DeepAI

Graph spanners and emulators are sparse structures that approximately preserve distances of the original graph. While there has been an extensive amount of work on additive spanners, so far little attention was given to weighted graphs. Only very recently <cit.> extended the classical +2 (respectively, +4) spanners for unweighted graphs of size O(n^3/2) (resp., O(n^7/5)) to the weighted setting, where the additive error is +2W (resp., +4W). This means that for every pair u,v, the additive stretch is at most +2W_u,v, where W_u,v is the maximal edge weight on the shortest u-v path (weights are normalized so that the minimum edge weight is 1). In addition, <cit.> showed a randomized algorithm yielding a +8W_max spanner of size O(n^4/3), here W_max is the maximum edge weight in the entire graph. In this work we improve the latter result by devising a simple deterministic algorithm for a +(6+ε)W spanner for weighted graphs with size O(n^4/3) (for any constant ε>0), thus nearly matching the classical +6 spanner of size O(n^4/3) for unweighted graphs. We also show a simple randomized algorithm for a +4W emulator of size Õ(n^4/3).

• 14 publications
• 2 publications
• 10 publications
03/17/2021

### On additive spanners in weighted graphs with local error

An additive +β spanner of a graph G is a subgraph which preserves distan...
04/05/2022

### All-Pairs Shortest Path Distances with Differential Privacy: Improved Algorithms for Bounded and Unbounded Weights

We revisit the problem of privately releasing the all-pairs shortest pat...
07/26/2019

### Almost Shortest Paths and PRAM Distance Oracles in Weighted Graphs

Let G=(V,E) be a weighted undirected graph with n vertices and m edges, ...
01/18/2022

### Optimal randomized quadrature for weighted Sobolev and Besov classes with the Jacobi weight on the ball

We consider the numerical integration INT_d(f)=∫_𝔹^df(x)w_μ(x)dx f...
12/01/2022

### Subquadratic Weighted Matroid Intersection Under Rank Oracles

Given two matroids ℳ_1 = (V, ℐ_1) and ℳ_2 = (V, ℐ_2) over an n-element i...
08/22/2021

### A Unified Framework for Hopsets and Spanners

Given an undirected graph G=(V,E), an (α,β)-spanner H=(V,E') is a subgra...
11/09/2022

### Lipschitz Continuous Algorithms for Graph Problems

It has been widely observed in the machine learning community that a sma...