Transcript slides

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
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
Range counting query: 5
Range reporting query
Classic Solutions
Space (words)
O(n1/2 + k)
Chazelle 1988
Range trees
O(n lg n)
O(lg n + k)
Chazelle 1988
O(n lgε n)
O(lg n + k)
O(lg n)
O(lg n + k lgε n)
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. 2000
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)
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(lgε 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 = 2 3 4 4 1 3 1 1 3 2 4 2 3…
Divide the grid into blocks of size lg2 n × t
A 2D array A: A[i,j] stores the result of dominance counting when
(i lg2 n+1, j) is given as the query point
Divide each block into subblocks of size lgλ n × t (0< λ < ε)
A 2D array B: B[i,j] stores, when (i lgλ 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 lgλ n points on a lgλ 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 xcoordinate can be removed using 2n+o(n) extra
 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
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 ×
(logσn + occ lg n)) time
 Our results: O(n lg σ) bits, O(m + lg n × (logσn +
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
Space: 3n lg n + o(n lg n) bits
 Time: O(m + occ lg n / lglg n)
Applications: Representing Small
Data: A sequence S of n numbers in [1..s],
where s = polylog (n)
 Ferragina et al. 2007
Space: nH0(S) + o(n) bits
Operations: rank/select in O(1) time
Our result:
New operation: Given a range of position [p1..p2] and
a range of values [v1..v2], retrieve the entries in
S[p1..p2] whose values are in [v1..v2]
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)
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.
Thank you!