Existence of k-ary Trees: Subtree Sizes, Heights and Depths

TL;DR

This paper investigates the computational complexity of determining the existence of k-ary trees based on various attribute sequences, providing polynomial algorithms for some cases and proving NP-Completeness for others, thereby clarifying the problem's difficulty boundaries.

Contribution

It introduces polynomial algorithms for verifying tree existence from depth and height sequences and establishes NP-Completeness results for subtree size sequences, advancing understanding of these problems.

Findings

Polynomial time algorithms for depth and height sequence verification

NP-Completeness of subtree size sequence existence problems

Identification of problem boundaries between easy and hard cases

Abstract

The rooted tree is an important data structure, and the subtree size, height, and depth are naturally defined attributes of every node. We consider the problem of the existence of a k-ary tree given a list of attribute sequences. We give polynomial time (O(nlog(n))) algorithms for the existence of a k-ary tree given depth and/or height sequences. Our most significant results are the Strong NP-Completeness of the decision problems of existence of k-ary trees given subtree sizes sequences. We prove this by multi-stage reductions from NUMERICAL MATCHING WITH TARGET SUMS. In the process, we also prove a generalized version of the 3-PARTITION problem to be Strongly NP-Complete. By looking at problems where a combination of attribute sequences are given, we are able to draw the boundary between easy and hard problems related to existence of trees given attribute sequences and enhance our understanding of where the difficulty lies in such problems.