The Power of Vertex Sparsifiers in Dynamic Graph Algorithms

12/18/2017
by   Gramoz Goranci, et al.
0

We introduce a new algorithmic framework for designing dynamic graph algorithms in minor-free graphs, by exploiting the structure of such graphs and a tool called vertex sparsification, which is a way to compress large graphs into small ones that well preserve relevant properties among a subset of vertices and has previously mainly been used in the design of approximation algorithms. Using this framework, we obtain a Monte Carlo randomized fully dynamic algorithm for (1+ε)-approximating the energy of electrical flows in n-vertex planar graphs with Õ(rε^-2) worst-case update time and Õ((r+n/√(r))ε^-2) worst-case query time, for any r larger than some constant. For r=n^2/3, this gives Õ(n^2/3ε^-2) update time and Õ(n^2/3ε^-2) query time. We also extend this algorithm to work for minor-free graphs with similar approximation and running time guarantees. Furthermore, we illustrate our framework on the all-pairs max flow and shortest path problems by giving corresponding dynamic algorithms in minor-free graphs with both sublinear update and query times. To the best of our knowledge, our results are the first to systematically establish such a connection between dynamic graph algorithms and vertex sparsification. We also present both upper bound and lower bound for maintaining the energy of electrical flows in the incremental subgraph model, where updates consist of only vertex activations, which might be of independent interest.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
06/05/2023

Fully-Dynamic All-Pairs Shortest Paths: Likely Optimal Worst-Case Update Time

The All-Pairs Shortest Paths (APSP) problem is one of the fundamental pr...
research
06/22/2021

Fully Dynamic Algorithms for Minimum Weight Cycle and Related Problems

We consider the directed minimum weight cycle problem in the fully dynam...
research
02/26/2018

Dynamic Effective Resistances and Approximate Schur Complement on Separable Graphs

We consider the problem of dynamically maintaining (approximate) all-pai...
research
12/01/2021

Faster Maxflow via Improved Dynamic Spectral Vertex Sparsifiers

We make several advances broadly related to the maintenance of electrica...
research
06/30/2020

Vertex guarding for dynamic orthogonal art galleries

Given an orthogonal polygon with orthogonal holes, we devise a dynamic a...
research
07/10/2019

Constant-Time Dynamic (Δ+1)-Coloring and Weight Approximation for Minimum Spanning Forest: Dynamic Algorithms Meet Property Testing

With few exceptions (namely, algorithms for maximal matching, 2-approxim...
research
05/05/2020

Fast Dynamic Cuts, Distances and Effective Resistances via Vertex Sparsifiers

We present a general framework of designing efficient dynamic approximat...

Please sign up or login with your details

Forgot password? Click here to reset