When Stuck, Flip a Coin

Many modern services need to routinely perform tasks on a large scale. This prompts us to consider the following question:

How can we design efficient algorithms for large-scale computation?

In this thesis, we focus on devising a general strategy to address the above question. Our approaches use tools from graph theory and convex optimization, and prove to be very effective on a number of problems that exhibit locality. A recurring theme in our work is to use randomization to obtain simple and practical algorithms.

The techniques we developed enabled us to make progress on the following questions:

• Parallel Computation of Approximately Maximum Matchings. We put forth a new approach to computing $O(1)$-approximate maximum matchings in the Massively Parallel Computation (MPC) model. In the regime in which the memory per machine is $\Theta(n)$, i.e., linear in the size of the vertex-set, our algorithm requires only $O((\log \log{n})^2)$ rounds of computations. This is an almost exponential improvement over the barrier of $\Omega(\log {n})$ rounds that all the previous results required in this regime.

• Parallel Computation of Maximal Independent Sets. We propose a simple randomized algorithm that constructs maximal independent sets in the MPC model. If the memory per machine is $\Theta(n)$ our algorithm runs in $O(\log \log{n})$ MPC-rounds. In the same regime, all the previously known algorithms required $O(\log{n})$ rounds of computation.

• Network Routing under Link Failures. We design a new protocol for stateless message-routing in $k$-connected graphs. Our routing scheme has two important features: (1) each router performs the routing decisions based only on the local information available to it; and, (2) a message is delivered successfully even if arbitrary $k-1$ links have failed. This significantly improves upon the previous work of which the routing schemes tolerate only up to $k/2 - 1$ failed links in $k$-connected graphs.

• Streaming Submodular Maximization under Element Removals. We study the problem of maximizing submodular functions subject to cardinality constraint $k$, in the context of streaming algorithms. In a regime in which up to $m$ elements can be removed from the stream, we design an algorithm that provides a constant-factor approximation for this problem. At the same time, the algorithm stores only $O(k \log^2{k} + m \log^3{k})$ elements. Our algorithm improves quadratically upon the prior work, that requires storing $O(k \cdot m)$ many elements to solve the same problem.

• Fast Recovery for the Separated Sparsity Model. In the context of compressed sensing, we put forth two recovery algorithms of nearly-linear time for the separated sparsity signals (that naturally model neural spikes). This improves upon the previous algorithm that had a quadratic running time. We also derive a refined version of the natural dynamic programming (DP) approach to the recovery of the separated sparsity signals. This DP approach leads to a recovery algorithm that runs in linear time for an important class of separated sparsity signals. Finally, we consider a generalization of these signals into two dimensions, and we show that computing an exact projection for the two-dimensional model is NP-hard.