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oai:drops-oai.dagstuhl.de:11541 2024-03-06T10:48:16Z ddc:004 open_access

Path and Ancestor Queries over Trees with Multidimensional Weight Vectors He, Meng Kazi, Serikzhan path queries range queries algorithms data structures theory 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Meng He and Serikzhan Kazi 2019 Is Part Of LIPIcs, Volume 149, 30th International Symposium on Algorithms and Computation (ISAAC 2019) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ISAAC.2019.45 urn:nbn:de:0030-drops-115415 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2019.45 eng https://creativecommons.org/licenses/by/3.0/legalcode