online interval skyline queries on time series icde 2009
DESCRIPTION
Introduction A power supplier need to analyze the consumption of different regions in the service area.TRANSCRIPT
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Online Interval Skyline Queries on Time Series
ICDE 2009
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Outline
Introduction Interval Skyline Query Algorithm
On-The-Fly (OTF) View-Materialization(VM)
Experiment Conclusion
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Introduction A power supplier need to analyze the consumption of
different regions in the service area.
![Page 4: Online Interval Skyline Queries on Time Series ICDE 2009](https://reader036.vdocument.in/reader036/viewer/2022062412/5a4d1acf7f8b9ab059970c29/html5/thumbnails/4.jpg)
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
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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
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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
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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
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On-The-Fly (OTF)
Candidate list {s2}
Time series s2 s3 s5 s1 s4Max 5 5 4 4 3
Maxmin[2:3] 1
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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
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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
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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
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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
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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)
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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)
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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
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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
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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)
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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)
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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
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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
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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
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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
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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
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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
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SDC5 4
2, 1, 3
3
2(4,4)
(5,1)
(3,2)(5,1)
(4,3,2)
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Experiment
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Conclusion
Interval Skyline Query
Radix priority search tree