Yun kuen cheung, aleksandar nikolov 1 overview in this lecture, we will introduce k wise independence and k wise independent hashing. Testing nonuniform kwise independent distributions over. Sampling graphs without forbidden subgraphs and unbalanced. Cs 6815 pseudorandomness and combinatorial constructions. Resources the main reference for the course will be scribed lecture notes. Denote t3,n the random variable on the space gn,p, which is equal to the number of triangles in a. We show that if the variables have no dominant value their minentropies are bounded away from zero then this joint. In this note we prove a large deviation bound on the sum of random variables with the following dependency structure. Our approach applies to several sums of weakly dependent random variables such as sums of martingale difference sequences, sums of kwise independent random variables and ustatistics.
Algorithmic construction of sets for krestrictions acm. We study the kwise independent relaxation of the usual model gn, p of random graphs where, as in this model, n labeled vertices are fixed and each edge is drawn with probability p, however, it is. We study the kwise independent relaxation of the usual model gn,p of random graphs where, as in this model, n labeled vertices are fixed and each edge is drawn with probability p, however, it is. In the construction 1 of pairwise independent random variables, notice that we can compute yu.
In this construction, yis a vector with oklogn elements chosen uniformly at random, and m has ncolumns. Pairwise independent random walks can be slightly unbounded. As our main result, we show that the problem is solvable with respect to some specially. We study the k wise independent relaxation of the usual model gn,p of random graphs where, as in this model, n labeled vertices are fixed and each edge is drawn with probability p, however, it is only required that the distribution of any subset of k edges is independent. Constructions of k wise almost independent permutations have been receiving a growing amount of attention in recent years. Derandomized constructions of kwise almost independent. We extend it into a visual variant of the k out of n secret sharing problem, in which a dealer provides a transparency to each one. Finally, we discuss some applications to the theory of random graphs. Let y be a random variable counting the number of independent sets of size k 0 in gn, p.
For the problem of distinguishing kwise independent distributions from those that are. Ganesh, university of bristol, 2015 we have so far seen a number of examples of random processes on networks, including the spread of information, competition between opinions. This work addresses krestriction problems, which unify combinatorial problems of the following type. A family of problems that have been studied in the context of various streaming. E is a k ramsey graph if it is an undirected graph on n vertices i. Random graphs were used by erdos 278 to give a probabilistic construction. A simple parallel algorithm for the maximal independent. If all the vertices of a graph have the same degree degv k, the graph.
Random variables that are fflbiased can be used to construct almost kwise independent random variables where ffl is a function of k. We formally show that there is an algorithm for dlog over. Another reason is that random graphs with olgnindependent edges 1 share many of the properties of truly random graphs. We also show a construction of computationally pseudorandom graphs that satisfy the 01 law of. Efficiently constructible huge graphs that preserve first. Testing kwise and almost kwise independence people. So we have shown that if we have a gthat is kunique, we have nkwise independent variables. Formally, when we are given a graph g and we say this is a random graph, we are wrong. The spectra of random matrices and random graphs have been extensively stud. Whp the maximum size of a clique or independent set in gn,12 is. 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. An introduction to random graphs, dependence graphs, and p.
On the probability of independent sets in random graphs. Hoeffdings inequality for sums of weakly dependent random. The concept of kwise independence was introduced by wegman and. However, unlike the case of k wise independent functions, the size of previously constructed families of such permutations is far from optimal. That is, we design a sampling algorithm for kwise independent graphs that supports. On the kindependence required by linear probing and minwise independence mihai p atra. Variables that are represented by neighboring vertices may be arbitrarily dependent, but collections of variables that form an independent set in g are t wise. This motivates studying which global attributes of random graphs are captured by their k wise independent counterparts. Such kwise independent graphs were used by naor, tromer and the second author 36 to ef. Pseudorandomness and combinatorial constructions cs286c.
The scheme is perfectly secure and very easy to implement. Simple constructions of almost kwise independent random. For example, if i have bits that are 4wise independent, can i say they are 3wise independent. May 31, 2008 in this note we prove a large deviation bound on the sum of random variables with the following dependency structure. Constructions of kwise almost independent permutations have been receiving a growing amount of attention in recent years. Universal hashing and kwise independent random variables via integer. On the kindependence required by linear probing and. It is also very simple to study these distributions in gnp,since the degree of each vertex is the sum of n1 independent random variables. It turns out that a random gis unique with high probability, so long as mis. The goal is to construct a short list of strings in. We study the kwise independent relaxation of the usual model gn,p of random graphs where, as in this model, n labeled vertices.
As our main result, we show that the problem is solvable with respect to some specially crafted kwise independent distribution over graphs. Simple constructions of almost k wise independent random variables noga alon oded goldreichy johan h astadz ren e peraltax february 22, 2002 abstract we present three alternative simple constructions of small proba. Before elaborating on properties of k wise independent graphs we provide some background on k wise independence, on properties of random graphs, and on the emulation of huge random graphs. A graph is said to be a k ramsey graph if it has no clique or independent set of size k. We study the kwise independent relaxation of the usual model gn,p of random graphs where, as in this model, n labeled vertices are fixed and each edge is drawn with probability p, however, it is only. Random graphs the markov random graphs of frank and strauss 1986 are a particular subclass of exponential random graph models in which a possible tie from i to j is assumed conditionally dependent3 only on other possible ties involving i andor j. Because of this key feature, kwise independent distributions are important concepts in probability, complexity, and algorithm design.
We study the kwise independent relaxation of the usual model gn,p of random graphs where, as in this model, n labeled vertices are fixed and each edge is drawn with probability p, however, it is only required that the distribution of any subset of k edges is independent. The simplest kwise independent hash function mapping positive integer x pdf download. This relaxation can be relevant in modeling phenomena where. We study the kwise independent relaxation of the usual model gn, p of random graphs where, as in this model, n labeled vertices are fixed and each edge is drawn with probability p, however, it is only required that the distribution of any subset of k edges is independent. We study the k wise independent relaxation of the usual model gn,p of random graphs where, as in this model, n labeled vertices are fixed and each edge is drawn with probability p, however, it. Lecture 1 kwise independence ubc computer science. So we have shown that if we have a gthat is kunique, we. Random variables that are fflbiased can be used to construct almost k wise independent random variables where ffl is a function of k. We also show a construction of computationally pseudo random graphs that satisfy the 01 law of random graphs note that in general, computational pseudorandomness does not imply such combinatorial properties. This paper gives a new method for reducing the size of families given by previous constructions. We study the k wise independent relaxation of the usual model gn, p of random graphs where, as in this model, n labeled vertices are fixed and each edge is drawn with probability p, however, it is only required that the distribution of any subset of k edges is independent. Randomized algorithms by rajeev motwani and prabhakar raghavan is a great source for the prerequisite probability theory background for this course expander graphs by shlomo hoory, nati linial, and avi wigderson. Constructions of almost kwise independent families of permutations, with optimal size up.
This is a brief overview of random graphs and the first and sec ond moment. We give lower bounds on the joint entropy of n pairwise independent random variables. The simplest k wise independent hash function mapping positive integer x random prime 2 611 will work and a i are some random positive integers less than p, a 0 0 2 wise independent hash. Kwise independent permutations, and uniformity for general group actions. Hoeffdings inequality, weakly dependent random variables, martin.
Simple constructions of almost k wise independent random variables. We extend it into a visual variant of the k out of n secret sharing problem, in which a dealer provides a transparency to each one of the n users. Our result applies to random uniform distributions over a large range of support sizes, and shows a tradeo. A probability distribution over 0,1n is kwise independent if its restriction to. Generating random graphs is an important method for investigating how likely or unlikely other network metrics are likely to occur given certain properties of the original graph. This relaxation can be relevant in modeling phenomena where only k wise independence is assumed to hold, and is.
Let y be a random variable counting the number of independent sets of size k 0 in. The stochastic block model and the expected degree distribution model fall within this framework. May 23, 2006 the scheme is perfectly secure and very easy to implement. By the induction hypothesis, tnfx0gare linearly independent, and they do not depend on r0, which x0does depend on. Random graphs alan frieze carnegie mellon university. The concept of kwise independence was introduced by wegman and carter 20.
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