1. What is the concept of pair visibility in d-choice balanced allocation algorithms?
Pair visibility, denoted as vis(i, j), refers to the number of rounds t in which a pair of distinct vertices {i, j} is contained in an edge of the t-th hypergraph, H(t). This concept is introduced to study the effect of the d-choice paradigm in different environments, such as dynamic graph and hypergraph models. The visibility of a pair helps in understanding the allocation process and the distribution of balls in bins. It plays a crucial role in analyzing the performance and efficiency of d-choice balanced allocation algorithms.
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2. What is the balanced allocation algorithm on dynamic graphs?
The balanced allocation algorithm on dynamic graphs proceeds in rounds, where in each round, the t-th ball chooses an edge of G(t) uniformly at random, and the ball is then placed in one of the bins incident to the edge with a lesser load, with ties broken randomly. This algorithm is used for regular dynamic graphs, where G(t) is t-regular for some positive integer t and all t = 1, 2, . . ., n. The algorithm ensures that the load is balanced across the bins, preventing any single bin from becoming overloaded.
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3. What is (a, m)-uniform allocation?
(a, m)-uniform allocation is a concept introduced in [25, Definition 3]. It refers to an allocation algorithm A that sequentially allocates n balls into n bins according to a mechanism. For a given constant a > 0 and ohm(n) = m n, A is considered (a, m)-uniform if the probability of ball t being allocated to bin i, given the allocations of balls 1 to t-1 by A, is equal to a n. This definition conditionally considers the allocations made by A up to the (t-1)-th ball. Berenbrink et al. [5] proposed an allocation algorithm B, where the t-th ball randomly selects an edge of H (t) = ([n], E t ) uniformly, and is allocated to an empty vertex (bin) of H t, with ties broken randomly. If H t does not contain an empty bin, the process fails. Lemma 17 and Lemma 18 further explore the properties and implications of (a, m)-uniform allocation in dynamic hypergraphs.
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4. What is the recursive construction in the Witness Graph?
The recursive construction starts with a bin containing l + c balls. It recovers edges for balls placed in bin r with height at least c. The children of r are bins with loads at least l + c - 1, l + c - 2, ..., c. The number of vertices generated by the recursive construction is f(l) = 2^l. The recursive procedure produces distinct children for each vertex with load at least c. The maximum load after n balls are allocated is at most log(2 log n) + O(1/e) with high probability.
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