1. Presented by: Duong, Huu Kinh Luan March 14 th, 2011

2.  Introduction  Background information  Problem Definition  Handling Technique  Algorithm  Experimental Results Authors of paper What is the problem? Why there is the problem ? Rank Based K-NN Related Papers on the same topic Top-k Properties Problem Definition Notations used Exact Algorithm Randomized Algorithm

3.  Authors of the paper Xuemin Lin - Professor The University of New South Wales The University of New South Wales PhD – C.S from the U. Queensland (Australia) in 1992U. Queensland Ying Zhang Research Fellow PhD – 01. 2008 Wenjie Zhang Post-doc research fellow PhD – 2010 Gaoping Zhu PhD Candidate Qianlu Lin PhD Candidate

4.  What is the problem? GPS TRACKING DEVICE SENSOR NETWORK

5.  What is the problem?

7.  G. Cormode, F. Li, and K. Yi “Semantic of ranking queries for probabilistic data and expected ranks”  R. Chen, L. Chen, J. Chen and X. Xie “Evaluating probability threshold k-nn queires over uncertain data”  V. Ljosa and A. K. Singh “Apla: Indexing arbitrary probability distributions”

8.  Introduction  Background information  Problem Definition  Handling Technique  Algorithm  Experimental Results

9. U U1U1 U3U3 U2U2 U4U4 q Set of objects: U = {U 1, U 2, …, U n }  U = {U 1, U 2, U 3, U 4 } Possible World: W = {u 1, u 2, u 3, …, u n }  W 1 = {U1, U2} Definition 1: Rank (Rank of an obj U in one possible world W)

10. Definition 2: Expected Rank Definition 3: Median Rank

11. Example: Show on board Possible Worlds? Rank for A? i.e. r(a 1 ), r(a 2 ), r(a 3 ) Expected rank for A? i.e. er(A) Median rank for A? i.e. mr(A)

12.  Top–K Query: Find k nearest neighbors for a given query q based on the expected (median) ranks of n objects.

13.  Top–K Properties: Exact-k: K-NN query answer should return exactly k objects Containment: (K+1)-NN should contain all objects in KNN Unique Ranking: The same object should not be listed multiple times in KNN Value invariance: The distance only determines the relative behavior of the object Stability: Making an item in the top-k list more likely or more important should not remove it from the list

14.  Top–K Properties: Proof that expected rank satisfies all 5 top-k properties  not this paper major concern.  It is done in the paper “Semantic of ranking queries for probabilistic data and expected ranks”, by G. Cormode, F. Li, and K. Yi

15.  Overcome previous paper’s difficulties: Pre-computed expected scores of objects Reduce the number of objects accessed Expected score might change upon different queries Approximation of KNN querie answer

16.  Introduction  Background information  Problem Definition  Handling Technique  Algorithm  Experimental Results

17. ! Lemma 2: Let u i and u j be the instances which determine the median rank and median distance of U respectively, we have r(u i ) = r(u j )

18.  Finding Minimal Set for Selection Problem (Using Bound Based Approach) Motivation for the Algorithm

19.  Introduction  Background information  Problem Definition  Handling Technique  Algorithm  Experimental Results

20. NotationMeaning UUncertain Object qQuery d - (U), d + (U)Minimal/Maximal possible distance between instances of U and q IInterval within which each instance has a possible distance from q between [d - (I), d + (I)] r - (I), r + (I)Minimal/maximal rank of an interval I er - (U), er + (U)Minimal/Maximal expected rank of U mr - (U),mr + (U)Minimal/Maximal median rank of U Urminmr - (U) or er - (U) Urmaxmr + (U) or er + (U)

21. Uncertain objects R-Tree query q also represented in R-Tree e(I) from d - (I) to d + (I)

22.  Example of calculating r - (I) and r + (I) smaller than Sum up for r - (I)

23.  Example of calculating r - (I) and r + (I) smaller than Sum up for r + (I)

24.  Exact Algorithm: acc rmin : accumulation of the probability values of the invervals {I of I} with d + (I)<=d U armin (d): accumulation of the probability values of the invervals {I of I U } with d + (I)<=d

25.  Exact Algorithm: Cost: Initial Procedure: O(nlogn + n p0 x c io ) One round: O(n x m log(n x m)) n: number of objects m: number of interval in 1 object h: max height of local R-Tree n po : number of IO n pi : number of IO in i th round c io : cost of each IO Total time cost: T = O( h x n x m log( n x m )) + n pi x c io (i:0:h)

26.  Randomized Algorithm: Sample the possible world such that the expected rank and median rank can be approximately computed in an efficient way.

27.  Randomized Algorithm: Estimate the expected rank of an object U where r i (U) is the rank of U in sample S i Recall:

28.  Randomized Algorithm: Find candidate objects C for the KNN query based on the global R-Tree Minimal/Maximal Expected rank for each object using Sweepline algorithm l and r --> value to prune or validate objects for the KNN query

29.  Randomized Algorithm – Cost: O(nlogn) O(logn) O(n’logn + n 1 x c io ) T = O(nlogn + n’logn + n 1 x c io )

30.  What is n’?

31.  Introduction  Background information  Problem Definition  Handling Technique  Algorithm  Experimental Results

36.  Comparision with the other paper: This paperThe other paper