Bottleneck paths and trees and deterministic graphical games

TY - GEN

T1 - Bottleneck paths and trees and deterministic graphical games

AU - Chechik, Shiri

AU - Kaplan, Haim

AU - Thorup, Mikkel

AU - Zamir, Or

AU - Zwick, Uri

N1 - Conference code: 33

PY - 2016/2/1

Y1 - 2016/2/1

N2 - Gabow and Tarjan showed that the Bottleneck Path (BP) problem, i.e., finding a path between a given source and a given target in a weighted directed graph whose largest edge weight is minimized, as well as the Bottleneck spanning tree (BST) problem, i.e., finding a directed spanning tree rooted at a given vertex whose largest edge weight is minimized, can both be solved deterministically in O(m log∗ n) time, where m is the number of edges and n is the number of vertices in the graph. We present a slightly improved randomized algorithm for these problems with an expected running time of O(mβ(m, n)), where β(m,n) = min{k ≥ 1 | log(k) n ≤ m/n} < log∗ n - log∗ (m/n) + 1. This is the first improvement for these problems in over 25 years. In particular, if m ≥ n log(k) n, for some constant k, the expected running time of the new algorithm is O(m). Our algorithm, as that of Gabow and Tarjan, work in the comparison model. We also observe that in the word-RAM model, both problems can be solved deterministically in O(m) time. Finally, we solve an open problem of Andersson et al., giving a deterministic O(m)-time comparison-based algorithm for solving deterministic 2-player turn-based zero-sum terminal payoff games, also known as Deterministic Graphical Games (DGG).

AB - Gabow and Tarjan showed that the Bottleneck Path (BP) problem, i.e., finding a path between a given source and a given target in a weighted directed graph whose largest edge weight is minimized, as well as the Bottleneck spanning tree (BST) problem, i.e., finding a directed spanning tree rooted at a given vertex whose largest edge weight is minimized, can both be solved deterministically in O(m log∗ n) time, where m is the number of edges and n is the number of vertices in the graph. We present a slightly improved randomized algorithm for these problems with an expected running time of O(mβ(m, n)), where β(m,n) = min{k ≥ 1 | log(k) n ≤ m/n} < log∗ n - log∗ (m/n) + 1. This is the first improvement for these problems in over 25 years. In particular, if m ≥ n log(k) n, for some constant k, the expected running time of the new algorithm is O(m). Our algorithm, as that of Gabow and Tarjan, work in the comparison model. We also observe that in the word-RAM model, both problems can be solved deterministically in O(m) time. Finally, we solve an open problem of Andersson et al., giving a deterministic O(m)-time comparison-based algorithm for solving deterministic 2-player turn-based zero-sum terminal payoff games, also known as Deterministic Graphical Games (DGG).

U2 - 10.4230/LIPIcs.STACS.2016.27

DO - 10.4230/LIPIcs.STACS.2016.27

M3 - Article in proceedings

SN - 978-3-95977-001-9

T3 - Leibniz International Proceedings in Informatics

SP - 1

EP - 13

BT - 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)

A2 - Ollinger, Nicolas

A2 - Vollmer, Heribert

PB - Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik GmbH

T2 - Symposium on Theoretical Aspects of Computer Science (STACS 2016)

Y2 - 17 February 2016 through 20 February 2016

ER -