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 Erdos-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(nlogn) 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.Comment: 19 pages, 1 figur