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### Online Interval Skyline Queries on Time Series

Outline

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
- 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
- Radix Priority Search Tree
- 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

5. s4.min=3<4=maxmin, s4 is discarded.

Return Sky={s2,s3,s5}

An Interval Skyline Query Algorithm

- Disadvantage

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

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.

- Advantages:
- Insertion in O(h)
- Deletion in O(h)
- Query in O(h)
- h: the height of the tree

Online Interval Skyline Query Algorithm

- Radix Priority Search Tree
- 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
- Radix Priority Search Tree
- 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”

- 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
- 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

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