Succinct Orthogonal Range Search Structures on a Grid with Applications to Text Indexing Prosenjit Bose, Carleton University Meng He, Unversity of Waterloo Anil Maheshwari and Pat Morin, Carleton University

2D Orthogonal Range Search  A fundamental geometric query problem  Data sets: A set, N, of n points in the plane  Query: Given an orthogonal query rectangle R, return information about the points in N∩R Orthogonal range counting queries Orthogonal range reporting queries  k: size of the output

Example Range counting query:Range reporting query5

Classic Solutions Data Structures Space (words) Time (counting) Time (reporting) R-treesO(n) kd-treesO(n)O(n 1/2 + k) Chazelle 1988O(n)O(lg n)O(lg n + k lg ε n) Range treesO(n lg n)O(lg n + k) Chazelle 1988O(n lg ε n)O(lg n + k)

Range Search on an n×n Grid  A special case: points coordinates are from [1..n]×[1..n] (rank space)  The general problem can be reduced to this special case using a standard approach Alstrup et al  Orthogonal range search structures in the rank space and succinct data structures

Background: Succinct Data Structures  What are succinct data structures (Jacobson 1989) Representing data structures using ideally information-theoretic minimum space Supporting efficient navigational operations  Why succinct data structures Large data sets in modern applications: textual, genomic, spatial or geometric

Succinct Orthogonal Range Search Structures in rank space  Wavelet Trees (Grossi et al. 2003) Space: n lg n + o (n lg n) bits Query time for orthogonal range search (Makinen and Navarro 2006):  Restriction: no points have the same x or y coordinates  Counting: O(lg n)  Reporting: O(k lg n)  Applications Space-efficient text indexes: Makinen and Navarro 2006, Chien et al. 2008

Support counting: an Overview  Reduce orthogonal range counting to Dominance counting  Design a succinct data structure supporting dominance counting on a narrow grid, i.e. an n×t grid where t = O(lg ε n) (0<ε<1). We also assume that each point has a distinct x-coordinate  Recursively divide the n×n grid into narrow grids and use the above structure at each level  Remove the restriction that each point has a distinct x-coordinate

Range counting on a Narrow Grid S = … Divide the grid into blocks of size lg 2 n × t A 2D array A: A[i,j] stores the result of dominance counting when (i lg 2 n+1, j) is given as the query point Divide each block into subblocks of size lg λ n × t (0< λ < ε) A 2D array B: B[i,j] stores, when (i lg λ n+1, j) is given as a query point, the result of dominance counting inside the block containing this point A table C that stores for each possible set of lg λ n points on a lg λ n × t grid and each query point in the grid, the result of dominance counting Space: n lg t + o(n) bits Time: O(1)

Range Counting on an n×n Grid Transform the original grid into a narrow grid by grouping y-coordinates into ranges of size n/t Construct orthogonal range search structures for this narrow grid and recurse Number of levels: log t n Space: n lg n + o(n lg n) bits Time: O(log t n)

More results  The restriction that each point has a distinct x- coordinate can be removed using 2n+o(n) extra bits  The support for range reporting is based on similar ideas but is more complicated  Our main result Space: n lg n + o (n lg n) bits Query time for orthogonal range  Counting: O(lg n / lg lg n)  Reporting: O(k lg n / lg lg n)

Applications: Substring Search  Notation: T-text, n-text size, σ-alphabet size P-pattern, m-pattern length occ-number of occurrences  Query: report the occurrences of P in T  Chien et al. 2008: O(n lg σ) bits, O(m + lg n × (log σ n + occ lg n)) time  Our results: O(n lg σ) bits, O(m + lg n × (log σ n + occ lg n) / lglg n) time

Applications: Position-Restricted Substring Search  Query: Given a pattern P and a range [i, j], how many times does P occur in T[i, j]?  Makinen and Navarro 2006 Space: 3n lg n + o(n lg n) bits Time: O(m + occ lg n)  Our results: Space: 3n lg n + o(n lg n) bits Time: O(m + occ lg n / lglg n)

Applications: Representing Small Integers  Data: A sequence S of n numbers in [1..s], where s = polylog (n)  Ferragina et al Space: nH 0 (S) + o(n) bits Operations: rank/select in O(1) time  Our result: New operation: Given a range of position [p 1..p 2 ] and a range of values [v 1..v 2 ], retrieve the entries in S[p 1..p 2 ] whose values are in [v 1..v 2 ] Time: O(1) for counting, O(1) per entry for reporting

Applications: A Restricted Versions of Range Search  Restriction: the query rectangle is defined by two points in the given point set  Notation: c: the number of bits required to encode the coordinates of a point  Space: cn + n lg n + o(n lg n) bits  Time: Counting: O (lg n / lglg n) Reporting: O(k lg n / lglg n)

Conclusions  We designed a succinct data structure for orthogonal range search on an n×n grid that provides more efficient support for both counting and reporting queries  This structure can be used to improve and extend previous results on succinct data structures, such as succinct text indexes and sequence representation.

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