Efficiently realizing interval sequences

AMOTZ BAR-NOY; KEERTI CHOUDHARY; DAVID PELEG; DROR RAWITZ

doi:10.1137/20m1326489

Efficiently realizing interval sequences

AMOTZ BAR-NOY, KEERTI CHOUDHARY, DAVID PELEG, DROR RAWITZ

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

Abstract

We consider the problem of realizable interval sequences. An interval sequence is comprised of n integer intervals [ai, bi] such that 0 ≤ ai ≤ bi ≤ n 1 and is said to be graphic/realizable if there exists a graph with degree sequence, say, D = (d1, . . . , dn), satisfying the condition ai ≤ di ≤ bi for each i in [1, n]. There is a characterization (also implying an O(n) verifying algorithm) known for realizability of interval sequences, which is a generalization of the ErdH os-Gallai characterization for graphic sequences. However, given any realizable interval sequence, there is no known algorithm for computing a corresponding graphic certificate in o(n2) time. In this paper, we provide an O(n log n) time algorithm for computing a graphic sequence for any realizable interval sequence. In addition, when the interval sequence is nonrealizable, we show how to find a graphic sequence having minimum deviation with respect to the given interval sequence in the same time. Finally, we consider variants of the problem, such as computing the most-regular graphic sequence and computing a minimum extension of a length p nongraphic sequence to a graphic one.

Original language	English
Pages (from-to)	2318-2337
Number of pages	20
Journal	SIAM Journal on Discrete Mathematics
Volume	34
Issue number	4
DOIs	https://doi.org/10.1137/20m1326489
State	Published - 2020

Bibliographical note

Publisher Copyright:
© 2020 Society for Industrial and Applied Mathematics.

Funding

\ast Received by the editors March 20, 2020; accepted for publication (in revised form) August 14, 2020; published electronically November 9, 2020. A preliminary version of this article appeared in ISAAC'19 [6]. https://doi.org/10.1137/20M1326489 Funding: This research was supported by US-Israel BSF grant 2018043, and by ARL Cooperative Grant, ARL Network Science CTA, W911NF-09-2-0053. \dagger Department of Computer and Information Science, Brooklyn College, Brooklyn, NY 11210 USA ([email protected]). \ddagger Department of Computer Science and Engineering, Indian Institute of Technology Delhi, New Delhi 110016, India ([email protected]). \S Applied Mathematics and Computer Science, Weizmann Institute of Science, Rehovot, 7610001, Israel ([email protected]). \P Faculty of Engineering, Bar Ilan University, Ramat Gan, 5290002, Israel ([email protected]).

Funders	Funder number
ARL Network Science CTA	W911NF-09-2-0053
US-Israel BSF	2018043
Association of Research Libraries

Keywords

Graph realization
Graphic sequence
Interval sequence

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10.1137/20m1326489

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abstract = "We consider the problem of realizable interval sequences. An interval sequence is comprised of n integer intervals [ai, bi] such that 0 ≤ ai ≤ bi ≤ n 1 and is said to be graphic/realizable if there exists a graph with degree sequence, say, D = (d1, . . . , dn), satisfying the condition ai ≤ di ≤ bi for each i in [1, n]. There is a characterization (also implying an O(n) verifying algorithm) known for realizability of interval sequences, which is a generalization of the ErdH os-Gallai characterization for graphic sequences. However, given any realizable interval sequence, there is no known algorithm for computing a corresponding graphic certificate in o(n2) time. In this paper, we provide an O(n log n) time algorithm for computing a graphic sequence for any realizable interval sequence. In addition, when the interval sequence is nonrealizable, we show how to find a graphic sequence having minimum deviation with respect to the given interval sequence in the same time. Finally, we consider variants of the problem, such as computing the most-regular graphic sequence and computing a minimum extension of a length p nongraphic sequence to a graphic one.",

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AU - CHOUDHARY, KEERTI

AU - PELEG, DAVID

AU - RAWITZ, DROR

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AB - We consider the problem of realizable interval sequences. An interval sequence is comprised of n integer intervals [ai, bi] such that 0 ≤ ai ≤ bi ≤ n 1 and is said to be graphic/realizable if there exists a graph with degree sequence, say, D = (d1, . . . , dn), satisfying the condition ai ≤ di ≤ bi for each i in [1, n]. There is a characterization (also implying an O(n) verifying algorithm) known for realizability of interval sequences, which is a generalization of the ErdH os-Gallai characterization for graphic sequences. However, given any realizable interval sequence, there is no known algorithm for computing a corresponding graphic certificate in o(n2) time. In this paper, we provide an O(n log n) time algorithm for computing a graphic sequence for any realizable interval sequence. In addition, when the interval sequence is nonrealizable, we show how to find a graphic sequence having minimum deviation with respect to the given interval sequence in the same time. Finally, we consider variants of the problem, such as computing the most-regular graphic sequence and computing a minimum extension of a length p nongraphic sequence to a graphic one.

KW - Graph realization

KW - Graphic sequence

KW - Interval sequence

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Efficiently realizing interval sequences

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