TY - GEN
T1 - Further results on colored range searching
AU - Chan, Timothy M.
AU - He, Qizheng
AU - Nekrich, Yakov
N1 - Publisher Copyright:
© Timothy M. Chan, Qizheng He, and Yakov Nekrich; licensed under Creative Commons License CC-BY 36th International Symposium on Computational Geometry (SoCG 2020).
PY - 2020/6/1
Y1 - 2020/6/1
N2 - We present a number of new results about range searching for colored (or “categorical”) data: 1. For a set of n colored points in three dimensions, we describe randomized data structures with O(n polylog n) space that can report the distinct colors in any query orthogonal range (axis-aligned box) in O(k polyloglog n) expected time, where k is the number of distinct colors in the range, assuming that coordinates are in {1,..., n}. Previous data structures require O(logloglognn + k) query time. Our result also implies improvements in higher constant dimensions. 2. Our data structures can be adapted to halfspace ranges in three dimensions (or circular ranges in two dimensions), achieving O(k log n) expected query time. Previous data structures require O(k log2 n) query time. 3. For a set of n colored points in two dimensions, we describe a data structure with O(n polylog n) space that can answer colored “type-2” range counting queries: report the number of occurrences of every distinct color in a query orthogonal range. The query time is O(logloglognn + k log log n), where k is the number of distinct colors in the range. Naively performing k uncolored range counting queries would require O(k logloglognn ) time. Our data structures are designed using a variety of techniques, including colored variants of randomized incremental construction (which may be of independent interest), colored variants of shallow cuttings, and bit-packing tricks.
AB - We present a number of new results about range searching for colored (or “categorical”) data: 1. For a set of n colored points in three dimensions, we describe randomized data structures with O(n polylog n) space that can report the distinct colors in any query orthogonal range (axis-aligned box) in O(k polyloglog n) expected time, where k is the number of distinct colors in the range, assuming that coordinates are in {1,..., n}. Previous data structures require O(logloglognn + k) query time. Our result also implies improvements in higher constant dimensions. 2. Our data structures can be adapted to halfspace ranges in three dimensions (or circular ranges in two dimensions), achieving O(k log n) expected query time. Previous data structures require O(k log2 n) query time. 3. For a set of n colored points in two dimensions, we describe a data structure with O(n polylog n) space that can answer colored “type-2” range counting queries: report the number of occurrences of every distinct color in a query orthogonal range. The query time is O(logloglognn + k log log n), where k is the number of distinct colors in the range. Naively performing k uncolored range counting queries would require O(k logloglognn ) time. Our data structures are designed using a variety of techniques, including colored variants of randomized incremental construction (which may be of independent interest), colored variants of shallow cuttings, and bit-packing tricks.
KW - Geometric data structures
KW - Random sampling
KW - Randomized incremental construction
KW - Range searching
KW - Word RAM
UR - http://www.scopus.com/inward/record.url?scp=85086502375&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85086502375&partnerID=8YFLogxK
U2 - 10.4230/LIPIcs.SoCG.2020.28
DO - 10.4230/LIPIcs.SoCG.2020.28
M3 - Conference contribution
AN - SCOPUS:85086502375
T3 - Leibniz International Proceedings in Informatics, LIPIcs
BT - 36th International Symposium on Computational Geometry, SoCG 2020
A2 - Cabello, Sergio
A2 - Chen, Danny Z.
PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
T2 - 36th International Symposium on Computational Geometry, SoCG 2020
Y2 - 23 June 2020 through 26 June 2020
ER -