online interval skyline queries on time series icde 2009

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], denot

ed by , if k [i : j], s[k] ≥ q[k]; and l [i : j], s[l] > ∀ ∈ ∃ ∈q[l].

Ex: Consider interval [1,2]

[ : ]i js q

[1,2]2 1s s

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 th

e 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.

' [ 2 : 1]c cW t w t

[ 1: ]c cW t w t

ct ct

ct

[ 1: ]c cW t w t

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 .

[ : ]i j W.min[ : ] .maxs i j q [ : ]i j

.max max [ ]k Wq q k

[ : ].min[ : ] min [ ]k i js i j s k

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]

[1: 3]W

On-The-Fly (OTF)

Candidate list {s2}

Time series s2 s3 s5 s1 s4Max 5 5 4 4 3

Maxmin[2:3] 1

On-The-Fly (OTF)

Candidate list {s2,s3}

Time series s2 s3 s5 s1 s4Max 5 5 4 4 3

Maxmin[2:3] 1 2

On-The-Fly (OTF)

Candidate list {s2,s3,s5}

Time series s2 s3 s5 s1 s4Max 5 5 4 4 3

Maxmin[2:3] 1 2 4

On-The-Fly (OTF)

Candidate list {s2,s3,s5}

Time series s2 s3 s5 s1 s4Max 5 5 4 4 3

Maxmin[2:3] 1 2 4 2

On-The-Fly (OTF)

Terminate and return candidate list

min[2 : 3] 4.maxMax s

Time series s2 s3 s5 s1 s4Max 5 5 4 4 3

Maxmin[2:3] 1 2 4 2 1

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

:[1 ~ 8]X

(2,1)

(4,6)

(1,4)

(3,2)

(5,8)

(8,5)

(6,3)

(7,7)

:[1 ~ 4]LX :[5 ~ 8]RX

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)

:[1 ~ 8]X

:[1 ~ 4]LX :[5 ~ 8]RX

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 Eac

h 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.

1ct w 1ct 0,..., 1w

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

Timestamps 1 2 3s1 4 3 2

[1: 3]W ' [2 : 4]W

45

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

Timestamps 1 2 3s1 3 2

[1: 3]W ' [2 : 4]W

45

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

Timestamps 1 2 3s1 5 3 2

[1: 3]W ' [2 : 4]W

45

' [2 : 4]W ' [2,1]W

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

[ : ]i j' '[ : ] [ : ]i j i j

[ : ]i j' '[ : ] [ : ]i j i j ' '[ : ]s NRSky i j

'[ : ]i js s ' ' '[ : ]s NRSky i j

[ : ]s Sky i j

View-Materialization(VM) Ex: Compute

Union the non-redundant interval skylines

s1=(2,5) s2=(1,5)

[3 : 4]Sky

[3 : 3] 3

[4 : 4] { 1, 2}[3 : 4] { 4}

s

s ss

[3:4]1 2s s 1 [3: 4]s Sky

SDC5 4

2, 1, 3

3

2(4,4)

(5,1)

(3,2)(5,1)

(4,3,2)

Experiment

Conclusion

Interval Skyline Query

Radix priority search tree