research-article

A New Approach to Incremental Cycle Detection and Related Problems

Authors:

Michael A. Bender,

Jeremy T. Fineman,

Robert E. TarjanAuthors Info & Claims

ACM Transactions on Algorithms (TALG), Volume 12, Issue 2

Article No.: 14, Pages 1 - 22

https://doi.org/10.1145/2756553

Published: 08 December 2015 Publication History

Abstract

We consider the problem of detecting a cycle in a directed graph that grows by arc insertions and the related problems of maintaining a topological order and the strong components of such a graph. For these problems, we give two algorithms, one suited to sparse graphs, the other to dense graphs. The former takes O(min {m^1/2, n^2/3}m) time to insert m arcs into an n-vertex graph; the latter takes O(n²log n) time. Our sparse algorithm is substantially simpler than a previous O(m^3/2)-time algorithm; it is also faster on graphs of sufficient density. The time bound of our dense algorithm beats the previously best time bound of O(n^5/2) for dense graphs. Our algorithms rely for their efficiency on vertex numberings weakly consistent with topological order: we allow ties. Bounds on the size of the numbers give bounds on running time.

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Cited By

Chen XShi JPeng YLin WWang SZhang W(2024)Minimum Strongly Connected Subgraph Collection in Dynamic GraphsProceedings of the VLDB Endowment10.14778/3648160.364817317:6(1324-1336)Online publication date: 3-May-2024
https://doi.org/10.14778/3648160.3648173
Zeng YFang YMa CZhou XLi K(2024)Efficient Distributed Hop-Constrained Path Enumeration on Large-Scale GraphsProceedings of the ACM on Management of Data10.1145/36392772:1(1-25)Online publication date: 26-Mar-2024
https://dl.acm.org/doi/10.1145/3639277
Chen LKyng RLiu YMeierhans SProbst Gutenberg MMohar BShinkar IO'Donnell R(2024)Almost-Linear Time Algorithms for Incremental Graphs: Cycle Detection, SCCs, s-t Shortest Path, and Minimum-Cost FlowProceedings of the 56th Annual ACM Symposium on Theory of Computing10.1145/3618260.3649745(1165-1173)Online publication date: 10-Jun-2024
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Index Terms

A New Approach to Incremental Cycle Detection and Related Problems
1. Mathematics of computing
  1. Discrete mathematics
    1. Graph theory
      1. Graph algorithms
2. Theory of computation
  1. Randomness, geometry and discrete structures

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Published In

cover image ACM Transactions on Algorithms

ACM Transactions on Algorithms Volume 12, Issue 2

February 2016

385 pages

ISSN:1549-6325

EISSN:1549-6333

DOI:10.1145/2846106

Editor:
Aravind Srinivasan
University of Maryland, USA

Issue’s Table of Contents

Copyright © 2015 ACM.

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Association for Computing Machinery

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Publication History

Published: 08 December 2015

Accepted: 01 April 2015

Revised: 01 February 2015

Received: 01 March 2014

Published in TALG Volume 12, Issue 2

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National Science Foundation
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U.S.-Israel Binational Science Foundation
NSF/CRA- sponsored CIFellows program
Distinguished Visitor program in the Stanford University Computer Science Department

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Cited By

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https://doi.org/10.14778/3648160.3648173
Zeng YFang YMa CZhou XLi K(2024)Efficient Distributed Hop-Constrained Path Enumeration on Large-Scale GraphsProceedings of the ACM on Management of Data10.1145/36392772:1(1-25)Online publication date: 26-Mar-2024
https://dl.acm.org/doi/10.1145/3639277
Chen LKyng RLiu YMeierhans SProbst Gutenberg MMohar BShinkar IO'Donnell R(2024)Almost-Linear Time Algorithms for Incremental Graphs: Cycle Detection, SCCs, s-t Shortest Path, and Minimum-Cost FlowProceedings of the 56th Annual ACM Symposium on Theory of Computing10.1145/3618260.3649745(1165-1173)Online publication date: 10-Jun-2024
https://dl.acm.org/doi/10.1145/3618260.3649745
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Karczmarz AMukherjee ASankowski PLeonardi SGupta A(2022)Subquadratic dynamic path reporting in directed graphs against an adaptive adversaryProceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing10.1145/3519935.3520058(1643-1656)Online publication date: 9-Jun-2022
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