Optimal Planar Orthogonal Skyline Counting Queries Gerth Stlting - - PowerPoint PPT Presentation

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 points k output

• rthogonal 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 Skyline Space (words) Query Space (words) Query Reporting n nlgε n nlgO(1) n  klgε n k + lglg n (k + lglg n) CLP11 ABR00 PT06 n nlglg n nlgε n nlg n/lglg n klgε n k(lglg n)2 klglg n + lg n/lglg n klglg n k + lg n/lglg n k + lg n/lglg n new NN12 new NN12 new DGKASK12 Counting n nlgO(1) n  lg n/lglg n lg n/lglg n JMS04 P07 nlg n nlg3 n/lglg n n nlgO(1) n  lg n lg n/lglg n lg n/lglg n (lg n/lglg n) DGKASK12 DGS13 new new

{ {

topmost point (x,y) y+1

Basic Geometry – Divide and Conquer

Basic Counting – Vertical Slab

1 1 1 2 2 2 3 3 2 2 3 3 4 3 rightmost topmost skyline count = 4 - 2 + 1 topmost rightmost 1 2 3 1 1 2 2 12 prefix sum = 8 Data Structure succinct prefix sum O(n) bits + succinct range maxima O(n) bits

Upper Bound

height lg n / lglg n degree lgε n Data Structure succinct fractional cascading on y O(nlg n) bits

Upper Bound – Multi-slab

1 lgO(ε) n points Block Btop 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 counting Space O(n) words + succinct stuff ...

Lower Bound – Skyline Counting

Reduction Reachability in the Butterfly Graph  Skyline Counting

Word size lgO(1) n bits, space O(nlgO(1) n)  (lg n / lglg n) query

t

000 001 010 011 100 101 110 111 000 001 010 011 100 101 110 111 001 010 011 100 101 110 111 001 010 011 100 101 110 111 000 000

s

[-, x]  [-, y] Butterfly Graph

Butterfly Graph

• dashed edges are deleted
• s-t paths are unique

t

000 001 010 011 100 101 110 111 000 001 010 011 100 101 110 111 001 010 011 100 101 110 111 001 010 011 100 101 110 111 000 000

s c a b rev(t) 011

100 101 110 000 001 100 011 010 101 110 111 111 000 001 010

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 Skyline Space (words) Query Space (words) Query Reporting n nlgε n nlgO(1) n  klgε n k + lglg n (k + lglg n) CLP11 ABR00 PT06 n nlglg n nlgε n nlg n/lglg n klgε n k(lglg n)2 klglg n + lg n/lglg n klglg n k + lg n/lglg n k + lg n/lglg n new NN12 new NN12 new DGKASK12 Counting n nlgO(1) n  lg n/lglg n lg n/lglg n JMS04 P07 nlg n nlg3 n/lglg n n nlgO(1) n  lg n lg n/lglg n lg n/lglg n (lg n/lglg n) DGKASK12 DGS13 new new

{ {