Shortest paths in fuzzy weighted graphs

Shortest paths in fuzzy weighted graphs

The task of finding shortest paths in weighted graphs is one of the archetypical problems encountered in the domain of combinatorial optimization and has been studied intensively over the past five decades. More recently, fuzzy weighted graphs, along with generalizations of algorithms for finding optimal paths within them, have emerged as an adequate modeling tool for prohibitively complex and/...

Approximate shortest paths in weighted graphs

We present an approximation algorithm for the all pairs shortest paths (APSP) problem in weighed graphs. Our algorithm solves the APSP problem for weighted directed graphs, with real (positive or negative) weights, up to an additive error of . For any pair of vertices u, v, the algorithm finds a path whose length is at most δ(u, v) + . The algorithm is randomized and runs in Õ(n) < O(n) time, w...

Deterministic Partially Dynamic Single Source Shortest Paths in Weighted Graphs

In this paper we consider the decremental single-source shortest paths (SSSP) problem, where given a graph G and a source node s the goal is to maintain shortest distances between s and all other nodes in G under a sequence of online adversarial edge deletions. In their seminal work, Even and Shiloach [JACM 1981] presented an exact solution to the problem in unweighted graphs with only O(mn) to...

Shortest Weighted Paths II

Lemma 1.1. Consider a (directed) weighted graph G = (V, E), w : E → R+ ∪ {0} with no negative edge weights, a source vertex s and an arbitrary distance value d ∈ R+ ∪ {0}. Let X = X ′ ∪ X ′′ , where X ′ = {v ∈ V : δ(s, v)< d} be the set of vertices that are less than d from s, X ′′ ⊆ {v ∈ V : δ(s, v) = d} be the set of vertices that are exactly d from s. Also, let d ′ = min{δ(s, u) : u ∈ V \ X ...

Single Source Shortest Paths on Weighted Co-Interval Graphs