Balancing Minimum Spanning and Shortest Path Trees

This paper give a simple linear-time algorithm that, given a weighted digraph, finds a spanning tree that simultaneously approximates a shortest-path tree and a minimum spanning tree. The algorithm provides a continuous trade-off: given the two trees and epsilon > 0, the algorithm returns a spanning tree in which the distance between any vertex and the root of the shortest-path tree is at most 1+epsilon times the shortest-path distance, and yet the total weight of the tree is at most 1+2/epsilon times the weight of a minimum spanning tree. This is the best tradeoff possible. The paper also describes a fast parallel implementation.Comment: conference version: ACM-SIAM Symposium on Discrete Algorithms (1993

Computing Shortest Paths in Series-Parallel Graphs in Logarithmic Space

Series-parallel graphs, which are built by repeatedly applying series or parallel composition operations to paths, play an important role in computer science as they model the flow of information in many types of programs. For directed series-parallel graphs, we study the problem of finding a shortest path between two given vertices. Our main result is that we can find such a path in logarithmic space, which shows that the distance problem for series-parallel graphs is L-complete. Previously, it was known that one can compute some path in logarithmic space; but for other graph types, like undirected graphs or tournament graphs, constructing some path between given vertices is possible in logarithmic space while constructing a shortest path is NL-complete

Complexity of the Temporal Shortest Path Interdiction Problem

In the shortest path interdiction problem, an interdictor aims to remove arcs of total cost at most a given budget from a directed graph with given arc costs and traversal times such that the length of a shortest s-t-path is maximized. For static graphs, this problem is known to be strongly NP-hard, and it has received considerable attention in the literature. While the shortest path problem is one of the most fundamental and well-studied problems also for temporal graphs, the shortest path interdiction problem has not yet been formally studied on temporal graphs, where common definitions of a "shortest path" include: latest start path (path with maximum start time), earliest arrival path (path with minimum arrival time), shortest duration path (path with minimum traveling time including waiting times at nodes), and shortest traversal path (path with minimum traveling time not including waiting times at nodes). In this paper, we analyze the complexity of the shortest path interdiction problem on temporal graphs with respect to all four definitions of a shortest path mentioned above. Even though the shortest path interdiction problem on static graphs is known to be strongly NP-hard, we show that the latest start and the earliest arrival path interdiction problems on temporal graphs are polynomial-time solvable. For the shortest duration and shortest traversal path interdiction problems, however, we show strong NP-hardness, but we obtain polynomial-time algorithms for these problems on extension-parallel temporal graphs

Floyd-Warshall Algorithm 1

Abstract: There are several applications in VLSI technology that require high-speed shortest-path computations. The shortest path is a path between two nodes (or points) in a graph such that the sum of the weights of its constituent edges is minimum. Floyd-Warshall algorithm provides fastest computation of shortest path between all pair of nodes present in the graph. With rapid advances in VLSI technology, Field Programmable Gate Arrays (FPGAs) are receiving the attention of the Parallel and High Performance Computing community. This paper gives implementation outcome of Floyd-Warshall algorithm to solve the all pairs shortest-paths problem for directed graph in Verilog

Topology Adaption for the Quantum Internet

In the quantum repeater networks of the quantum Internet, the varying stability of entangled quantum links makes dynamic topology adaption an emerging issue. Here we define an efficient topology adaption method for quantum repeater networks. The model assumes the random failures of entangled links and several parallel demands from legal users. The shortest path defines a set of entangled links for which the probability of stability is above a critical threshold. The scheme is utilized in a base-graph of the overlay quantum network to provide an efficient shortest path selection for the demands of all users of the network. We study the problem of entanglement assignment in a quantum repeater network, prove its computational complexity, and show an optimization procedure. The results are particularly convenient for future quantum networking, quantum-Internet, and experimental long-distance quantum communications.Comment: 17 pages, Journal-ref: Quant. Inf. Proc. (2018

Parallel Graph Decompositions Using Random Shifts

We show an improved parallel algorithm for decomposing an undirected unweighted graph into small diameter pieces with a small fraction of the edges in between. These decompositions form critical subroutines in a number of graph algorithms. Our algorithm builds upon the shifted shortest path approach introduced in [Blelloch, Gupta, Koutis, Miller, Peng, Tangwongsan, SPAA 2011]. By combining various stages of the previous algorithm, we obtain a significantly simpler algorithm with the same asymptotic guarantees as the best sequential algorithm