Efficiently realizing interval sequences

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.