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...ande, and Mehta [1]. The use of deterministic differential equations to model random processes was first studied by Kurtz, who gave a general purpose theorem for continuous-time jump Markov processes =-=[15]-=-. A discrete-time theorem tailored for random graphs was given by Wormald, which we use in this paper [20, 21]. The differential equation method has been used to study a variety of graph properties in...

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...ing. The ranking algorithm of Karp, Vazirani, and Vazirani picks a random permutation of bins up front and matches each ball to its unmatched neighboring bin that is ranked highest in the permutation =-=[13]-=-. ranking guarantees a matching size at least 1 − 1/e of the size of the maximum matching in expectation, which is the best possible worst-case guarantee. ∗Research supported in part by NSF grant 1029...

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...tudied by Kurtz, who gave a general purpose theorem for continuous-time jump Markov processes [15]. A discrete-time theorem tailored for random graphs was given by Wormald, which we use in this paper =-=[20, 21]-=-. The differential equation method has been used to study a variety of graph properties including k-cores, independent sets, and greedy packing on hypergraphs [18, 20, 21]. It was also used to analyze...

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...as received significant attention due to applications in Internet advertising as well as under streaming models of computation, which places limits on memory utilization for processing large datasets =-=[16, 19, 9]-=- . From a worst-case perspective, it is well known that the greedy algorithm, which matches each ball to a random unmatched neighboring bin (if possible), always achieves a matching size that is at le...

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...mald, which we use in this paper [20, 21]. The differential equation method has been used to study a variety of graph properties including k-cores, independent sets, and greedy packing on hypergraphs =-=[18, 20, 21]-=-. It was also used to analyze a load balancing scenario similar to ours by Mitzenmacher [17]. Early studies of matchings on random graphs focused on determining the existence of perfect matchings; Erd...

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...twell [4]. Their result (specifically Theorem 14 in [4]) is stated in terms of the size of the largest independent set for a bipartite graph, which by König’s theorem bounds the maximum matching size =-=[14]-=-. Theorem 6. (Bollobás and Brightwell [4]) Let µ ∗ (n, n, c/n) denote the size of the maximum matching on the graph G(n, n, c/n) where p = c/n. Then a.a.s., µ ∗ (n, n, c/n) n ≤ 2 − γ∗ + γ∗ + γ ∗ γ∗ c ...

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...as received significant attention due to applications in Internet advertising as well as under streaming models of computation, which places limits on memory utilization for processing large datasets =-=[16, 19, 9]-=- . From a worst-case perspective, it is well known that the greedy algorithm, which matches each ball to a random unmatched neighboring bin (if possible), always achieves a matching size that is at le...

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... 8]. One of the first studies of greedy matchings on random graphs, as well as the first to employ the differential equation method (specifically via Kurtz’s theorem), was the work of Karp and Sipser =-=[12, 15]-=-. They considered ordinary sparse graphs,specifically the G(n, p) model 1 with p = c/n. The Karp-Sipser algorithm first matches all vertices with degree one until there are no such vertices remaining...

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.... This results in a matching size that is within 1 − o(1) of the maximum matching. The algorithm was studied more in depth by Aronson, Frieze, and Pittel, who gave sharper error bounds on performance =-=[2]-=-. Simpler greedy matching algorithms for ordinary graphs were studied by Dyer, Frieze, and Pittel [6]. Again for the G(n, p) model with p = c/n, they looked at the greedy algorithm which picks random ...

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...yzed by Karp, Vazirani and Vazirani, who introduced the ranking algorithm and showed that it obtains a competitive ratio of 1 − 1/e [13]. Simpler proofs of the ranking algorithm have since been found =-=[5, 3]-=-. A 1 − 1/e competitive algorithm is also known for vertex-weighted online bipartite matching, which was given by Aggarwal, Goel, Karande, and Mehta [1]. The use of deterministic differential equation...

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... variety of graph properties including k-cores, independent sets, and greedy packing on hypergraphs [18, 20, 21]. It was also used to analyze a load balancing scenario similar to ours by Mitzenmacher =-=[17]-=-. Early studies of matchings on random graphs focused on determining the existence of perfect matchings; Erdős and Rényi showed that the threshold for the existence of a perfect matching occurs when t...

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...as received significant attention due to applications in Internet advertising as well as under streaming models of computation, which places limits on memory utilization for processing large datasets =-=[16, 19, 9]-=- . From a worst-case perspective, it is well known that the greedy algorithm, which matches each ball to a random unmatched neighboring bin (if possible), always achieves a matching size that is at le...

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... the ranking algorithm have since been found [5, 3]. A 1 − 1/e competitive algorithm is also known for vertex-weighted online bipartite matching, which was given by Aggarwal, Goel, Karande, and Mehta =-=[1]-=-. The use of deterministic differential equations to model random processes was first studied by Kurtz, who gave a general purpose theorem for continuous-time jump Markov processes [15]. A discrete-ti...

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...tence of isolated vertices in our analysis, which is unlikely to be realistic for many applications. This could be resolved by imposing a restriction on the minimum degree of vertices, as was done in =-=[11]-=-. Unbalanced bipartite graphs (i.e. with more vertices on one side) are likely to be encountered in practice – our approach can be used in this situation, but less is known about expected maximum matc...

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...studied more in depth by Aronson, Frieze, and Pittel, who gave sharper error bounds on performance [2]. Simpler greedy matching algorithms for ordinary graphs were studied by Dyer, Frieze, and Pittel =-=[6]-=-. Again for the G(n, p) model with p = c/n, they looked at the greedy algorithm which picks random edges, as well as what they refer to as modified greedy, which first picks a random vertex and then s...

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...ld [21]. We believe that this is one of the most simple applications of Wormald’s theorem. In fact, our proofs are nearly as simple as the worst-case proof for ranking of Devanur, Jain, and Kleinberg =-=[5]-=-. In the remainder of this section we briefly state our main results for the oblivious and greedy algorithms and discuss related work. The analysis of oblivious and greedy is given in Section 2. We ex...

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...in e−cw ( 1 − c2 ) w n ≤ ( 1 − c ) nw n ≤ e−cw . The result follows by rearranging terms and using cwe−cw ≤ 1/e. □ For bounding the maximum matching size, we use a result from Bollobás and Brightwell =-=[4]-=-. Their result (specifically Theorem 14 in [4]) is stated in terms of the size of the largest independent set for a bipartite graph, which by König’s theorem bounds the maximum matching size [14]. The...

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...ed on determining the existence of perfect matchings; Erdős and Rényi showed that the threshold for the existence of a perfect matching occurs when the graph is likely to have a minimum degree of one =-=[7, 8]-=-. One of the first studies of greedy matchings on random graphs, as well as the first to employ the differential equation method (specifically via Kurtz’s theorem), was the work of Karp and Sipser [12...

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...e same can be said regarding a result given in [2]. A similar observation was also made by Frieze with respect to differential equations for the Karp-Sipser algorithm on bipartite and ordinary graphs =-=[10]-=-. Note that it is not sufficient to simply argue that these properties hold for the mere fact that bipartite graphs have twice as many vertices; there are indeed important structural differences betwe...