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.
In the parallel setting, @cite_12 showed that a constant bound on the gap is possible with @math communication rounds. Adler et.al. @cite_22 consider parallel balls and bins with multiple rounds. They present analysis for @math bound on the gap (for @math ) using @math rounds of communication.
@cite_22 @cite_12
1985526537 2131709403 1885032915 2478193891
We prove an optimal @math lower bound on the randomized communication complexity of the much-studied gap-hamming-distance problem. As a consequence, we obtain essentially optimal multipass space lower bounds in the data stream model for a number of fundamental problems, including the estimation of frequency moments. The gap-hamming-distance problem is a communication problem, wherein Alice and Bob receive @math -bit strings @math and @math , respectively. They are promised that the Hamming distance between @math and @math is either at least @math or at most @math , and their goal is to decide which of these is the case. Since the formal presentation of the problem by Indyk and Woodruff [Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science, 2003, pp. 283--289], it had been conjectured that the naive protocol, which uses @math bits of communication, is asymptotically optimal. The conjecture was shown to be true in several special cases, e.g., when the communication is de... The Gap-Hamming-Distance problem arose in the context of proving space lower bounds for a number of key problems in the data stream model. In this problem, Alice and Bob have to decide whether the Hamming distance between their @math -bit input strings is large (i.e., at least @math ) or small (i.e., at most @math ); they do not care if it is neither large nor small. This @math gap in the problem specification is crucial for capturing the approximation allowed to a data stream algorithm. Thus far, for randomized communication, an @math lower bound on this problem was known only in the one-way setting. We prove an @math lower bound for randomized protocols that use any constant number of rounds. As a consequence we conclude, for instance, that @math -approximately counting the number of distinct elements in a data stream requires @math space, even with multiple (a constant number of) passes over the input stream. This extends earlier one-pass lower bounds, answering a long-standing open question. We obtain similar results for approximating the frequency moments and for approximating the empirical entropy of a data stream. In the process, we also obtain tight @math lower and upper bounds on the one-way deterministic communication complexity of the problem. Finally, we give a simple combinatorial proof of an @math lower bound on the one-way randomized communication complexity. Multiround algorithms are now commonly used in distributed data processing systems, yet the extent to which algorithms can benefit from running more rounds is not well understood. This paper answers this question for a spectrum of rounds for the problem of computing the equijoin of @math relations. Specifically, given any query @math with width @math , intersection width @math , input size @math , output size @math , and a cluster of machines with @math memory available per machine, we show that: (1) @math can be computed in @math rounds with @math communication cost. (2) @math can be computed in @math rounds with @math communication cost. Intersection width is a new notion of queries and generalized hypertree decompositions (GHDs) of queries we introduce to capture how connected the adjacent cyclic components of the GHDs are. We achieve our first result by introducing a distributed and generalized version of Yannakakis's algorithm, called GYM. GYM takes as input any GHD of @math with width @math and depth @math , and computes @math in @math rounds and @math communication cost. We achieve our second result by showing how to construct GHDs of @math with width @math and depth @math . We describe another technique to construct GHDs with longer widths and shorter depths, demonstrating a spectrum of tradeoffs one can make between communication and the number of rounds. This paper addresses the cornerstone family of in distributed computing, and investigates the curious gap between randomized and deterministic solutions under bandwidth restrictions. Our main contribution is in providing tools for derandomizing solutions to local problems, when the @math nodes can only send @math -bit messages in each round of communication. We combine bounded independence, which we show to be sufficient for some algorithms, with the method of conditional expectations and with additional machinery, to obtain the following results. Our techniques give a deterministic maximal independent set (MIS) algorithm in the CONGEST model, where the communication graph is identical to the input graph, in @math rounds, where @math is the diameter of the graph. The best known running time in terms of @math alone is @math , which is super-polylogarithmic, and requires large messages. For the CONGEST model, the only known previous solution is a coloring-based @math -round algorithm, where @math is the maximal degree in the graph. On the way to obtaining the above, we show that in the model, which allows all-to-all communication, there is a deterministic MIS algorithm that runs in @math rounds. , where @math is the maximum degree. When @math , the bound improves to @math and holds also for @math -coloring. In addition, we deterministically construct a @math -spanner with @math edges in @math rounds. For comparison, in the more stringent CONGEST model, the best deterministic algorithm for constructing a @math -spanner with @math edges runs in @math rounds.