Escaping a Grid by Edge-Disjoint Paths

4.1 Edge Disjoint Paths

Problem Statement: Given a directed graphG and a set of terminal pairs {(s1, t1), (s2, t2), · · · , (sk, tk)}, our goal is to connect as many pairs as possible using non edge intersecting paths. Edge disjoint paths problem is NP-Complete and is closely related to the multicommodity flow problem. In fact integer multicommodity flow is a generalization of this problem. We describe a greedy approx...

Reconstructing edge-disjoint paths

From edge-disjoint paths to independent paths

Let f(k) denote the maximum such that every simple undirected graph containing two vertices s, t and k edge-disjoint s–t paths, also contains two vertices u, v and f(k) independent u–v paths. Here, a set of paths is independent if none of them contains an interior vertex of another. We prove that f(k) = ( k if k ≤ 2, and 3 otherwise. Since independent paths are edge-disjoint, it is clear that f...

Reconstructing edge-disjoint paths faster

For a simple undirected graph with n vertices and m edges, we consider a data structure that given a query of a pair of vertices u, v and an integer k ≥ 1, it returns k edge-disjoint uv-paths. The data structure takes Õ(n) time to build, using O( √ mn log n) space, and each query takes O( √ kn) time, which is optimal and beats the previous query time of O(knα(n)).

Edge-disjoint paths in a grid bounded by two nested rectangles

This paper presents an algorithm for finding edge-disjoint paths in a given plane grid bounded by two nested rectangles. A pair of vertices on the boundary of the same rectangle are designated as terminals for each of the paths. The number of terminals lying on each boundary vertex is determined by the degree of the boundary vertex. Every vertex of degree 2 has either 0 or 2 terminals lying on ...