eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-06-08
28:1
28:15
10.4230/LIPIcs.SoCG.2020.28
article
Further Results on Colored Range Searching
Chan, Timothy M.
1
https://orcid.org/0000-0002-8093-0675
He, Qizheng
1
Nekrich, Yakov
2
Department of Computer Science, University of Illinois at Urbana-Champaign, IL, USA
Department of Computer Science, Michigan Technological University, Houghton, MI, USA
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((log n)/(log log n) + 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 log²n) 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((log n)/(log log n) + 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 (log n)/(log log n)) 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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol164-socg2020/LIPIcs.SoCG.2020.28/LIPIcs.SoCG.2020.28.pdf
Range searching
geometric data structures
randomized incremental construction
random sampling
word RAM