ﻻ يوجد ملخص باللغة العربية
In the decremental single-source shortest paths (SSSP) problem, the input is an undirected graph $G=(V,E)$ with $n$ vertices and $m$ edges undergoing edge deletions, together with a fixed source vertex $sin V$. The goal is to maintain a data structure that supports shortest-path queries: given a vertex $vin V$, quickly return an (approximate) shortest path from $s$ to $v$. The decremental all-pairs shortest paths (APSP) problem is defined similarly, but now the shortest-path queries are allowed between any pair of vertices of $V$. Both problems have been studied extensively since the 80s, and algorithms with near-optimal total update time and query time have been discovered for them. Unfortunately, all these algorithms are randomized and, more importantly, they need to assume an oblivious adversary. Our first result is a deterministic algorithm for the decremental SSSP problem on weighted graphs with $O(n^{2+o(1)})$ total update time, that supports $(1+epsilon)$-approximate shortest-path queries, with query time $O(|P|cdot n^{o(1)})$, where $P$ is the returned path. This is the first $(1+epsilon)$-approximation algorithm against an adaptive adversary that supports shortest-path queries in time below $O(n)$, that breaks the $O(mn)$ total update time bound of the classical algorithm of Even and Shiloah from 1981. Our second result is a deterministic algorithm for the decremental APSP problem on unweighted graphs that achieves total update time $O(n^{2.5+delta})$, for any constant $delta>0$, supports approximate distance queries in $O(loglog n)$ time; the algorithm achieves an $O(1)$-multiplicative and $n^{o(1)}$-additive approximation on the path length. All previous algorithms for APSP either assume an oblivious adversary or have an $Omega(n^{3})$ total update time when $m=Omega(n^{2})$.
In the decremental $(1+epsilon)$-approximate Single-Source Shortest Path (SSSP) problem, we are given a graph $G=(V,E)$ with $n = |V|, m = |E|$, undergoing edge deletions, and a distinguished source $s in V$, and we are asked to process edge deletion
Let $G = (V,E,w)$ be a weighted, digraph subject to a sequence of adversarial edge deletions. In the decremental single-source reachability problem (SSR), we are given a fixed source $s$ and the goal is to maintain a data structure that can answer pa
We study the decremental All-Pairs Shortest Paths (APSP) problem in undirected edge-weighted graphs. The input to the problem is an $n$-vertex $m$-edge graph $G$ with non-negative edge lengths, that undergoes a sequence of edge deletions. The goal is
We consider the parameterized complexity of the problem of tracking shortest s-t paths in graphs, motivated by applications in security and wireless networks. Given an undirected and unweighted graph with a source s and a destination t, Tracking Shor
In this paper, we study the single-source shortest-path (SSSP) problem with positive edge weights, which is a notoriously hard problem in the parallel context. In practice, the $Delta$-stepping algorithm proposed by Meyer and Sanders has been widely