research-article

Public Access

A Data Structure for Nearest Common Ancestors with Linking

Author:

Harold N. GabowAuthors Info & Claims

ACM Transactions on Algorithms (TALG), Volume 13, Issue 4

Article No.: 45, Pages 1 - 28

https://doi.org/10.1145/3108240

Published: 19 September 2017 Publication History

Abstract

Consider a forest that evolves via link operations that make the root of one tree the child of a node in another tree. Intermixed with link operations are nca operations, which return the nearest common ancestor of two given nodes when such exists. This article shows that a sequence of m such nca and link operations on a forest of n nodes can be processed online in time O(mα (m,n)+n). This was previously known only for a restricted type of link operation.

The special case where a link only extends a tree by adding a new leaf occurs in Edmonds’ algorithm for finding a maximum weight matching on a general graph. Incorporating our algorithm into the implementation of Edmonds’ algorithm in [9] achieves time O(n(m + nlog n)) for weighted matching, an arguably optimum asymptotic bound (n and m are the number of vertices and edges, respectively). Our data structure also provides a simple alternative implementation of the incremental-tree set merging algorithm of Gabow and Tarjan [11].

References

[1]

A. V. Aho, J. E. Hopcroft, and J. D. Ullman. 1976. On finding lowest common ancestors in trees. SIAM J. Comput. 5 (1976), 115--132

[2]

M. A. Bender and M. Farach-Colton. 2000. The LCA problem revisited. Latin American Theoretical INformatics (LATIN) 2000 (2000), 88--94.

Digital Library

[3]

O. Berkman and U. Vishkin. 1993. Recursive star-tree parallel data structure. SIAM J. Comput. 22, 2 (1993), 221--242.

Digital Library

[4]

R. Cole and R. Hariharan. 2005. Dynamic LCA queries on trees. SIAM J. Comput. 34, 4 (2005), 894--923.

Digital Library

[5]

T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein. 2001. Introduction to Algorithms (2nd ed.). McGraw--Hill, New York.

Digital Library

[6]

J. Edmonds. 1965. Maximum matching and a polyhedron with 0,1-vertices. J. Res. Nat. Bur. Standards 69B (1965), 125--130.

[7]

H. N. Gabow. 1985. A scaling algorithm for weighted matching on general graphs. In Proc. 26th Annual Symp. on Found. of Comp. Sci. IEEE Computer Society, Long Beach, CA, 90--100.

Digital Library

[8]

H. N. Gabow. 1990. Data structures for weighted matching and nearest common ancestors with linking. In Proc. 1st Annual ACM-SIAM Symp. on Disc. Algorithms. SIAM, Philadelphia, PA, 434--443.

Digital Library

[9]

H. N. Gabow. 2016. Data structures for weighted matching and extensions to b-matching and f-factors. (2016). Available at https://arxiv.org/abs/1611.07541.

[10]

H. N. Gabow and M. Stallmann. 1985. Efficient algorithms for graphic matroid intersection and parity. In Automata, Languages and Programming: 12th Colloquium; Lecture Notes in Computer Science 194, W. Brauer (Ed.). Springer-Verlag, 210--220.

Digital Library

[11]

H. N. Gabow and R. E. Tarjan. 1985. A linear--time algorithm for a special case of disjoint set union. J. Comp. and System Sci. 30, 2 (1985), 209--221.

[12]

D. Gusfield. 1999. Algorithms on Strings, Trees, and Sequences. Cambridge University Press, New York.

Digital Library

[13]

D. Harel and R. E. Tarjan. 1984. Fast algorithms for finding nearest common ancestors. SIAM J. Comput. 13, 2 (1984), 338--355.

Digital Library

[14]

J. A. La Poutré. 1990. New techniques for the union-find problem. In Proc. 1st Annual ACM-SIAM Symp. on Disc. Algorithms. SIAM, Philadelphia, PA, 54--63.

Digital Library

[15]

B. Schieber and U. Vishkin. 1988. On finding lowest common ancestors: Simplification and parallelization. SIAM J. Comput. 17, 6 (1988), 1253--1262.

Digital Library

[16]

A. Schrijver. 2003. Combinatorial Optimization: Polyhedra and Efficiency. Springer, New York.

[17]

D. D. Sleator and R. E. Tarjan. 1983. A data structure for dynamic trees. J. Comp. and System Sci. 26 (1983), 362--391.

Digital Library

[18]

B. Smyth. 2003. Computing Patterns in Strings. Pearson Education Limited, Harlow, England.

[19]

R. E. Tarjan. 1979. Applications of path compression on balanced trees. J. ACM 26, 4 (1979), 690--715.

Digital Library

[20]

R. E. Tarjan. 1983. Data Structures and Network Algorithms. SIAM, Philadelphia, PA.

Digital Library

Cited By

Merino AMütze T(2024)Traversing Combinatorial 0/1-Polytopes via OptimizationSIAM Journal on Computing10.1137/23M161201953:5(1257-1292)Online publication date: 9-Sep-2024
https://doi.org/10.1137/23M1612019
Merino AMütze T(2023)Traversing combinatorial 0/1-polytopes via optimization2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS57990.2023.00076(1282-1291)Online publication date: 6-Nov-2023
https://doi.org/10.1109/FOCS57990.2023.00076
Gabow H(2023)Blocking Trails for f-factors of MultigraphsAlgorithmica10.1007/s00453-023-01126-y85:10(3168-3213)Online publication date: 13-May-2023
https://dl.acm.org/doi/10.1007/s00453-023-01126-y
Show More Cited By

Index Terms

A Data Structure for Nearest Common Ancestors with Linking
1. Mathematics of computing
  1. Discrete mathematics
    1. Graph theory
      1. Trees
2. Theory of computation
  1. Design and analysis of algorithms
    1. Data structures design and analysis
    2. Graph algorithms analysis

Recommendations

Nearest common ancestors: a survey and a new distributed algorithm
SPAA '02: Proceedings of the fourteenth annual ACM symposium on Parallel algorithms and architectures

Several papers describe linear time algorithms to preprocess a tree, such that one can answer subsequent nearest common ancestor queries in constant time. Here, we survey these algorithms and related results. A common idea used by all the algorithms for ...
Labeling schemes for nearest common ancestors through minor-universal trees
SODA '18: Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

Preprocessing a tree for finding the nearest common ancestor of two nodes is a basic tool with multiple applications. Quite a few linear-space constant-time solutions are known and the problem seems to be well-understood. This is however not so clear if ...
Optimal Pointer Algorithms for Finding Nearest Common Ancestors in Dynamic Trees

We consider the problem of finding the nearest common ancestor of two given nodes x and y (denoted by nca(x,y)) in a dynamic forest of rooted trees. Interspersed with nca-queries are online commands link(x,y), where x, but not necessarily y, is a tree ...

Comments

Information & Contributors

Information

Published In

cover image ACM Transactions on Algorithms

ACM Transactions on Algorithms Volume 13, Issue 4

October 2017

333 pages

ISSN:1549-6325

EISSN:1549-6333

DOI:10.1145/3143522

Editor:
Aravind Srinivasan
University of Maryland, USA

Issue’s Table of Contents

Copyright © 2017 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 19 September 2017

Accepted: 01 May 2017

Received: 01 December 2014

Published in TALG Volume 13, Issue 4

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Research-article
Research
Refereed

Funding Sources

NSF

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

6
Total Citations
View Citations
530
Total Downloads

Downloads (Last 12 months)91
Downloads (Last 6 weeks)11

Reflects downloads up to 15 Oct 2024

Other Metrics

View Author Metrics

Citations

Cited By

Merino AMütze T(2024)Traversing Combinatorial 0/1-Polytopes via OptimizationSIAM Journal on Computing10.1137/23M161201953:5(1257-1292)Online publication date: 9-Sep-2024
https://doi.org/10.1137/23M1612019
Merino AMütze T(2023)Traversing combinatorial 0/1-polytopes via optimization2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS57990.2023.00076(1282-1291)Online publication date: 6-Nov-2023
https://doi.org/10.1109/FOCS57990.2023.00076
Gabow H(2023)Blocking Trails for f-factors of MultigraphsAlgorithmica10.1007/s00453-023-01126-y85:10(3168-3213)Online publication date: 13-May-2023
https://dl.acm.org/doi/10.1007/s00453-023-01126-y
Gabow H(2018)Data Structures for Weighted Matching and Extensions to b-matching and f-factorsACM Transactions on Algorithms10.1145/318336914:3(1-80)Online publication date: 22-Jun-2018
https://dl.acm.org/doi/10.1145/3183369
Sankararaman SEfrat ARamasubramanian SAgarwal P(2014)On channel-discontinuity-constraint routing in wireless networksAd Hoc Networks10.1016/j.adhoc.2011.04.01113(153-169)Online publication date: 1-Feb-2014
https://dl.acm.org/doi/10.1016/j.adhoc.2011.04.011
Bae JRathinam S(2010)Approximation algorithms for multiple terminal, Hamiltonian path problemsOptimization Letters10.1007/s11590-010-0252-46:1(69-85)Online publication date: 27-Oct-2010
https://doi.org/10.1007/s11590-010-0252-4
van LEEUWEN J(1990)Graph AlgorithmsAlgorithms and Complexity10.1016/B978-0-444-88071-0.50015-1(525-631)Online publication date: 1990
https://doi.org/10.1016/B978-0-444-88071-0.50015-1

View Options

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Get Access

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Article

Media

Figures

Other

Tables

View Issue’s Table of Contents