An &Ohgr;(n2 log n) lower bound to the shortest paths problem
STOC '77 Proceedings of the ninth annual ACM symposium on Theory of computing
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The following problem was recently raised by C. William Gear [1]: Let F(x1,x2,...,xn) &equil; &Sgr;i≤j a'ijxixj + &Sgr;i bixi +c be a quadratic form in n variables. We wish to compute the point x→(0) &equil; (x1(0),...,xn(0)), at which F achieves its minimum, by a series of adaptive functional evaluations. It is clear that, by evaluating F(x→) at 1/2(n+1)(n+2)+1 points, we can determine the coefficients a'ij,bi,c and thereby find the point x→(0). Gear's question is, “How many evaluations are necessary?” In this paper, we shall prove that O(n2) evaluations are necessary in the worst case for any such algorithm.