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Space-Efficient Online Computation of Quantile Summaries

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### Space-Efficient Online Computation of Quantile Summaries

SIGMOD 01

Michael Greenwald & Sanjeev Khanna

Presented by ellery

Outline

- Introduction
- The summary data structure
- Operation and algorithm
- Tree representation
- Analysis and experimental result
- Conclusion

Introduction

- Space-efficient computation of quantile summaries of very large data sets in a single pass.
- Quantile queries: Given a quantile, , return the value whose rank is N

sorting

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 10, 10, 11, 11, 11, 12

0.5 quantile returns element ranked 8 ( 0.5*16)

which is 8

0.75 quantile returns element ranked 12 (0.75*16)

which is 10

Requirements

- Explicit & tunable a priori guarantees on the precision of the approximation
- As small a memory footprint as possible
- Online:Single pass over the data
- Data Independent Performance: guarantees should be unaffected by arrival order, distribution of values, or cardinality of observations.
- Data Independent Setup: no a priori knowledge required about data set (size, range, distribution, order).

ε- approximate

- A quantile summary for a data sequence is ε- approximate if, for any given rank r, it returns a value whose rank r’ is guaranteed to be within the interval [r -εN , r + εN ]

Example : A data stream with 100 elements,

0.5 – quantile with ε= 0.1 returns a value v.

The true rank of v is within [40,60]

The Summary Data Structure

- Let rmin(v) and rmax(v) denote the lower and upper bounds on the rank of v
- Each tuple ti = (vi , gi ,Δi)

Query

- Sketch S isε- approximate, That is for each ψ (0,1] , there is a (vi , rmin(vi), rmax(vi)) in S such that
- vi is our answer for ψ-quantile

Corollary

- If at any time n, the summary S(n) satisfies the property that

then we can answer any ψ-quantile query to within an εn precision.

Overview of Summary Data Structure

= .29

r = N = 522

.01, N=1800

{28,7}

{15,2}

{10,1}

- Quantile = .29? Compute r and choose best vi

192

201

204

[529,536]

[539,540]

[501,503]

Overview of Summary Data Structure

.01, N=1800

{28,7}

{15,2}

{10,1}

- If (rmax(vi+1) - rmin(vi)) ≦ 2N, then -approximate summary.
- Our goal: always maintain this property.
- Tuple formulation of this rule: gi + I ≦ 2N

2N=36

192

204

201

[529,536]

[539,540]

[501,503]

Overview of Summary Data Structure

.01, N=1800

{28,7}

{15,2}

{10,1}

- Goal: always maintain -approximate summary(rmax(vi+1) - rmin(vi)) = (gi + I) ≦ 2N
- Insert new observations into summary

2N=36

192

204

201

[539,540]

[529,536]

[501,503]

Overview of Summary Data Structure

.01, N=1800

{28,7}

{15,2}

{10,1}

- Goal: always maintain -approximate summary(rmax(vi+1) - rmin(vi)) = (gi + I) ≦ 2N
- Insert new observations into summary

2N=36

197

192

204

201

[502,536]

[501,503]

[529,536]

[539,540]

Overview of Summary Data Structure

.01, N=1801

{28,7}

{15,2}

{1,34}

{10,1}

- Goal: always maintain -approximate summary (rmax(vi+1) - rmin(vi)) = (gi + I) ≦ 2N
- Insert new observations into summary
- Insert tuple before the ith tuple. gnew = 1; new = gi + I - 1;

2N=36.02

197

192

204

201

[502,536]

[530,537]

[540,541]

[501,503]

Overview of Summary Data Structure

.01, N=1801

{28,7}

{15,2}

{1,34}

{10,1}

- Goal: always maintain -approximate summary (rmax(vi+1) - rmin(vi)) = (gi + I) ≦ 2N
- Insert new observations into summary
- Delete all “superfluous” entries.

2N=36.02

197

192

204

201

[502,536]

[540,541]

[530,537]

[501,503]

Overview of Summary Data Structure

.01, N=1801

{28,7}

{15,2}

{1,34}

{10,1}

- Goal: always maintain -approximate summary (rmax(vi+1) - rmin(vi)) = (gi + I) ≦ 2N
- Insert new observations into summary
- Delete all “superfluous” entries.

2N=36.02

192

204

201

[530,537]

[540,541]

[501,503]

Overview of Summary Data Structure

.01, N=1801

{29,7}

{15,2}

{10,1}

- Goal: always maintain -approximate summary (rmax(vi+1) - rmin(vi)) = (gi + I) ≦ 2N
- Insert new observations into summary
- Delete all “superfluous” entries. gi = gi + gi-1

2N=36.02

192

204

201

[530,537]

[540,541]

[501,503]

Overview of Summary Data Structure

.01, N=1801

{29,7}

{15,2}

{10,1}

2N=36.02

- Insert: gnew = 1; new = gi + I - 1;
- Delete: gi = gi + gi-1

192

204

201

[530,537]

[540,541]

[501,503]

Terminology

- Full tuple: A tuple is full if gi + I = 2N
- Full tuple pair: A pair of tuples is full if deleting the left-hand tuple would overfill the right one
- Capacity: number of observations that can be counted by gi before the tuple becomes full. (=2N - I)

General strategy will be to delete tuples with small capacity and preserve tuples with large capacity.

Operations

- Insert(v)：Find the smallest i, such that

, and insert

- Delete(vi)：to delete from S, replace and by the new tuple
- Compress()：from right to left, merge all mergeable pair.

Tree Representation

-range Capacity Band0-7 8-15 38-11 4-7 212-13 2-3 114 1 0

.001, N=7,000

2N=14

- Group tuples with similar capacities into bands
- First (least index) node to the right with higher capacity band becomes parent.

0

0

0

3

1

2

1

1

1

0

3

0

1

2

3

1

2

0

1

1

3

Tree Representation

-range Capacity Band0-7 8-15 38-11 4-7 212-13 2-3 114 1 0

.001, N=7,000

2N=14

- Group tuples with similar capacities into bands
- First (least index) node to the right with higher capacity band becomes parent.

3

3

3

3

0

0

0

1

2

1

1

1

0

0

1

2

1

2

0

1

1

3

3

3

2

2

2

0

0

0

1

1

1

1

0

0

1

1

0

1

1

Tree Representation-range Capacity Band0-7 8-15 38-11 4-7 212-13 2-3 114 1 0

.001, N=7,000

- Group tuples with similar capacities into bands
- First (least index) node to the right with higher capacity band becomes parent.

2N=14

3

3

3

3

2

2

2

1

1

1

1

1

1

1

1

0

0

0

0

0

0

Tree Representation-range Capacity Band0-7 8-15 38-11 4-7 212-13 2-3 114 1 0

.001, N=7,000

2N=14

- Group tuples with similar capacities into bands
- First (least index) node to the right with higher capacity band becomes parent.

Operation (compress)

General strategy: delete tuples with small capacity and preserve tuples with large capacity.

1) Deletion cannot leave descendants unmerged --- it must delete entire subtrees

2) Deletion can only merge a tuple with small capacity into a tuple with similar or larger capacity.

3) Deletion cannot create an over-full tuple (i.e with g+ > floor(2N))

Experimental Result

- Measurement:
- |S|
- Observed (vs. desired ) : max, avg, and for 16 representative quantiles
- Optimal max observed
- Compared 3 algorithms
- MRL
- Preallocated (1/3 number of stored observations as MRL)
- Adaptive: allocate a new quantile only when observed error is about to exceed desired

Conclusion

- Better worst-case behavior than previous algorithms
- It does not require a priori knowledge of the parameter N

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