Aligning sequences via an evolutionary tree: complexity and approximation
STOC '94 Proceedings of the twenty-sixth annual ACM symposium on Theory of computing
SIAM Journal on Computing
Algorithms on strings, trees, and sequences: computer science and computational biology
Algorithms on strings, trees, and sequences: computer science and computational biology
On the approximability of the Steiner tree problem in phylogeny
Discrete Applied Mathematics - Special volume on computational molecular biology DAM-CMB series volume 2
When Hamming Meets Euclid: The Approximability of Geometric TSP and Steiner Tree
SIAM Journal on Computing
Beyond Steiner's Problem: A VLSI Oriented Generalization
WG '89 Proceedings of the 15th International Workshop on Graph-Theoretic Concepts in Computer Science
Tighter Bounds for Graph Steiner Tree Approximation
SIAM Journal on Discrete Mathematics
Computing steiner minimum trees in Hamming metric
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
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Given a set $S$ of vertices in a connected graph $G$, the classic Steiner tree problem asks for the minimum number of edges of a connected subgraph of $G$ that contains $S$. We study this problem in the hypercube. Given a set $S$ of vertices in the $n$-dimensional hypercube $Q_n$, the Steiner cost of $S$, denoted by $cost(S)$, is the minimum number of edges among all connected subgraphs of $Q_n$ that contain $S$. We obtain the following results on $cost(S)$. Let $\epsilon$ be any given small, positive constant, and set $k=|S|$. (1) [upper bound] For every set $S$ we have $cost(S) c_1$, then $cost(S) (\frac{1}{3} - \epsilon)kn$. Thus for $k$ in this range with $k\to \infty$, the upper bound (1) is asymptotically tight. We also show that for fixed $k$, as $n\to \infty$, almost always a random family of $k$ vertices in $Q_n$ satisfies $[\frac{k}{3}+\frac{2}{9}(-1+(-\frac{1}{2})^k)] n - \sqrt{n\ln n}\leq cost (S)\leq [\frac{k}{3}+\frac{2}{9}(-1+(-\frac{1}{2})^k)] n + \sqrt{n\ln n}.$