Distributed Online Tracking Mingwang Tang, Feifei Li , Yufei Tao 1
Motivation and challenge Sensor network LBS service 2
Motivation and challenge Sensor network LBS service Observation: Large distributed data are being generated continuously in many application domains. 2
Motivation and challenge Sensor network LBS service Observation: Large distributed data are being generated continuously in many application domains. Challenge: How to track a function computed over distributed values in online fasion continuously? 2
A concrete example: SAMOS Project 3
A concrete example: SAMOS Project Observation 1: Each ship observes a sequence of values over time, e.g., wind direction, wind speed, sound speed. Observation 2: Tracking a user defined function (e.g. the max wind speed) exactly at every time instance over such distributed data is communication-expensive. 3
At the University of Utah: the MesoWest project Close to 40,000 stations across the US, http://mesowest.org > 10 billion readings and counting 4
Problem formulation m distributed observers { s 1 , . . . , s m } , connected to a tracker T using a network topology. 5
Problem formulation m distributed observers { s 1 , . . . , s m } , connected to a tracker T using a network topology. f i ( t ) represents the value observed at s i at time t . 5
Problem formulation m distributed observers { s 1 , . . . , s m } , connected to a tracker T using a network topology. f i ( t ) represents the value observed at s i at time t . f ( t ) = f ( f 1 ( t ) , f 2 ( t ) , . . . , f m ( t )) for some function f at tracker T . 5
Problem formulation m distributed observers { s 1 , . . . , s m } , connected to a tracker T using a network topology. f i ( t ) represents the value observed at s i at time t . f ( t ) = f ( f 1 ( t ) , f 2 ( t ) , . . . , f m ( t )) for some function f at tracker T . tracking f ( t ) is expensive! 5
Problem formulation m distributed observers { s 1 , . . . , s m } , connected to a tracker T using a network topology. f i ( t ) represents the value observed at s i at time t . f ( t ) = f ( f 1 ( t ) , f 2 ( t ) , . . . , f m ( t )) for some function f at tracker T . tracking f ( t ) is expensive! Tracker T : maintain g ( t ) ∈ [ f ( t ) − ∆ , f ( t ) + ∆] for some error threshold ∆ at any time instance t 5
Some concrete instantiations base case: centralized setting 6
Some concrete instantiations base case: centralized setting a chain topology 6
Some concrete instantiations a broom topology 7
Some concrete instantiations a simple tree topology 7
Some concrete instantiations a general tree topology 7
State-of-the-art result Centralized setting: a tracker with one observer. 8
State-of-the-art result Centralized setting: a tracker with one observer. Ke Yi and Qin Zhang: Multi-Dimentional Online Tracking. SODA Conference 2009 8
State-of-the-art result Centralized setting: a tracker with one observer. Ke Yi and Qin Zhang: Multi-Dimentional Online Tracking. SODA Conference 2009 f ( t ) g ( t ) t f (0) native method: unbounded competitive ratio with respect to the messages by optimal offline method. 8
State-of-the-art result Centralized setting: a tracker with one observer. Ke Yi and Qin Zhang: Multi-Dimentional Online Tracking. SODA Conference 2009 f ( t ) g ( t ) f (0) > ∆ t native method: unbounded competitive ratio with respect to the messages by optimal offline method. 8
State-of-the-art result Centralized setting: a tracker with one observer. Ke Yi and Qin Zhang: Multi-Dimentional Online Tracking. SODA Conference 2009 f ( t ) g ( t ) > ∆ f (0) > ∆ t native method: unbounded competitive ratio with respect to the messages by optimal offline method. 8
State-of-the-art result Centralized setting: a tracker with one observer. Ke Yi and Qin Zhang: Multi-Dimentional Online Tracking. SODA Conference 2009 f ( t ) C + ∆ C C − ∆ t native method: unbounded competitive ratio with respect to the messages by optimal offline method. 8
State-of-the-art result Centralized setting: a tracker with one observer. Ke Yi and Qin Zhang: Multi-Dimentional Online Tracking. SODA Conference 2009 f ( t ) C + ∆ C C − ∆ t native method: unbounded competitive ratio with respect to the messages by optimal offline method. 8
State-of-the-art result Centralized setting: a tracker with one observer. Ke Yi and Qin Zhang: Multi-Dimentional Online Tracking. SODA Conference 2009 f ( t ) g ( t ) ∆ t S ∆ native method: unbounded competitive ratio with respect to the messages by optimal offline method. OptTrack method: OptTrack is an O (log ∆) competitive online algorithm to track any f : Z → Z within ∆. 8
State-of-the-art result Centralized setting: a tracker with one observer. Ke Yi and Qin Zhang: Multi-Dimentional Online Tracking. SODA Conference 2009 f ( t ) g ( t ) ∆ t S ∆ native method: unbounded competitive ratio with respect to the messages by optimal offline method. OptTrack method: OptTrack is an O (log ∆) competitive online algorithm to track any f : Z → Z within ∆. 8
State-of-the-art result Centralized setting: a tracker with one observer. Ke Yi and Qin Zhang: Multi-Dimentional Online Tracking. SODA Conference 2009 f ( t ) g ( t ) S ∆ t S ∆ native method: unbounded competitive ratio with respect to the messages by optimal offline method. OptTrack method: OptTrack is an O (log ∆) competitive online algorithm to track any f : Z → Z within ∆. 8
Chain online tracking: f : Z + → Z h relay nodes g ( t ) ∈ [ f ( t ) − ∆ , f ( t ) + ∆] observer tracker f ( t ) g ( t ) = g h +1 ( t ) 9
Chain online tracking: f : Z + → Z h relay nodes g ( t ) ∈ [ f ( t ) − ∆ , f ( t ) + ∆] observer tracker f ( t ) g ( t ) = g h +1 ( t ) 9
Chain online tracking: f : Z + → Z h relay nodes g ( t ) ∈ [ f ( t ) − ∆ , f ( t ) + ∆] ∆ ∆ ∆ ∆ h +1 h +1 h +1 h +1 observer tracker f ( t ) g ( t ) = g h +1 ( t ) ChainTrackA : distributed ∆ averagely among h + 1 centralized tracking instances. 9
Chain online tracking: f : Z + → Z h relay nodes g ( t ) ∈ [ f ( t ) − ∆ , f ( t ) + ∆] ∆ h +1 ∆ 3 ∆ 1 ∆ 2 observer tracker f ( t ) g ( t ) = g h +1 ( t ) � h +1 i =1 ∆ i = ∆ ChainTrackA : distributed ∆ averagely among h + 1 centralized tracking instances. ChainTrackR : distributed ∆ Randomly among h + 1 centralized tracking instances. 9
Chain online tracking: f : Z + → Z h relay nodes g ( t ) ∈ [ f ( t ) − ∆ , f ( t ) + ∆] ∆ h +1 ∆ 3 ∆ 1 ∆ 2 observer tracker f ( t ) g ( t ) = g h +1 ( t ) � h +1 i =1 ∆ i = ∆ ChainTrackA : distributed ∆ averagely among h + 1 centralized tracking instances. unbounded competitive ratio ChainTrackR : distributed ∆ Randomly among h + 1 unbounded competitive ratio centralized tracking instances. 9
Chain online tracking: f : Z + → Z h relay nodes g ( t ) ∈ [ f ( t ) − ∆ , f ( t ) + ∆] 0 0 0 ∆ observer tracker f ( t ) g ( t ) = g h +1 ( t ) ChainTrackA : distributed ∆ averagely among h + 1 centralized tracking instances. unbounded competitive ratio ChainTrackR : distributed ∆ Randomly among h + 1 unbounded competitive ratio centralized tracking instances. ChainTrackO : achieves a O ( log ∆) competitive ratio 9
Chain online tracking: f : Z + → Z h relay nodes g ( t ) ∈ [ f ( t ) − ∆ , f ( t ) + ∆] 0 0 0 ∆ observer tracker f ( t ) g ( t ) = g h +1 ( t ) ChainTrackA : distributed ∆ averagely among h + 1 centralized tracking instances. unbounded competitive ratio ChainTrackR : distributed ∆ Randomly among h + 1 unbounded competitive ratio centralized tracking instances. ChainTrackO : achieves a O ( log ∆) competitive ratio Competitive ratio: w.r.t offline optimal 9
Distributed: good competitive ratio is impossible! Broom Model General-tree Model g ( t ) ∈ [ f ( t ) − ∆ , f ( t ) + ∆] g ( t ) ∈ [ f ( t ) − ∆ , f ( t ) + ∆] tracker T tracker T h relay nodes s i 10
Distributed: good competitive ratio is impossible! Broom Model General-tree Model g ( t ) ∈ [ f ( t ) − ∆ , f ( t ) + ∆] g ( t ) ∈ [ f ( t ) − ∆ , f ( t ) + ∆] tracker T tracker T h relay nodes s i consider f = max: f ( t ) = max { f 1 ( t ) , . . . , f m ( t ) } . 10
Distributed: good competitive ratio is impossible! Broom Model General-tree Model g ( t ) ∈ [ f ( t ) − ∆ , f ( t ) + ∆] g ( t ) ∈ [ f ( t ) − ∆ , f ( t ) + ∆] tracker T tracker T h relay nodes s i consider f = max: f ( t ) = max { f 1 ( t ) , . . . , f m ( t ) } . Observation: by knowing the future of data at all sites, an offline method can “communicate” between leaf nodes for free. 10
Performance metric of an online algorithm Centralized and chain model: cost ( A , I ) cratio ( A ) = max I ∈I cost ( offline , I ) competitive ratio 11
Performance metric of an online algorithm Centralized and chain model: cost ( A , I ) cratio ( A ) = max I ∈I competitive ratio cost ( offline , I ) for a class A of algorithms, A ∗ Tree model: I is the optimal algorithm on an input I cost ( A , I ) ratio ( A , I ) = cost ( A ∗ I , I ) ratio ( A ) = max I ∈I ratio ( A , I ) 11
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