1. Covering Uncertain Points in a Tree
Haitao Wang and Jingru Zhang
Utah State University
WADS 2017, St. John’s, Canada

2. An uncertain point in a tree T
An uncertain point 𝑷 𝒊 has multiple possible locations on T with probabilities
𝑝 𝑖2
0.2
𝑝 𝑖1
0.4
0.3
𝑝 𝑖5
𝑝 𝑖4
0.1
𝑝 𝑖3
0.1

3. An uncertain point
The expected distance from 𝑷 𝒊 to a point x is 𝐸𝑑 𝑃 𝑖 ,𝑥 = 𝜔 𝑖 ∙ 𝑗=1 𝑚 𝑓 𝑖𝑗 ∙𝑑(𝑥, 𝑝 𝑖𝑗 )
𝑓 𝑖3
𝑝 𝑖3
𝑓 𝑖2 x
𝑓 𝑖5
𝑓 𝑖1
𝑝 𝑖2
𝑝 𝑖5
𝑝 𝑖1
𝑓 𝑖4
𝑝 𝑖4

4. Problem definition T 𝑄= 𝑞 1 , 𝑞 2 𝑞 1 𝑞 2
Input: a tree T a set 𝑷 of n uncertain points and each 𝑷 𝒊 has 𝒎 𝒊 locations a covering range 𝝀 Goal: Find a set 𝑄 of a minimum number of centers on T such that max 1≤𝑖≤𝑛 𝐸𝑑( 𝑃 𝑖 ,𝑄)≤𝝀 where 𝐸𝑑 𝑃 𝑖 ,𝑄 = min 𝑞∈𝑄 𝐸𝑑( 𝑃 𝑖 ,𝑞) T
𝑄= 𝑞 1 , 𝑞 2
𝑞 1
𝑞 2 𝑃 𝑃 𝑃

5. An application The k-center of uncertain points on a tree
Input: a tree T and a set 𝑷 of uncertain points
Goal: Find a set 𝑄 of at most k centers to minimize
max 1≤𝑖≤𝑛 𝐸𝑑( 𝑃 𝑖 ,𝑄)

6. Related works
Covering certain points in a tree: Time: 𝑂 𝑛 --- O. Kariv and S. Hakimi (1979) The k-center problem of uncertain points on a line: Time: 𝑂 𝑚𝑛 log 𝑚𝑛 +𝑛 log 𝑘 log 𝑛 --- Wang and Zhang (2014) The one-center problem of uncertain points in a tree: Time: 𝑂 𝑚𝑛 --- Wang and Zhang (2015)

7. Our result
The problem has not been studied before Our result: 𝑂( 𝑇 +𝑀 log 𝑀 2 ) where 𝑀 is the number of locations of all uncertain points The k-center problem: O(|T|+n2 log n log M + M log3 M)

8. Problem definition T 𝑄= 𝑞 1 , 𝑞 2 𝑞 1 𝑞 2
Input: a tree T a set 𝑷 of n uncertain points and each 𝑷 𝒊 with 𝒎 𝒊 locations a covering range 𝝀 Goal: Find a set 𝑄 of a minimum number of centers on T such that max 1≤𝑖≤𝑛 𝐸𝑑( 𝑃 𝑖 ,𝑄)≤𝝀 T
𝑄= 𝑞 1 , 𝑞 2
𝑞 1
𝑞 2 𝑃 𝑃 𝑃

9. 𝐸𝑑 𝑝 𝑖 , 𝑃 𝑖 ≤𝐸𝑑 𝑥, 𝑃 𝑖 for all 𝑥∈𝑇
The Median 𝑝 𝑖 of 𝑃 𝑖
The median 𝑝 𝑖 of an uncertain point 𝑃 𝑖 is a vertex of T such that 𝐸𝑑 𝑥, 𝑃 𝑖 is minimized for all 𝑥∈𝑇. 𝑇
𝑝 𝑖1
𝑝 𝑖2
𝑝 𝑖
𝑝 𝑖3
𝑝 𝑖4
𝑝 𝑖5
𝐸𝑑 𝑝 𝑖 , 𝑃 𝑖 ≤𝐸𝑑 𝑥, 𝑃 𝑖 for all 𝑥∈𝑇
Observation: 𝐸𝑑 𝑥, 𝑃 𝑖 is monotonically increasing if x is moving away from pi

10. The medians-spanning tree
The Medians-Spanning Tree 𝑇 𝑚 is the minimum connected subtree of T that spans all medians.
𝑇 𝑚
𝑝 6 𝑇
𝑝 3
𝑝 1
𝑝 2
𝑝 4
𝑝 5

11. An observation 𝑇 𝑇 𝑚 All centers are in the medians-spanning tree of T
𝑞 1
𝑞 2
𝑝 3
𝑝 1
𝑝 2
𝑝 4
𝑝 5

12. A greedy algorithm 𝑇 𝑚 𝑝 6 𝑞 𝑝 7 𝑝 3 𝑝 1 𝑝 2 𝑝 8 𝑝 4 𝑝 5 𝑝 9 𝑝 10
Place centers on 𝑇 𝑚 from bottom to top in the post-order traversal
𝑇 𝑚
𝑟 --- root of 𝑇 𝑚
𝑝 6
Initially, all medians are active. 𝑞
𝑝 7
𝑝 3
𝑝 1
𝑝 2
𝑝 8
𝑝 4
𝑝 5
𝑝 9
𝑝 10
Deactivation: make medians whose uncertain points are covered by 𝑞 inactive

13. Processing a leaf 𝑇 𝑚 ∙∙∙∙∙∙ 𝑟 𝑐 𝑐 𝑝 𝑗 e 𝑝 𝑖 𝑝 𝑖+1
Test whether a center is necessary on e for active 𝑃 𝑖
Compute c on 𝜋( 𝑝 𝑖 ,r) with
𝐸𝑑 𝑐, 𝑃 i =𝜆 𝑟
𝑇 𝑚
𝜋( 𝑝 𝑖 ,r)
c is on e? Y N
𝐸𝑑 𝑐, 𝑃 i =𝜆 𝑐
∙∙∙∙∙∙
Put a center at c for 𝑃 𝑖
𝑃 𝑖 served by a center not on e 𝑐 e
𝑝 𝑗
Report all uncertain points covered by 𝑐
𝑝 𝑖
𝑝 𝑖+1
c --- a candidate center
Deactivation

14. Processing an internal node
Test whether a center is necessary on e for active 𝑃 𝑖1 , 𝑃 𝑖2 , ∙∙∙, 𝑃 𝑖𝑥
Data Structure 𝐴 1
Range-status-query 𝑟
Check whether 𝑻 v has active medians
𝑇 𝑚
𝜋(v,r) Y
Data Structure 𝐴 2
Compute c on 𝜋(v,r) with max 𝑖1≤𝑖≤𝑖𝑥 𝐸𝑑 𝑐, 𝑃 i =𝜆
Candidate center 𝑐
Candidate-center-query 𝑐 e v
c is on e?
Data Structure 𝐴 3 N Y
𝑻 v
Coverage-report-query
Severed by a center not on e
Active medians 𝑝 𝑖1 , 𝑝 𝑖2 , ∙∙∙, 𝑝 𝑖𝑥
Report all uncertain points covered by 𝑐
Deactivation

15. Designing data structures---Major challenge
Preprocessing time
Range-status query
Deletion
𝑂(𝑀)
𝑂( log 𝑛 )
Data Structure 𝐴 1
Preprocessing time
Candidate-center-query
Deletion
𝑂(𝑀 log 𝑀 +𝑛 log 2 𝑀)
𝑂( log 𝑛 )
Data Structure 𝐴 2
Preprocessing time
Coverage-report-query
Deletion
𝑂(𝑀 log 2 𝑀)
𝑂(𝑘 log 𝑛 + log 𝑀 log 𝑛 )
𝑂( 𝑚 𝑖 log 𝑀 log 2 𝑛)
Data Structure 𝐴 3
amortized

16. A connector-bounded centroid decomposition
𝑇 2 (𝑐)
𝑇 1 (𝑐) 𝒄
c---the centroid of T
max⁡{ 𝑇 1 𝑐 , | 𝑇 2 (𝑐)|}≤ 2 𝑇 3
Decomposition Tree 𝛤 T
c --- connector of 𝑇 1 𝑐 and 𝑇 2 𝑐
𝑇 1 (𝑐)
𝑇 2 (𝑐)

17. A connector-bounded centroid decomposition
𝑇 2 ( 𝑐 1 )
𝑇 1 (𝑐) 𝒄 c
𝑇 2 (𝑐)
𝑇 1 ( 𝑐 1 )
Decomposition Tree 𝛤 T
𝑇 1 (𝑐)
𝑇 2 (𝑐)
c, 𝑐 connectors of 𝑇 1 ( 𝑐 1 )
𝑐 connectors of 𝑇 2 ( 𝑐 1 )
𝑇 1 ( 𝑐 1 )
𝑇 2 ( 𝑐 1 )

18. A connector-bounded centroid decomposition
𝑇 1 ( 𝑐 1 )
𝑇 13 ( 𝑐 4 )
Every subtree of 𝛤 has at most two connectors.
𝑇 1 ( 𝑐 3 )
𝒄 𝟏 c
𝒄 𝟒
Decomposition Tree 𝛤
𝑇 11 ( 𝑐 4 ) 𝒄 c c
𝒄 𝟑 T c
𝑇 12 ( 𝑐 4 )
𝑇 1 (𝑐)
𝑇 2 (𝑐)
𝑇 2 ( 𝑐 3 )
𝑐 4 is the vertex such that 𝑇 1 ( 𝑐 3 ) can be decomposed to three subtrees that contain c, 𝑐 1 , 𝑐 3 , respectively.
𝑇 1 ( 𝑐 1 )
𝑇 2 ( 𝑐 1 )
Decompose 𝑇 1 ( 𝑐 3 ) at 𝑐 4
𝑇 11 ( 𝑐 4 )
𝑇 12 ( 𝑐 4 )
𝑇 13 ( 𝑐 4 )
𝑇 2 ( 𝑐 3 )

19. The decomposition tree Γ of T
Each node 𝑢 of Γ has at most four children
𝑇 1
𝑇 2
Each node 𝑢 of Γ
--- A subtree 𝑇(𝑢) of 𝑇
Every 𝑇(𝑢) has at most two connectors
𝑇 3
𝑇 4
𝑇 5
𝑇 6
Height: 𝑂( log 𝑀 )
# of nodes: 𝑂(𝑀) 𝑢
𝑇(𝑢)
∙∙∙∙∙∙
∙∙∙∙∙∙

20. Data structure 𝐴 3 for coverage-report-query
Decomposition Tree Γ
For every node 𝑢 of Γ
𝐿 𝑢 --- List of all such 𝑃 𝑖 𝑢′
𝑇(𝑢′)
𝐸𝑑(𝑥,𝑃 𝑖 ) --- Expected distance functions of all 𝑃 𝑖 ∈𝐿(𝑢) with respect to x∈𝑇(𝑢) 𝑢
𝑇(𝑢)
𝑡 𝑢 = |𝐿 𝑢 |
∙∙∙∙∙∙
Coverage-report-query in 𝑇(𝑢)
--- 𝑂(𝑘 log 𝑡 𝑢 + log 𝑡 𝑢 ) time
For any point x∈𝑇(𝑢),
Report all 𝑃 𝑖 ∈𝐿(𝑢) covered by 𝑥
--- all 𝑃 𝑖 ∈𝐿 𝑢 with 𝐸𝑑(𝑥,𝑃 𝑖 )≤𝜆
𝑃 𝑖 --- no location in 𝑇(𝑢) but in 𝑇( 𝑢 ′ )

21. Coverage-report-query in 𝑇(𝑢)
𝑐 2
Observation: for each 𝑃 𝑖 ∈𝐿(𝑢)
𝐸𝑑 𝑥, 𝑃 𝑖 = 𝜔 𝑖 ( 𝑎 𝑥 + 𝑑 𝑖 𝑏 𝑥 + 𝐷 𝑖 )
a plane in ℝ 3
𝑇(𝑢)
𝑣 𝑥
for each 𝑃 𝑖 ∈𝐿(𝑢), 𝐸𝑑 𝑥, 𝑃 𝑖 ≤𝝀
defines a half-plane in the plane z = 𝝀
𝑐 1
𝑏 𝑥
𝑎 𝑥
𝑥= 𝑎 𝑥 , 𝑏 𝑥
Report all bounding lines above 𝑥
--- report all 𝑃 𝑖 ∈𝐿(𝑢) with 𝐸𝑑 𝑥, 𝑃 𝑖 ≤λ
𝑎 𝑥 --- the distance of 𝑥 to the closest
vertex 𝑣 𝑥 on 𝜋( 𝑐 1 , 𝑐 2 )
𝑏 𝑥 --- the distance of 𝑣 𝑥 to 𝑐 1
𝐸𝑑 𝑥, 𝑃 𝑖1 ≤𝝀
𝑏 𝑥
𝐸𝑑 𝑥, 𝑃 𝑖2 ≤𝝀
L --- the set of 𝑡 𝑢 bounding lines 𝑥
𝐸𝑑 𝑥, 𝑃 𝑖3 ≤𝝀
𝑎 𝑥

22. Coverage-report-query in 𝑇(𝑢)
𝑐 2
𝑇(𝑢)
Report bounding lines of 𝐿 above 𝑥
𝑐 1
Build half-plane range reporting data structure on 𝐿 in 𝑂( 𝑇 𝑢 + 𝑡 𝑢 log 𝑡 𝑢 ) time
𝑥= 𝑎 𝑥 , 𝑏 𝑥
Half-plane range reporting query for 𝑥
𝜑(𝑢)
1. Coverage-report-query in 𝑇(𝑢) 𝑂(𝑘 log 𝑡 𝑢 + log 𝑡 𝑢 ) time
Report all 𝑃 𝑖 ∈𝐿(𝑢) with 𝐸𝑑 𝑥, 𝑃 𝑖 ≤λ for any 𝑥∈𝑇 𝑢
2. Removing any 𝑃 𝑖 from 𝜑(𝑢) 𝑂( log 𝑡 𝑢 ) time

23. Coverage-report-query on 𝐴 3
𝑇(𝑢), 𝐿(𝑢), 𝜑 𝑢 𝝅
Any point 𝒙 on T
A path 𝝅 in Υ 𝑢
𝑂 log 𝑀 nodes --- subtrees 𝑇(𝑢)
∙∙∙∙∙∙
Coverage-report-query at 𝒙 in every 𝑇(𝑢) 𝒙
Report all indices i with 𝐸𝑑(𝑥,𝑃 𝑖 )≤𝜆
--- 𝑂(𝑘 log 𝑛 + log 𝑀 log 𝑛 ) time

24. Candidate-center-query Coverage-report-query
Data structures
Preprocessing time
Range-status query
Deletion
𝑂(𝑀)
𝑂( log 𝑛 )
Data Structure 𝐴 1
Preprocessing time
Candidate-center-query
Deletion
𝑂(𝑀 log 𝑀 +𝑛 log 2 𝑀)
𝑂( log 𝑛 )
Data Structure 𝐴 2
Preprocessing time
Coverage-report-query
Deletion
𝑂(𝑀 log 2 𝑀)
𝑂(𝑘 log 𝑛 + log 𝑀 log 𝑛 )
𝑂( 𝑚 𝑖 log 𝑀 log 𝑛 )
Data Structure 𝐴 3

25. Thank you! Q&A