K-ary Clustering with Optimal Leaf Ordering for Gene Expression Data
WABI '02 Proceedings of the Second International Workshop on Algorithms in Bioinformatics
Optimal leaf ordering of complete binary trees
Journal of Discrete Algorithms
Lower bounds for maximum parsimony with gene order data
RCG'05 Proceedings of the 2005 international conference on Comparative Genomics
Visual triangulation of network-based phylogenetic trees
VISSYM'04 Proceedings of the Sixth Joint Eurographics - IEEE TCVG conference on Visualization
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Due to a printer's error, incorrect figures were published as part of this paper in Mathematics of Operations Research, Vol. 23, No. 3, August 1998, pp. 613-623. The paper is reprinted correctly below: Let D = (dij) be the n × n distance matrix of a set of n cities {1, 2,...,n}, and let T be a PQ-tree with node degree bounded by d that represents a set Π(T) of permutations over {1, 2,...,n}. We show how to compute for D in O(2dn3) time the shortest travelling salesman tour contained in Π(T). Our algorithm may be interpreted as a common generalization of the well-known Held and Karp dynamic programming algorithm for the TSP and of the dynamic programming algorithm for finding the shortest pyramidal TSP tour. A consequence of our result is that the shortcutting phase of the "twice around the tree" heuristic for the Euclidean TSP can be optimally implemented in polynomial time. This contradicts a statement of Papadimitriou and Vazirani as published in 1984.