# Analysis and Design of Algorithms

May 8th, 2015
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Assignment 3 to Compute the given sums

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Analysis and Design of AlgorithmsTK3043 : Analysis and Design of AlgorithmsAssignment 31. Compute the following sums:a. ?Answer:=?=u-1+1= (n + 1) - 3 + 1=n+1-2=n-2b. ?Answer:=?= [1 + 2] + ? + n=?+ (n + 1) - (1 + 2)=?+ (n + 1) - 3=?+n -2= n(n + 1) + (n - 2)2= n2 + n + (n - 2)2= n2 + 3n - 42c. ?Answer:?=?=?= n (n+1) (2n + 1) + n (n+1)62= (n - 1) (n -1 + 1) (2 ( n -1) +1) + (n - 1) (n - 1 + 1) 62= (n - 1) (n) (2n - 2 + 1) + (n - 1) (n)622= (n - n) (2n - 1) + (n - 1) (n)62= (n3 - n2 - 2n2 + n) + (n2 - n)62= 2n3 - 3n2 + n + n2 - n62= 2n3 - 3n2 + n + 3n3 - 3n6= 2n3 - 2n6= n3 - n32=n -132. Consider the following algorithm.Algorithm Mystery( n)//Input: A nonnegative integer nS?0for i ? 1 to n doS?S+i*ireturn Sa. What does this algorithm compute?Answer:2S(n) = ?b. What is its basic operation?Answer:Multiplicationc. How many times is the basic operation executed?Answer:At least one timed. What is the efficiency class of this algorithm?Answer:C(n) = n ?? (n)e. Suggest an improvement or a better algorithm altogether and indicate its efficiency class. If you cannot do it, try to prove that, in fact, it cannot be done. Answer:?2= n (n + 1) (2n + 1)63. Consider the following algorithm.Algorithm Secret(A[0..n ? 1])//Input: An array A[0..n ? 1] of n real numbersminval ? A[0]; maxval ? A[0]for i ? 1 to n ? 1 doif A[i] < minvalminval ? A[i]if A[i] > maxvalmaxval ? A[i]return maxval ? minval

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