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


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}


Maxmin[2:3] 1 2

On-The-Fly (OTF)

Candidate list {s2,s3,s5}


Maxmin[2:3] 1 2 4

On-The-Fly (OTF)

Candidate list {s2,s3,s5}


Maxmin[2:3] 1 2 4 2

On-The-Fly (OTF)

Terminate and return candidate list

min[2 : 3] 4.maxMax s


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)


:[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


(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


(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


Radix priority search tree

online interval skyline queries on time series icde 2009

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