Lecture12 - Divide and Conquer CSE 421 Algorithms Richard...

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1 CSE 421 Algorithms Richard Anderson Lecture 12 Recurrences and Divide and Conquer Divide and Conquer Recurrence Examples T(n) = 2 T(n/2) + cn – O(n log n) T(n) = T(n/2) + cn – O(n) More useful facts: – log k n = log 2 n / log 2 k – k log n = n log k T(n) = aT(n/b) + f(n) Recursive Matrix Multiplication Multiply 2 x 2 Matrices: | r s | | a b| |e g| | t u| | c d| | f h| r = ae + bf s = ag + bh t = ce + df u = cg + dh A N x N matrix can be viewed as a 2 x 2 matrix with entries that are (N/2) x (N/2) matrices. The recursive matrix multiplication algorithm recursively multiplies the (N/2) x (N/2) matrices and combines them using the equations for multiplying 2 x 2 matrices = Recursive Matrix Multiplication How many recursive calls are made at each level? How much work in combining the results? What is the recurrence?
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2 What is the run time for the recursive Matrix Multiplication Algorithm? • Recurrence: T(n) = 4T(n/2) + cn T(n) = 2T(n/2) + n 2 T(n) = 2T(n/2) + n 1/2 Recurrences Three basic behaviors – Dominated by initial case – Dominated by base case – All cases equal – we care about the depth
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