Efficiently Monitoring Top-k Pairs over Sliding Windows

Computer Science and Engineering Presented By: Zhitao Shen 1 Joint work with Muhammad Aamir Cheema 1 , Xuemin Lin 21 , Wenjie Zhang 1 , Haixun Wang 3 1 The University of New South Wales, Australia 2 East China Normal University 3 Microsoft Research Asia

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Introduction

• Top-k Pairs Query: Given a scoring function score() that computes the score of a pair of objects, return k pairs of objects with the smallest scores.

• • Examples: k closest pairs queries k furthest pairs queries • Top-k Pairs against sliding windows Given a data stream, return top-k pairs among the most recent N objects.

• Applications Wireless sensor network, stock market, traffic monitoring and transaction monitoring

Motivation

No existing work for general pairs queries over sliding windows Select a.id, b.id from trans a, trans b where a.id <> b.id and a.account = b.account

order by |a.time - b.time| - dist(a.loc, b.loc) Example: limit k – Query the transaction pairs that have small time difference but the locations are far away.

203-13845 10:15:20 New York $1000 203-13845 10:18:10 L.A.

$1000

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Problem Definitions (Preliminaries)

Sliding Windows – A sliding window contains most recent N objects of the data stream. – The number of pairs is N(N – 1) / 2 The age of a pair depends on the older object.

older . . . . .

newer o 7 o 6 o 5 o 4 o 3 o 2 o 1

Lower bound runtime cost : O(N) for each new object Lower bound storage cost : O(N)

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Contributions

• • • Unified framework First to study top-k pairs queries over sliding windows.

Support arbitrarily complex scoring functions Support efficient queries for any window size n ≤ N and any k ≤ K Storage requirement Skyband maintenance cost for each object Answering top-k pairs Lower bound O(N) O(N) Expected cost for our algorithms O(N) + O(K log(N/K)) for each scoring function O(N (log (log N) + log K)) O(k) O(log(log n) + log K + k)

Preliminaries

o 4 o 3 o 2 o 1 o 0 Map all the pairs to an age –score space Top-2 pairs p 1 (o 0 , o 1 )  (p 1 .age, p 1 .score)  (1, 3)

Task1 : how we efficiently maintain the K-skyband Task2 : how we use the K skyband to efficiently obtain top-k pairs against any sliding window n

≤ N p 1 p p 2 3 p p 6 p 4 5 p p p p 10 9 8 7 p 2 dominates p 5 because p 2 .score < p 5 .score and p 2 expires no later than p 5 .

Expected size of skyband is O(K log(N/K)) Naive: O(N |SKB|) for checking all N-1 pairs Our: O(N log|SKB|) 1 2 3 4 Age K-skyband [Papadias et al., TODS05] keeps the minimum set for the candidate results.

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Efficient Skyband Maintenance

Can we find a boundary between the skyband points and non-skyband points?

How can we efficiently compute the K-staircase and K-skyband?

p 1 p 7 Update the K-staircase and K-skyband in O(|SKB| log K)), p 2 s 2 p 5 s 1 s 2

K-staircase

p 1 K-staircase p 6 Check if a pair is dominated by K-skyband in O(log |SKB|) time for each new pair by doing binary search.

p 3 s 1 p 4 p 5 2-skyband Age

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Efficient Query Answering

Can we do better for any sliding window size n < N?

p 2 p 3 p 1 p 4 p 5 p 6 p 7 p 8 2-skyband Age

Use Priority Search Tree to index the skyband points

p 1 6 p 5 3 p 2 9 p 7 1 p 6 4 p 3 8 p 8 2 p 4 5

Priority Search Tree Self-balancing tree Efficient 3-sides range query

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Efficient Query Answering

Our contribution: Retrieve top-k pairs in the 1-sided range.

Any window size = n < N

p 2 p 3 p 1 p 4 p 5 p 6 p 7 p 8 2-skyband Age

An algorithm similar to post-order traversal costs O(log|SKB| + k)

p 1 6 p 5 3 p 2 9 p 7 1 p 6 4 p 8 2 p 4 5

Priority Search Tree

p 3 8

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What else in the paper?

Efficient continuous queries on the skyband.

• Continuously monitoring the top-k results for any fixed k (k ≤ K) and n (n ≤ N).

• Amortized O(k/n (log |SKB| + k)) time per update.

Optimization on monotonic scoring functions.

• Handling the k-closest pairs, k-furthest pairs queries.

• Applying Threshold Algorithm on sorted lists • Improving the number of considered pairs for each new object from N to (d+1) N d/(d+1) K 1/(d+1)

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Experimental Settings

Real dataset.

– Sensor data in the Intel research lab – 2.3 million records.

score(o x , o y )  | o x .temp

| o x .time

o y .time

o y .temp

| | o x .humidity

| o y .humidity

| Synthetic data.

– Uniform, correlated and anti-correlated distributions.

– 2 million objects – Closest and furthest pairs in Manhattan distance

Experiments (Overall Cost on real data)

SCase: our algorithm using K-staircase to maintain the skyband.

Naïve: maintains kN pairs and sort them on their scores.

LB: shows lower bound cost

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Varying K Varying N (in thousands)

Experiments (Query Answering)

Linear: scan the skyband points to find the top-k pairs.

Snapshot: our snapshot query algorithm.

Continuous: our continuous query algorithm.

LB: an algorithm to obtain top-k results in O(k) time.

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Varying K Varying |Q| (in thousands)

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Conclusion:

• First to study a broad class of top-k pairs queries over sliding windows.

• We present efficient algorithms and show that the performance of our algorithm is reasonably close to the lower bound cost. • We provide extensive experiment results on both real and synthetic data sets to show the efficiency and scalability of the proposed algorithms.

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Question and Answer

Thank You!

Any Questions?

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Related Work

Top-k Query Processing • Fagin’s Algorithm (FA), threshold Algorithm (TA), no-random access (NRA) Top-k Pairs Queries Processing • k-closest pairs queries • k-furthest pairs queries • Top-k pairs queries [Cheema et al., ICDE’11] Data Stream Processing • Top-k query processing over data stream [Mouratidis et al., SIGMOD’06] • k-nearest neighbour queries [Böhm et al., ICDE’07]

Experiments (Skyband Maintenance algorithm)

Basic: maintening algorithm without K-staircase SCase: our algorithm using K-staircase to maintain the skyband.

TA: Optimized algorithm for monotonic scoring functions.

LB: show lower bound cost

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Varying K # of attributes