1. The 𝜏-Skyline for Uncertain Data
Haitao Wang
(Utah State University)
Wuzhou Zhang
Presented by
Matt Gibson
(Duke University)
(University of Texas
at San Antonio)

2. Skyline 𝑝 dominates 𝑞, denoted as 𝑝≽𝑞, if 𝑥 𝑝 ≥𝑥(𝑞) and 𝑦 𝑝 ≥𝑦(𝑞)
Here a point dominates itself for simplicity of discussion
Given a set 𝑃 of (exact) points, a point 𝑝∈𝑃 is a skyline point of 𝑃 if 𝑝 is not dominated by any other point of 𝑃.

3. Uncertain Data 𝒫={ 𝑃 1 , …, 𝑃 𝑛 } : a set of 𝑛 uncertain points
𝑃 𝑖 = 𝑝 𝑖1 , …, 𝑝 𝑖𝑘
Pr[ 𝑃 𝑖 =𝑝 𝑖𝑗 ]= 𝑤 𝑖𝑗
𝑗=1 𝑘 𝑤 𝑖𝑗 =1
For a location 𝑝∈ 𝑃 𝑖 , we also use 𝑤(𝑝) to denote the probability of 𝑃 𝑖 being at 𝑝.
Assume: 𝑃 𝑖 ’s are pairwise independent
Set 𝑚=𝑛𝑘.
0.2
0.3
0.4
0.1

4. 𝜏-Skyline of 𝑃 𝑖
Given a point 𝑞, the probability that 𝑃 𝑖 dominates 𝑞:
𝛿 𝑖 𝑞 = 𝑝∈ 𝑃 𝑖 , 𝑝≽𝑞 𝑤(𝑝)
Given 𝜏∈ 0, 1 , the 𝜏-skyline region of 𝑃 𝑖 :
𝑅 𝑖 ={𝑞∣ 𝛿 𝑖 𝑞 ≤𝜏}
i.e., the set of points 𝑞 such that the probability of 𝑃 𝑖 dominating 𝑞 is at most 𝜏.
The 𝜏-skyline of 𝑃 𝑖 , denoted by 𝜋 𝑖 , is the boundary of 𝑅 𝑖 .
0-skyline of 𝑃 𝑖 corresponds to the (conventional) skyline of 𝑃 𝑖 .

5. 𝜏-Skyline of 𝒫
Given 𝜏∈ 0, 1 , the 𝜏-skyline region of 𝒫= 𝑃 1 , …, 𝑃 𝑛 :
𝑅={𝑞∣ 𝛿 𝑖 𝑞 ≤𝜏, ∀𝑖≤𝑛}
i.e., the set of points 𝑞 such that the probability of any 𝑃 𝑖 dominating 𝑞 is at most 𝜏.
The 𝜏-skyline of 𝒫, denoted by 𝜋, is the boundary of 𝑅.
0-skyline of 𝒫 corresponds to the (conventional) skyline of 𝑖=1 𝑛 𝑃 𝑖 .
The 𝜏-skyline probability of each 𝑃 𝑖 of 𝒫 is defined to be the probability of 𝑃 𝑖 lying inside the 𝜏-skyline region of 𝒫.

6. Related Work 𝜌-skyline
Given 𝑞 and 𝒫={ 𝑃 1 , …, 𝑃 𝑛 } , the skyline probability of 𝑞: 𝛼 𝑞,𝒫 = 𝑖=1 𝑛 (1− 𝛿 𝑖 (𝑞) )
The skyline probability of 𝑃 𝑖 :
𝛼 𝑃 𝑖 ,𝒫 = 𝑗=1 𝑘 𝑤 𝑖𝑗 𝛼( 𝑝 𝑖𝑗 , 𝒫 ≠𝑖 )
where 𝒫 ≠𝑖 =𝒫− 𝑃 𝑖 .
Given a parameter 𝜌, the goal is to compute 𝜌-skyline of 𝒫:
{ 𝑃 𝑖 ∣𝛼 𝑃 𝑖 ,𝒫 ≥𝜌}
First sub-quadratic: 𝑂( 𝑚 5/3 poly(log 𝑛)) [Atallah et al. 2011]
𝑂( 𝑚 3/2 ), and 𝑂(𝑚𝑘log 𝑚) when 𝑘≪𝑛 [Afshani et al. 2011]
Heuristic algorithms [Pei et al. 2007]

7. Relate Work (cnt.) Other variants of probabilistic skylines 𝐾-skyband
[Lian et al. 2008] [Zhang et al. 2013]
𝐾-skyband
Given a set 𝑃 of 𝑛 (exact) points and a parameter 𝐾≤𝑛, the 𝐾-skyband of 𝑃 asks for the set of points in 𝑃 which are dominated by at most 𝐾 points of 𝑃.
0-skyband of 𝑃 corresponds the conventional skyline of 𝑃.
Our 𝜏-skyline of 𝑃 𝑖 can be used to answer the weighted skyband of 𝑃 𝑖 .
As a byproduct, we obtain the first algorithm for computing the skyband of a set of weighted points!

8. Our Results We first show that
The 𝜏-skyline 𝜋 𝑖 of 𝑃 𝑖 has complexity 𝑂 𝑘 , and can be computed in 𝑂(𝑘log 𝑘) time.
Using this as a subroutine:
The 𝜏-skyline 𝜋 of 𝒫 has complexity 𝑂 𝑚 , and can be computed in 𝑂(𝑚log 𝑚) time, where 𝑚=𝑛𝑘.
After which,
The 𝜏-skyline probabilities of all 𝑃 𝑖 ’s can be computed in 𝑂(𝑚log 𝑚) time.
Our method is very simple and easy to implement!

9. Our Algorithm
We first compute, for each 𝑃 𝑖 of 𝒫, the 𝜏-skyline 𝜋 𝑖 of 𝑃 𝑖 , in 𝑂(𝑘log 𝑘) time.
Sweeping
horizontally
vertically
Two movements:
move downwards
move rightwards
Zig-Zag path
We then show that 𝜋 is the upper envelope of 𝜋 1 , …, 𝜋 𝑛 , and can be computed in 𝑂(𝑚log 𝑚) time, where 𝑚=𝑛𝑘.

10. Computing the 𝜏-skyline 𝜋 𝑖 of 𝑃 𝑖
The goal is to find the set of points 𝑞 such that the probability of 𝑃 𝑖 dominating 𝑞 is at most 𝜏.
Sweeping in both horizontal and vertical directions!
As a preprocessing step, we sort all the locations of 𝑃 𝑖 into two sorted lists 𝐿 𝑥 and 𝐿 𝑦 :
𝐿 𝑥 by increasing 𝑥-coordinate
𝐿 𝑦 by decreasing 𝑦-coordinate
Maintain two invariants during the sweeping process

11. Two Invariants During Sweeping
We say that 𝑞′ strictly dominates 𝑞, denoted as 𝑞 ′ ≻𝑞, if
𝑥 𝑞 ′ >𝑥 𝑞 and 𝑦 𝑞 ′ >𝑦(𝑞)
Given a point 𝑞, the probability that 𝑃 𝑖 strictly dominates 𝑞:
𝛿 𝑖 + 𝑞 = 𝑝∈ 𝑃 𝑖 , 𝑝≻𝑞 𝑤(𝑝)
Let 𝑞 be any point on the 𝜏-skyline 𝜋 𝑖 of 𝑃 𝑖 .
Two invariants:
𝛿 𝑖 𝑞 >𝜏
𝛿 𝑖 + 𝑞 ≤𝜏

12. Let’s Do The Sweeping!
Initially, we sweep vertically along the line 𝑥=−∞ by scanning the sorted list 𝐿 𝑦 to find 𝑞=(−∞, 𝑦(𝑝)) for some 𝑝∈ 𝐿 𝑦 such that two invariants hold. (trivial)
Then we sweep horizontally.
An event happens if
Either 𝑥 𝑞 =𝑥(𝑝) for some 𝑝∈ 𝐿 𝑥
Or 𝑦 𝑞 =𝑦(𝑝) for some 𝑝∈ 𝐿 𝑦
At each event, we decide to move rightwards or downwards.
Terminate when either 𝐿 𝑥 or 𝐿 𝑦 becomes empty.

13. Analysis We have 𝑂 𝑘 events: 𝐿 𝑥 + 𝐿 𝑦 =2𝑘 At each event, 𝑂(1) time:
Two very simple invariant checks
We update 𝛿 𝑖 (𝑞) and 𝛿 𝑖 + (𝑞)
𝑂 𝑘 events also implies 𝑂 𝑘 complexity of 𝜋 𝑖
Correctness is also easy to verify (see our paper)
Theorem:
The 𝜏-skyline 𝜋 𝑖 of 𝑃 𝑖 has complexity 𝑂 𝑘 , and can be computed in 𝑂(𝑘log 𝑘) time.

14. Computing the 𝜏-skyline 𝜋 of 𝒫
An easy observation is that the 𝜏-skyline region 𝑅 of 𝒫 is the common intersection of the 𝜏-skyline region 𝑅 1 ,…, 𝑅 𝑛 , i.e.,
𝑅= 𝑖=1 𝑛 𝑅 𝑖
And the 𝜏-skyline 𝜋 of 𝒫 is simply the upper envelope of 𝜋 1 , …, 𝜋 𝑛 . Equivalently, 𝜋 is the (conventional) skyline of the 𝑂(𝑛𝑘) turning points of all the 𝜋 𝑖 ’s.
Theorem:
The 𝜏-skyline 𝜋 of 𝒫 has complexity 𝑂 𝑚 , and can be computed in 𝑂(𝑚log 𝑚) time, where 𝑚=𝑛𝑘.