eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2019-11-28
47:1
47:15
10.4230/LIPIcs.ISAAC.2019.47
article
Efficiently Realizing Interval Sequences
Bar-Noy, Amotz
1
Choudhary, Keerti
2
Peleg, David
2
Rawitz, Dror
3
City University of New York (CUNY), USA
Weizmann Institute of Science, Rehovot, Israel
Bar Ilan University, Ramat-Gan, Israel
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol149-isaac2019/LIPIcs.ISAAC.2019.47/LIPIcs.ISAAC.2019.47.pdf
Graph realization
graphic sequence
interval sequence