Ranking and unranking trees with given degree sequences

TL;DR

This paper introduces efficient algorithms for ranking, unranking, and randomly generating Cayley trees with specified degree sequences or multisets, handling large combinatorial classes with optimized computational complexity.

Contribution

It presents the first efficient algorithms for ranking and unranking degree-restricted Cayley trees with proven complexity bounds.

Findings

Algorithms operate in O(n^2 log n) bit operations.

Largest ranks are of order n! for trees with n vertices.

Methods enable uniform random generation of degree-restricted trees.

Abstract

In this paper, we provide algorithms to rank, unrank, and randomly generate certain degree-restricted classes of Cayley trees. Specifically, we consider classes of trees that have a given degree sequence or a given multiset of degrees. If the underlying set of trees have n vertices, then the largest ranks involved in each case are of order n! so that it takes O(nlog(n)) bits just to write down the ranks. Our ranking and unranking algorithms for these degree-restricted classes are as efficient as can be expected since we show that they require O(n^2log(n)) bit operations if the underlying trees have n vertices.