Fixed Partial Match Queries in Quadtrees
Several recent papers in the literature have addressed the analysis of the cost P_{n,q} of partial match search for a given fixed query q - that has s out of K specified coordinates - in different multidimensional data structures. Indeed, detailed asymptotic estimates for the main term in the expected cost P_{n,q} = E {P_{n,q}} in standard and relaxed K-d trees are known (for any dimension K and any number s of specified coordinates), as well as stronger distributional results on P_{n,q} for standard 2-d trees and 2-dimensional quadtrees. In this work we derive a precise asymptotic estimate for the main order term of P_{n,q} in quadtrees, for any values of K and s, 0 < s < K, under the assumption that the limit of P_{n,q}/n^alpha when n -> infty exists, where alpha is the exponent of n in the expected cost of a random partial match query with s specified coordinates in a random K-dimensional quadtree.
Quadtree
Partial match queries
Associative queries
Multidimensional search
Analysis of algorithms
Theory of computation~Data structures design and analysis
Theory of computation~Design and analysis of algorithms
20:1-20:18
Regular Paper
This work has been partially supported by funds from the Spanish Ministery of Economy, Industry and Competitiviness (MINECO) and the European Union (FEDER) under grant GRAMM (TIN2017-86727-C2-1-R), and by funds from the Catalan Government (AGAUR) under grant 2017SGR 786.
Amalia
Duch
Amalia Duch
Universitat Politècnica de Catalunya
https://orcid.org/0000-0003-4371-1286
Gustavo
Lau
Gustavo Lau
Universitat Politècnica de Catalunya
https://orcid.org/0000-0002-3460-9186
Conrado
Martínez
Conrado Martínez
Universitat Politècnica de Catalunya
https://orcid.org/0000-0003-1302-9067
10.4230/LIPIcs.AofA.2018.20
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Amalia Duch, Gustavo Lau, and Conrado Martínez
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