Abstract—We introduce a simple erasure recovery algorithm for codes derived from cascades of sparse bipartite graphs and analyze the algorithm by analyzing a corresponding discrete-time random process. As a result, we obtain a simple criterion involving the fractions of nodes of different degrees on both sides of the graph which is necessary and sufficient for the decoding process to finish successfully with high probability. By carefully designing these graphs we can construct for any given rate and any given real number a family of linear codes of rate which can be encoded in time proportional to ��@I A times their block length. Furthermore, a codeword can be recovered with high probability from a portion of its entries of length @IC A or more. The recovery algorithm also runs in time proportional to ��@I A. Our algorithms have been implemented and work well in practice; various implementation issues are discussed. Index Terms—Erasure channel, large deviation analysis, lowdensity parity-check codes. I.

Suppose that n balls are placed into n bins, each ball being placed into a bin chosen independently and uniformly at random. Then, with high probability, the maximum load in any bin is approximately log n log log n . Suppose instead that each ball is placed sequentially into the least full of d bins chosen independently and uniformly at random. It has recently been shown that the maximum load is then only log log n log d +O(1) with high probability. Thus giving each ball two choices instead of just one leads to an exponential improvement in the maximum load. This result demonstrates the power of two choices, and it has several applications to load balancing in distributed systems. In this thesis, we expand upon this result by examining related models and by developing techniques for stu...

The k -core of a graph is the largest subgraph with minimum degree at least k . For the Erdos-R'enyi random graph G(n; m) on n vertices, with m edges, it is known that a giant 2-core grows simultaneously with a giant component, that is when m is close to n=2 . We show that for k 3 , with high probability, a giant k -core appears suddenly when m reaches c k n=2 ; here c k = min ?0 = k () and k () = PfPoisson() k \Gamma 1g . In particular, c 3 3:35 . We also demonstrate that, unlike the 2-core, when a k -core appears for the first time it is very likely to be giant, of size p k ( k )n . Here k is the minimum point of = k () and p k ( k ) = PfPoisson( k ) kg . For k = 3 , for instance, the newborn 3-core contains about 0:27n vertices. Our proofs are based on the probabilistic analysis of an edge deletion algorithm that always finds a k -core if the graph has one. 1991 Mathematics Subject Classification. Primary 05C80, 05C85, 60C05; Secondary 60F10, 60G42, 60J10.

ITo motivate this survey, we begin with a simple problem that demonstrates a powerful fundamental idea. Suppose that n balls are thrown into n bins, with each ball choosing a bin independently and uniformly at random. Then the maximum load, or the largest number of balls in any bin, is approximately log n= log log n with high probability. Now suppose instead that the balls are placed sequentially, and each ball is placed in the least loaded of d 2 bins chosen independently and uniformly at random. Azar, Broder, Karlin, and Upfal showed that in this case, the maximum load is log log n= log d + (1) with high probability [ABKU99]. The important implication of this result is that even a small amount of choice can lead to drastically different results in load balancing. Indeed, having just two random choices (i.e.,...

We introduce a new set of probabilistic analysis tools based on the analysis of And-Or trees with random inputs. These tools provide a unifying, intuitive, and powerful framework for carrying out the analysis of several previously studied random processes of interest, including random loss-resilient codes, solving random k-SAT formula using the pure literal rule, and the greedy algorithm for matchings in random graphs. In addition, these tools allow generalizations of these problems not previously analyzed to be analyzed in a straightforward manner. We illustrate our methodology on the three problems listed above. 1 Introduction We introduce a new set of probabilistic analysis tools related to the amplification method introduced by [12] and further developed and used in [13, 5]. These tools provide a unifying, intuitive, and powerful framework for carrying out the analysis of several previously studied random processes of interest, including the random loss-resilient codes introduced ...

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It is well known that simple randomized load balancing schemes can balance load effectively while incurring only a small overhead, making such schemes appealing for practical systems. In this paper, we provide new analyses for several such dynamic randomized load balancing schemes. Our work extends a previous analysis of the supermarket model, a model that abstracts a simple, efficient load balancing scheme in the setting where jobs arrive at a large system of parallel processors. In this model, customers arrive at a system of n servers as a Poisson stream of rate #n, # < 1, with service requirements exponentially distributed with mean 1. Each customer chooses d servers independently and uniformly at random from the n servers, and is served according to the First In First Out (FIFO) protocol at the choice with the fewest customers. For the supermarket model, it has been shown that using d = 2 choices yields an exponential improvement in the expected time a customer spends in the syst...

We formulate some simple conditions under which a Markov chain may be approximated by the solution to a differential equation, with quantifiable error probabilities. The role of a choice of coordinate functions for the Markov chain is emphasised. The general theory is illustrated in three examples: the classical stochastic epidemic, a population process model with fast and slow variables, and core-finding algorithms for large random hypergraphs. 1

Let X be a set of n Boolean variables and denote by C(X) the set of all 3-clauses over X, i.e. the set of all 8 n 3 possible disjunctions of three distinct, non-complementary literals from variables in X. Let F (n; m) be a random 3-SAT formula formed by selecting, with replacement, m clauses uniformly at random from C(X) and taking their conjunction. The satisfiability threshold conjecture asserts that there exists a constant r3 such that as n !1, F (n; rn) is satis able with probability that tends to 1 if r r3 . Experimental evidence suggests r3 4:2. We prove r3 > 3:145 improving over the previous best lower bound r3 > 3:003 due to Frieze and Suen. For this, we introduce a satisfiability heuristic that works iteratively, permanently setting the value of a pair of variables in each round. The framework we...

We introduce a natural k-coloring algorithm and analyze its performance on random graphs with con-stant expected degree c (Gn,p=c/n). For k = 3 our re-sults imply that almost all graphs with n vertices and 1.923 n edges are 3-colorable. This improves the lower bound on the threshold for random 3-colorability significantly and settles the last case of Q long-standing open question of Bollobas [5]. We also provide a tight asymptotic analysis of the algorithm. We show that for all k 2 3, if c 5 klnk- 3/2k then the algorithm almost surely succeeds, while for any E> 0, and k sufficiently large, if c 2 (1 + E)k In k then the algorithm almost surely fails. The analysis is based on the use of differential equations to approximate the mean path of certain Markov chains.