How to prove pairwise independence of a family of hash. Pairwise independent hash functions 1 hash functions the goal of hash functions is to map elements from a large domain to a small one. Yun kuen cheung, aleksandar nikolov 1 overview in this lecture, we will introduce kwise independence and kwise independent hashing. A pairwiseindependent hash family is a set of functions h h. In the next section, we discuss how this is accomplished. For alternative, we can use the \universal hash functions or kwise independent hash functions, which can save randomness while having the same running time for hashing algorithms. A set hof hash functions is said to be a strong universal. Because of this, hash functions chosen from a strongly 2universal family are also known as pairwise independent hash functions. Then whatever the parity of the sum of the rst js jj 1bits of s j is the sum of this number a and z will be 0, resp. However, clearly they are not jointly independent, since z can explicitly be determined by knowing x and y.
Moreover, the idea of pairwise independence can be generalized. First, we extend the notion of a minwise independent family of hash functions by defining a dkminwise independent family of hash functions. Suppose that we have such a pairwise independent family h, such that every function in h. Sublinear time and space algorithms 2018b lecture 4 amplifying success and hash functions robert krauthgamer 1 amplifying success probability to amplify the success probability of algorithm countmin in general case, we use median of. A family of hash functions h from u to v is said to be kuniversal if, for any elements x1,x2. Unfortunately, such hash functions are not practical. Loosely speaking, universal families of hashing functions consist of functions operating on the same domainrange pair so that a function uniformly selected in the family maps each pair of points in a pairwise independent and uniform manner. Before we move on, here is another construction of pairwise independent random variables taking values in 0,1n which may in some instances be more useful than the family in claim 9. As a consequence, pairwise independent hash families 2. M is a prime and m iui so how do i show that the family is pairwise independent. They are generally based on modular arithmetic constraints of the form ax b. Let hbe a family of hash functions, we say his approximate pairwise independent if for all distinct x 1. We wish the set of functions to be of small size while still behaving similarly to the set of all functions when we pick a member at random. Recall that a pairwise independent family of hash functions satis es p hhx 1 y.
One neat thing about this example is that, in addition to all variables being pairwise independent, the associativity of xor means that theyre also interchangeable. Pairwise independence is not the same as complete independence. Whenever we write h 2h, we shall assume the uniform distribution. R is called a family of pairwise independent hash functions if for di erent x 1. Pairwise independence the following proposition, which we will frequently apply together with chebyshevs inequality, is a key to why pairwise independence is so useful. In this paper we address this gap in the complexity theory by proposing the notion of localitypreserving hash functions for generalpurpose parallel computa tion. The leftover hash lemma shows us how to explicitly construct an extractor from a family of pairwise independent functions h. For theoretical analysis of hashing, there have been two main approaches. A family of hash functions his universal if for every h2h, and for all x6 y2u, pr. We use the method of defered decissions to show that y j is a uniform bit. Suppose that we have such a pairwise independent family h, such that every function in h can be representedusingasmallamountofbitssay,o logn andsuchthateveryfunctioninhcanbecomputed eciently. Choosing an independent hash function, given hash function value. One simple way to construct a family of hash functions mapping.
Recursive ngram hashing is pairwise independent, at best. Definition 2 pair wise independent family of hash functions a family of hash functions his called pairwise independent if 8x 6 y 2d and 8a 1. Lecture 5 1 overview 2 pairwise independent hash functions. Last time we discussed a class of pairwise independent hash functions over nite elds. N mgis called a pairwise independent family of hash functions if for all i6 j2n and any k. Iterated hash functions process strings recursively, one character at a time. While hashing by irreducible polynomials is pairwise independent, our implementations either run in time o n or use an exponential amount of memory.
Since x and y are defined in the same way, z must also be independent of y. We will very frequently use 2universal and pairwise independent hash function families but we will see that larger independence will also sometimes be useful. We exhibit a universal family of hash functions that can be performed in. We now formalize this notion in the following definition. Here we focus on the family of linear hash functions of the form hx signxw, with w. 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. A natural candidate is a pairwise independent hash family, for we are simply seeking to minimize collisions, and collisions are pairwise events, so the statistics will be the same. Fourier analysis of hash functions for inference tra of many boolean functions are well studied in theoretical computer science, learning theory and computational social choice odonnell, 2003, this theoretical bridge allows us to quickly make predictions about the statisti. Pairwise independent hash functions in java stack overflow. Lowdensity parity constraints for hashingbased discrete. The analysis of the collision probabilities in the countmin sketch looks remarkably similar to the analysis of collision probabilities in a chained hash table which only requires a family of universal hash functions, not pairwise independent hash functions, and i cant spot the difference in the analyses. Such families allow good average case performance in randomized algorithms or data structures, even if the input data is.
Sublinear time and space algorithms 2018b lecture 4. A small approximately minwise independent family of hash functions piotr indyk1 departmentofcomputerscience,stanforduniversity,stanford,california94305 email. I have found many descriptions of pairwise independent hash functions for fixedlength bitvectors based on random linear functions. The extractor uses a random hash function h r has its seed and keeps this seed in the output of the extractor. By exhausting all 2lgn npossibilities of the pairwise independent random bits, and choosing the one which gives the largest jev 0. Finally, as a usage example, we show how to apply those hash functions to the. I want to prove pairwise independence of a family of hash functions, but i dont know where to start. Why does the countmin sketch require pairwise independent. We prove that recursive hash families cannot be more than pairwise independent.
Definition 2 pairwise independent family of hash functions a family of hash functions h. Lowdensity parity constraints for hashingbased discrete integration stefano ermon, carla p. Pairwise independent random walks can be slightly unbounded. Localitypreserving hash functions for general purpose. Michael mitzenmachery salil vadhanz abstract hashing is fundamental to many algorithms and data structures widely used in practice. U r is said to be pairwise independent, if for any two distinct elements x1 x2. It is known that lgnbits su ce to generating npairwise independent random bits see example 5. Let h be a family of hash functions, we say h is pairwise inde pendent if for all distinct x1,x2.
A small approximately minwise independent family of hash. The rst such hash function worth considering is the universal families and the strong unversal families of hash functions. Pairwise hash functions that are independent from each other. The most popular data independent approach to generate those hash functioniscalledlocalitysensitivehashinglsh23,9. For example, consider following set of three pairwiseindependent binary variables u 1,2,3,t 0,1,t 2, where each row gives an assignment to the three variables and the associated probability. To update item iby a quantity c i, c i is added to one element in each row, where the element in row j is determined by the hash function h j. We present the efficient implementation of a family. 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.
Low compute and fully parallel computer vision with. A family of hash functions h is called weakly universal if for any pair of distinct elements x1,x2. Pdf 2014 in recent years, a number of probabilistic inference and counting techniques have been proposed that exploit pairwise independent hash. Feature learning based deep supervised hashing with.
V 1j, we have a deterministic 1 2approximation to maxcut. Intuitively, this means that the probability of a hash collision with a specific element is small, even if the output of the hash function for that element is known. Definition 2 pairwise independent family of hash functions a family of hash functions his called pairwise independent if 8x 6 y 2d and 8a 1. More generally, if a family is strongly kuniversal and we choose a hash function from. In computer science, a family of hash functions is said to be kindependent or kuniversal if selecting a function at random from the family guarantees that the hash codes of any designated k keys are independent random variables see precise mathematical definitions below. The univeral hash family is a family of hash functions h fhjh.
Many universal families are known for hashing integers, vectors, strings. Pairwise independence is sometimes called strong universality. A family of problems that have been studied in the context of various streaming algorithms are generalizations of the fact that the expected maximum distance of a 4wise independent random walk on a line over n steps is ovn. Thatis, ifhisafunctionchosenuniformlyatrandomfromh, thentherandomvariablesh x andh y are uniformlydistributed andpairwiseindependent. As a more scalable alternative, we make hashing by cyclic polynomials pairwise independent by ignoring n1 bits. Ideally, i would have some object universalfamily representing the family which would return me objects with a method hash which hashes integers. In mathematics and computing, universal hashing refers to selecting a hash function at random. Note that if we consider the random seed as being a string of bits that we must query to hash our values, then to hash a family of nvalues using the above schemes.
The three are not independent, but they are pairwise. Pairwise independence and derandomization ias school of. Im looking for a quick and easy way to use a universal family of pairwise independent hash functions in my java projects. Pdf lowdensity parity constraints for hashingbased. I need to use a hash function which belongs to a family of kwise independent hash functions. However, it is also true that, as long as we consider only speci. Introduction to pairwise independent hashing weizmann institute of.
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