Near-Optimal Range Reporting Structures for Categorical Data

Abstract

Range reporting on categorical (or colored) data is a well-studied generalization of the classical range reporting problem in which each of the N input points has an associated color (category). A query then asks to report the set of colors of the points in a given rectangular query range, which may be far smaller than the set of all points in the query range.We study two-dimensional categorical range reporting in both the word-RAM and I/O-model. For the I/O-model, we present two alternative data structures for three-sided queries. The first answers queries in optimal O(lg B N + K/B) I/Os using O(N lg * N ) space, where K is the number of distinct colors in the output, B is the disk block size, and lg * N is the iterated logarithm of N . Our second data structure uses linear space and answers queries in O(lg B N + lg (h) N + K/B) I/Os for any constant integer h ≥ 1. Here lg (1) N = lg N and lg (h) N = lg(lg (h−1) N ) when h > 1. Both solutions use only comparisons on the coordinates. We also show that the lg B N terms in the query costs can be reduced to optimal lg lg B U when the input points lie on a U × U grid and we allow word-level manipulations of the coordinates. We further reduce the query time to just O(1) if the points are given on an N × N grid. Both solutions also lead to improved data structures for four-sided queries. For the word-RAM, we obtain optimal data structures for three-sided range reporting, as well as improved upper bounds for four-sided range reporting.Finally, we show a tight lower bound on onedimensional categorical range counting using an elegant reduction from (standard) two-dimensional range counting.

Categorical range maxima queries

“…Further, using standard techniques [23,18] in conjunction with results in Theorem 2, Theorem 3, we obtain following results for (two dimensional) four-sided color reporting problem. Although this improves the known results of the problem [18], the output set may contain multiple (at most two) copies of the same color. Here k is the output size, log * n is the iterated logarithm of n and B is the block size.…”

“…The first external memory result for this problem was given by Nekrich [23]. His results on this problem were further improved by Larsen and Walderveen [18], where they presented an O(nh)-word data structure with O(log (h) n + k B ) query cost, k being the output size, 1 ≤ h ≤ log * n, log (h) n = log log (h−1) n and log (1) n = log n. Thus by choosing h = log * n, an I/O-optimal structure can be obtained. On the other-hand, a linear space structure can be obtained by choosing h = O(1).…”

“…The problem has been extensively studied in computational geometry and database communities [14,12,6,20,16,23,17,18].…”

“…The data structures described in [23,18] have a limitation that can compromise their usefulness in some situations: the list of colors in the output set may contain several (yet constant) occurrences of the same color. Eliminating such duplicates (in the current settings) needs extra I/Os (sorting is inevitable in these solutions, which makes these results less-optimal in terms of query I/Os).…”

Connectivity Oracles for Graphs Subject to Vertex Failures

“…Each of the three methods uses a different, more sophisticated orthogonal range searching structure. In Section 6.1 we show howÕ(d 3 ) time can be achieved with a 2D colored (aka categorical) range searching structure [58]. Section 6.2 uses a 2D range counting [22] data structure, and Section 6.3 uses a 3D range emptiness data structure [20].…”