We consider the online stochastic matching problem proposed by Feldman et al. [4] as a model of display ad allocation. We are given a bipartite graph; one side of the graph corresponds to a fixed set of bins and the other side represents the set of possible ball types. At each time step, a ball is sampled independently from the given distribution and it needs to be matched upon its arrival to an empty bin. The goal is to maximize the size of the matching. We present an online algorithm for this problem with a competitive ratio of 0.702. Before our result, algorithms with a competitive ratio better than 1 − 1/e were known under the assumption that the expected number of arriving balls of each type is integral. A key idea of the algorithm is to collect statistics about the decisions of the optimum offline solution using Monte Carlo sampling and use those statistics to guide the decisions of the online algorithm. We also show that no online algorithm can have a competitive ratio better than 0.823. 1

The performance of a dynamic dictionary is measured mainly by its update time, lookup time, and space consumption. In terms of update time and lookup time there are known constructions that guarantee constant-time operations in the worst case with high probability, and in terms of space consumption there are known constructions that use essentially optimal space. In this paper we settle two fundamental open problems: • We construct the first dynamic dictionary that enjoys the best of both worlds: we present a two-level variant of cuckoo hashing that stores n elements using (1+ϵ)n memory words, and guarantees constant-time operations in the worst case with high probability. Specifically, for any ϵ = Ω((log log n / log n) 1/2) and for any sequence of polynomially many operations, with high probability over the randomness of the initialization phase, all operations are performed in constant time which is independent of ϵ. The construction is based on augmenting cuckoo hashing with a “backyard ” that handles a large fraction of the elements, together with a de-amortized perfect hashing scheme for eliminating the dependency on ϵ.

Let h> w> 0 be two fixed integers. Let H be a random hypergraph whose hyperedges are uniformly of size h. To w-orient a hyperedge, we assign exactly w of its vertices positive signs with respect to the hyperedge, and the rest negative. A (w, k)-orientation of H consists of a w-orientation of all hyperedges of H, such that each vertex receives at most k positive signs from its incident hyperedges. When k is large enough, we determine the threshold of the existence of a (w, k)-orientation of a random hypergraph. The (w, k)-orientation of hypergraphs is strongly related to a general version of the off-line load balancing problem. The graph case, when h = 2 and w = 1, was solved recently by Cain, Sanders and Wormald and independently by Fernholz and Ramachandran, thereby settling a conjecture made by Karp and Saks. Motivated by a problem of cuckoo hashing, the special hypergraph case with w = k = 1, was solved in three separate preprints dating from October 2009, by Frieze and Melsted, by Fountoulakis and

Let h> w> 0 be two fixed integers. Let H be a random hypergraph whose hyperedges are all of cardinality h. To w-orient a hyperedge, we assign exactly w of its vertices positive signs with respect to the hyperedge, and the rest negative. A (w, k)-orientation of H consists of a w-orientation of all hyperedges of H, such that each vertex receives at most k positive signs from its incident hyperedges. When k is large enough, we determine the threshold of the existence of a (w, k)-orientation of a random hypergraph. The (w, k)-orientation of hypergraphs is strongly related to a general version of the off-line load balancing problem. The graph case, when h = 2 and w = 1, was solved recently by Cain, Sanders and Wormald and independently by Fernholz and Ramachandran, which settled a conjecture of Karp and Saks. 1

This paper considers large scale distributed content service platforms, such as peer-to-peer video-on-demand systems. Such systems feature two basic resources, namely storage and bandwidth. Their efficiency critically depends on two factors: (i) content replication within servers, and (ii) how incoming service requests are matched to servers holding requestedcontent. Toinformthecorrespondingdesignchoices, we make the following contributions. Wefirstshowthat, for underloadedsystems, so-called proportional content placement with a simple greedy strategy formatchingrequeststoserversensuresfullsystemefficiency provided storage size grows logarithmically with the system size. However, for constant storage size, this strategy undergoes a phase transition with severe loss of efficiency as system load approaches criticality. To better understand the role of the matching strategy in this performance degradation, we characterize the asymptotic system efficiency under an optimal matching policy. Our analysis shows that –in contrast to greedy matching– optimal matching incurs an inefficiency that is exponentially small in the server storage size, even at critical system loads. It further allows a characterization of content replication policies that minimize the inefficiency. These optimal policies, which differ markedly from proportional placement, have a simple structure which makes them implementable in practice. On the methodological side, our analysis of matching performance uses the theory of local weak limits of random graphs, and highlights a novel characterization of matching numbers in bipartite graphs, which may both be of independent interest.

A h-uniform hypergraph H = (V,E) is called (ℓ,k)orientable if there exists an assignment of each hyperedge e ∈ E to exactly ℓ of its vertices v ∈ e such that no vertex is assigned more than k hyperedges. Let Hn,m,h be a hypergraph, drawn uniformly at random from the set of all h-uniform hypergraphs with n vertices and m edges. In this paper, we determine the threshold of the existence of a (ℓ,k)-orientation of Hn,m,h for k ≥ 1 and h> ℓ ≥ 1, extending recent results motivated by applications such as cuckoo hashing or load balancing with guaranteed maximum load. Our proof combines the local weak convergence of sparse graphs and a careful analysis of a Gibbs measure on spanning subgraphs with degree constraints. It allows us to deal with a much broader class than the uniform hypergraphs.

The XOR-satisfiability (XORSAT) problem requires finding an assignment of n Boolean variables that satisfymexclusiveOR(XOR)clauses, wherebyeachclause constrains a subset of the variables. We consider random XORSAT instances, drawn uniformly at random from the ensemble of formulae containing n variables and m clauses of size k. This model presents several structural similarities to other ensembles of constraint satisfaction problems, such as k-satisfiability (k-SAT). For many of these ensembles, as the number of constraints per variable grows, the set of solutions shatters into an exponential number of well-separated components. This phenomenon appears to be related to the difficulty of solving random instances of such problems. We prove a complete characterization of this clustering phase transition for random k-XORSAT. In particular we prove that the clustering threshold is sharp and determine its exact location. We prove that the set of solutions has large conductance below this threshold and that each of the clusters has large conductance above the same threshold. Our proof constructs a very sparse basis for the set of solutions (or the subset within a cluster). This construction is achieved through a low complexity iterative algorithm. 1

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Cuckoo hashing with a stash is a robust high-performance hashing scheme that can be used in many real-life applications. It complements cuckoo hashing by adding a small stash storing the elements that cannot fit into the main hash table due to collisions. However, the exact required size of the stash and the tradeoff between its size and the memory over-provisioning of the hash table are still unknown. We settle this question by investigating the equivalent maximum matching size of a random bipartite graph, with a constant left-side vertex degree d = 2. Specifically, we provide an exact expression for the expected maximum matching size and show that its actual size is close to its mean, with high probability. This result relies on decomposing the bipartite graph into connected components, and then separately evaluating the distribution of the matching size in each of these components. In particular, we provide an exact expression for any finite bipartite graph size and also deduce asymptotic results as the number of vertices goes to infinity. We also extend our analysis to cases where only part of the left-side vertices have a degree of 2; as well as to the case where the set of right-size vertices is partitioned into two subsets, and each

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