Given a graph G = (V,E) and k source-sink pairs {(s1, t1), . . . , (sk, tk)} with each si, ti ∈ V , the Min-Sum Disjoint Paths problem asks to find k disjoint paths connecting all the source-sink pairs with minimized total length, while the Min-Max Disjoint Paths problem asks for k disjoint paths connecting all the sourcesink pairs with minimized length of the longest path. We show that the wei...

In 1988, Leighton, Maggs, and Rao showed that for any network and any set of packets whose paths through the network are xed and edge-simple, there exists a schedule for routing the packets to their destinations in O(c+d) steps using constant-size queues, where c is the congestion of the paths in the network, and d is the length of the longest path. The proof, however, used the Lovv asz Local L...

An st-orientation of a biconnected undirected graph defines a directed graph with no cycles, a single source s and a single sink t. Given an undirected graph G as input, linear-time algorithms have been proposed for computing an st-orientation of G. Such an orientation is useful especially in graph drawing algorithms which use it at their first stage [23]. Namely, before they process the origin...

We have developed an instrumentation scheme that enables programs written in multithreaded languages to compute a critical path at runtime. Our scheme gives not only the length (execution time) of the critical path but also the lengths and locations of all the subpaths making up the critical path. Although the scheme is like Cilk's algorithm in that it uses a \longest path" computation, it allo...

Activity networks are used to plan projects by showing precedence relationships between the various activities that constitute such projects [1]. An activity network is a special case of a directed graph in which the nodes (or vertices) represent points in time and the arcs (or edges) represent activities with time values modeling activity durations. Figure 1 shows F1 an example of a stochastic...

This paper presents a method of using mutual information to improve the recognition algorithm of unknown Chinese words, it can resolve the complexity of weight settings and the increasing garbage strings caused by the omnisegmentation of fragments that affected the efficiency of unknown Chinese words recognition existed in the literature[7]. The process of the method is as following: first, seg...

In this work, we obtain the following new results: – Given a tree T = (V, E) with a length function ` : E → R and a weight function w : E → R, a positive integer k, and an interval [L, U ], the Weight-Constrained k Longest Paths problem is to find the k longest paths among all paths in T with weights in the interval [L,U ]. We show that the Weight-Constrained k Longest Paths problem has a lower...

In this paper, we investigate the star graph Sn with faulty vertices and/or edges from the graph theoretic point of view. We show that between every pair of vertices with different colors in a bicoloring of Sn, n ≥ 4, there is a fault-free path of length at least n!− 2fv − 1, and there is a path of length at least n!− 2fv − 2 joining a pair of vertices with the same color, when the number of fa...

The detour index of a connected graph is defined as the sum of the detour distances (lengths of longest paths) between unordered pairs of vertices of the graph. We give bounds for the detour index, determine the graphs with minimum and maximum detour indices respectively in the class of n-vertex unicyclic graphs with cycle length r, where 3 ≤ r ≤ n−2, and the graphs with the first, the second a...

In 1988, Leighton, Maggs, and Rao showed that for any network and any set of packets whose paths through the network are xed and edge-simple, there exists a schedule for routing the packets to their destinations in O(c + d) steps using constant-size queues, where c is the congestion of the paths in the network, and d is the length of the longest path. The proof, however, used the Lovv asz Local...