Balanced allocation of online balls-into-bins has long been an active area of research for efficient load balancing and hashing applications.There exists a large number of results in this domain for different settings, such as parallel allocations parallel , multi-dimensional allocations multi , weighted balls weight etc. For sequential multi-choice allocation, where @math balls are thrown into @math bins with each ball choosing @math (constant) bins independently uniformly at random, the maximum load of a bin is @math with high probability heavily_load . This offers the current best known allocation scheme. However, for @math , the gap reduces to @math soda08 .A similar constant gap bound has been established for parallel allocations with @math communication rounds lenzen . In this paper we propose a novel multi-choice allocation algorithm, ( @math ) achieving a constant gap with a high probability for the sequential single-dimensional online allocation problem with constant @math . We achieve a maximum load of @math with high probability for constant @math choice scheme with constant number of retries or rounds per ball. We also show that the bound holds even for an arbitrary large number of balls, @math . Further, we generalize this result to (i) the weighted case, where balls have weights drawn from an arbitrary weight distribution with finite variance, (ii) multi-dimensional setting, where balls have @math dimensions with @math randomly and uniformly chosen filled dimension for @math , and (iii) the parallel case, where @math balls arrive and are placed parallely in the bins. We show that the gap in these case is also a constant w.h.p. (independent of @math ) for constant value of @math with expected constant number of retries per ball.
Kunal et.al. @cite_21 prove that for weighted balls (weight distribution with finite fourth moment) and @math , the expected gap is independent of the number of balls and is less than @math , where @math depends on the weight distribution. They first prove the weak gap theorem which says that w.h.p @math . Since in the weighted case the @math choice process is not dominated by the one choice process, they prove the weak gap theorem via a potential function argument. Then, the is proved. While in @cite_10 the short memory theorem is proven via coupling, @cite_21 uses similar coupling arguments but defines a different distance function and use a sophisticated argument to show that the coupling converges.
@cite_21 @cite_10
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We investigate balls-and-bins processes where m weighted balls areplaced into n bins using the "power of two choices" paradigm,whereby a ball is inserted into the less loaded of two randomly chosen bins. The case where each of the m balls has unit weight had been studied extensively. In a seminal paper Azar et.al. showed that when m=n the most loaded bin has Θ(log log n) balls with high probability. Surprisingly, thegap in load between the heaviest bin and the average bin does not increase with m and was shown by tobe Θ(log log n) with high probability for arbitrarily large m. We generalize this result to the weighted case where balls have weights drawn from an arbitrary weight distribution. We show that aslong as the weight distribution has finite second moment andsatisfies a mild technical condition, the gap between the weight of the heaviest bin and the weight of the average bin is independent ofthe number balls thrown. This is especially striking whenconsidering heavy tailed distributions such as Power-Law andLog-Normal distributions. In these cases, as more balls are thrown,heavier and heavier weights are encountered. Nevertheless with high probability, the imbalance in the load distribution does notincrease. Furthermore, if the fourth moment of the weight distribution is finite, the expected value of the gap is shown to beindependent of the number of balls. Suppose m balls are sequentially thrown into n bins where each ball goes into a random bin. It is well-known that the gap between the load of the most loaded bin and the average is Θ (√mlog n n), for large m. If each ball goes to the lesser loaded of two random bins, this gap dramatically reduces to Θ (log log n) independent of m. Consider now the following "(1 + β)-choice" process for some parameter β ∈ (0, 1): each ball goes to a random bin with probability (1 - β) and the lesser loaded of two random bins with probability β. How does the gap for such a process behave? Suppose that the weight of each ball was drawn from a geometric distribution. How is the gap (now defined in terms of weight) affected? In this work, we develop general techniques for analyzing such balls-into-bins processes. Specifically, we show that for the (1 + β)-choice process above, the gap is Θ(log n β), irrespective of m. Moreover the gap stays at Θ(log n β) in the weighted case for a large class of weight distributions. No non-trivial explicit bounds were previously known in the weighted case, even for the 2-choice paradigm. A proof is provided of a strong law of large numbers for a one-dimensional random walk in a dynamic random environment given by a supercritical contact process in equilibrium. The proof is based on a coupling argument that traces the space-time cones containing the infection clusters generated by single infections and uses that the random walk eventually gets trapped inside the union of these cones. For the case where the local drifts of the random walk are smaller than the speed at which infection clusters grow, the random walk eventually gets trapped inside a single cone. This in turn leads to the existence of regeneration times at which the random walk forgets its past. The latter are used to prove a functional central limit theorem and a large deviation principle. The qualitative dependence of the speed, the volatility and the rate function on the infection parameter is investigated, and some open problems are mentioned. We prove a quantitative version of the Gibbard-Satterthwaite theorem. We show that a uniformly chosen voter profile for a neutral social choice function f of q ≥ 4 alternatives and n voters will be manipulable with probability at least 10−4∈2 n −3 q −30, where ∈ is the minimal statistical distance between f and the family of dictator functions. Our results extend those of [11], which were obtained for the case of 3 alternatives, and imply that the approach of masking manipulations behind computational hardness (as considered in [4,6,9,15,7]) cannot hide manipulations completely. Our proof is geometric. More specifically it extends the method of canonical paths to show that the measure of the profiles that lie on the interface of 3 or more outcomes is large. To the best of our knowledge our result is the first isoperimetric result to establish interface of more than two bodies.