Improved Deterministic Connectivity in Massively Parallel Computation

A long line of research about connectivity in the Massively Parallel Computation model has culminated in the seminal works of Andoni et al. [FOCS\u2718] and Behnezhad et al. [FOCS\u2719]. They provide a randomized algorithm for low-space MPC with conjectured to be optimal round complexity O(log D + log log_{m/n} n) and O(m) space, for graphs on n vertices with m edges and diameter D. Surprisingly, a recent result of Coy and Czumaj [STOC\u2722] shows how to achieve the same deterministically. Unfortunately, however, their algorithm suffers from large local computation time. We present a deterministic connectivity algorithm that matches all the parameters of the randomized algorithm and, in addition, significantly reduces the local computation time to nearly linear. Our derandomization method is based on reducing the amount of randomness needed to allow for a simpler efficient search. While similar randomness reduction approaches have been used before, our result is not only strikingly simpler, but it is the first to have efficient local computation. This is why we believe it to serve as a starting point for the systematic development of computation-efficient derandomization approaches in low-memory MPC

Improved Deterministic Connectivity in Massively Parallel Computation

A long line of research about connectivity in the Massively Parallel Computation model has culminated in the seminal works of Andoni et al. [FOCS'18] and Behnezhad et al. [FOCS'19]. They provide a randomized algorithm for low-space MPC with conjectured to be optimal round complexity $O(\log D + \log \log_{\frac m n} n)$ and $O(m)$ space, for graphs on $n$ vertices with $m$ edges and diameter $D$. Surprisingly, a recent result of Coy and Czumaj [STOC'22] shows how to achieve the same deterministically. Unfortunately, however, their algorithm suffers from large local computation time. We present a deterministic connectivity algorithm that matches all the parameters of the randomized algorithm and, in addition, significantly reduces the local computation time to nearly linear. Our derandomization method is based on reducing the amount of randomness needed to allow for a simpler efficient search. While similar randomness reduction approaches have been used before, our result is not only strikingly simpler, but it is the first to have efficient local computation. This is why we believe it to serve as a starting point for the systematic development of computation-efficient derandomization approaches in low-memory MPC

Deterministic Massively Parallel Symmetry Breaking for Sparse Graphs

We consider the problem of designing deterministic graph algorithms for the model of Massively Parallel Computation (MPC) that improve with the sparsity of the input graph, as measured by the standard notion of arboricity. For the problems of maximal independent set (MIS), maximal matching (MM), and vertex coloring, we improve the state of the art as follows. Let λ denote the arboricity of the n-node input graph with maximum degree δ. MIS and MM We develop a low-space MPC algorithm that deterministically reduces the maximum degree to poly(λ) in O(log log n) rounds, improving and simplifying the randomized O(log log n)-round poly(max(λ, log n))-degree reduction of Ghaffari, Grunau, Jin [DISC'20]. Our approach when combined with the state-of-the-art O(log Δ+ log log n)-round algorithm by Czumaj, Davies, Parter [SPAA'20, TALG'21] leads to an improved deterministic round complexity of O(log λ + log log n). The above MIS and MM algorithm however works in the setting where the global memory allowed, i.e., the number of machines times the local memory per machine, is superlinear in the input size. We extend them to obtain the first low-space MIS and MM algorithms that work with linear global memory. Specifically, we show that both problems can be solved in deterministic time O(log λ · log logλ n), and even in O(log log n) time for graphs with arboricity at most logO(1) log n. In this setting, only a O(log2 log n)-running time bound for trees was known due to Latypov and Uitto [ArXiv'21]. Vertex Coloring We present a O(1)-round deterministic algorithm for the problem of O(λ)-coloring in the linear-memory regime of MPC, with relaxed global memory of n · poly(λ). This matches the round complexity of the state-of-the-art randomized algorithm by Ghaffari and Sayyadi [ICALP'19] and significantly improves upon the deterministic O(λϵ )-round algorithm by Barenboim and Khazanov [CSR'18]. Our algorithm solves the problem after just one single graph partitioning step, in contrast to the involved local coloring simulations of the above state-of-the-art algorithms. Using O(n + m) global memory, we derive a O(log λ)-round algorithm by combining the constant-round (Δ+ 1)-list-coloring algorithm by Czumaj, Davies, Parter [PODC'20, SIAM J. Comput.'21] with that of Barenboim and Khazanov