Online Interval Skyline Queries on Time Series ICDE 2009

Outline Introduction Interval Skyline Query Algorithm  On-The-Fly (OTF)  View-Materialization(VM) Experiment Conclusion

Introduction A power supplier need to analyze the consumption of different regions in the service area.

Interval Skyline Query A time series s consists of a set of (timestamp, value) pairs. (Ex: A={(1,4) (2,3)} ) Dominance Relation  Time series s is said to dominate time series q in interval [i : j], denoted by, if ∀ k ∈ [i : j], s[k] ≥ q[k]; and ∃ l ∈ [i : j], s[l] > q[l].  Ex: Consider interval [1,2]

Interval Skyline Query Let be the most recent timestamp. We call interval the base interval.  Whenever a new timestamp +1 comes, the oldest one −w+1 expires.  Consequently, the base interval becomes Problem Definition: Given a set of time series S such that each time series is in the base interval, we want to maintain a data structure D such that any interval skyline queries in interval [i:j] W can be answered efficiently using D.

On-The-Fly (OTF) The on the fly method keeps the minimum and maximum values for each time series. Lemma: For two time series p,q and interval if then s dominates q in. 

On-The-Fly (OTF) Iteravively process the time series in S in their max value descending order Ex: Consider Let us Compute the skyline in interval [2,3]

On-The-Fly (OTF) Candidate list {s2} Time seriess2s3s5s1s4 Max55443 Maxmin[2:3]1

On-The-Fly (OTF) Candidate list {s2,s3} Time seriess2s3s5s1s4 Max55443 Maxmin[2:3]12

On-The-Fly (OTF) Candidate list {s2,s3,s5} Time seriess2s3s5s1s4 Max55443 Maxmin[2:3]124

On-The-Fly (OTF) Candidate list {s2,s3,s5} Time seriess2s3s5s1s4 Max55443 Maxmin[2:3]1242

On-The-Fly (OTF) Terminate and return candidate list Time seriess2s3s5s1s4 Max55443 Maxmin[2:3]12421

Online Interval Skyline Query Answering Radix priority search tree (2,1) (4,6) (1,4) (3,2) (5,8) (8,5) (6,3) (7,7)

Online Interval Skyline Query Answering Radix priority search tree (2,1) (4,6) (1,4) (3,2) (5,8) (8,5) (6,3) (7,7)

Online Interval Skyline Query Answering Radix priority search tree (2,1) (4,6) (1,4) (3,2) (5,8) (8,5) (6,3) (7,7)

Online Interval Skyline Query Answering Radix priority search tree (2,1) (4,6) (1,4) (3,2) (5,8) (8,5) (6,3) (7,7)

Online Interval Skyline Query Answering Radix priority search tree (2,1) (4,6) (1,4) (3,2) (5,8) (8,5) (6,3) (7,7)

Online Interval Skyline Query Answering Radix priority search tree (2,1) (4,6) (1,4) (3,2) (5,8) (8,5) (6,3) (7,7)

Online Interval Skyline Query Answering Maintaining a Radix Priority Search Tree for Each Time Series  To process a time series, we use the time dimension (i.e the timestamps) as the binary tree dimension X and data values as the heap dimension Y.  Since the base interval W always consists of w timestamps represent w consecutive natural number. Apply the module w operation Domain of X is and will map the same timestamp.

Online Interval Skyline Query Answering Ex: and w=3 When the base interval becomes Timestamps123 s

Online Interval Skyline Query Answering Ex: and w=3 When the base interval becomes Timestamps123 s

Online Interval Skyline Query Answering Ex: and w=3 When the base interval becomes = [1,1] and [2,3] Timestamps123 s

View-Materialization(VM) Non-redundant skyline time series in interval [i:j]  (1) s is in the skyline interval  (2) s is not in the skyline in any subinterval Lemma: Give a time series s and an interval if for all interval such that, for any time series then

View-Materialization(VM) Ex: Compute  Union the non-redundant interval skylines s1=(2,5) s2=(1,5)

SDC 5 4 2, 1, (4,4) (5,1) (3,2) (5,1) (4,3,2)

Experiment

Conclusion Interval Skyline Query Radix priority search tree