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 SpaceQueryUpdate Chazelle (1988)O(n)O(lg n) JáJá (2004)*O(n)O(lg n / lglg n) Chazelle (1988)O(n)O(lg 2 n) Nekrich (2009)O(n)O((lg n / lglg n) 2 )O(lg 4+ε n) (0<ε<1) Our resultO(n)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(i 1, j 1, i 2, j 2 ): the sum of numbers in A[i 1..i 2, i 2.. j 2 ] 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.

Dynamic Range Sum: An Example range_sum(2, 3, 3, 6) =25insert(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(lg λ n), 0 < λ < 1 modify(i, j, δ): |δ| ≤ lg n insert and delete: no support  Our solution Space: O(lg 1+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(lg γ 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 r th occurrence of α in S range_count(p 1, p 2, v 1, v 2 ): number of entries in S[p 1.. p 2 ] whose values are in the range [v 1.. v 2 ]. 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: nH 0 + 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  T x : 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  T y : A weight balanced B-tree (Arge and Vitter 2003) constructed over all the y-coordinates Branching factor d = Θ(lg ε 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 T y, construct a sequence S v : Each entry of S v corresponds to a point whose y-coordinate is in the range represented by node v S v [i] corresponds to the point with the i th smallest x-coordinate among all these points S v [i] indicates which child of v contains the y-coordinate of the above point  For each level m, construct a sequence L m [1..n] of integers from [1..4d] by concatenating the all the S v ’s constructed at level m  L m : 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: [x 1..x 2 ] × [y 1..y 2 ]  Use T x 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 T y whose ranges contain y 1 and y 2  Perform range_count on S v 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 T y as weight balanced B- tree allows us to amortize the updating cost of subsequences of L m ’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: nH 0 + o(n lg σ) + O(w) bits Time (including range_count): O(──── ( ──── + 1)) lg σ lg lg n lg 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

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