
Deterministic Algorithms for Decremental Shortest Paths via Layered Core Decomposition
In the decremental singlesource shortest paths (SSSP) problem, the inpu...
read it

NearOptimal Algorithms for Reachability, StronglyConnected Components and Shortest Paths in Partially Dynamic Digraphs
In this thesis, we present new techniques to deal with fundamental algor...
read it

Deterministic Algorithms for Decremental Approximate Shortest Paths: Faster and Simpler
In the decremental (1+ϵ)approximate SingleSource Shortest Path (SSSP) ...
read it

NearOptimal Decremental SSSP in Dense Weighted Digraphs
In the decremental SingleSource Shortest Path problem (SSSP), we are gi...
read it

Decremental APSP in Directed Graphs Versus an Adaptive Adversary
Given a directed graph G = (V,E), undergoing an online sequence of edge ...
read it

Deterministic Decremental SSSP and Approximate MinCost Flow in AlmostLinear Time
In the decremental singlesource shortest paths problem, the goal is to ...
read it

A Model for Ant Trail Formation and its Convergence Properties
We introduce a model for ant trail formation, building upon previous wor...
read it
Deterministic Decremental Reachability, SCC, and Shortest Paths via Directed Expanders and Congestion Balancing
Let G = (V,E,w) be a weighted, digraph subject to a sequence of adversarial edge deletions. In the decremental singlesource reachability problem (SSR), we are given a fixed source s and the goal is to maintain a data structure that can answer pathqueries s ↣ v for any v ∈ V. In the more general singlesource shortest paths (SSSP) problem the goal is to return an approximate shortest path to v, and in the SCC problem the goal is to maintain strongly connected components of G and to answer path queries within each component. All of these problems have been very actively studied over the past two decades, but all the fast algorithms are randomized and, more significantly, they can only answer path queries if they assume a weaker model: they assume an oblivious adversary which is not adaptive and must fix the update sequence in advance. This assumption significantly limits the use of these data structures, most notably preventing them from being used as subroutines in static algorithms. All the above problems are notoriously difficult in the adaptive setting. In fact, the stateoftheart is still the Even and Shiloach tree, which dates back all the way to 1981 and achieves total update time O(mn). We present the first algorithms to break through this barrier: 1) deterministic decremental SSR/SCC with total update time mn^2/3 + o(1) 2) deterministic decremental SSSP with total update time n^2+2/3+o(1). To achieve these results, we develop two general techniques of broader interest for working with dynamic graphs: 1) a generalization of expanderbased tools to dynamic directed graphs, and 2) a technique that we call congestion balancing and which provides a new method for maintaining flow under adversarial deletions. Using the second technique, we provide the first nearoptimal algorithm for decremental bipartite matching.
READ FULL TEXT
Comments
There are no comments yet.