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Extra info for 14.Computers
A thief breaks into a house that has n items of values v1, . , vn and integer weights w1, . , wn. His knapsack, however, can carry only a total weight of M. How should he choose the items so that the value of his booty is maximized? This is called the 0-1 knapsack problem to distinguish it from a variation where the thief is allowed to take a fraction of an item. Of course, this problem is applicable in many other situations, more important and wholesome. There is an easy dynamic programming algorithm for this problem.
An array with only one element is the base case of the recursion and is already sorted. The key step in this algorithm is the partitioning. , a random element) as the pivot and shuffling A so that elements smaller than the pivot lie in the left subarray and those larger than it lie in the right one. Strassen’s Matrix Multiplication Algorithm This algorithm, due to Strassen (5), is a very famous application of the divide-and-conquer technique. The naive algorithm to multiply two n ϫ n matrices requires time ⌰(n3) because n multiplications are required to compute each of the n2 entries in the product matrix.
Many variations on the basic model are equivalent for computability and are polynomially equivalent for complexity measures.