Succinct Data Structures for Families of Interval Graphs

We consider the problem of designing succinct data structures for interval graphs with n vertices while supporting degree, adjacency, neighborhood and shortest path queries in optimal time. Towards showing succinctness, we first show that at least nlog2n-2nlog2log2n-O(n) $$n\log _2{n} - 2n\log _2\lo...

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Bibliographic Details
Published inAlgorithms and Data Structures Vol. 11646; pp. 1 - 13
Main Authors Acan, Hüseyin, Chakraborty, Sankardeep, Jo, Seungbum, Satti, Srinivasa Rao
Format Book Chapter
LanguageEnglish
Published Switzerland Springer International Publishing AG 01.01.2019
Springer International Publishing
SeriesLecture Notes in Computer Science
Subjects
Interval graphs
Space efficient data structures
Succinct encoding
Online AccessGet full text
ISBN3030247651
9783030247652
ISSN0302-9743
1611-3349
DOI10.1007/978-3-030-24766-9_1

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Summary:We consider the problem of designing succinct data structures for interval graphs with n vertices while supporting degree, adjacency, neighborhood and shortest path queries in optimal time. Towards showing succinctness, we first show that at least nlog2n-2nlog2log2n-O(n) $$n\log _2{n} - 2n\log _2\log _2 n - O(n)$$ bits. are necessary to represent any unlabeled interval graph G with n vertices, answering an open problem of Yang and Pippenger [Proc. Amer. Math. Soc. 2017]. This is augmented by a data structure of size nlog2n+O(n) $$n\log _2{n} +O(n)$$ bits while supporting not only the above queries optimally but also capable of executing various combinatorial algorithms (like proper coloring, maximum independent set etc.) on interval graphs efficiently. Finally, we extend our ideas to other variants of interval graphs, for example, proper/unit, k-improper interval graphs, and circular-arc graphs, and design succinct data structures for these graph classes as well along with supporting queries on them efficiently.
Bibliography:Original Abstract: We consider the problem of designing succinct data structures for interval graphs with n vertices while supporting degree, adjacency, neighborhood and shortest path queries in optimal time. Towards showing succinctness, we first show that at least nlog2n-2nlog2log2n-O(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\log _2{n} - 2n\log _2\log _2 n - O(n)$$\end{document} bits. are necessary to represent any unlabeled interval graph G with n vertices, answering an open problem of Yang and Pippenger [Proc. Amer. Math. Soc. 2017]. This is augmented by a data structure of size nlog2n+O(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\log _2{n} +O(n)$$\end{document} bits while supporting not only the above queries optimally but also capable of executing various combinatorial algorithms (like proper coloring, maximum independent set etc.) on interval graphs efficiently. Finally, we extend our ideas to other variants of interval graphs, for example, proper/unit, k-improper interval graphs, and circular-arc graphs, and design succinct data structures for these graph classes as well along with supporting queries on them efficiently.
ISBN:3030247651
9783030247652
ISSN:0302-9743
1611-3349
DOI:10.1007/978-3-030-24766-9_1