By Dempe S.
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Our purpo,se in this discussion is to gain some intuitive feeling for how divide-and-conquer algorithms achieve efficiency, not to do detailed analysis of the algorithms. Indeed, the particular recurrences that we’ve just solved are sufficient to describe the performance of most of the algorithms that we’ll be studying, and we’ll simply be referring back to them. Matrix Multiplication The most famous application of the divide-and-conquer technique to an arithmetic problem is Strassen’s method for matrix multiplication.
Below we’ll look more closely at the savings achieved by this method. For the example given above, with p(x) = 1 +x +3x2 -4x3 and q(x) = 1 + 2x - 5x2 - 3x3, we have Q(X) = (1+ x)(1 + 2x) = I + 3x + 2x2, Q(X) = (3 -4x)(-5 - 3x) = -15 + 11x + 12x2, T,(X) = (4 - 3x)(-4 - x) =: -16 +8x + 3x2. -h(x) = -2 - 6x - 11x2, and the product is computed as p(x)q(x) = (1 + 3x + 2x2) + (-2 -6x - 11x2)x2 + (-15 + 11x + 12x2)x4 = 1+3x - 6x3 - 26x4 + 11x5 t 12x6. This divide-and-conquer approach solves a polynomial multiplication problem of size N by solving three subproblems of size N/2, using some polynomial addition to set up the subproblems and to combine their solutions.
A notrvery-random sequence of integers between 0 and 380. Any initial value can be used to get the random number generator started with no particular effect except of course that different initial values will give rise to different random sequences. Often, it is not necessary to store the whole sequence as in the program above. Rather, we simply maintain a global variable a, initialized with some value, then updated by the computation a:=(a*b+l) mod m. In Pascal (and many other programming languages) we’re still one step away from a working implementation because we’re not allowed to ignore overflow: it’s defined to be an error condition that can lead to unpredictable results.