Abstract
In this paper we study succinct data structures for onedimensional color reporting and color counting problems.
We are given a set of n points with integer coordinates in the range [1,m] and every point is assigned a color from the set {1,...\sigma}.
A color reporting query asks for the list of distinct colors that occur in a query interval [a,b] and a color counting query asks for the number of distinct colors in [a,b].
We describe a succinct data structure that answers approximate color counting queries in O(1) time and uses \mathcal{B}(n,m) + O(n) + o(\mathcal{B}(n,m)) bits,
where \mathcal{B}(n,m) is the minimum number of bits required to represent an arbitrary set of size n from a universe of m elements. Thus we show, somewhat counterintuitively,
that it is not necessary to store colors of points in order to answer approximate color counting queries.
In the special case when points are in the rank space (i.e., when n=m), our data structure needs only O(n) bits.
Also, we show that \Omega(n) bits are necessary in that case.
Then we turn to succinct data structures for color reporting.
We describe a data structure that uses \mathcal{B}(n,m) + nH_d(S) + o(\mathcal{B}(n,m)) + o(n\lg\sigma) bits and answers queries in O(k+1) time,
where k is the number of colors in the answer, and nH_d(S) (d=\log_\sigma n) is the dth order empirical entropy of the color sequence. Finally, we consider succinct color reporting under restricted updates. Our dynamic data structure uses nH_d(S)+o(n\lg\sigma) bits and supports queries in O(k+1) time.
BibTeX  Entry
@InProceedings{elzein_et_al:LIPIcs:2017:8209,
author = {Hicham ElZein and J. Ian Munro and Yakov Nekrich},
title = {{Succinct Color Searching in One Dimension}},
booktitle = {28th International Symposium on Algorithms and Computation (ISAAC 2017)},
pages = {30:130:11},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770545},
ISSN = {18688969},
year = {2017},
volume = {92},
editor = {Yoshio Okamoto and Takeshi Tokuyama},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2017/8209},
URN = {urn:nbn:de:0030drops82096},
doi = {10.4230/LIPIcs.ISAAC.2017.30},
annote = {Keywords: Succinct Data Structures, Range Searching, Computational Geometry}
}
Keywords: 

Succinct Data Structures, Range Searching, Computational Geometry 
Collection: 

28th International Symposium on Algorithms and Computation (ISAAC 2017) 
Issue Date: 

2017 
Date of publication: 

07.12.2017 