Shietung Peng

A 2-approximation algorithm for genome rearrangements by reversals and transpositions

Recently, a new approach to analyze genomes evolving which is based on comparision of gene orders versus traditional comparision of DNA sequences was proposed (Sankoff et al. 1992). The approach is based on the global rearrangements (e.g., inversions and transpositions of fragments). Analysis of genomes evolving by inversions and transpositions leads to a combinatorial problem of sorting by reversals and transpositions, i.e., sorting of a permutation using reversals and transpositions of arbitrary fragments. We study sorting of signed permutations by reversals and transpositions, a problem which adequately models genome rearrangements, as the genes in DNA are oriented. We establish a lower bound and give a 2-approximation algorithm for the problem.

Node-to-set disjoint paths problem in star graphs

Abstract Given a node s and a set T = t1,…, tk of k nodes in a k-connected graph, the node-to-set disjoint paths problem is to find k node-disjoint paths pi: s → ti, 1 ⩽ i ⩽ k. In this paper, we give two O(n2) time algorithms for the node-to-set disjoint paths problem in n-dimensional star graphs Gn which are (n − 1)-connected. We first give a simple algorithm which finds n − 1 node-disjoint paths of length at most d(Gn) + 3, where d(G n ) = ⌊ 3(n − 1) 2 ⌋ is the diameter of Gn. Then, we refine the algorithm to find n − 1 node-disjoint paths of length at most d(Gn) + 2. A lower bound on the length of the paths for the above problem in Gn is d(Gn) + 1.

Unicast in hypercubes with large number of faulty nodes

Unicast in computer/communication networks is a one-to-one communication between a source node s and a destination node t. We propose three algorithms which find a nonfaulty routing path between s and t for unicast in the hypercube with a large number of faulty nodes. Given the n-dimensional hypercube H/sub n/ and a set F of faulty nodes, node u/spl epsiv/ H/sub n/ is called k-safe if u has at least k nonfaulty neighbors. The H/sub n/ is called k-safe if every node of H/sub n/ is k-safe. It has been known that for 0/spl les/k/spl les/n/2, a k-safe H/sub n/ is connected if |F|/spl les/2/sup k/(n-k)-1. Our first algorithm finds a nonfaulty path of length at most d(s,t)+4 in O(n) time for unicast between 1-safe s and t in the H/sub n/ with |F|/spl les/2n-3, where d(s,t) is the distance between s and t. The second algorithm finds a nonfaulty path of length at most d(s,t)+6 in O(n) time for unicast in the 2-safe H/sub n/ with |F|/spl les/4n-9. The third algorithm finds a nonfaulty path of length at most d(s,t)+O(k/sup 2/) in time O(|F|+n) for unicast in the k-safe H/sub n/ with |F|/spl les/2/sup k/(n-k)-1 (0/spl les/k/spl les/n/2). The time complexities of the algorithms are optimal. We show that in the worst case, the length of the nonfaulty path between s and t in a k-safe H/sub n/ with |F|/spl les/2/sup k/(n-k)-1 is at least d(s,t)+2(k+1) for 0/spl les/k/spl les/n/2. This implies that the path lengths found by the algorithms for unicast in the 1-safe and 2-safe hypercubes are optimal.

Efficient Algorithms for Finding a Core of a Tree with a Specified Length

Acoreof a graphGis a pathPinGthat is central with respect to the property of minimizingd(P)=?v?V(G)d(v,P), whered(v,P) is the distance from vertexvto pathP. This paper presents efficient algorithms for finding a core of a tree with a specified length. The sequential algorithm runs inO(nlogn) time, wherenis the size of the tree. The parallel algorithm runs inO(log2n) time usingO(n) processors on an EREW PRAM model.

An Efficient Algorithm for the k-Pairwise Disjoint Paths Problem in Hypercubes

A graph G(V, E) (|V|?2k) satisfies property Ak if, given k pairs of distinct nodes (s1, t1), ?, (sk, tk) of V(G), there are k mutually node-disjoint paths, one connecting si and ti for each i, 1?i?k. A necessary condition for any graph to satisfy Ak is that it is (2k?1)-connected. Hypercubes are important interconnection topologies for parallel computation and communication networks. It has been known that hypercubes of dimension n (which are n-connected) satisfy A?n/2?. In this paper we give an algorithm which, given k=?n/2? pairs of distinct nodes (s1, t1), ?, (sk, tk) in the n-dimensional hypercube, finds the k disjoint paths of length at most n+?logn?+1 in O(n2log*n) time.

Node-to-set and set-to-set cluster fault tolerant routing in hypercubes

Abstract We study node-to-set and set-to-set fault tolerant routing problems in n -dimensional hypercubes H n . Node-to-set routing problem is that given a node s and a set of nodes T ={ t 1 ,…, t k }, finds k node-disjoint paths s → t i ,1⩽ i ⩽ k . Set-to-set routing problem is that given two sets of nodes S ={ s 1 ,…, s k } and T ={ t 1 ,…, t k }, finds k node-disjoint paths, each path connects a node of S and a node of T . From Mengers theorem, it is known that these two problems in H n can tolerate at most n − k arbitrary faulty nodes. In this paper, we prove that both routing problems can tolerate n − k arbitrary faulty subgraphs (called cluster) of diameter 1. For 2⩽ k ⩽ n , we show that, in the presence of at most n − k faulty clusters of diameter at most 1, the k paths of length at most n +2 for node-to-set routing in H n can be found in O( kn ) optimal time and the k paths of length at most n + k +2 for set-to-set routing in H n can be found in O (kn log k) time. The upper bound n +2 on the length of the paths for node-to-set routing in H n is optimal.

K-pairwise cluster fault tolerant routing in hypercubes

In this paper, we introduce a general fault tolerant routing problem, cluster fault tolerant routing, which is a natural extension of the well studied node fault tolerant routing problem. A cluster is a connected subgraph of a graph G, and a cluster is faulty if all nodes in it are faulty. In cluster fault tolerant routing (abbreviated as CFT routing), we are interested in the number of faulty clusters and the size of the clusters that an interconnection network can tolerate for certain routing problems. As a case study, we investigate the following k-pairwise CFT routing in n-dimensional hypercubes H/sub n/: Given a set of faulty clusters and k distinct nonfaulty node pairs (s/sub 1/, t/sub 1/), ..., (s/sub k/, t/sub k/) in H/sub n/, find k fault-free node-disjoint paths s/sub i//spl rarr/t/sub i/, 1/spl les/i/spl les/k. We show that H/sub n/ can tolerate n-2 faulty clusters of diameter one, plus one faulty node for the k-pairwise CFT routing with k=1. For n/spl les/4 and 2/spl les/k/spl les/[n/2], we prove that H/sub n/ can tolerate n-2k+1 faulty clusters of diameter one for the k-pairwise CFT routing. We also give an O(kn log n) time algorithm which finds the k paths for the mentioned problem. Our algorithm implies an O(n/sup 2/ log n) time algorithm for the k-pairwise node-disjoint paths problem in H/sub n/, which improves the previous result of O(n/sup 3/ log n).

Optimal Algorithms for Node-to-Node Fault Tolerant Routing in Hypercubes

In this paper, we give an algorithm which, given at most n - 1 faulty nodes and non-faulty nodes s and t in the n-dimensional hypercube, H n , finds a fault-free path s → t of length at most d(s, t) + 2 in O(n) time, where d(s, t) is the distance between s and t in H n . Using this algorithm as a subroutine, we present another algorithm which, given at most 2n - 3 faulty nodes such that the faulty nodes can be covered by n - 1 subgraphs of diameter 1, finds a fault-free path s → t of length at most d(s, t) + 4 in O(n) time. The algorithms are optimal in the sense that both the upper bounds on the length of s → t and the time complexity are optimal.

Node-to-node cluster fault tolerant routing in star graphs

In this paper, we proved (n-2,2) is an OCPT number of G n for node-to-node CFT routing. Our result shows that G n has very good node fault tolerant properties. We also give an algorithm which finds fault-free paths of length d(G n )+8 and of lenght at most d(G n )+6 in O(|F|+n) time and O(n 2 ) time, respectively, for any tolerable set F in the above problem. The problem of finding the optimal length of the fault-free path for node-to-node CFT routing in G n is open. It is interesting to find OCFT number for star graphs for d(F)>2. Investigating CFT routing properties and designing efficient CFT routing algorithms for other interconnection networks are certainly worth further research attention.

A Simple Optimal Parallel Algorithm for a Core of a Tree

Abstract A core of a tree T = (V, E) is a path in T which minimizes ∑v∈Vd(v, P), where d(v, P), the distance from a vertex v to path P, is defined as minu∈Pd(v, u). We present an optimal parallel algorithm to find a core of T in O(log n) time using O(n/log n) processors on an EREW PRAM machine, where n is the number of vertices of tree T.