Random Sampling on Big Data: Techniques and Applications Ke Yi Hong - - PowerPoint PPT Presentation

Random Sampling on Big Data: Techniques and Applications

Ke Yi

Hong Kong University of Science and Technology yike@ust.hk

“Big Data” in one slide

The 3 V’s:

 Volume  Velocity  Variety – Integers, real numbers – Points in a multi-dimensional space – Records in relational database – Graph-structured data

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Dealing with Big Data

 The first approach: scale up / out the computation  Many great technical innovations: – Distributed/parallel systems – Simpler programming models

• MapReduce, Pregel, Dremel, Spark…
• BSP

– Failure tolerance and recovery – Drop certain features: ACID, CAP, noSQL  This talk is not about this approach!

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Downsizing data

 A second approach to computational scalability:

scale down the data!

– A compact representation of a large data set – Too much redundancy in big data anyway – What we finally want is small: human readable analysis / decisions – Necessarily gives up some accuracy: approximate answers – Examples: samples, sketches, histograms, various transforms

• See tutorial by Graham Cormode for other data summaries

 Complementary to the first approach – Can scale out computation and scale down data at the same time – Algorithms need to work under new system architectures

• Good old RAM model no longer applies

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Outline for the talk

 Simple random sampling – Sampling from a data stream – Sampling from distributed streams – Sampling for range queries  Not-so-simple sampling – Importance sampling: Frequency estimation on distributed data – Paired sampling: Medians and quantiles – Random walk sampling: SQL queries (joins)  Will jump back and forth between theory and practice

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Simple Random Sampling

 Sampling without replacement – Randomly draw an element – Don’t put it back – Repeat s times  Sampling with replacement – Randomly draw an element – Put it back – Repeat s times  Trivial in the RAM model

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The statistical difference is very small, for 𝑜 ≫ 𝑡

Random Sampling from a Data Stream

 A stream of elements coming in at high speed  Limited memory  Need to maintain the sample continuously  Applications – Data stored on disk – Network traffic

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Reservoir Sampling

 Maintain a sample of size 𝑡 drawn (without replacement)

from all elements in the stream so far

 Keep the first 𝑡 elements in the stream, set 𝑜 ← 𝑡  Algorithm for a new element – 𝑜 ← 𝑜 + 1 – With probability 𝑡/𝑜, use it to replace an item in the current

sample chosen uniformly at random

– With probability 1 − 𝑡/𝑜, throw it away  Perhaps the first “streaming” algorithm

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[Waterman ??; Knuth’s book]

Correctness Proof

 By induction on 𝑜 – 𝑜 = 𝑡: trivially correct – Assume each element so far is sampled with probability 𝑡/𝑜 – Consider 𝑜 + 1:

• The new element is sampled with probability 𝑡

𝑜+1

• Any element in the current sample is sampled with probability

𝑡

𝑜 ⋅ 1 − 𝑡 𝑜+1 + 𝑡 𝑜+1 ⋅ 𝑡−1 𝑡

=

𝑡 𝑜+1. Yeah!

 This is a wrong (incomplete) proof  Each element being sampled with probability 𝑡 𝑜 is not a sufficient

condition of random sampling

– Counter example: Divide elements into groups of 𝑡 and pick one

group randomly

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Reservoir Sampling Correctness Proof

 Many “proofs” found online are actually wrong – They only show that each item is sampled with probability 𝑡/𝑜 – Need to show that every subset of size 𝑡 has the same

probability to be the sample

 Correct proof relates with the Fisher-Yates shuffle

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a b c d b a c d a b c d b c a d b d a c s = 2

Sampling from Distributed Streams

 One coordinator and 𝑙 sites  Each site can communicate with

the coordinator

 Goal: Maintain a random sample

• f size 𝑡 over the union of all

streams with minimum communication

 Difficulty: Don’t know 𝑜, so can’t

run reservoir sampling algorithm

 Key observation: Don’t have to

know 𝑜 in order to sample!

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[Cormode, Muthukrishnan, Yi, Zhang, PODS’10, JACM’12] [Woodruff, Tirthapura, DISC’11]

Reduction from Coin Flip Sampling

 Flip a fair coin for each element until we get “1”  An element is active on a level if it is “0”  If a level has ≥ 𝑡 active elements, we can draw a sample from

those active elements

 Key: The coordinator does not want all the active elements,

which are too many!

– Choose a level appropriately

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The Algorithm

 Initialize 𝑗 ← 0  In round 𝑗: – Sites send in every item w.p. 2−𝑗

(This is a coin-flip sample with prob. 2−𝑗)

– Coordinator maintains a lower sample and a higher sample:

each received item goes to either with equal prob. (The lower sample is a sample with prob. 2−(𝑗+1))

– When the lower sample reaches size 𝑡, the coordinator

broadcasts to advance to round 𝑗 ← 𝑗 + 1

– Discard the upper sample – Split the lower sample into a new lower sample and a higher

sample

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Communication Cost of Algorithm

 Communication cost of each round: 𝑃(𝑙 + 𝑡) – Expect to receive 𝑃(𝑡) sampled items before round ends – Broadcast to end round: 𝑃(𝑙)  Number of rounds: 𝑃(log 𝑜) – In each round, need Θ(𝑡) items being sampled to end round – Each item has prob. 2−𝑗 to contribute: need Θ(2𝑗𝑡) items  Total communication: 𝑃( 𝑙 + 𝑡 log 𝑜) – Can be improved to 𝑃(𝑙 log𝑙/𝑡𝑜 + 𝑡 log 𝑜) – A matching lower bound  Sliding windows

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Random Sampling for Range Queries

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[Christensen, Wang, Li, Yi, Tang, Villa, SIGMOD’15 Best Demo Award]

Online Range Sampling

 Problem Definition: Preprocess a set of points in the plane, so

that for any range query, we can return samples (with or without replacement) drawn from all points in the range until user termination.

 Parameters: – 𝑜: data size – 𝑟: query size – 𝑡: sample size

(not known beforehand)

– 𝑜 ≫ 𝑟 ≫ 𝑡

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 Naïve solutions: – Query then sample: 𝑃 𝑔 𝑜 + 𝑟 – Sample then query: 𝑃

𝑡𝑜 𝑟

(store data in random order)

 New solution: 𝑃 𝑔 𝑡𝑜 𝑟

+ 𝑡

[Wang, Christensen, Li, Yi, VLDB’16]

𝑔(𝑦): # canonical nodes in tree of size 𝑦, between log 𝑦 and 𝑦

Indexing Spatial Data

 Numerous spatial indexing structures in the literature

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R-tree

RS-tree

 Attach a sample to node 𝑣 drawn from leaves below 𝑣 – Total space: 𝑃(𝑜) – Construction time: 𝑃(𝑜)

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RS-tree: A 1D Example

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 4 5 7 9 12 14 16 3 8 12 14 7 14 5 Report: Active nodes 5

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RS-tree: A 1D Example

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 4 5 7 9 12 14 16 3 8 12 14 7 14 5 Report: 5 Active nodes

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RS-tree: A 1D Example

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 4 5 7 9 12 14 16 3 8 12 14 7 14 5 Report: 5 Active nodes 7

Pick 7 or 14 with equal prob.

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RS-tree: A 1D Example

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 4 5 7 9 12 14 16 3 8 12 14 7 14 5 Report: 5 7 Active nodes

Pick 3, 8, or 14 with prob. 1:1:2

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RS-tree: A 1D Example

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 4 5 7 9 12 14 16 3 8 12 14 7 14 5 Report: 5 7 Active nodes

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RS-tree: A 1D Example

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 4 5 7 9 12 14 16 3 8 12 14 7 14 5 Report: 5 7 Active nodes 12

Pick 3, 8, or 12 with equal prob

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Not-So-Simple Random Sampling

When simple random sampling is not optimal/feasible

Frequency Estimation on Distributed Data

 Given: A multiset 𝑇 of 𝑜 items drawn from the universe [𝑣] – For example: IP addresses of network packets  𝑇 is partitioned arbitrarily and stored on 𝑙 nodes – Local count 𝑦𝑗𝑘: frequency of item 𝑗 on node 𝑘 – Global count 𝑧𝑗 = 𝑦𝑗𝑘

𝑘

 Goal: Estimate 𝑧𝑗 with additive error 𝜁𝑜 for all 𝑗 – Can’t hope for relative error for all 𝑧𝑗 – Heavy hitters are estimated well

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[Huang, Yi, Liu, Chen, INFOCOM’11]

Frequency Estimation: Standard Solutions

 Local heavy hitters

– Let 𝑜𝑘 = 𝑦𝑗𝑘

𝑗

be the data size at node 𝑘

– Node 𝑘 sends in all items with frequency ≥ 𝜁𝑜𝑘 – Total error is at most 𝜁𝑜𝑘 = 𝜁𝑜

𝑘

– Communication cost: 𝑃(𝑙/𝜁)

 Simple random sampling

– A simple random sample of size 𝑃(1/𝜁2) can be used to estimate the

frequency of any item with error 𝜁𝑜

• Extra log factor for all items

– Algorithm

• Coordinator first gets 𝑜𝑘 for all 𝑘
• Decides how many samples to get from each 𝑘
• Get the samples from the nodes

– Communication cost: 𝑃(𝑙 + 1/𝜁2)

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Importance Sampling

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Horvitz–Thompson estimator:

𝑌𝑗𝑘 = 𝑦𝑗𝑘 𝑕 𝑦𝑗𝑘 𝑗𝑔 𝑦𝑗𝑘 𝑡𝑏𝑛𝑞𝑚𝑓𝑒 𝑓𝑚𝑡𝑓

Estimator for global count 𝑧𝑗: 𝑍

𝑗 = 𝑌𝑗,1 + ⋯ + 𝑌𝑗,𝑙

Importance Sampling: What is a good 𝒉(𝒚)?

 Natural choice: 𝑕1 𝑦 = 𝑙 𝜁𝑜 𝑦 – More precisely: 𝑕1 𝑦 = max

𝑙 𝜁𝑜 𝑦, 1

– Can show: 𝑊𝑏𝑠 𝑍

𝑗 = 𝑃

𝜁𝑜 2 for any 𝑗

– Communication cost: 𝑃

𝑙/𝜁

– This is (worst-case) optimal  Interesting discovery: 𝑕2 𝑦 = 𝑕1 𝑦 2 – Also has 𝑊𝑏𝑠 𝑍

𝑗 = 𝑃

𝜁𝑜 2 for any 𝑗

– Also has communication cost 𝑃

𝑙/𝜁 in the worst case

– But can be much lower than 𝑕1(𝑦) on some inputs

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𝒉𝟑 𝒚 is Instance-Optimal

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Median and Quantiles (order statistics)

 Exact quantiles: 𝐺−1() for 0 <  < 1, 𝐺  : CDF  Approximate version: tolerate answer between 𝐺−1( −

) … 𝐺−1( + )

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Estimating Median by Random Sampling

 Simple random sampling – An 𝜁-approximation needs a sample

• f size Θ(1/𝜁2)

 Paired Sampling – Divide data into chunks of size 𝑡 = 𝑃(1/𝜁) – Sort each chunk – Do binary merges into one chunk – Each merge takes odd-positioned

• r even-positioned elements with

equal probability

– Similar ideas used in discrepancy methods  This needs 𝑃(𝑜 log 𝑡) time, how is it useful?

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1 5 6 7 8 2 3 4 9 10 1 3 5 7 9

+

Application 1: Streaming Computation

 Can merge chunks up as items arrive

in the stream

 At any time, keep at most 𝑃 log 𝑜

chunks

 Space: 𝑃(1/𝜁 ⋅ log 𝑜) – Can be improved to 𝑃(1/𝜁 ⋅ log(1/𝜁)) by combining with

random sampling

– Can find all quantiles [Felber, Ostrovsky, ’15]  Reservoir sampling needs 𝑃(1/𝜁2) space  Best deterministic algorithm needs 𝑃(1/𝜁 ⋅ log 𝑜) space

[Greenwald, Khana, ’01]

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[Wang, Luo, Yi, Cormode, SIGMOD’13]

Application 2: Distributed Data

 Data stored on 𝑙 nodes  Each node reduces its data to 𝑃 1/𝜁 𝑙 using paired

sampling, and send to coordinator

 The coordinator reduces all the data received to a size of

𝑃(1/𝜁) using paired sampling

 Communication cost: 𝑃( 𝑙/𝜁)  Looks familiar?

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Generalization: -approximations

 A sample that preserves “density” of point sets – For any range (e.g., a circle), – |fraction of sample points − fraction of all points|≤ 𝜁 – Simple random sample needs size 𝑃

(1/𝜁2)

– Paired sampling yields size 𝑃

(1/𝜁2𝑒/(𝑒+1))

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𝜁-approximations on distributed data

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[Huang, Yi, FOCS’14]

Database Workloads

 Transactional (OLTP) – Deduct 𝑦 dollars from account A, credit 𝑦 dollars to account B – Challenge: Efficiency and correctness (ACID)  Analytical (OLAP) – Large fraction of data – Many tables – Complex conditions – Challenge: Efficiency – Correctness?

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Complex Analytical Queries (TPC-H)

SELECT SUM(l_extendedprice * (1 - l_discount)) FROM customer, lineitem, orders, nation, region WHERE c_custkey = o_custkey AND l_orderkey = o_orderkey AND l_returnflag = 'R' AND c_nationkey = n_nationkey AND n_regionkey = r_regionkey AND r_name = 'ASIA'

This query finds the total revenue loss due to returned orders in a given region.

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Online Aggregation [Haas, Hellerstein, Wang, SIGMOD’97]

SELECT ONLINE SUM(l_extendedprice * (1 - l_discount)) FROM customer, lineitem, orders, nation, region WHERE c_custkey = o_custkey AND l_orderkey = o_orderkey AND l_returnflag = 'R' AND c_nationkey = n_nationkey AND n_regionkey = r_regionkey AND r_name = 'ASIA' WITHTIME 60000 CONFIDENCE 95 REPORTINTERVAL 1000

Confidence interval: Pr 𝑍 − 𝜁 < 𝑍 < 𝑍 + 𝜁 > 0.95

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𝑍 𝑍 − 𝜁 𝑍 + 𝜁

Ripple Join [Haas, Hellerstein, SIGMOD’99]

 Store tuples in each table in random order  In each step – Reads the next tuple from a table in a round-robin fashion – Join with sampled tuples from other tables  Works well for full Cartesian product – But most joins are sparse …

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A Running Example

Nation CID US 1 US 2 China 3 UK 4 China 5 US 6 China 7 UK 8 Japan 9 UK 10

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OrderID ItemID Price 4 301 $2100 2 304 $100 3 201 $300 4 306 $500 3 401 $230 1 101 $800 2 201 $300 5 101 $200 4 301 $100 2 201 $600 BuyerID OrderID 4 1 3 2 1 3 5 4 5 5 5 6 3 7 5 8 3 9 7 10

What’s the total revenue of all orders from customers in China?

𝑂: size of each table, e.g., 109 𝑜: # tuples taken from each table 𝑡: # estimators, e.g., 103 𝑜3 ⋅ 1 𝑂2 = 𝑡 𝑜 = 𝑂2/3𝑡1/3 = 107

Join as a Graph

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𝑆1 𝑆2 𝑆3

Conceptual only Never materialized

Join as a Graph

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𝑆1 𝑆2 𝑆3

Conceptual only Never materialized

Join as a Graph

Nation CID US 1 US 2 China 3 UK 4 China 5 US 6 China 7 UK 8 Japan 9 UK 10

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OrderID ItemID Price 4 301 $2100 2 304 $100 3 201 $300 4 306 $500 3 401 $230 1 101 $800 2 201 $300 5 101 $200 4 301 $100 2 201 $600 BuyerID OrderID 4 1 3 2 1 3 5 4 5 5 5 6 3 7 5 8 3 9 7 10

Sampling by Random Walks

Nation CID US 1 US 2 China 3 UK 4 China 5 US 6 China 7 UK 8 Japan 9 UK 10

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OrderID ItemID Price 4 301 $2100 2 304 $100 3 201 $300 4 306 $500 3 401 $230 1 101 $800 2 201 $300 5 101 $200 4 301 $100 2 201 $600 BuyerID OrderID 4 1 3 2 1 3 5 4 5 5 5 6 3 7 5 8 3 9 7 10

Sampling by Random Walks

Nation CID US 1 US 2 China 3 UK 4 China 5 US 6 China 7 UK 8 Japan 9 UK 10

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OrderID ItemID Price 4 301 $2100 2 304 $100 3 201 $300 4 306 $500 3 401 $230 1 101 $800 2 201 $300 5 101 $200 4 301 $100 2 201 $600 BuyerID OrderID 4 1 3 2 1 3 5 4 5 5 5 6 3 7 5 8 3 9 7 10

Sampling by Random Walks

Nation CID US 1 US 2 China 3 UK 4 China 5 US 6 China 7 UK 8 Japan 9 UK 10

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OrderID ItemID Price 4 301 $2100 2 304 $100 3 201 $300 4 306 $500 3 401 $230 1 101 $800 2 201 $300 5 101 $200 4 301 $100 2 201 $600 BuyerID OrderID 4 1 3 2 1 3 5 4 5 5 5 6 3 7 5 8 3 9 7 10

Sampling by Random Walks

Nation CID US 1 US 2 China 3 UK 4 China 5 US 6 China 7 UK 8 Japan 9 UK 10

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OrderID ItemID Price 4 301 $2100 2 304 $100 3 201 $300 4 306 $500 3 401 $230 1 101 $800 2 201 $300 5 101 $200 4 301 $100 2 201 $600 BuyerID OrderID 4 1 3 2 1 3 5 4 5 5 5 6 3 7 5 8 3 9 7 10

Unbiased estimator:

$𝟔𝟏𝟏 𝐭𝐛𝐧𝐪𝐦𝐣𝐨𝐡 𝐪𝐬𝐩𝐜. = $𝟔𝟏𝟏 𝟐/𝟒⋅𝟐/𝟓⋅𝟐/𝟒

𝑂: size of each table size, e.g., 109 𝑜: # tuples taken from each table = # random walks 𝑡: # estimators, e.g., 103 𝑜 = 𝑡 = 103

Walk Plan Optimization

 Structure of the data graph  Selection predicates – Starting table: use index – Table in the middle: reject random walk  Data distribution – Non-uniformity

may not be a bad thing!

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𝑆1 𝑆2 𝑆3

1 1 1 1 5 6 3

𝑆1 𝑆2 𝑊𝑏𝑠 𝑆1 → 𝑆2 < 𝑊𝑏𝑠 𝑆2 → 𝑆1 𝑆1 𝑆2

5 6 3 1 1 1 1

𝑊𝑏𝑠 𝑆1 → 𝑆2 > 𝑊𝑏𝑠 𝑆2 → 𝑆1

Walk Plan Optimizer

 Enumerate all plans  Conduct ~ 100 trial random walks using each plan  Measure the variance of each plan  Select the best plan  All trials runs are still useful

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Convergence Comparison

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[Li, Wu, Yi, Zhao, SIGMOD’16 Best Paper Award]

Wander Join in PostgreSQL

Logarithmic growth due to B-tree lookup to find random neighbours

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Running on Insufficient Memory (4GB)

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 Insufficient memory incurs a heavy, one-time penalty  Growth is still logarithmic  Fundamentally: Random sampling at odds with hard disks – But does it matter? Spark, In-Memory DB, RAM cloud… – The algorithm is embarrassingly parallel Turbo DBO [Dobra, Jermaine, Rusu, Xu, VLDB’09]

Accuracy Achieved in 1/10 Time of Full Join

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Wander Join vs Ripple Join

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Wander Join Ripple Join Sampling methodology Independent but non-uniform Uniform but non-independent Index needed? Yes Index or random storage Confidence interval computation Easy, 𝑃(𝑜) time Complicated, 𝑃(𝑜𝑙) time 𝑙: # tables Convergence time (20GB data, 3 tables) ~ 3s ~ 50s Scalability Logarithmic Slightly less than linear System implementation PostgreSQL (finished) Oracle (in progress) SparkSQL (in progress) Informix (internal project) DBO

Online Aggregation vs Data Cube

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Online Aggregation Data Cube Queries Online, ad hoc Offline, fixed Latency Seconds Hours, then milliseconds Query mode One at a time Batch Accuracy Small error No error Data schema Any (relational, graph) Multidimensional cube Work with OLTP Integrated Separate Target scenario Online, ad hoc, interactive data analytics Monthly report

Thank you!

Index Ripple Join [Lipton, Naughton, Schneider, SIGMOD’90]

Nation CID US 1 US 2 China 3 UK 4 China 5 US 6 China 7 UK 8 Japan 9 UK 10

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OrderID ItemID Price 4 301 $2100 2 304 $100 3 201 $300 4 306 $500 3 401 $230 1 101 $800 2 201 $300 5 101 $200 4 301 $100 2 201 $600 BuyerID OrderID 4 8 3 5 1 3 5 4 5 2 5 3 3 7 5 1 3 9 7 10

Sampling from a B-tree [Olken, ’93]

 Sampling from an aggregate (ranked) B-tree is easy  But – incurs heavy cost for transactions – need to modify existing B-tree implementations

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4 2 3

Rejection Sampling [Olken, ’93]

 Imagine each node has maximum fanout  Reject as soon as it walks out of bound

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Non-Uniform Sampling

 As long as we can compute the sampling probability, wander

join still works!

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1 3 ⋅ 4 1 3 ⋅ 4 1 3 ⋅ 4 1 3 ⋅ 4 1 3 ⋅ 2 1 3 ⋅ 3 1 3 ⋅ 3 1 3 ⋅ 3 1 3 ⋅ 2

Compare with BlinkDB [Agarwal, Mozafari, Panda, Milner, Madden,

Stoica, ’13]

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Wander Join BlinkDB Methodology Query  Sampling Sampling  Query Sampling method Random walks Stratified sampling Joins supported Any Big table joining a small table (no sampling on small table) Error Reduce over time Fixed Data schema Any (relational, graph) Star / snowflake Work with OLTP Integrated Separate Group-by support Unbalanced Balanced