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Perfectly balanced allocation
 in Proceedings of the 7th International Workshop on Randomization and Approximation Techniques in Computer Science, Princeton, NJ, 2003, Lecture Notes in Comput. Sci. 2764
, 2003
"... Abstract. We investigate randomized processes underlying load balancing based on the multiplechoice paradigm: m balls have to be placed in n bins, and each ball can be placed into one out of 2 randomly selected bins. The aim is to distribute the balls as evenly as possible among the bins. Previousl ..."
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Abstract. We investigate randomized processes underlying load balancing based on the multiplechoice paradigm: m balls have to be placed in n bins, and each ball can be placed into one out of 2 randomly selected bins. The aim is to distribute the balls as evenly as possible among the bins. Previously, it was known that a simple process that places the balls one by one in the least loaded bin can achieve a maximum load of m/n + Θ(log log n) with high probability. Furthermore, it was known that it is possible to achieve (with high probability) a maximum load of at most ⌈m/n ⌉ +1using maximum flow computations. In this paper, we extend these results in several aspects. First of all, we show that if m ≥ cn log n for some sufficiently large c, thenaperfect distribution of balls among the bins can be achieved (i.e., the maximum load is ⌈m/n⌉) with high probability. The bound for m is essentially optimal, because it is known that if m ≤ c ′ n log n for some sufficiently small constant c ′ , the best possible maximum load that can be achieved is ⌈m/n ⌉ +1with high probability. Next, we analyze a simple, randomized load balancing process based on a local search paradigm. Our first result here is that this process always converges to a best possible load distribution. Then, we study the convergence speed of the process. We show that if m is sufficiently large compared to n,thenno matter with which ball distribution the system starts, if the imbalance is ∆, then the process needs only ∆·n O(1) steps to reach a perfect distribution, with high probability. We also prove a similar result for m ≈ n, and show that if m = O(n log n / log log n), then an optimal load distribution (which has the maximum load of ⌈m/n ⌉ +1) is reached by the random process after a polynomial number of steps, with high probability.
The korientability thresholds for Gn,p
 In SODA ’07: Proceedings of the eighteenth annual ACMSIAM symposium on Discrete algorithms
, 2007
"... We prove that, for k ≥ 2, the korientability threshold for the random graph Gn,p coincides with the threshold at which the (k + 1)core has average degree 2k. The proof involves the analysis of a heuristic algorithm that attempts to find a korientation of the random graph. The korientation thresh ..."
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We prove that, for k ≥ 2, the korientability threshold for the random graph Gn,p coincides with the threshold at which the (k + 1)core has average degree 2k. The proof involves the analysis of a heuristic algorithm that attempts to find a korientation of the random graph. The korientation threshold has several applications including offline balanced allocation with a limit of k on maximum binsize, perfect hashing with a limit of k on maximum chainlength, and concurrent access to parallel memories through redundancy, 1
and Blocked Cuckoo Hashing
, 2005
"... We study a particular aspect of the balanced allocation paradigm (also known as the “twochoices paradigm”): constant sized bins, packed as tightly as possible. Let d ≥ 1 be fixed, and assume there are m bins of capacity d each. To each of n ≤ dm balls two possible bins are assigned at random. How cl ..."
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We study a particular aspect of the balanced allocation paradigm (also known as the “twochoices paradigm”): constant sized bins, packed as tightly as possible. Let d ≥ 1 be fixed, and assume there are m bins of capacity d each. To each of n ≤ dm balls two possible bins are assigned at random. How close can dm/n = 1 + ε be to 1 so that with high probability each ball can be put into one of the two bins assigned to it, without any bin overflowing? We show that ε> (2/e) d−1 is sufficient. If a new ball arrives with two new randomly assigned bins, we wish to rearrange some of the balls already present in order to accommodate the new ball. We show that on average it takes constant time to rearrange the balls to achieve this, for ε> γ · β d, for some constants γ> 0, β < 1. An alternative way to describe the problem is in data structure language. Generalizing cuckoo hashing (Pagh and Rodler, 2001), we consider a hash table with m positions, each representing a bucket of capacity d ≥ 1. Keys are assigned to buckets by two fully random hash functions. How many keys can be placed in these bins, if key x may go to bin h1(x) or to bin h2(x)? Our results lead to an implementation of a dynamic dictionary that accommodates n keys in m = (1 + ε)n/d buckets of size d = O(log(1/ε)), so that key x resides in bucket h1(x) or h2(x). If d ≥ 1 + 3.26 · ln(1/ε), then for a lookup operation only two hash functions have to be evaluated and two contiguous segments of d memory cells have to be inspected. The expected time for inserting a new key is constant, for some d = O(log(1/ε)).