Multi-Dimensional Online Tracking

Transcription

1 Multi-Dimensional Online Tracking Ke Yi and Qin Zhang Hong Kong University of Science & Technology SODA 2009 January 4-6,

2 A natural problem Bob: tracker f(t) g(t) Alice: observer (t, g(t)) t 2-1

3 A natural problem Bob: tracker f(t) g(t) Alice: observer t (t, g(t)) message format at time t now : (t now, g(t now )). communicate to guarantee that at t now f(t now ) g(t last ) t last : the last time Bob got informed. 2-2

4 A natural problem Bob: tracker f(t) g(t) Alice: observer Our Goal: minimize communication, in terms of competitive ratios. For t (t, g(t)) message format at time t now : (t now, g(t now )). communicate to guarantee that at t now f(t now ) g(t last ) t last : the last time Bob got informed. 2-3

5 A natural problem Bob: tracker f(t) g(t) Alice: observer Our Goal: minimize communication, in terms of competitive ratios. For t 1. f : Z + Z 2. f : Z + Z d 3. with prediction (t, g(t)) message format at time t now : (t now, g(t now )). communicate to guarantee that at t now f(t now ) g(t last ) t last : the last time Bob got informed. 2-4

6 Motivation Wireless sensors Monitoring the temperature 1D case 3-1

7 Motivation Wireless sensors Transmission of Data is the biggest source of battery drain! Monitoring the temperature 1D case 3-2

8 Motivation Wireless sensors Location-based services Transmission of Data is the biggest source of battery drain! Monitoring the temperature 1D case Keep track of the user s location 2D case 3-3

9 Motivation Wireless sensors Location-based services Transmission of Data is the biggest source of battery drain! Monitoring the temperature 1D case Publish/subscribe system Keep track of the user s location 2D case Subscribers register (potentially the same) queries at the publisher; results (a set of items) change over time high-d case 3-4

10 Motivation Wireless sensors Location-based services Transmission of Data is the biggest source of battery drain! Monitoring the temperature 1D case Publish/subscribe system Bandwidth consumption is the main concern! Keep track of the user s location 2D case Subscribers register (potentially the same) queries at the publisher; results (a set of items) change over time high-d case 3-5

11 Naive solution fails Consider tracking the function f : Z + Z, and require an absolute error of at most. The natural solution is to 1. first communicate f(0) to Bob. 2. every time f(t) has changed by more than since the last communication, Alice updates Bob with the current f(t). 4-1

12 Naive solution fails Consider tracking the function f : Z + Z, and require an absolute error of at most. The natural solution is to 1. first communicate f(0) to Bob. 2. every time f(t) has changed by more than since the last communication, Alice updates Bob with the current f(t). Unbounded competitive ratio! 4-2

13 Naive solution fails Consider tracking the function f : Z + Z, and require an absolute error of at most. The natural solution is to 1. first communicate f(0) to Bob. 2. every time f(t) has changed by more than since the last communication, Alice updates Bob with the current f(t). Unbounded competitive ratio! 2 SOL =, OPT = 1! f(t) t 4-3

14 Our Results problem comp. ratio running time 1-dim O(log ) O(1) d-dim O(d 2 log(d )) poly(d, log ) 1-dim + prediction O(log( T )) poly(, T ) Results for online tracking. T : length of the tracking period. 5-1

15 Our Results problem comp. ratio running time 1-dim O(log ) O(1) d-dim O(d 2 log(d )) poly(d, log ) 1-dim + prediction O(log( T )) poly(, T ) Results for online tracking. T : length of the tracking period. Prediction. Allow to send prediction functions (e.g. linear functions) instead of only a single value every time. OPT also uses the same family of functions. f(t) g(t) line prediction 5-2

16 Related research domains Communication complexity Alice (has x) Our case. compute f(x, y) 1. Alice: observer, Bob: tracker. 2. Inputs arrive online, only seen by Alice. Bob (has y), x, y are given offline. 6-1

17 Related research domains Communication complexity Alice (has x) Our case. compute f(x, y) 1. Alice: observer, Bob: tracker. 2. Inputs arrive online, only seen by Alice. Data streams Small space. Our case: communication cost. Bob (has y), x, y are given offline. 6-2

18 One dimension General idea to track f : Z + Z. Divide the whole tracking period into rounds, and show that A OPT must communicate once in each round, while our algorithm communicates at most, say, k times competitive ratio k. 7-1

19 One dimension (Cont.) The Algorithm to track f : Z + Z 8-1

20 One dimension (Cont.) The Algorithm to track f : Z + Z f(t) S S t 8-2

21 One dimension (Cont.) The Algorithm to track f : Z + Z f(t) S S t The Analysis If A OPT hasn t sent a message in the current round, then its last message must be included in S. The cardinality of S decreases by half whenever Algorithm 1 sends a message. 8-3

22 One dimension (Cont.) The Algorithm to track f : Z + Z f(t) S S t The Analysis If A OPT hasn t sent a message in the current round, then its last message must be included in S. The cardinality of S decreases by half whenever Algorithm 1 sends a message. O(log ) -competitive 8-4

23 One dimension (Cont.) The Algorithm to track f : Z + Z f(t) S S t The Analysis If A OPT hasn t sent a message in the current round, then its last message must be included in S. The cardinality of S decreases by half whenever Algorithm 1 sends a message. O(log ) -competitive Also tight! 8-5

24 One dimension (Cont.) The Algorithm to track f : Z + Z Real range unbounded! f(t) S S t The Analysis If A OPT hasn t sent a message in the current round, then its last message must be included in S. The cardinality of S decreases by half whenever Algorithm 1 sends a message. O(log ) -competitive Also tight! 8-6

25 High dimensions The general idea follows from 1D Divide the whole tracking period into rounds, and show that the competitive ratio in each round is k. 9-1

26 High dimensions The general idea follows from 1D Divide the whole tracking period into rounds, and show that the competitive ratio in each round is k. The framework of one round 1. At time t = t start, initialize a set S = S 0. (Many choices of S 0 ) 2. In each iteration in the while loop, we first pick a median from S as g(t now ) and send it to Bob. 3. When f deviates from g(t last ) by more than, we cut S as S S ball(f(t now ), ). 4. When S becomes empty, we can terminate the round. 9-2

27 High dimensions (cont.) Initialize S = S 0 f(t start )/g(tnow ) 10-1

28 High dimensions (cont.) S S ball(f(t now ), ) g(tnow ) f(tnow ) : f(tnow ) g(t last ) > 10-2

29 High dimensions (cont.) S S ball(f(t now ), ) g(tnow ) f(tnow ) 10-3

30 High dimensions (cont.) f(tnow ) S =! Start next round 10-4

31 High dimensions (cont.) The key property of S. If S becomes empty at some time step, then A OPT must have communicated once in the current round. 11-1

32 High dimensions (cont.) The key property of S. If S becomes empty at some time step, then A OPT must have communicated once in the current round. Two main issues left How to choose the initial set S 0 so that above property is met? 2. How to pick the median so that we can have small competitive ratios? 11-2

33 Take one, Tukey medians Choices for two issues Set S 0 = C 2 C 3... C d+1 C l : be the collection of centers of the smallest enclosing balls of every l points in Ball(f(t start ), 2 ) Z d. Send the Tukey median of S at every triggering. 12-1

34 Take one, Tukey medians Choices for two issues Set S 0 = C 2 C 3... C d+1 C l : be the collection of centers of the smallest enclosing balls of every l points in Ball(f(t start ), 2 ) Z d. Send the Tukey median of S at every triggering. Definition of the Tukey medians Any halfspace containing Tukey median v also contains at least n points d+1 1 where n is the cardinality of the point set. v 12-2

35 Tukey medians (cont.) Analysis of competitive ratio (ρ) S 0 = O ( d ( e( 4 +1) d d+1 ) d+1 ) S deceases by a factor of at least 1/(d + 1) at every triggering of communication ρ = log 1+ 1 d S 0 = O(d 3 log ) 13-1

36 Tukey medians (cont.) Analysis of competitive ratio (ρ) S 0 = O ( d ( e( 4 +1) d d+1 ) d+1 ) S deceases by a factor of at least 1/(d + 1) at every triggering of communication ρ = log 1+ 1 d S 0 = O(d 3 log ) However, the running time is exponential in d :( 1. S 0 is too large. 2. Computing Tukey medians in high-d (even approximately) is hard. 13-2

37 Volume cutting O(d 2 log d )-competitive; running time polynomial in d and log. 14-1

38 Volume cutting O(d 2 log d )-competitive; running time polynomial in d and log. Send the (approximate) centroids of a convex set containing S. 14-2

39 Volume cutting O(d 2 log d )-competitive; running time polynomial in d and log. Send the (approximate) centroids of a convex set containing S. Use some geometry to bound the number of cuts performed until S

40 Volume cutting O(d 2 log d )-competitive; running time polynomial in d and log. Send the (approximate) centroids of a convex set containing S. Use some geometry to bound the number of cuts performed until S 1. Techniques similar to those used in convex programming to find the last point of S if exists. 14-4

41 With prediction We consider the case that algorithms are allowed to send a linear function to predict the future trend of f 15-1

42 With prediction We consider the case that algorithms are allowed to send a linear function to predict the future trend of f Ideas. 1. Still follow the general framewok. 2. Cutting in the parametric space. A line l passing (0, q 0 ), (t 1, q 1 ) a point (q 0, q 1 ) in 2D. 15-2

43 With prediction We consider the case that algorithms are allowed to send a linear function to predict the future trend of f Ideas. 1. Still follow the general framewok. 2. Cutting in the parametric space. A line l passing (0, q 0 ), (t 1, q 1 ) a point (q 0, q 1 ) in 2D. Competitive ratio: O(log( T )). T : length of the tracking period, 15-3

44 Open problems and future directions Generalize our techniques to multiple observers. What if the length of the messages is considered (in the high- D) case? Lower bounds for high dimensional tracking. Online tracking in other metric spaces. 16-1

45 Open problems and future directions Generalize our techniques to multiple observers. Need a stronger model. What if the length of the messages is considered (in the high- D) case? Lower bounds for high dimensional tracking. Online tracking in other metric spaces. 16-2

46 The End T HAN K YOU Q and A 17-1