# 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.

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