Space Efficient Data Structures for Dynamic Orthogonal

Range Counting

Meng He and J. Ian Munro

University of Waterloo

Dynamic Orthogonal Range Counting A fundamental geometric query problem

Definitions Data sets: a set P of n points in the plane Query: given an axis-aligned query rectangle R,

compute the number of points in P∩R Update: insertion or deletion of a point

Applications Geometric data processing (GIS, CAD) Databases

Example

Classic Solutions and Our Result

Space Query Update

Chazelle (1988) O(n) O(lg n)

JáJá (2004)* O(n) O(lg n / lglg n)

Chazelle (1988) O(n) O(lg2 n) O(lg2 n)

Nekrich (2009) O(n) O((lg n / lglg n)2) O(lg4+ε n) (0<ε<1)

Our result O(n) O((lg n / lglg n)2) O((lg n / lglg n)2)

Matches the lower bound under the group model Pătraşcu (2007)

* For integer coordinates.

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

A novel and unusual way of using succinct data structures (this paper) Matching the storage cost of standard data structures Improving the time efficiency

Dynamic Range Sum Data

A 2D array A[1..r, 1..c] of numbers Operations

range_sum(i1, j1, i2, j2): the sum of numbers in A[i1..i2, i2.. j2] modify(i, j, δ): A[i, j] ← A[i, j] + δ insert(j): insert a 0 between A[i, j-1] and A[i, j] for i = 1, 2, …, r. delete(j): delete A[i, j] for for i = 1, 2, …, r. To perform this, A[i, j]

must be 0 for all i. Restrictions on r, c and δ and operations supported may

apply.

0

0

00

5

Dynamic Range Sum: An Example

8 2 9 5 4

9 0 7 3 1

1 5 3 10 -2

2 9 1 8 0

5 12 0 3 1

0 0 4 2 8

3 5 4 1 04 1 0 18 5

range_sum(2, 3, 3, 6) = 25 insert(6)

delete(6)range_sum(2, 3, 3, 7) = 30

modify(2, 6, 5)

modify(2, 6, -5)

Dynamic Range Sum in a small 2D Array Assumptions and restrictions

Word size w: Ω(lg n) Each number: nonnegative, O(lg n) bits rc = O(lgλ n) , 0 < λ < 1 modify(i, j, δ): |δ| ≤ lg n insert and delete: no support

Our solution Space: O(lg1+λ n) bits, with an o(n)-bit universal table Time: modify and range_sum in O(1) time Generalization of the 1D array version (Raman et al. 2001) Deamortization is interesting

Range Sum in a Narrow 2D Array Assumptions and restrictions

b = O(w): number of bits required to encode each number “Narrow”: r = O(lgγ c), 0 < λ < 1 |δ| ≤ lg c

Our results Space: O(rcb + w) bits, with an O(c lg c)-bit buffer Operations: O(lg c / lg lg c) time

A generalization of the solution to CSPSI problem based on B trees (He and Munro 2010), using our small 2D array structure on each B-tree node

Range Counting in Dynamic Integer Sequences

Notation Integer range: [1..σ] Sequence: S[1..n]

Operations: access(x): S[x] rank(α, x): number of occurrences of α in S[1..x] select(α, r): position of the rth occurrence of α in S range_count(p1, p2, v1, v2): number of entries in S[p1..

p2] whose values are in the range [v1.. v2]. insert(α, i): insert α between S[i-1] and S[i] delete(i): delete S[i] from S

Range Counting in Integer Sequences: An Example

S = 5,5,2,5,3,1,3,4,7,6,4,1,2,2,5,8

rank(5, 8) = 3

select(2, 3) = 14

range_count(6, 12, 2, 6) =

4

Range Counting in Sequences of Small Integers

Restrictions σ = O(lg ρ n) for any constant 0 < ρ < 1

Our result Space: nH0 + o(n lg σ) + O(w) bits Time: O(lg n / lglg n)

This is achieved by combining: Our solution to range sum on narrow 2D arrays A succinct dynamic string representation (He and Munro

2010)

Dynamic Range Counting: An Augmented Red Black Tree

Tx: A red black tree storing all the x-coordinates

Each node also stores the number of its descendants

Purpose: conversions between real x-coordinates and rank space in O(lg n) time

Dynamic Range Counting: A Range Tree

Ty: A weight balanced B-tree (Arge and Vitter 2003) constructed over all the y-coordinates Branching factor d = Θ(lgε n) for constant 0 < ε < 1 Leaf parameter: 1 The levels are numbered 0, 1, … from top to bottom

Essentially a range tree Each node represents a range of y-coordinates

Choice of weight balanced B-tree: amortizing a rebuilding cost

Dynamic Range Counting: A Wavelet Tree Ideas from generalized wavelet trees (Ferragina et al. 2006)

For each node v of Ty, construct a sequence Sv: Each entry of Sv corresponds to a point whose y-coordinate is in

the range represented by node v Sv [i] corresponds to the point with the ith smallest x-coordinate

among all these points Sv [i] indicates which child of v contains the y-coordinate of the

above point

For each level m, construct a sequence Lm[1..n] of integers from [1..4d] by concatenating the all the Sv’s constructed at level m

Lm : stored as dynamic sequences of small integers Space: O(n lg d + w) bits per level, O(n) words overall

Range Counting Queries Query range: [x1..x2] × [y1..y2]

Use Tx to convert the query x-range to a range in rank space

Perform a top-down traversal to locate the (up to two) leaves in Ty whose ranges contain y1 and y2

Perform range_count on Sv for each node v visited in the above traversal

Sum up the query results to get the answer Time: O(lg n / lglg n) per level, O(lg n / lglg n)

levels

Insertions and Deletions More complicated: splits and merges; changes

to child ranks

The choice of storing Ty as weight balanced B-tree allows us to amortize the updating cost of subsequences of Lm’s

Additional techniques supporting batch updating of integer sequences are also developed

Our Results Dynamic Orthogonal Range Counting

Space: O(n) words Time: O((lg n / lglg n)2)

Points on a U×U grid Space: O(n) words Time (worst-case): O(lg n lg U / (lg lg n)2)

Succinct representations of dynamic integer sequences Space: nH0 + o(n lg σ) + O(w) bits Time (including range_count):O(──── ( ──── + 1))lg σ

lg lg nlg n

lg lg n

Conclusions Results

The best result for dynamic orthogonal range counting Same problem for points on a grid The first succinct representations of dynamic integer sequences

supporting range counting Two preliminary results on dynamic range sum

Techniques The first that combines wavelet trees with range trees Deamortization on 2D arrays

Future work Lower bound Use techniques from succinct data structures to improve

standard data structures

Thank you!