Path and Ancestor Queries on Trees with Multidimensional Weight Vectors

TL;DR

This paper introduces efficient data structures for answering complex path and ancestor queries on trees with multidimensional weights, generalizing orthogonal range queries and achieving near-optimal tradeoffs.

Contribution

It presents novel algorithms and data structures for path and ancestor queries on trees with multidimensional weights, extending orthogonal range query techniques.

Findings

Achieves near-optimal query and space tradeoffs for ancestor dominance reporting.

First to solve path successor problem for one-dimensional weights.

Provides efficient solutions for path counting and reporting problems.

Abstract

We consider an ordinal tree $T$ on $n$ nodes, with each node assigned a $d$-dimensional weight vector $\pnt{w} \in \{1,2,\ldots,n\}^d,$ where $d \in \mathbb{N}$ is a constant. We study path queries as generalizations of well-known {\textit{orthogonal range queries}}, with one of the dimensions being tree topology rather than a linear order. Since in our definitions $d$ only represents the number of dimensions of the weight vector without taking the tree topology into account, a path query in a tree with $d$-dimensional weight vectors generalize the corresponding $(d+1)$-dimensional orthogonal range query. We solve {\textit{ancestor dominance reporting}} problem as a direct generalization of dominance reporting problem, %in time $\O((\lg^{d-1} n)/(\lg\lg n)^{d-2}+k)$ in time $\O(\lg^{d-1}{n}+k)$ %and space of $\O(n(\lg n)^{d-1}/(\lg \lg n)^{d-2})$ words, and space of $\O(n\lg^{d-2}n)$ words, where $k$ is the size of the output, for $d \geq 2.$ We also achieve a tradeoff of $\O(n\lg^{d-2+\eps}{n})$ words of space, with query time of $\O((\lg^{d-1} n)/(\lg\lg n)^{d-2}+k),$ for the same problem, when $d \geq 3.$ We solve {\textit{path successor problem}} in $\O(n\lg^{d-1}{n})$ words of space and time $\O(\lg^{d-1+\eps}{n})$ for $d \geq 1$ and an arbitrary constant $\eps > 0.$ We propose a solution to {\textit{path counting problem}}, with $\O(n(\lg{n}/\lg\lg{n})^{d-1})$ words of space and $\O((\lg{n}/\lg\lg{n})^{d})$ query time, for $d \geq 1.$ Finally, we solve {\textit{path reporting problem}} in $\O(n\lg^{d-1+\eps}{n})$ words of space and $\O((\lg^{d-1}{n})/(\lg\lg{n})^{d-2}+k)$ query time, for $d \geq 2.$ These results match or nearly match the best tradeoffs of the respective range queries. We are also the first to solve path successor even for $d = 1$.