The longest common subsequence (LCS) problem is one of the fundamental problems in string algorithms. A constrained version, the repetitionfree longest common subsequence (RFLCS) problem, additionally requires that each character may appear at most once in the solution sequence. In sharp contrast to LCS, RFLCS is NP-hard and even APX-hard. Previous work has shown that multivalued decision diagrams (MDDs) provide an effective tool for solving the RFLCS problem to optimality.

Expander decompositions of graphs have significantly advanced the understanding of many classical graph problems and led to numerous fundamental theoretical results. However, their adoption in practice has been hindered due to their inherent intricacies and large hidden factors in their asymptotic running times. Here, we introduce the first practically efficient algorithm for computing expander decompositions and their hierarchies and demonstrate its effectiveness and utility by incorporating it as the core component in a novel solver for the normalized cut graph clustering objective.

Clustering is a fundamental problem in unsupervised machine learning with many applications in data analysis. Popular clustering algorithms such as Lloyd's algorithm and $k$-means++ can take $\Omega(ndk)$ time when clustering $n$ points in a $d$-dimensional space (represented by an $n\times d$ matrix $X$) into $k$ clusters. On massive datasets with moderate to large $k$, the multiplicative $k$ factor can become very expensive. We introduce a simple randomized clustering algorithm that provably runs in expected time $O(\mathsf{nnz}(X) + n\log n)$ for arbitrary $k$. Here $\mathsf{nnz}(X)$ is the total number of non-zero entries in the input dataset $X$, which is upper bounded by $nd$ and can be significantly smaller for sparse datasets. We prove that our algorithm achieves approximation ratio $\widetilde{O}(k^4)$ on any input dataset for the $k$-means objective, and our experiments show that the quality of the clusters found by our algorithm is usually much better than this worst-case bound. We use our algorithm for $k$-means clustering and for coreset construction; our experiments show that it gives a new tradeoff between running time and cluster quality compared to previous state-of-the-art methods for these tasks. Our theoretical analysis is based on novel results of independent interest. We show that the approximation ratio achieved after a random one-dimensional projection can be lifted to the original points and that $k$-means++ seeding can be implemented in expected time $O(n\log n)$ in one dimension.

Motivated by fairness requirements in communication networks, we introduce a natural variant of the online paging problem, called min-max paging, where the objective is to minimize the maximum number of faults on any page. While the classical paging problem, whose objective is to minimize the total number of faults, admits k-competitive deterministic and O(log k)-competitive randomized algorithms, we show that min-max paging does not admit a c(k)-competitive algorithm for any function c. Specifically, we prove that the randomized competitive ratio of min-max paging is Ω(log(n)) and its deterministic competitive ratio is Ω(k log(n)/log(k)), where n is the total number of pages ever requested. We design a fractional algorithm for paging with a more general objective - minimize the value of an n-variate differentiable convex function applied to the vector of the number of faults on each page. This gives an O(log(n) log(k))-competitive fractional algorithm for min-max paging. We show how to round such a fractional algorithm with at most a k factor loss in the competitive ratio, resulting in a deterministic O(k log(n) log(k))-competitive algorithm for min-max paging. This matches our lower bound modulo a poly(log(k)) factor. We also give a randomized rounding algorithm that results in a O(log2 n log k)-competitive algorithm.