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In this thesis, we present new techniques to deal with fundamental algorithmic graph problems where graphs are directed and partially dynamic, i.e. undergo either a sequence of edge insertions or deletions: - Single-Source Reachability (SSR), - Strongly-Connected Components (SCCs), and - Single-Source Shortest Paths (SSSP). These problems have recently received an extraordinary amount of attention due to their role as subproblems in various more complex and notoriously hard graph problems, especially to compute flows, bipartite matchings and cuts. Our techniques lead to the first near-optimal data structures for these problems in various different settings. Letting $n$ denote the number of vertices in the graph and by $m$ the maximum number of edges in any version of the graph, we obtain - the first randomized data structure to maintain SSR and SCCs in near-optimal total update time $tilde{O}(m)$ in a graph undergoing edge deletions. - the first randomized data structure to maintain SSSP in partially dynamic graphs in total update time $tilde{O}(n^2)$ which is near-optimal in dense graphs. - the first deterministic data structures for SSR and SCC for graphs undergoing edge deletions, and for SSSP in partially dynamic graphs that improve upon the $O(mn)$ total update time by Even and Shiloach from 1981 that is often considered to be a fundamental barrier.
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 distances between $s$ and all other nodes in $G$ under a sequence of online adversar
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
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 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