Optimal Planar Orthogonal Skyline Counting Queries

Gerth Stølting Brodal and Kasper Green Larsen

Aarhus University

14th Scandinavian Workshop on Algorithm Theory, Copenhagen, Denmark, July 3, 2014

n pointsk output

orthogonal query range

dominated by 4 points

skyline count = 5

not dominated = skyline point

Assumptions coordinates { 0, 1, ... , n-1 } Unit cost RAM with word size w = (log n)

Results

Orthogonal range SkylineSpace

(words)Query Space

(words)Query

Reportingn

nlgε nnlgO(1) n

klgε nk + lglg n

(k + lglg n)

CLP11ABR00PT06

nnlglg n

nlgε n

nlg n/lglg n

klgε nk(lglg n)2

klglg n + lg n/lglg n

klglg nk + lg n/lglg nk + lg n/lglg n

newNN12new

NN12new

DGKASK12

Counting nnlgO(1) n

lg n/lglg nlg n/lglg n

JMS04P07

nlg nnlg3 n/lglg n

nnlgO(1) n

lg nlg n/lglg nlg n/lglg n

(lg n/lglg n)

DGKASK12DGS13

newnew

{{

topmost point(x,y)

y+1

Basic Geometry – Divide and Conquer

recurse

recurse

Basic Counting – Vertical Slab

11

1

2

2

2

3

3

223

34

3

rightmost

topmost

skyline count = 4 - 2 + 1

topmostrightmost

0

10

02

03

010

1

022

12

prefix sum = 8Data Structure

succinct prefix sumO(n) bits

+succinct

range maximaO(n) bits

Upper Bound

height lg n / lglg n

degree lgε n

Data Structure

succinct fractional cascading on y

O(nlg n) bits

Upper Bound – Multi-slab

1

lgO(ε) npoints

BlockBtop

Bbottom

degree lgε n

2

3

4

5

right

right

top

top

1 3+ tabulation ( blocks have o(lg n) bit signatures )4 5+ single slab queries ( succinct prefix sum )2 lg2ε n multi-slab structures using lglg n bits per block ( succinct prefix sum, range maxima )

Upper Bound – Summary

O(lg n / lglg n) orthogonal skyline countingSpace O(n) words

+ succinct stuff ...

Lower Bound – Skyline CountingReductionReachability in the Butterfly Graph Skyline Counting

Word size lgO(1) n bits, space O(nlgO(1) n) (lg n / lglg n) query

t

000 001 010 011 100 101 110 111

000 001 010 011 100 101 110 111

001 010 011 100 101 110 111

001 010 011 100 101 110 111000

000

s

[-, x] [-, y]

Butterfly Graph

Butterfly Graph dashed edges are deleted s-t paths are unique

t

000 001 010 011 100 101 110 111

000 001 010 011 100 101 110 111

001 010 011 100 101 110 111

001 010 011 100 101 110 111000

000

s

c

a

b

rev(t) 011

100

101

110

000 001 100011010 101 110 111

111

000

001

010

s

a

b

c

2-sided Skyline Range Counting depth of edge aspect ratio of rectangle edge = 1 point, deleted edge = 2 points

Results

Orthogonal range SkylineSpace

(words)Query Space

(words)Query

Reportingn

nlgε nnlgO(1) n

klgε nk + lglg n

(k + lglg n)

CLP11ABR00PT06

nnlglg n

nlgε nnlg n/lglg n

klgε nk(lglg n)2

klglg n + lg n/lglg n

klglg nk + lg n/lglg nk + lg n/lglg n

newNN12new

NN12new

DGKASK12

Counting nnlgO(1) n

lg n/lglg nlg n/lglg n

JMS04P07

nlg nnlg3 n/lglg n

nnlgO(1) n

lg nlg n/lglg nlg n/lglg n

(lg n/lglg n)

DGKASK12DGS13

newnew

{{

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