1. 1 Summarizing Data using Bottom-k Sketches Edith Cohen AT&T Haim Kaplan Tel Aviv University

2. 2 The basic application setup There is a universe of items, each has a weight Keep a sketch of k items, so that you can estimate the weight of any subpopulation from the sketch. (also other aggregates, weight functions) Example: Items are flows going through a router w(i) is the number of packets of flow i. Queries supported by the sketch are estimates on the total weight of flows of a particular port, particular size, particular destination IP address, etc. w(i 1 )w(i 2 )w(i 3 )

3. 3 Application setup : Coordinated sketches for multiple subsets Universe I of weighted items and a set of subsets over that universe. Keep a size-k sketch of each subset, so that you can support both queries within a subset and queries on subset relations such as aggregates over union, or intersection, resemblance…. These sketches are coordinated so that subset relations queries can be supported better.

4. 4 …Sketches for multiple subsets Example applications: Items are features and subsets are documents. Estimate similarity of documents. Items are documents and subsets are features. Estimate “correlation” of features. Items are files and subsets are neighborhoods in a p2p network. Estimate total size of distinct items in a neighborhood. Items are goods and subsets are consumers. Estimate marketing cost of all goods in a subset of consumers.

5. 5 Application setup : All-distances sketches Items are located in a metric space ( data stream with time stamp or sequence number, network with distance ). All-distances sketch of a location v is a compact encoding of the sketches of ALL neighborhoods of v. Efficient time-decaying and spatially-decaying aggregation For each query distance d, the sketch of the set of items within distance d from v can be retrieved from the all-distances sketch v

6. 6 All-distances sketches

7. 7 One approach: k-mins sketches Each item i draws a rank r(i) from an exponential distribution with parameter w(i) (-ln u/w(i) u ∊ [0,1] ) Pick the item with the smallest rank to your set Repeat k times (possibly concurrently) Equivalent to weighted sampling of k items with replacement, (convenient in distributed settings).  k-mins sketch (Cohen 97) ( (i 2,r (1) (i 2 ), (i 2,r (k) (i 2 ) ) (i 5,r (2) (i 5 ),….

8. 8 …… k-mins sketches Multiple subsets: same r (1),…,r (k) for all subsets. All-distances: encode min rank at any distance (0,0.8), (2,0.6), (10,0.5), (15,0.2) Estimators (examples): The fraction of blue items among the k in the sketch is an unbiased estimate of their fraction in the population (k-1)/sum(k min ranks) is an unbiased estimate of the weight of the set (error decreases with k) Multiple subsets: fraction of “common” coordinates is an unbiased estimator for resemblance (ratio of intersection and union) x k

9. 9 Applications of k-mins sketches Graph theory: Estimate the size of the transitive closure of a directed graph and of neighborhoods without explicitly computing them (Cohen ’94) Sensor networks: Estimating # of items, variance, in a neighborhood of a sensor (Cohen, Kaplan, SIGMOD’04), aggregation where weights decay with distance Streaming: aggregation where weights decay with time (Cohen, Strauss, PODS’03) Databases: Estimate the size of the “join” before actually computing it (Lewis, Cohen SODA’97) Data mining: Estimate resemblance of Web pages and Web sites (Broder 97, Bharat, Broder 99,…) Databases: Estimate association rules (CDFGM TKDE ’01)

10. 10 Our approach: Bottom-k sketches Each item appears once – this is equivalent to weighted sampling without replacement Each item i draws a rank r(i) from some distribution that depends on its weight, say -ln u/w(i) where u ∊ [0,1] (exponential with PDF w(i)e -w(i)x ) Pick the k smallest-ranked items to your sketch. Multiple subsets: pick the k items with smallest ranks in the subset for the sketch of the subset.

11. 11 Advantages of Bottom-k sketches Intuitively the sample is more informative, in particular for Zipf-like distribution, where there are few large weights. Often, more efficient to compute Provide more accurate estimates Bottom-k sketches can be used instead of k-mins sketches in almost every application

12. 12 Bottom-k sketches can replace k- mins sketches in most applications Plain sketches for explicitly-represented subsets: Bottom-k sketches can be computed much more efficiently. All-distance sketches: Bottom-k sketches can be computed as efficiently ? Open: Euclidean plane

13. 13 Multiple subsets with explicit representation Items are processed one by one, sketch is updated when a new item is processed: k-mins: The new item draws a vector of k random numbers We compare each coordinate to the “winner” so far in that coordinate and update if the new one wins  O(k) time to test and update bottom-k: The new item draws one random number We compare the number to the k smallest so far and update if smaller  O(1) time to test O(log k) time to update

14. 14 … Multiple subsets with explicit representation The number of tests = sum of the sizes of subsets. The number of updates is “generally” logarithmic (depends on item-weights distribution and the order items are processed) and about the same for k-mins and bottom-k sketches. (Precise analysis in the paper) k-mins: O(k) time to test and update bottom-k: O(1) time to test O(log k) time to update

15. 15 All distances bottom-k sketches Data structures are more complex than all- distances k-mins sketches -- maintain k smallest items in a single rank assignment instead of one smallest in k independent assignments. We analyze the number of operations for Constructing all-distances k-mins and bottom-k sketches for different orders in which items are “processed”, weight distributions, and relations of weight and location. Querying sketches  The number of operations is comparable for both types of sketches.

16. 16 Bottom-k sketches provide more accurate estimates The red line shows an estimate based on k-mins sketch All other lines are various estimators that use bottom-k sketch

17. 17 Estimating with bottom-k sketches (by “mimicking”) We can produce from a bottom-k sketch S a distribution D(S) on k-mins sketches. Drawing S and then a k-mins sketch from D(S) is equivalent to drawing a k-mins sketch  Can use all known estimators for k-mins sketch  We can do even better by taking expectation over D(S) or drawing from D(S) multiple times bottom-k sketches S k-mins sketches D(S)

18. 18 Maximum likelihood estimate using bottom-k sketches Lemma: Over sketches with item weights (w 1,w 2,…w k ) The rank differences are independent, each distributed exponentially r 1 – r 0 r 2 – r 1... r k -r k-1 exp. with parameter W= w(I) exp. with parameter W-w 1 exp. with parameter W-w 1 -w 2 -...-w k-1 The probability that we see the particular differences is The ML estimate is the W which maximizes this

19. 19 Adjusted weights estimators Old technique by Horvitz and Thompson: Let p i be the probability that item i is sampled. Assign to item i an adjusted weight of a(i)=w(i)/p i if i is sampled and a(i)=0 otherwise. Then  a(i) is an unbiased estimator of w(i) To estimate a subset J we sum up the adjusted weights of the elements of the sketch that are in J

20. 20 Subspace conditioning Problem: p i may be impossible or hard to compute from the information in the sketch. Partition the probability space into subspaces and apply the technique within each subset. p i is now the probability that w i is in the sketch conditioned on the fact that the sketch is from a particular subspace. Our idea:

21. 21 Rank conditioning estimators Partition the sketches according to the ( k+1 )- smallest rank ( r k+1 ) ( k th smallest rank among all other items) Then p i is the probability that the rank of i is smaller than r k+1 Lemma: E(a(i)a(j)) = w(i)w(j) (cov(i,j)=0) i<>j  the variance of the estimator for a subset J is the sum of the variances of the adjusted weights of the items in J

22. 22 Reducing the variance Use a coarser partition of the probability space to do the conditioning Lemma: The variance is smaller for a coarser partition. Typically p i gets somewhat harder to compute

23. 23 Priority sampling (Duffield,Lund,Thorup) fits nicely in the rank conditioning framework: These are bottom-k sketches with ranks drawn as follows: Item i with weight w(i) gets rank u/w(i) where u ∊ [0,1] Item i is in the sample if u/w(i) < r k+1. This happens with probability min{w(i) r k+1,1} So the adjusted weight of i is a(i)=w(i)/min{w(i) r k+1,1} = max{w(i), 1/ r k+1 } Rank conditioning adjusted weights:

24. 24 Other weight functions We can estimate aggregates with respect to other weight functions, such as, number of distinct items, size distribution…. For a numeric property h(i), h(i)a(i)/w(i) is an unbiased estimator of h(i). Therefore, is an unbiased estimator of h(J)

25. 25 Other aggregates Selectivity of a subpopulation Approximate quantiles Variance and higher moments Weighted random sample ……

26. 26 Summary: b ottom-k sketches Useful in many application setups: –Sketch a single set –Coordinated sketches of multiple subsets –All-distances sketches A better alternative to k-mins sketches We facilitate use of bottom-k sketches: –Data structures, all-distances sketches –Analyze construction and query cost –Estimators, variance, confidence intervals [also in further work]