Online Interval Skyline Queries on Time Series

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Online Interval Skyline Queries on Time Series. Bin Jiang, Jian Pei. Outline. Problem Definition An On-the-fly Method Interval Skyline Query Answering Algorithm Online Interval Skyline Query Algorithm Radix Priority Search Tree A View-Materialization Method

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### Online Interval Skyline Queries on Time Series

Bin Jiang, Jian Pei

Outline
• Problem Definition
• An On-the-fly Method
• Interval Skyline Query Answering Algorithm
• Online Interval Skyline Query Algorithm
• A View-Materialization Method
• Non-redundant skyline time series---NRSky[i:j]
• Experiments
Problem Definition
• Notions
• Time Series: A time series s consists of a set of ( value, timestamp) pairs.Here we denote the value of s at timestamp I by s[i], and s as a sequence of values s[1],s[2],…
• Time Interval: a range in time, denoted as [i : j]. We write

if ; if .

Some Notions in This Paper

Problem Definition
• Interval Skyline
• Given a set S of time series and interval[i:j], the interval skyline is the set of time series that are not dominated by any other time series in [i:j], denoted by

Suppose S={S1, S2, S3}

S1 and S2 are in Sky[16:22], while S3 is doninated by S2.

S2

S1

S3

Problem Definition
• Interval Skyline

Property 1:If there exist timestamps k1,…,kl(i≤k1<…<kl≤j) such that

and s is the only such a time series, then

time series is in .

Problem Definition
• 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 can be answered efficiently using D.
• Methods
• An On-The-Fly Method
• Original Interval Skyline Query Algorithm
• Online Interval Skyline Query Algorithm
• A View-Materialization Method
Outline
• Problem Definition
• An On-the-fly Method
• Interval Skyline Query Answering Algorithm
• Online Interval Skyline Query Algorithm
• A View-Materialization Method
• Non-redundant skyline time series---NRSky[i:j]
• Experiments
An Interval Skyline Query Algorithm
• Idea

Using the maximum value and minimum value of the time series, we can determine the domination of some time series without checking the details.

An Interval Skyline Query Algorithm
• Algorithm
• Set current Skyline Set Sky is null;
• Sort the time series in a list L in the descending order of their maximum value;
• Set the maximum value of the minimum value of the time series in Sky
• For each time series s that satisfies in L, determine whether it can dominate or be dominated by time series in Sky; If it can not be dominated:
• add it into Sky ;
• delete its dominance in Sky ;
• update ;
• Return Sky;
An Interval Skyline Query Algorithm
• Example

Goal: compute the skyline in interval [2:3]

Steps:

1. s2->Sky, maxmin =1

2. s3->Sky, maxmin =2

3. s5->Sky, maxmin =4

4. s5->s1, s1 is discarded, maxmin =4

Return Sky={s2,s3,s5}

An Interval Skyline Query Algorithm

Checking the max value for each time series and the min[i:j] for the query interval [i:j] is costly.

• Improvement Idea
• Utilize Radix Priority Search Tree to maintain the min[i:j]
• Use a sketch to keep the max value for each time series
Online Interval Skyline Query Algorithm

Radix Priority Search Tree is a two-dimensional data structure, a hybrid of a heap on one dimension and a binary search tree on the other dimension.

• Insertion in O(h)
• Deletion in O(h)
• Query in O(h)
• h: the height of the tree
Online Interval Skyline Query Algorithm
• Build
• Use the timestamps as the binary tree dimension X and the data value as the heap dimension Y;
• Map W into a fixed domain of X, {0,1,...,w-1};
• The height of the tree is O(logw)
• Update →

One insertion s[ ]

One deletion s[ ]

: the most recent timestamp

Maintain max values Using Sketches
• Sketches
• A pair (v,t) is maintained if no other pair (v1,t1) such that v1>v, t1>t;
• These pairs form the skyline of points in the interval;
• The expected number of points in the skyline is O(logw);
• With the sketches, finding the maximum value in W costs O(1) time ;

W=[1,3]

Sketches : (4,1),(3,2),(2,3)

W=[1,4]

Sketches : (5,4)

Online Interval Skyline Query Algorithm
• Complexity
• Space
• Radix priority search tree O(w)
• Sketch of the max values O(logw)

Total: O(nw)

• Time
• Radix priority search tree O(logw)
• Sketch of the max values O(logw)

Total: O(nlogw)

Outline
• Problem Definition
• An On-the-fly Method
• Interval Skyline Query Answering Algorithm
• Online Interval Skyline Query Algorithm
• A View-Materialization Method
• Non-redundant skyline time series---NRSky[i:j]
• Experiments
A View-Materialization Method
• Non-redundant interval skylines

A time series s is called a non-redundant skyline time series in interval [i:j] if

• S is in the skyline in interval[i:j]
• S is not in the skyline in any subinterval[i׳:j׳] [i:j]

It can be proved by pigeonhole principle, if there are more than w skyline intervals, at least two of them will share the same starting timestamps, then one of them is not a minimum skyline interval.

A View-Materialization Method
• Idea

Suppose all non-redundant interval skylines are materialized, we can union all these skylines over all intervals in [i:j] and remove those fail Lemma 2.

• Algorithm
A View-Materialization Method
• Example

W= [2:4]

Goal: compute the interval skyline in [3:4]

Steps:

1. s3->Sky

2. s4->Sky

3. s1->Sky(s2 is dominated by s1)

Return Sky={s1,s3,s4}

How to maintain the non-redundant skylines ?

Maintain Non-Redundant Interval Skylines
• Step1
• Use the on-the-fly algorithm to obtain the interval skyline in the new interval W׳.
• Find possible false negatives .
Maintain Non-Redundant Interval Skylines
• Step2-Shared Divide-and-Conquer Algorithm
• This algorithm is an extension of the divide-and conquer algorithm(DC).
• In SDC, a space is defined as a time interval. Each timestamp represents a dimension.
• The related spaces(intervals) are organized as a path, eg. [j:j],[j-1,j],...,[i,j](i<j).
Divide-and-Conquer Algorithm

Merge Step

Divide Step

S12

S22

B

B

S1

S2

B

P4

P4

P3

P3

P3

P1

P1

P1

mB

P5

P5

P5

P2

P2

P2

S11

S21

mA

mA

A

A

A

SDC Algorithm
• Comparisons
• Results
Maintain Non-Redundant Interval Skylines
• Step3-Remove “redundant time series”
Outline
• Problem Definition
• An On-the-fly Method
• Interval Skyline Query Answering Algorithm
• Online Interval Skyline Query Algorithm
• A View-Materialization Method
• Non-redundant skyline time series---NRSky[i:j]
• Experiments
Experiments
• Parameters
Experiments
• Synthetic Data Sets
• Data Sets Properties
• Query Efficiency
Experiments
• Synthetic Data Sets
• Update Efficiency
• Space Cost
Experiments
• Stock Data Sets
• Query Time