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not exactly approximate algorithms for big data

NOT EXACTLY! APPROXIMATE ALGORITHMS FOR BIG DATA FANGJIN YANG - PowerPoint PPT Presentation

Sep 24, 2022 •454 likes •1.12k views

NOT EXACTLY! APPROXIMATE ALGORITHMS FOR BIG DATA FANGJIN YANG DRUID COMMITTER METAMARKETS NELSON RAY QUANTITATIVE ANALYST GOOGLE OVERVIEW THE PROBLEM MANAGE DATA COST EFFICIENTLY THE DATA DEALING WITH EVENT STREAMS SIMPLIFYING

  1. NOT EXACTLY! APPROXIMATE ALGORITHMS FOR BIG DATA FANGJIN YANG · DRUID COMMITTER · METAMARKETS NELSON RAY · QUANTITATIVE ANALYST · GOOGLE

  2. OVERVIEW THE PROBLEM MANAGE DATA COST EFFICIENTLY THE DATA DEALING WITH EVENT STREAMS SIMPLIFYING STORAGE DATA SUMMARIZATION FINDING UNIQUES HYPERLOGLOG ESTIMATING DISTRIBUTION APPROXIMATE HISTOGRAMS

  3. THE PROBLEM

  4. Real-time Bidding Fangjin Yang & Nelson Ray 2014

  5. PROBLEMS ‣ Storing/processing billions of rows is expensive ‣ Reduce storage, improve performance ‣ Reduce storage by throwing away information ‣ Throwing away information reduces accuracy Fangjin Yang & Nelson Ray 2014

  6. THE DATA

  7. THE DATA Timestamp Bid Price 2013-10-28T02:13:43Z 1.19 2013-10-28T02:14:21Z 0.05 2013-10-28T02:55:32Z 1.04 2013-10-28T03:07:28Z 0.16 2013-10-28T03:13:43Z 1.03 2013-10-28T04:18:19Z 0.15 2013-10-28T05:36:34Z 0.01 2013-10-28T05:37:59Z 1.03 Fangjin Yang & Nelson Ray 2014

  8. DATA SUMMARIZATION Timestamp Bid Price Timestamp Revenue Number of Prices 2013-10-28T02:13:43Z 1.19 2013-10-28T02 2013-10-28T02:14:21Z 0.05 2.28 3 2013-10-28T02:55:32Z 1.04 2013-10-28T03:07:28Z 0.16 2013-10-28T03 1.19 2 2013-10-28T03:13:43Z 1.03 2013-10-28T04 0.15 1 2013-10-28T04:18:19Z 0.15 2013-10-28T05:36:34Z 0.01 2013-10-28T05 1.04 2 2013-10-28T05:37:59Z 1.03 Fangjin Yang & Nelson Ray 2014

  9. COMBINING SUMMARIZATIONS Timestamp Revenue Number of Prices Timestamp Revenue Number of Prices 2013-10-28T02 2.28 3 2013-10-28 2013-10-28T03 1.19 2 4.66 8 2013-10-28T04 0.15 1 2013-10-28T05 1.04 2 Fangjin Yang & Nelson Ray 2014

  10. Fangjin Yang & Nelson Ray 2014

  11. SUMMARIZATION SUMMARY ‣ Throw away information about individual events ‣ Drastically reduce storage and improve query speed • On average, 40x reduction in storage on with our own data ‣ We’ve lost info about individual prices ‣ Data summarization is not always trivial Fangjin Yang & Nelson Ray 2014

  12. CASE STUDY 1

  13. CASE STUDY 1 ‣ Problem: determine unique number of elements in a set ‣ Use case: measuring number of unique users DATA BIG DATA Fangjin Yang & Nelson Ray 2014

  14. EXACT SOLUTION ‣ Store every single username (in a Java HashSet) ‣ No loss of information, no accuracy tradeoff Fangjin Yang & Nelson Ray 2014

  15. HASHSET Timestamp Username Timestamp Usernames 2013-10-28T02:13:43Z user1 2013-10-28T02 2013-10-28T02:14:21Z user2 {user1, user2} 2013-10-28T02:55:32Z user1 2013-10-28T03:07:28Z user4 2013-10-28T03 {user4, user97} 2013-10-28T03:13:43Z user97 2013-10-28T04 {user2} 2013-10-28T04:18:19Z user2 2013-10-28T05:36:34Z user9834 {user9834, 2013-10-28T05 user97} 2013-10-28T05:37:59Z user97 Fangjin Yang & Nelson Ray 2014

  16. HASHSET Usernames Timestamp Usernames Timestamp 2013-10-28T02 {user1, user2} {user1, user2, 2013-10-28 2013-10-28T03 {user4, user97} user4, user97, user9834} 2013-10-28T04 {user2} {user9834, 2013-10-28T05 user97} Fangjin Yang & Nelson Ray 2014

  17. EXACT SOLUTION ‣ Storage/Computation: O(# uniques) ‣ We’re not throwing away any information about usernames ‣ Accuracy: 100% Fangjin Yang & Nelson Ray 2014

  18. INFEASIBLE STORAGE ‣ High cardinality user dimensions == infeasible storage • Storage cost for 10^9 unique elements == ~48GB of storage Fangjin Yang & Nelson Ray 2014

  19. CARDINALITY ESTIMATION ‣ Plenty of literature • Linear Counting • Count-Min Sketch • LogLog Fangjin Yang & Nelson Ray 2014

  20. HYPERLOGLOG ‣ Storage: 1.5 KB ( for cardinalities 10^9 and above) • 99.999997% decrease in storage size ‣ Computation: O(1) ( for cardinalities < ~10^10) ‣ Accuracy: 97% Fangjin Yang & Nelson Ray 2014

  21. HASH FUNCTIONS ‣ Maps value in one space (generally larger) to another value in another space (generally smaller) String 0001 HashFn Fangjin Yang & Nelson Ray 2014

  22. WHAT MAKES A GOOD HASH FUNCTION? ‣ Bits of output value are independent and have an equal probability of occurring (50%) String 50% Probability 0xxx HashFn String 50% Probability 1xxx HashFn Fangjin Yang & Nelson Ray 2014

  23. HASHING TWO STRINGS user1 0xxx HashFn user2 1xxx HashFn Fangjin Yang & Nelson Ray 2013

  24. THE NEXT BIT String 00xx 25% Probability HashFn String 10xx 25% Probability HashFn String 25% Probability 01xx HashFn String 25% Probability 11xx HashFn Fangjin Yang & Nelson Ray 2013

  25. HASHING 4 STRINGS user1 00xx HashFn user2 10xx HashFn user3 01xx HashFn user4 11xx HashFn Fangjin Yang & Nelson Ray 2013

  26. HYPERLOGLOG ‣ What about 001x? • If we hashed one string, 12.5% chance this could occur • If we hashed 8 strings, one of them should be this value ‣ What about 000001…x? • Extremely unlikely to occur if we only hashed one string Fangjin Yang & Nelson Ray 2013

  27. HYPERLOGLOG ‣ Looks at distribution of bits of hashed values ‣ Cares about the position of the left most ‘1’ bit ‣ 1000 -> position == 1 ‣ 0100 -> position == 2 ‣ 0011 -> position == 3 Fangjin Yang & Nelson Ray 2014

  28. HYPERLOGLOG ‣ Stores the max position of the left-most ‘1’ bit of hashed values ‣ User1 —> hash —> 1000 (position == 1) ‣ User2 —> hash —> 0100 (position == 2) ‣ User3 —> hash —> 0011 (position == 3) ‣ HLL will store position == 3 Fangjin Yang & Nelson Ray 2014

  29. HYPERLOGLOG Fangjin Yang & Nelson Ray 2014

  30. HYPERLOGLOG ACCURACY String 00xx HashFn String 10xx HashFn String 25% Probability 01xx HashFn String 11xx HashFn Fangjin Yang & Nelson Ray 2013

  31. HYPERLOGLOG ‣ If we fed the stream through a second hash function, we’d have a second independent estimate ‣ Adding more hash functions gives us more independent estimates that we can combine together for a lower variance estimate ‣ This is expensive because we have to hash the same data n times Fangjin Yang & Nelson Ray 2014

  32. HYPERLOGLOG ‣ Instead we can split the stream ‣ Estimate the cardinality of each sub-stream ‣ For each sub-stream ‣ Store the maximum over the positions of the leftmost '1' bit for hashed values of the sub-stream Fangjin Yang & Nelson Ray 2014

  33. HYPERLOGLOG Buckets -INF -INF -INF -INF Fangjin Yang & Nelson Ray 2014

  34. HYPERLOGLOG Buckets user1 01xxx...x 2 HashFn -INF -INF -INF Fangjin Yang & Nelson Ray 2014

  35. HYPERLOGLOG Buckets user1 01xxx...x 2 HashFn user4 01xxx...x 2 HashFn user12 01xxx...x 2 HashFn user7 1xxxx...x 1 HashFn Fangjin Yang & Nelson Ray 2014

  36. HYPERLOGLOG Buckets user6 001xx...x 2 -> 3 HashFn 2 2 1 Fangjin Yang & Nelson Ray 2014

  37. DETERMINING FINAL CARDINALITY Buckets 3 11.00 2 MATH 2 1 Fangjin Yang & Nelson Ray 2014

  38. HYPERLOGLOG Timestamp Buckets 2013-10-28T02 [3, 2, 2, 1] 2013-10-28T03 [1, 2, 1, 2] 2013-10-28T04 [2, 1, 4, 1] 2013-10-28T05 [2, 2, 3, 1] Fangjin Yang & Nelson Ray 2014

  39. HYPERLOGLOG Timestamp HLL Object 2013-10-28 [3, 2, 4, 2] Fangjin Yang & Nelson Ray 2014

  40. Fangjin Yang & Nelson Ray 2014

  41. RESULTS Fangjin Yang & Nelson Ray 2014

  42. CASE STUDY 2

  43. CASE STUDY 2 ‣ Problem: determine distribution of values ‣ Use case: quantiles and histograms ‣ Hourly truncation Fangjin Yang & Nelson Ray 2014

  44. THE DATA Timestamp Bid Price 2013-10-28T02:13:43Z 1.19 2013-10-28T02:14:21Z 0.05 2013-10-28T02:55:32Z 1.04 2013-10-28T03:07:28Z 0.16 2013-10-28T03:13:43Z 1.03 2013-10-28T04:18:19Z 0.15 2013-10-28T05:36:34Z 0.01 2013-10-28T05:37:59Z 1.03 Fangjin Yang & Nelson Ray 2014

  45. EXACT SOLUTION Bid Price Timestamp Timestamp Bid Prices 2013-10-28T02:13:43Z 1.19 2013-10-28T02 2013-10-28T02:14:21Z 0.05 [1.19, 0.05, 1.04] 2013-10-28T02:55:32Z 1.04 2013-10-28T03:07:28Z 0.16 2013-10-28T03 [0.16, 1.03] 2013-10-28T03:13:43Z 1.03 2013-10-28T04:18:19Z 0.15 2013-10-28T04 [0.15] 2013-10-28T05:36:34Z 0.01 2013-10-28T05 [0.01, 1.03] 2013-10-28T05:37:59Z 1.03 Fangjin Yang & Nelson Ray 2014

  46. EXACT SOLUTION Timestamp Bid Prices Timestamp Bid Prices 2013-10-28T02 [1.19, 0.05, 1.04] 2013-10-28 [1.19, 0.05, 1.04, 0.16, 2013-10-28T03 [0.16, 1.03] 1.03, 0.15, 0.01, 1.03] 2013-10-28T04 [0.15] 2013-10-28T05 [0.01, 1.03] Fangjin Yang & Nelson Ray 2014

  47. EXACT SOLUTION ‣ Arrays of values ‣ Storage: Linear ‣ Computation: Linear ‣ Accuracy: 100% ‣ Problem: Storing raw values can often be more expensive than storing the rest of the row. ‣ Solution: Store an approximate representation! Fangjin Yang & Nelson Ray 2014

  48. APPROXIMATE HISTOGRAMS ‣ “A Streaming Parallel Decision Tree Algorithm” ‣ Yael Ben-Haim & Elad Tom-Tov ‣ Storage: Sublinear/Linear ‣ Computation: Sublinear/Linear ‣ Accuracy: pretty good Fangjin Yang & Nelson Ray 2014

  49. RAW DATA • 40 Prices: 3.46, 5.37, 5.62, 5.87, 6.21, 6.79, 7.11, 7.36, 7.55, 7.64, 7.89, 7.9, 8.07, 8.44, 8.62, 8.78, 8.87, 9.03, 9.24, 9.36, 9.58, 9.59, 9.81, 10.31, 10.35, 10.39, 10.47, 10.77, 10.93, 11.04, 11.1, 13.1, 13.27, 13.29, 13.87, 14.29, 14.51, 14.9, 15.75, 17.07 Fangjin Yang & Nelson Ray 2013

  50. RAW DATA Fangjin Yang & Nelson Ray 2013

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