We consider the Similarity Sketching problem: Given a universe rus " t0, . . . , u´1u we want a random function S mapping subsets A Ď rus into vectors SpAq of size t, such that similarity is preserved. More precisely: Given sets A, B Ď rus, define X i " rSpAqris " SpBqriss and X " ř iPrts X i . We want to have ErXs " t¨JpA, Bq, where JpA, Bq " |A X B|{|A Y B| and furthermore to have strong concentration guarantees (i.e. Chernoffstyle bounds) for X. This is a fundamental problem which has found numerous applications in data mining, large-scale classification, computer vision, similarity search, etc. via the classic MinHash algorithm. The vectors SpAq are also called sketches.The seminal tˆMinHash algorithm uses t random hash functions h 1 , . . . , h t , and stores pmin aPA h 1 pAq, . . . , min aPA h t pAqq as the sketch of A. The main drawback of MinHash is, however, its Opt¨|A|q running time, and finding a sketch with similar properties and faster running time has been the subject of several papers. Addressing this, Li et al. [NIPS'12] introduced one permutation hashing (OPH), which creates a sketch of size t in Opt`|A|q time, but with the drawback that possibly some of the t entries are "empty" when |A| " Optq. One could argue that sketching is not necessary in this case, however the desire in most applications is to have one sketching procedure that works for sets of all sizes. Therefore, filling out these empty entries is the subject of several follow-up papers initiated by Shrivastava and Li [ICML'14]. However, these "densification" schemes fail to provide good concentration bounds exactly in the case |A| " Optq, where they are needed.In this paper we present a new sketch which obtains essentially the best of both worlds. That is, a fast Opt log t`|A|q expected running time while getting the same strong concentration bounds as MinHash. Our new sketch can be seen as a mix between sampling with replacement and sampling without replacement. We demonstrate the power of our new sketch by considering popular applications in large-scale classification with linear SVM as introduced by Li et al. [NIPS'11] as well as approximate similarity search using the LSH framework of Indyk and Motwani [STOC'98]. In particular, for the j 1 , j 2 -approximate similarity search problem on a collection of n sets we obtain a data-structure with space usage Opn 1`ρ`ř APC |A|q and Opn ρ log n`|Q|q expected time for querying a set Q compared to a Opn ρ log n¨|Q|q expected query time of the classic result of Indyk and Motwani.

We consider the hash function hpxq " ppax`bq mod pq mod n where a, b are chosen uniformly at random from t0, 1, . . . , p´1u. We prove that when we use hpxq in hashing with chaining to insert n elements into a table of size n the expected length of the longest chain isÕ`n 1{3˘. The proof also generalises to give the same bound when we use the multiply-shift hash function by Dietzfelbinger et al.