Browsing by Author "Fischer, Manuela"

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  • Deterministic (1+)-approximate maximum matching with poly(1/) passes in the semi-streaming model and beyond
    (2022-09-06) Fischer, Manuela; Mitrović, Slobodan; Uitto, Jara
     A4 Artikkeli konferenssijulkaisussa
    We present a deterministic (1+ϵ)-approximate maximum matching algorithm in poly(1/ϵ) passes in the semi-streaming model, solving the long-standing open problem of breaking the exponential barrier in the dependence on 1/ϵ. Our algorithm exponentially improves on the well-known randomized (1/ϵ)O(1/ϵ)-pass algorithm from the seminal work by McGregor [APPROX05], the recent deterministic algorithm by Tirodkar with the same pass complexity [FSTTCS18]. Up to polynomial factors in 1/ϵ, our work matches the state-of-the-art deterministic (logn / loglogn) · (1/ϵ)-pass algorithm by Ahn and Guha [TOPC18], that is allowed a dependence on the number of nodes n. Our result also makes progress on the Open Problem 60 at sublinear.info. Moreover, we design a general framework that simulates our approach for the streaming setting in other models of computation. This framework requires access to an algorithm computing a maximal matching and an algorithm for processing disjoint ( 1 / ϵ)-size connected components. Instantiating our framework in CONGEST yields a (logn, 1/ϵ) round algorithm for computing (1+ϵ)-approximate maximum matching. In terms of the dependence on 1/ϵ, this result improves exponentially state-of-the-art result by Lotker, Patt-Shamir, and Pettie [LPSP15]. Our framework leads to the same quality of improvement in the context of the Massively Parallel Computation model as well.
  • Exponential Speedup over Locality in MPC with Optimal Memory
    (2022-10-17) Balliu, Alkida; Brandt, Sebastian; Fischer, Manuela; Latypov, Rustam; Maus, Yannic; Olivetti, Dennis; Uitto, Jara
     A4 Artikkeli konferenssijulkaisussa
    Locally Checkable Labeling (LCL) problems are graph problems in which a solution is correct if it satisfies some given constraints in the local neighborhood of each node. Example problems in this class include maximal matching, maximal independent set, and coloring problems. A successful line of research has been studying the complexities of LCL problems on paths/cycles, trees, and general graphs, providing many interesting results for the LOCAL model of distributed computing. In this work, we initiate the study of LCL problems in the low-space Massively Parallel Computation (MPC) model. In particular, on forests, we provide a method that, given the complexity of an LCL problem in the LOCAL model, automatically provides an exponentially faster algorithm for the low-space MPC setting that uses optimal global memory, that is, truly linear. While restricting to forests may seem to weaken the result, we emphasize that all known (conditional) lower bounds for the MPC setting are obtained by lifting lower bounds obtained in the distributed setting in tree-like networks (either forests or high girth graphs), and hence the problems that we study are challenging already on forests. Moreover, the most important technical feature of our algorithms is that they use optimal global memory, that is, memory linear in the number of edges of the graph. In contrast, most of the state-of-the-art algorithms use more than linear global memory. Further, they typically start with a dense graph, sparsify it, and then solve the problem on the residual graph, exploiting the relative increase in global memory. On forests, this is not possible, because the given graph is already as sparse as it can be, and using optimal memory requires new solutions.