A tandem duplication random loss (TDRL) operation duplicates a contiguous segment of genes, followed by the random loss of one copy of each of the duplicated genes. Although the importance of this operation is founded by several recent biological studies, it has been investigated only rarely from a theoretical point of view. Of particular interest are sorting TDRLs which are TDRLs that, when applied to a permutation representing a genome, reduce the distance towards another given permutation. The identification of sorting genome rearrangement operations in general is a key ingredient of many algorithms for reconstructing the evolutionary history of a set of species. In this paper we present methods to compute all sorting TDRLs for two given gene orders. In addition, a closed formula for the number of sorting TDRLs is derived and further properties of sorting TDRLs are investigated. It is also shown that the theoretical findings are useful for identifying unique sorting TDRL scenarios for mitochondrial gene orders.

We study the problem of uniformly partitioning the edge set of a tree with n edges into k connected components, where k n. The objective is to minimize the ratio of the maximum to the minimum number of edges of the subgraphs in the partition. We show that, for any tree and k 4, there exists a k-split with ratio at most two. For general k, we propose a simple algorithm that finds a k-split with ratio at most three in O(n log k) time. Experimental results on random trees are also shown.

In this paper, we are concerned with the problem of deploying two servers in a tree network, where each server may fail with a given probability. Once a server fails, the other server will take full responsibility for the services. Here, we assume that the servers do not fail simultaneously. In the backup 2-center problem, we want to deploy two servers at the vertices such that the expected distance from a farthest vertex to the closest functioning server is minimum. In the backup 2-median problem, we want to deploy two servers at the vertices such that the expected sum of distances from all vertices to the set of functioning servers is minimum. We propose an O(n)-time algorithm for the backup 2-center problem and an O(n log n)-time algorithm for the backup 2-median problem, where n is the number of vertices in the given tree network.

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