Efficiently realizing interval sequences

Amotz Bar-Noy; Keerti Choudhary; David Peleg; Dror Rawitz

doi:10.4230/LIPIcs.ISAAC.2019.47

Efficiently realizing interval sequences

Amotz Bar-Noy, Keerti Choudhary, David Peleg, Dror Rawitz

Bar-Ilan University - The Alexander Kofkin Faculty of Engineering

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

4 Scopus citations

Abstract

We consider the problem of realizable interval-sequences. An interval sequence comprises 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 ∈ [1, n]. There is a characterisation (also implying an O(n) verifying algorithm) known for realizability of interval-sequences, which is a generalization of the Erdös-Gallai characterisation 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 non-realizable, 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 non-graphic sequence to a graphic one.

Original language	English
Title of host publication	30th International Symposium on Algorithms and Computation, ISAAC 2019
Editors	Pinyan Lu, Guochuan Zhang
Publisher	Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)	9783959771306
DOIs	https://doi.org/10.4230/LIPIcs.ISAAC.2019.47
State	Published - Dec 2019
Event	30th International Symposium on Algorithms and Computation, ISAAC 2019 - Shanghai, China Duration: 8 Dec 2019 → 11 Dec 2019

Publication series

Name	Leibniz International Proceedings in Informatics, LIPIcs
Volume	149
ISSN (Print)	1868-8969

Conference

Conference	30th International Symposium on Algorithms and Computation, ISAAC 2019
Country/Territory	China
City	Shanghai
Period	8/12/19 → 11/12/19

Bibliographical note

Publisher Copyright:
© Amotz Bar-Noy, Keerti Choudhary, David Peleg, and Dror Rawitz; licensed under Creative Commons License CC-BY

Funding

Funding US-Israel BSF grant 2018043; Army Research Laboratory Cooperative Grant, ARL Network Science CTA, W911NF-09-2-0053. US-Israel BSF grant 2018043; Army Research Laboratory Cooperative Grant, ARL Network Science CTA, W911NF-09- 2-0053.

Funders	Funder number
ARL Network Science CTA	W911NF-09-2-0053
Bloom's Syndrome Foundation	2018043
Army Research Laboratory
United States-Israel Binational Science Foundation

Keywords

Graph realization
Graphic sequence
Interval sequence

Access to Document

10.4230/LIPIcs.ISAAC.2019.47

Cite this

Bar-Noy, A., Choudhary, K., Peleg, D., & Rawitz, D. (2019). Efficiently realizing interval sequences. In P. Lu, & G. Zhang (Eds.), 30th International Symposium on Algorithms and Computation, ISAAC 2019 Article 47 (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 149). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.ISAAC.2019.47

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title = "Efficiently realizing interval sequences",

abstract = "We consider the problem of realizable interval-sequences. An interval sequence comprises 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 ∈ [1, n]. There is a characterisation (also implying an O(n) verifying algorithm) known for realizability of interval-sequences, which is a generalization of the Erd{\"o}s-Gallai characterisation 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 non-realizable, 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 non-graphic sequence to a graphic one.",

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Bar-Noy, A, Choudhary, K, Peleg, D & Rawitz, D 2019, Efficiently realizing interval sequences. in P Lu & G Zhang (eds), 30th International Symposium on Algorithms and Computation, ISAAC 2019., 47, Leibniz International Proceedings in Informatics, LIPIcs, vol. 149, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 30th International Symposium on Algorithms and Computation, ISAAC 2019, Shanghai, China, 8/12/19. https://doi.org/10.4230/LIPIcs.ISAAC.2019.47

Efficiently realizing interval sequences. / Bar-Noy, Amotz; Choudhary, Keerti; Peleg, David et al.
30th International Symposium on Algorithms and Computation, ISAAC 2019. ed. / Pinyan Lu; Guochuan Zhang. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2019. 47 (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 149).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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AU - Bar-Noy, Amotz

AU - Choudhary, Keerti

AU - Peleg, David

AU - Rawitz, Dror

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N2 - We consider the problem of realizable interval-sequences. An interval sequence comprises 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 ∈ [1, n]. There is a characterisation (also implying an O(n) verifying algorithm) known for realizability of interval-sequences, which is a generalization of the Erdös-Gallai characterisation 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 non-realizable, 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 non-graphic sequence to a graphic one.

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KW - Interval sequence

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Efficiently realizing interval sequences

Abstract

Publication series

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Bibliographical note

Funding

Keywords

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