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# Path and Ancestor Queries over Trees with Multidimensional Weight Vectors

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LIPIcs.ISAAC.2019.45.pdf
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## Cite As

Meng He and Serikzhan Kazi. Path and Ancestor Queries over Trees with Multidimensional Weight Vectors. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 45:1-45:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.ISAAC.2019.45

## Abstract

We consider an ordinal tree T on n nodes, with each node assigned a d-dimensional weight vector w in {1,2,...,n}^d, where d in N is a constant. We study path queries as generalizations of well-known {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 {ancestor dominance reporting} problem as a direct generalization of dominance reporting problem, in time O(lg^{d-1}{n}+k) and space of O(n lg^{d-2}n) words, where k is the size of the output, for d >= 2. We also achieve a tradeoff of O(n lg^{d-2+epsilon}{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 >= 3. We solve {path successor problem} in O(n lg^{d-1}{n}) words of space and time O(lg^{d-1+epsilon}{n}) for d >= 1 and an arbitrary constant epsilon > 0. We propose a solution to {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 >= 1. Finally, we solve {path reporting problem} in O(n lg^{d-1+epsilon}{n}) words of space and O((lg^{d-1}{n})/(lg lg{n})^{d-2}+k) query time, for d >= 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.

## Subject Classification

##### ACM Subject Classification
• Information systems → Data structures
• Theory of computation → Data structures design and analysis
• Information systems → Multidimensional range search
##### Keywords
• path queries
• range queries
• algorithms
• data structures
• theory

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