Efficiently Realizing Interval Sequences
We consider the problem of realizable interval-sequences. An interval sequence comprises of n integer intervals [a_i,b_i] such that 0 <= a_i <= b_i <= n-1, and is said to be graphic/realizable if there exists a graph with degree sequence, say, D=(d_1,...,d_n) satisfying the condition a_i <= d_i <= b_i, for each i in [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(n^2) 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.
Graph realization
graphic sequence
interval sequence
Mathematics of computing~Graph algorithms
Mathematics of computing~Enumeration
47:1-47:15
Regular Paper
US-Israel BSF grant 2018043; Army Research Laboratory Cooperative Grant, ARL Network Science CTA, W911NF-09- 2-0053.
Amotz
Bar-Noy
Amotz Bar-Noy
City University of New York (CUNY), USA
Keerti
Choudhary
Keerti Choudhary
Weizmann Institute of Science, Rehovot, Israel
David
Peleg
David Peleg
Weizmann Institute of Science, Rehovot, Israel
Dror
Rawitz
Dror Rawitz
Bar Ilan University, Ramat-Gan, Israel
10.4230/LIPIcs.ISAAC.2019.47
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Amotz Bar-Noy, Keerti Choudhary, David Peleg, and Dror Rawitz
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