5 research outputs found
Deterministic parallel algorithms for bilinear objective functions
Many randomized algorithms can be derandomized efficiently using either the
method of conditional expectations or probability spaces with low independence.
A series of papers, beginning with work by Luby (1988), showed that in many
cases these techniques can be combined to give deterministic parallel (NC)
algorithms for a variety of combinatorial optimization problems, with low time-
and processor-complexity.
We extend and generalize a technique of Luby for efficiently handling
bilinear objective functions. One noteworthy application is an NC algorithm for
maximal independent set. On a graph with edges and vertices, this
takes time and processors, nearly
matching the best randomized parallel algorithms. Other applications include
reduced processor counts for algorithms of Berger (1997) for maximum acyclic
subgraph and Gale-Berlekamp switching games.
This bilinear factorization also gives better algorithms for problems
involving discrepancy. An important application of this is to automata-fooling
probability spaces, which are the basis of a notable derandomization technique
of Sivakumar (2002). Our method leads to large reduction in processor
complexity for a number of derandomization algorithms based on
automata-fooling, including set discrepancy and the Johnson-Lindenstrauss
Lemma
Derandomizing Local Distributed Algorithms under Bandwidth Restrictions
This paper addresses the cornerstone family of local problems in distributed computing, and investigates the curious gap between randomized and deterministic solutions under bandwidth restrictions.
Our main contribution is in providing tools for derandomizing solutions to local problems, when the n nodes can only send O(log n)-bit messages in each round of communication. We combine bounded independence, which we show to be sufficient for some algorithms, with the method of conditional expectations and with additional machinery, to obtain the following results.
First, we show that in the Congested Clique model, which allows all-to-all communication, there is a deterministic maximal independent set (MIS) algorithm that runs in O(log^2 Delta) rounds, where Delta is the maximum degree. When Delta=O(n^(1/3)), the bound improves to O(log Delta).
Adapting the above to the CONGEST model gives an O(D log^2 n)-round deterministic MIS algorithm, where D is the diameter of the graph. Apart from a previous unproven claim of a O(D log^3 n)-round algorithm, the only known deterministic solutions for the CONGEST model are a coloring-based O(Delta + log^* n)-round algorithm, where Delta is the maximal degree in the graph, and a 2^O(sqrt(log n log log n))-round algorithm, which is super-polylogarithmic in n.
In addition, we deterministically construct a (2k-1)-spanner with O(kn^(1+1/k) log n) edges in O(k log n) rounds in the Congested Clique model. For comparison, in the more stringent CONGEST model, where the communication graph is identical to the input graph, the best deterministic algorithm for constructing a (2k-1)-spanner with O(kn^(1+1/k)) edges runs in O(n^(1-1/k)) 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
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]