
FullyDynamic AllPairs Shortest Paths: Improved WorstCase Time and Space Bounds
Given a directed weighted graph G=(V,E) undergoing vertex insertions and...
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The idemetric property: when most distances are (almost) the same
We introduce the idemetric property, which formalises the idea that most...
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Efficient Stepping Algorithms and Implementations for Parallel Shortest Paths
In this paper, we study the singlesource shortestpath (SSSP) problem w...
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Algorithms and Hardness for Diameter in Dynamic Graphs
The diameter, radius and eccentricities are natural graph parameters. Wh...
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New Techniques and FineGrained Hardness for Dynamic NearAdditive Spanners
Maintaining and updating shortest paths information in a graph is a fund...
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The Power of Vertex Sparsifiers in Dynamic Graph Algorithms
We introduce a new algorithmic framework for designing dynamic graph alg...
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Dynamic Longest Increasing Subsequence and the ErdösSzekeres Partitioning Problem
In this paper, we provide new approximation algorithms for dynamic varia...
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Dynamic Approximate Shortest Paths and Beyond: Subquadratic and WorstCase Update Time
Consider the following distance query for an nnode graph G undergoing edge insertions and deletions: given two sets of nodes I and J, return the distances between every pair of nodes in I× J. This query is rather general and captures several versions of the dynamic shortest paths problem. In this paper, we develop an efficient (1+ϵ)approximation algorithm for this query using fast matrix multiplication. Our algorithm leads to answers for some open problems for SingleSource and AllPairs Shortest Paths (SSSP and APSP), as well as for Diameter, Radius, and Eccentricities. Below are some highlights. Note that all our algorithms guarantee worstcase update time and are randomized (Monte Carlo), but do not need the oblivious adversary assumption. Subquadratic update time for SSSP, Diameter, Centralities, ect.: When we want to maintain distances from a single node explicitly (without queries), a fundamental question is to beat trivially calling Dijkstra's static algorithm after each update, taking Θ(n^2) update time on dense graphs. It was known to be improbable for exact algorithms and for combinatorial anyapproximation algorithms to polynomially beat the Ω(n^2) bound (under some conjectures) [Roditty, Zwick, ESA'04; Abboud, V. Williams, FOCS'14]. Our algorithm with I={s} and J=V(G) implies a (1+ϵ)approximation algorithm for this, guaranteeing Õ(n^1.823/ϵ^2) worstcase update time for directed graphs with positive real weights in [1, W]. With ideas from [Roditty, V. Williams, STOC'13], we also obtain the first subquadratic worstcase update time for (5/3+ϵ)approximating the eccentricities and (1.5+ϵ)approximating the diameter and radius for unweighted graphs (with small additive errors). [...]
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