Graph sparsification for derandomizing massively parallel computation with low space

Massively Parallel Computation (MPC) is an emerging model which distills core aspects of distributed and parallel computation. It was developed as a tool to solve (typically graph) problems in systems where input is distributed over many machines with limited space. Recent work has focused on the regime in which machines have sublinear (in n, number of nodes in the input graph) space, with randomized algorithms presented for the fundamental problems of Maximal Matching and Maximal Independent Set. There are, however, no prior corresponding deterministic algorithms. A major challenge in the sublinear space setting is that the local space of each machine may be too small to store all the edges incident to a single node. To overcome this barrier we introduce a new graph sparsification technique that deterministically computes a low-degree subgraph with additional desired properties: degrees in the subgraph are sufficiently small that nodes’ neighborhoods can be stored on single machines, and solving the problem on the subgraph provides significant global progress towards solving the problem for the original input graph. Using this framework to derandomize the well-known randomized algorithm of Luby [SICOMP’86], we obtain O(log(\Delta) + loglog(n))- round deterministic MPC algorithms for solving the fundamental problems of Maximal Matching and Maximal Independent Set with O(n epsilon) space on each machine for any constant epsilon > 0. Based on the recent work of Ghaffari et al. [FOCS’18], this additive O(loglog(n)) factor is conditionally essential. These algorithms can also be shown to run in O(log(\Delta)) rounds in the closely related model of CONGESTED CLIQUE, improving upon the state-of-the-art bound of O(log^2(\Delta)) rounds by Censor-Hillel et al. [DISC’17]

Distributed Symmetry Breaking on Power Graphs via Sparsification

In this paper, we present efficient distributed algorithms for classical symmetry breaking problems, maximal independent sets (MIS) and ruling sets, in power graphs. We work in the standard CONGEST model of distributed message passing, where the communication network is abstracted as a graph $G$. Typically, the problem instance in CONGEST is identical to the communication network $G$, that is, we perform the symmetry breaking in $G$. In this work, we consider a setting where the problem instance corresponds to a power graph $G^k$, where each node of the communication network $G$ is connected to all of its $k$-hop neighbors. Our main contribution is a deterministic polylogarithmic time algorithm for computing $k$-ruling sets of $G^k$, which (for $k>1$) improves exponentially on the current state-of-the-art runtimes. The main technical ingredient for this result is a deterministic sparsification procedure which may be of independent interest. On top of being a natural family of problems, ruling sets (in power graphs) are well-motivated through their applications in the powerful shattering framework [BEPS JACM'16, Ghaffari SODA'19] (and others). We present randomized algorithms for computing maximal independent sets and ruling sets of $G^k$ in essentially the same time as they can be computed in $G$. We also revisit the shattering algorithm for MIS [BEPS JACM'16] and present different approaches for the post-shattering phase. Our solutions are algorithmically and analytically simpler (also in the LOCAL model) than existing solutions and obtain the same runtime as [Ghaffari SODA'16]

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 (Δ + 1)-coloring in low-space MPC

We present a deterministic O(log log log n)-round low-space Massively Parallel Computation (MPC) algorithm for the classical problem of (Δ + 1)-coloring on n-vertex graphs. In this model, every machine has sublinear local space of size n^b for any arbitrary constant b ∈ (0, 1). Our algorithm works under the relaxed setting where each machine is allowed to perform exponential local computations, while respecting the n^b space and bandwidth limitations. Our key technical contribution is a novel derandomization of the ingenious (Δ + 1)-coloring local algorithm by Chang-Li-Pettie (STOC 2018, SIAM J. Comput. 2020). The Chang-Li-Pettie algorithm runs in T(n) = poly(log log n) rounds, which sets the state-of-the-art randomized round complexity for the problem in the local model. Our derandomization employs a combination of tools, notably pseudorandom generators (PRG) and bounded-independence hash functions. The achieved round complexity of O(log log log n) rounds matches the bound of log(T(n)), which currently serves an upper bound barrier for all known randomized algorithms for locally-checkable problems in this model. Furthermore, no deterministic sublogarithmic low-space MPC algorithms for the (Δ + 1)-coloring problem have been known before

Optimal (degree+1)-Coloring in Congested Clique

We consider the distributed complexity of the (degree+1)-list coloring problem, in which each node u of degree d(u) is assigned a palette of d(u) + 1 colors, and the goal is to find a proper coloring using these color palettes. The (degree+1)-list coloring problem is a natural generalization of the classical (Δ + 1)-coloring and (Δ + 1)-list coloring problems, both being benchmark problems extensively studied in distributed and parallel computing. In this paper we settle the complexity of the (degree+1)-list coloring problem in the Congested Clique model by showing that it can be solved deterministically in a constant number of rounds

Optimal (Degree+1)-Coloring in Congested Clique

We consider the distributed complexity of the (degree+1)-list coloring problem, in which each node u of degree d(u) is assigned a palette of d(u)+1 colors, and the goal is to find a proper coloring using these color palettes. The (degree+1)-list coloring problem is a natural generalization of the classical (?+1)-coloring and (?+1)-list coloring problems, both being benchmark problems extensively studied in distributed and parallel computing. In this paper we settle the complexity of the (degree+1)-list coloring problem in the Congested Clique model by showing that it can be solved deterministically in a constant number of rounds

Simple, deterministic, constant-round coloring in the congested clique

We settle the complexity of the (\Delta+1)-coloring and (\Delta+1)-list coloring problems in the CONGESTED CLIQUE model by presenting a simple deterministic algorithm for both problems running in a constant number of rounds. This matches the complexity of the recent breakthrough randomized constant-round (\Delta+1)-list coloring algorithm due to Chang et al. (PODC’19), and significantly improves upon the state-of-the-art O(log(\Delta))-round deterministic (\Delta+1)-coloring bound of Parter (ICALP’18). A remarkable property of our algorithm is its simplicity. Whereas the state-of-the-art randomized algorithms for this problem are based on the quite involved local coloring algorithm of Chang et al. (STOC’18), our algorithm can be described in just a few lines. At a high level, it applies a careful derandomization of a recursive procedure which partitions the nodes and their respective palettes into separate bins. We show that after O(1) recursion steps, the remaining uncolored subgraph within each bin has linear size, and thus can be solved locally by collecting it to a single node. This algorithm can also be implemented in the Massively Parallel Computation (MPC) model provided that each machine has linear (in n) the number of nodes in the input graph) space. We also show an extension of our algorithm to the MPC regime in which machines have sublinear space: we present the first deterministic (\Delta+1)-list coloring algorithm designed for sublinear-space MPC, which runs in O(log(\Delta + loglog(n)) rounds

Optimal (degree+1)-coloring in congested clique

We consider the distributed complexity of the (degree+1)-list coloring problem, in which each node u of degree d(u) is assigned a palette of d(u) + 1 colors, and the goal is to find a proper coloring using these color palettes. The (degree+1)-list coloring problem is a natural generalization of the classical (Δ + 1)-coloring and (Δ + 1)-list coloring problems, both being benchmark problems extensively studied in distributed and parallel computing. In this paper we settle the complexity of the (degree+1)-list coloring problem in the Congested Clique model by showing that it can be solved deterministically in a constant number of rounds