Fast on-line/off-line algorithms for optimal reinforcement of a network and its connections with principal partition

Abstract

The problem of computing the strength and performing optimal reinforcement for an edge-weighted graph G(V, E,w;) is well-studied [1,2,3,6,7,9]. In this paper, we present fast (sequential linear time and parallel logarithmic time) on-line algorithms for optimally reinforcing the graph when the reinforcement material is available continuosly online. These are first on-line algortithms for this problem. Although we invest some time in preprocessing the graph before the start of our algorithms, it is also shown that the output of our on-line algorithms is as good as that of the off-line algorithms, making our algorithms viable alternatives to the fastest off-line algorithms in situtations when a sequence of more than O(|V |) reinforcement problems need to be solved. In such a situation the time taken for preprocessing the graph is less that the time taken for all the invocations of the fastest off-line algorithms. Thus our algorithms are also efficient in the general sense. The key idea is to make use of the theory of Principal Partition of a Graph. Our results can be easily generalized to the general setting of principal partition of nondecreasing submodular functions.