Fast Deterministic Fully Dynamic Distance Approximation

11/05/2021
by   Jan van den Brand, et al.
0

In this paper, we develop deterministic fully dynamic algorithms for computing approximate distances in a graph with worst-case update time guarantees. In particular we obtain improved dynamic algorithms that, given an unweighted and undirected graph G=(V,E) undergoing edge insertions and deletions, and a parameter 0 < ϵ≤ 1, maintain (1+ϵ)-approximations of the st distance of a single pair of nodes, the distances from a single source to all nodes ("SSSP"), the distances from multiple sources to all nodes ("MSSP”), or the distances between all nodes ("APSP"). Our main result is a deterministic algorithm for maintaining (1+ϵ)-approximate single-source distances with worst-case update time O(n^1.529) (for the current best known bound on the matrix multiplication coefficient ω). This matches a conditional lower bound by [BNS, FOCS 2019]. We further show that we can go beyond this SSSP bound for the problem of maintaining approximate st distances by providing a deterministic algorithm with worst-case update time O(n^1.447). This even improves upon the fastest known randomized algorithm for this problem. At the core, our approach is to combine algebraic distance maintenance data structures with near-additive emulator constructions. This also leads to novel dynamic algorithms for maintaining (1+ϵ, β)-emulators that improve upon the state of the art, which might be of independent interest. Our techniques also lead to improvements for randomized approximate diameter maintenance.

READ FULL TEXT

Please sign up or login with your details

Forgot password? Click here to reset