2-wise independent hash function pdf

Let the third random variable z be equal to 1 if exactly one of those coin tosses resulted in heads, and 0 otherwise. So, if our algorithm is ok with using a hash function that provides only a kwise independence guarantee for some small k as was, for example. P h2h hi1 j1 hik jk 1 bk note that we can store h 2 h in memory with log2 jhj bits. These lecture notes show that linear probing takes expected constant time if the hash function is 5independent. It can be shown that we dont need h to be uniform, a olog nwise. The property required from his 2wise independence, informally a hash function family is 2 wise independent if the hash value hx provides no information about hy. One can identify \somewhat random key sets s, often quite natural, sometimes somewhat construed, for which 2wise independent classes behave badly. Lecture 1 kwise independence ubc computer science. To illustrate, the rst few levels of our tree would look like. Computational complexity of universal hashing request pdf. Such families allow good average case performance in randomized algorithms or data structures, even if the input data is. Once we reach rlevels, output the concatenation of all leaves, which is a lengthrsbit string. A scalable and nearly uniform generator of sat witnesses aug 20, 20 nsf site visit 20 motivators.

This \ 2wise independence turns out to be su cient to guarantee many of the properties possessed by fully random hash functions 3, 20. That is, for any i,jand values x,y, it holds that pr x i x. Ejaj jsj2 r varjaj ojsj2 r both of these still hold under 2independence the bounds on jajfollow from chebyshevs inequality d. To achieve unbounded simulationsoundness, we move from a 2wise independent hash function to an af. We generally refer to 2wise independent families of functions as pairwise independent. Families with 2 wise pairwise independence have useful analytic properties in pram simulation 2, 24, 28, 31 and may be adequate for all practical purposes. Input is arbitrary, but the hash function is random. One can see this by expanding and applying linearity of expectation. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields.

One example of such a family h is the set of all functions mapping a to b. N a prime number exhibits a similar behavior as the class in a, again in the case where the key set is relatively dense in u. An optimal algorithm for the distinct elements problem daniel m. M is a prime and m iui so how do i show that the family is pairwise independent.

Or in other words, h acts like a completely random function when restricted to any k elements of u. It is common to think of h as a hash function, because it is a randomlike function. An optimal algorithm for the distinct elements problem. How to find a 2wise independent hash family that is not 3wise independent. Advanced algorithms ii sublinear algorithms rutgers.

How to prove pairwise independence of a family of hash. We keep track in memory of the ksmallest hash evaluations. Another widely studied case is that of k wise independent permutations of nelements. We saw in class a construction of 2wise independent hash functions.

Definition 2 pairwise independent family of hash functions a family of hash functions h. Building lossy trapdoorfunctions from lossy encryption. The simplest kwise independent hash function mapping positive integer x 0 2wise independent hash. Geometric sampling of streams suppose we have a substitute that gives us t as a 32approximation to t. On the kindependence required by linear probing and minwise. We can store a hash function from this family in ologqk oklogq space. Return to the notation of section iiic, and recall that u is the universe of all possible packets, v is the characteristic vector of the stream of packets, and w. Another widely studied case is that of kwise independent permutations of nelements. One can identify \somewhat random key sets s, often quite natural, sometimes somewhat construed, for which 2 wise independent classes behave badly. Regularity of lossy rsa on subdomains and its applications.

Obtaining a kwise independent hash function stack overflow. A useful lemma concerning the almost uniform coverage of pairwise independent hash functions is the following. Incremental randomized sketching for online kernel learning formulated an explicit feature mapping using a nystrom. We want to design a hash function family h from a field f to. Pairwise independence does not imply mutual independence, as shown by the following example attributed to s. M for m n3 from a 2wise independent hash family the idea here is to discretize 0. The usage of xorbased hash functions allows us to represent a cell as conjunction of a boolean formula in conjunctive normal form cnf and xor constraints, and a sat solver is invoked to enumerate solutions inside a randomly. On risks of using cuckoo hashing with simple universal. A scalable and nearly uniform generator of sat witnesses.

When applied to ndimensional vectors, the transform is speci. However, the converse is not true as shown by the following example. This is true even when the two hash functions use di. Great ideas in algorithms problem set 3, due 528, 2015. However, we show in this paper that linear probing using a 2wise independent hash function may have expected logarithmic cost per operation. Here are some links comparing the quality of general purpose hash functions. In this work, we show that if enc is a lossy encryption with messages at least 1bit longer than randomness, and h is a pairwise independent hash function, then x 7. Countsketch and tensorsketch we start by describing the countsketch transform charikar et al.

Families with 2wise pairwise independence have useful analytic properties in pram simulation 2, 24, 28, 31 and may be adequate for all practical purposes. One can identify \somewhat random key sets s, often quite natural, sometimes somewhat construed, for which 2. Ex jfj 1 k jfj 1 2 applying markovs inequality, we have prx jfj 1 1 2 the probability that all the excesses at ai. This family of hash functions is 2wise independent. A family h of functions mapping a into b is kwise independent. Therefore, if we choose m s2, the expected number of collisions will be smaller than 1. The proof in 3 proceeds by bounding pki d utstsuk 2 pki d utstsukl 2 l using trace inequalities. The original technique for constructing kindependent hash functions, given by carter and wegman, was to select a large prime number p, choose k random numbers modulo p, and use these numbers as the coefficients of a polynomial of degree k. However, specifying a random hash function requires onlogb bits of storage, the truth table must be stored to evaluate the hash function.

On the kindependence required by linear probing and. For our problem, we will only need a 2wise independent hash function h. This paper focuses on proving bounds for these pairwise independent hash families. The algorithm estimates the frequency of item x j up to. Mcgregor, vu, icdt 17 i a 2 approx for set cover in 2 passes and omn space. Incremental randomized sketching for online kernel learning. Typically, to obtain the required guarantees, we would need not just one function, but a family of functions, where we would use randomness to sample a hash function from this family. Kane department of mathematics stanford university. V is kwise independent if for any distinct u 1u k 2u and any v 1v k 2v pr h2hhu i v i for all i 1 jvjk. Computer science stack exchange is a question and answer site for students, researchers and practitioners of computer science. In computer science, a family of hash functions is said to be kindependent or k universal if. Localitypreserving hash functions for general purpose. Let his a hash function family from the domain dto the range r. Finally, linear probing, a wellknown implementation of hash tables, was shown to take o 1 expected update time with any 5wise independent hash function but was shown to take.

Using an engineered hash function like murmur will maximize the quality of the distribution, and minimize the number of collisions, but it offers no other guarantee. Kane stanford distinct elements february 2014 21 30. A set of n wise random variables is really just a way to sample one row. A set of nwise random variables is really just a way to sample one row. For k 2 the group of invertible a ne transformations x7.

X 2 be uniform independent random variables taking values in f0. This property provides good bounds for the distribution of the. More generally, researchers have considered kwise independence where any kvalues of the hash function are independent. One such family is the set of all degreek 1 polynomials in f qab. Let lsbx be the location of the least signi cant bit of x which is a 1. Local algorithm from a 2wise independent hash function and choose the hash functions across the rounds independently. For instance, hash chaining takes constant expected time even with a 2independent hash function, because the expected time to perform a. However, we study the problem of bounding the position throughout the random walk, by providing comparable moment bounds. Suppose x and y are two independent tosses of a fair coin, where we designate 1 for heads and 0 for tails. As an illustrative example, the notion of a 2wise independent family of hash functions assures the independence of each pair of elements. T is a randomized function that provides the guarantee that, for any kdistinct. We present a di erent, simpler proof by 2, which leverages the machinery of.

Hash functions with similar properties are called kwise independent and pairwise independent hash functions, respectively. Lectures, 14 1 streaming algorithms eecs at uc berkeley. Further, this bound is achievable when m divides n. On risks of using cuckoo hashing with simple universal hash. Having seen these examples, we will just assume that we have access to some 2 wise independent hash families, which will let us store in lgn bits. I want to prove pairwise independence of a family of hash functions, but i dont know where to start. Intuitively, it simulates a random hash function if we take only kvalues. Lecture 17 1 introduction 2 kwise independent hash functions. For this theorem to hold, hneeds to be a 2wise independent hash function, and.

Lecture 5 1 overview 2 pairwise independent hash functions. Set cover and max coverage andrew mcgregor i a 12 approx for max k coverage in ok space. Pairwise independent hash functions 1 hash functions the goal of hash functions is to map elements from a large domain to a small one. Here, we will generalize this to 4wise independence. We show this is possible via approaches based on 47. Cryptographic hash functions a hash function maps a message of an arbitrary length to a mbit output output known as the fingerprint or the message digest if the message digest is transmitted securely, then changes to the message can be detected a hash is a manytoone function, so collisions can happen. Also, if k 1 k 2, k 1wise independence implies k 2wise independence. In computer science, a family of hash functions is said to be kindependent or kuniversal if. Building lossy trapdoor functions from lossy encryption. The memory requirement can be reduced by choosing hfrom a hash function family hof small size having good independence properties. Show that this new lca still outputs a correct mis with high probability and that its runtime for each probe is o log lognand it needs poly.