Online interval skyline queries on time series
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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.

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Outline
Outline

  • Introduction

  • Interval Skyline Query

  • Algorithm

    • On-The-Fly (OTF)

    • View-Materialization(VM)

  • Experiment

  • Conclusion


Introduction
Introduction

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


Interval skyline query
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 query1
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
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 otf1
On-The-Fly (OTF)

Iteravively process the time series in S in their max value descending order

Ex:

Consider

Let usCompute the skyline in interval [2,3]


On the fly otf candidate list s2
On-The-Fly (OTF)Candidate list {s2}


On the fly otf candidate list s2 s3
On-The-Fly (OTF)Candidate list {s2,s3}


On the fly otf candidate list s2 s3 s5
On-The-Fly (OTF)Candidate list {s2,s3,s5}


On the fly otf candidate list s2 s3 s51
On-The-Fly (OTF)Candidate list {s2,s3,s5}


On the fly otf terminate and return candidate list
On-The-Fly (OTF)Terminate and return candidate list


Online interval skyline query answering
Online Interval Skyline Query Answering

  • Radix priority search tree

(5,8)

(7,7)

(4,6)

(8,5)

(1,4)

(6,3)

(3,2)

(2,1)


Online interval skyline query answering1
Online Interval Skyline Query Answering

  • Radix priority search tree

(5,8)

(7,7)

(4,6)

(8,5)

(1,4)

(6,3)

(3,2)

(2,1)


Online interval skyline query answering2
Online Interval Skyline Query Answering

  • Radix priority search tree

(5,8)

(7,7)

(4,6)

(8,5)

(1,4)

(6,3)

(3,2)

(2,1)


Online interval skyline query answering3
Online Interval Skyline Query Answering

  • Radix priority search tree

(5,8)

(7,7)

(4,6)

(8,5)

(1,4)

(6,3)

(3,2)

(2,1)


Online interval skyline query answering4
Online Interval Skyline Query Answering

  • Radix priority search tree

(5,8)

(7,7)

(4,6)

(8,5)

(1,4)

(6,3)

(3,2)

(2,1)


Online interval skyline query answering5
Online Interval Skyline Query Answering

  • Radix priority search tree

(5,8)

(7,7)

(4,6)

(8,5)

(1,4)

(6,3)

(3,2)

(2,1)


Online interval skyline query answering6
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 answering7
Online Interval Skyline Query Answering

  • Ex: and w=3

    When the base interval becomes


Online interval skyline query answering8
Online Interval Skyline Query Answering

  • Ex: and w=3

    When the base interval becomes


Online interval skyline query answering9
Online Interval Skyline Query Answering

  • Ex: and w=3

    When the base interval becomes

    =

    [1,1] and [2,3]


View materialization vm
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 vm1
View-Materialization(VM)

  • Ex: Compute

    • Union the non-redundant

      interval skylines

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


Online interval skyline queries on time series
SDC

5 4

3

2, 1, 3

2

(4,4)

(5,1)

(3,2)

(5,1)

(4,3,2)



Conclusion
Conclusion

  • Interval Skyline Query

  • Radix priority search tree