Brief Announcement

Abstract

We give a randomness-efficient Massively Parallel Computation (MPC) algorithm for deciding whether an undirected graph is connected. For Connectivity on n-vertex, m-edge graphs whose components have diameter at most D = 2 o(log n/log log n) , our algorithm runs in R = O(log D + log log m/n n) rounds and uses a total of (log n) O (R) random bits, O(m) machines, and n 1−Ω(1) space per machine with good probability. 1 Our algorithm achieves a superpolynomial saving in randomness complexity as compared to the breakthrough algorithm of Andoni et al. (FOCS '18) and the subsequent improvement by Behnezhad et al. (FOCS '19). Our algorithm has the same round complexity as that of Behnezhad et al., but uses more total space.Our Connectivity algorithm is an instantiation of a general method we develop for converting randomized algorithms in the PRAM model to highly randomness-efficient MPC algorithms. We show that for k = o(log n/log log n) and p = n O (1) , any time-k pprocessor randomized PRAM algorithm computing a function on n input bits can be converted to an equivalent strongly sublinear MPC algorithm with O(k) rounds and only a total of (log n) O (k ) random bits. Our Connectivity algorithm follows from applying this method to the recent CRCW PRAM algorithm of Liu, Tarjan, and Zhong (SPAA '20).Our approach is based on the design of a pseudorandom generator for PRAM algorithms. The analysis of our generator is built on classic and influential results in circuit complexity (Håstad '86; Nisan and Wigderson '88), which we generalize from the setting of small-depth circuits to the more powerful setting of PRAM algorithms. The parameters that we achieve are optimal given the current state of the art in complexity theory, in the sense that further improvements will imply P NC 1 . CCS CONCEPTS• Mathematics of computing → Graph algorithms; • Theory of computation → Massively parallel algorithms; Pseudorandomness and derandomization.1 With good probability means with probability at least 1 − 1/poly((m log n)/n), which is the same as in Liu, Tarjan, and Zhong (SPAA '20).

Deterministic Massively Parallel Connectivity

“…All these results are randomized and we are not aware of any prior 𝑜 (log 𝑛)-rounds deterministic MPC algorithm in the low local space regime. Still, a recent related work [22] shows that one can slightly reduce the randomness used in [13]: (log 𝑛) 𝑂 (log 𝐷+log log 𝑚/𝑛 𝑛) random bits suffice to obtain an MPC algorithm with local space S = 𝑂 (𝑛 𝛿 ) and M = 𝑂 (𝑛 + 𝑚) machines (and so with global space 𝑂 ((𝑛 + 𝑚) • 𝑛 𝛿 ), which is larger than the linear global space 𝑂 (𝑛 + 𝑚) used in [13]) that determines graph connectivity in 𝑂 (log 𝐷 + log log 𝑚/𝑛 𝑛) rounds with probability 1 − 1/poly((𝑚 log 𝑛)/𝑛).…”