We study the fully dynamic All-Pairs Shortest Paths (APSP) problem in undirected edge-weighted graphs. Given an n-vertex graph G with non-negative edge lengths, that undergoes an online sequence of edge insertions and deletions, the goal is to support approximate distance queries and shortest-path queries. We provide a deterministic algorithm for this problem, that, for a given precision parameter є, achieves approximation factor (loglogn)2O(1/є3), and has amortized update time O(nєlogL) per operation, where L is the ratio of longest to shortest edge length. Query time for distance-query is O(2O(1/є)· logn· loglogL), and query time for shortest-path query is O(|E(P)|+2O(1/є)· logn· loglogL), where P is the path that the algorithm returns. To the best of our knowledge, even allowing any o(n)-approximation factor, no adaptive-update algorithms with better than Θ(m) amortized update time and better than Θ(n) query time were known prior to this work. We also note that our guarantees are stronger than the best current guarantees for APSP in decremental graphs in the adaptive-adversary setting.
In order to obtain these results, we consider an intermediate problem, called Recursive Dynamic Neighborhood Cover (RecDynNC), that was formally introduced in [Chuzhoy, STOC ’21]. At a high level, given an undirected edge-weighted graph G undergoing an online sequence of edge deletions, together with a distance parameter D, the goal is to maintain a sparse D-neighborhood cover of G, with some additional technical requirements. Our main technical contribution is twofolds. First, we provide a black-box reduction from APSP in fully dynamic graphs to the RecDynNC problem. Second, we provide a new deterministic algorithm for the RecDynNC problem, that, for a given precision parameter є, achieves approximation factor (loglogm)2O(1/є2), with total update time O(m1+є), where m is the total number of edges ever present in G. This improves the previous algorithm of [Chuzhoy, STOC ’21], that achieved approximation factor (logm)2O(1/є) with similar total update time. Combining these two results immediately leads to the deterministic algorithm for fully-dynamic APSP with the guarantees stated above.
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Factorization and pseudofactorization of weighted graphs
For unweighted graphs, finding isometric embeddings of a graph G is closely related to decompositions of G into Cartesian products of smaller graphs. When G is isomorphic to a Cartesian graph product, we call the factors of this product a factorization of G. When G is isomorphic to an isometric subgraph of a Cartesian graph product, we call those factors a pseudofactorization of G. Prior work has shown that an unweighted graph’s pseudofactorization can be used to generate a canonical isometric embedding into a product of the smallest possible pseudofactors. However, for arbitrary weighted graphs,
which represent a richer variety of metric spaces, methods for finding isometric embeddings or determining their existence remain elusive, and indeed pseudofactorization and factorization have not previously been extended to this context. In this work, we address the problem of finding the factorization and pseudofactorization of a weighted graph G, where G satisfies the property that every edge constitutes a shortest path between its endpoints. We term such graphs minimal graphs, noting that every graph can be made minimal by removing edges not affecting its path metric. We generalize pseudofactorization and factorization to minimal graphs and develop new proof techniques that extend the previously proposed
algorithms due to Graham and Winkler [Graham and Winkler, ’85] and Feder [Feder, ’92] for pseudofactorization and factorization of unweighted graphs. We show that any n-vertex, m-edge graph with positive integer edge weights can be factored in O(m2) time, plus the time to find all pairs shortest paths (APSP) distances in a weighted graph, resulting in an overall running time of O(m2+n2 log log n) time. We also show that a pseudofactorization for such a graph can be computed in O(mn) time, plus the time to solve APSP, resulting in an O(mn + n2 log log n) running time.
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- Award ID(s):
- 1956054
- PAR ID:
- 10410644
- Date Published:
- Journal Name:
- Discrete applied mathematics
- ISSN:
- 0166-218X
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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