Efficiently Realizing Interval Sequences

TL;DR

This paper introduces an efficient $O(n \,\log\, n)$ algorithm for constructing a graphic sequence from a realizable interval sequence, improving over previous methods and addressing related problems.

Contribution

It presents the first $O(n \,\log\, n)$ algorithm for generating a graphic sequence from any realizable interval sequence, and extends to non-realizable cases and variants.

Findings

Developed an $O(n \,\log\, n)$ algorithm for graphic sequence construction.

Provided methods for minimal deviation sequences in non-realizable cases.

Addressed variants like most regular sequences and minimal extensions.

Abstract

We consider the problem of realizable interval-sequences. An interval sequence comprises of $n$ integer intervals $[a_i,b_i]$ such that $0\leq a_i \leq b_i \leq n-1$, and is said to be graphic/realizable if there exists a graph with degree sequence, say, $D=(d_1,\ldots,d_n)$ satisfying the condition $a_i \leq d_i \leq b_i$, 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 Erdos-Gallai characterisation for graphic sequences. However, given any realizable interval-sequence, there is no known algorithm for computing a corresponding graphic certificate in $o(n^2)$ 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.