense graphs to handle edge weights in [1, W ] with total update timeÕ(n 2 log W ).…”

“…They also presented an algorithm for sparse graphs [BC17] with update timeÕ(n 1.25 √ m) =Õ(mn 3/4 ). Bernstein [Ber17] later extended the algorithm for dense graphs to handle edge weights in [1, W ] with total update timeÕ(n 2 log W ). Each of these algorithms can answer queries for distance estimates in constant time but their fundamental technique of contracting vertices in dense components of the graphs makes it impossible to retrieve a corresponding path.…”