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COMMENTS using barcode printer for none control to generate, create none image in none applications. Microsoft Office Official Website 173. This i none none s often called the sum rule for big-Os. 176.

This is often called the product rule for big-Os.. 4 . Recurrence Relations Recurrence relations are a useful tool for the analysis of recursive algorithms, as we will see later in Section 6.2. The problems in this chapter are intended to develop skill in solving recurrence relations.

. SIMPLE RECURRENCES T (1) = 1, and for all n 2, T (n) = 3T (n 1) + 2. T (1) = 8, and for all n 2, T (n) = 3T (n 1) 15. T (1) = 2, and for all n 2, T (n) = T (n 1) + n 1.

T (1) = 3, and for all n 2, T (n) = T (n 1) + 2n 3. T (1) = 1, and for all n 2, T (n) = 2T (n 1) + n 1. T (1) = 5, and for all n 2, T (n) = 2T (n 1) + 3n + 1.

T (1) = 1, and for all n 2 a power of 2, T (n) = 2T (n/2) + 6n 1. T (1) = 4, and for all n 2 a power of 2, T (n) = 2T (n/2) + 3n + 2. T (1) = 1, and for all n 2 a power of 6, T (n) = 6T (n/6) + 2n + 3.

T (1) = 3, and for all n 2 a power of 6, T (n) = 6T (n/6) + 3n 1. T (1) = 3, and for all n 2 a power of 3, T (n) = 4T (n/3) + 2n 1. T (1) = 2, and for all n 2 a power of 3, T (n) = 4T (n/3) + 3n 5.

T (1) = 1, and for all n 2 a power of 2, T (n) = 3T (n/2) + n2 n. T (1) = 4, and for all n 2 a power of 2, T (n) = 3T (n/2) + n2 2n + 1. T (1) = 1, and for all n 2 a power of 2, T (n) = 3T (n/2) + n 2.

37. Solve the f none none ollowing recurrences exactly. 202. 203.

204. 205. 206.

207. 208. 209.

210. 211. 212.

213. 214. 215.

216.. 38 217. 218. 219.

220.. Chap. 4. Recurrence Relations T (1) = 1, and for all n 2 a power of 2, T (n) = 3T (n/2) + 5n 7. T (1) = 1, and for all n 2 a power of 3, T (n) = 4T (n/3) + n2 . T (1) = 1, and for all n 2 a power of 3, T (n) = 4T (n/3) + n2 7n + 5.

T (1) = 1, and for n 4 a power of 4, T (n) = T (n/4) + n + 1.. MORE DIFFICULT RECURRENCES Suppose 0 & lt; , < 1, where + = 1. Let T (1) = 1, and for all n 1, T (n) = T ( n) + T ( n) + cn, for some c IN. Prove that T (n) = O(n log n).

You may make any necessary assumptions about n. The Fibonacci numbers Fn for n 0 are de ned recursively as follows: F0 = 0, F1 = 1, and for n 2, Fn = Fn 1 +Fn 2 . Prove by induction on n that Fn = ( n n )/ 5, where = (1 + 5)/2, and = (1 5)/2.

. Let X(n) be none none the number of di erent ways of parenthesizing the product of n values. For example, X(1) = X(2) = 1, X(3) = 2 (they are (xx)x and x(xx)), and X(4) = 5 (they are x((xx)x), x(x(xx)), (xx)(xx), ((xx)x)x, and (x(xx))x). 223.

Prove that if n 2, then X(n) = 1; and otherwise. X(n) =. X(k) X(n k). 224. 225..

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