One of the first things you learn in calculus is how to minimize a differentiable function such as y = ax2 + bx + c, where a >0. The Minimum Spanning Tree Problem, on the other hand, is a minimization problem of a very different flavor: there are now just a finite number of possibilities for how the minimum might be achieved—rather than a continuum of possibilities—and we are interested in how to perform the computation without having to exhaust this (huge) finite number of possibilities.
One can ask what happens when these two minimization issues are brought together, and the following question is an example of this. Suppose we have a connected graph G = (V, E). Each edge e now has a time-varying edge cost given by a function fe :R→R. Thus, at time t, it has cost fe(t). We’ll assume that all these functions are positive over their entire range. Observe that the set of edges constituting the minimum spanning tree of G may change over time. Also, of course, the cost of the minimum spanning tree of G becomes a function of the time t; we’ll denote this function cG(t). A natural problem then becomes: find a value of t at which cG(t) is minimized.
Suppose each function fe is a polynomial of degree 2: fe(t) = aet2 + bet + ce, where ae >0. Give an algorithm that takes the graph G and the values {(ae, be, ce) : e ε E} and returns a value of the time t at which the minimum spanning tree has minimum cost. Your algorithm should run in time polynomial in the number of nodes and edges of the graph G. You may assume that arithmetic operations on the numbers {(ae, be, ce)} can be done in constant time per operation.
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