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We consider the problem of center-based clustering in low-dimensional Euclidean spaces under the perturbation stability assumption. An instance is $alpha$-stable if the underlying optimal clustering continues to remain optimal even when all pairwise distances are arbitrarily perturbed by a factor of at most $alpha$. Our main contribution is in presenting efficient exact algorithms for $alpha$-stable clustering instances whose running times depend near-linearly on the size of the data set when $alpha ge 2 + sqrt{3}$. For $k$-center and $k$-means problems, our algorithms also achieve polynomial dependence on the number of clusters, $k$, when $alpha geq 2 + sqrt{3} + epsilon$ for any constant $epsilon > 0$ in any fixed dimension. For $k$-median, our algorithms have polynomial dependence on $k$ for $alpha > 5$ in any fixed dimension; and for $alpha geq 2 + sqrt{3}$ in two dimensions. Our algorithms are simple, and only require applying techniques such as local search or dynamic programming to a suitably modified metric space, combined with careful choice of data structures.
We show that the edit distance between two strings of length $n$ can be computed within a factor of $f(epsilon)$ in $n^{1+epsilon}$ time as long as the edit distance is at least $n^{1-delta}$ for some $delta(epsilon) > 0$.
Let $P$ be a path graph of $n$ vertices embedded in a metric space. We consider the problem of adding a new edge to $P$ so that the radius of the resulting graph is minimized, where any center is constrained to be one of the vertices of $P$. Previous
Diffusion is a fundamental graph procedure and has been a basic building block in a wide range of theoretical and empirical applications such as graph partitioning and semi-supervised learning on graphs. In this paper, we study computationally effici
We consider the problem of efficiently scheduling jobs with precedence constraints on a set of identical machines in the presence of a uniform communication delay. In this setting, if two precedence-constrained jobs $u$ and $v$, with ($u prec v$), ar
Given a graph $G=(V,E)$ and an integer $k ge 1$, a $k$-hop dominating set $D$ of $G$ is a subset of $V$, such that, for every vertex $v in V$, there exists a node $u in D$ whose hop-distance from $v$ is at most $k$. A $k$-hop dominating set of minimu