TY - GEN

T1 - Efficiently realizing interval sequences

AU - Bar-Noy, Amotz

AU - Choudhary, Keerti

AU - Peleg, David

AU - Rawitz, Dror

N1 - Publisher Copyright:
© Amotz Bar-Noy, Keerti Choudhary, David Peleg, and Dror Rawitz; licensed under Creative Commons License CC-BY

PY - 2019/12

Y1 - 2019/12

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.

AB - 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.

KW - Graph realization

KW - Graphic sequence

KW - Interval sequence

UR - http://www.scopus.com/inward/record.url?scp=85076341173&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.ISAAC.2019.47

DO - 10.4230/LIPIcs.ISAAC.2019.47

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AN - SCOPUS:85076341173

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 30th International Symposium on Algorithms and Computation, ISAAC 2019

A2 - Lu, Pinyan

A2 - Zhang, Guochuan

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 30th International Symposium on Algorithms and Computation, ISAAC 2019

Y2 - 8 December 2019 through 11 December 2019

ER -