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Homework 3
CS 3343 – Fall 2006 assigned September 7, 2006 Tom Bylander, Instructor due September 15, 2006
Your solutions must be submitted as a document to WebCT.
- (20 pts.) Indicate the order of growth (big theta) of the following sums.
(a)
∑^ n i=
1 /i
(b)
∑^ n i=
log 2 i
(c)
∑^ n i=
i
(d)
∑^ n i=
i log 2 i
(e)
∑^ n i=
i^2
(f)
∑^ n i=
i^3
(g)
∑^ n i=
i^4
(h)
∑^ n i=
2 i
(i)
∑^ n i=
i 2 i
(j)
∑^ n i=
i!
- (20 pts.) Consider the following algorithm.
algorithm Mystery(A[0..n − 1]) // Input: An array A for i ← 0 to n − 1 do for j ← 0 to n − 1 do if A[i] 6 = A[j] then return false return true
(a) What does this algorithm compute? (b) What is its basic operation?
1
(c) How many times is the basic operation executed? (worst-case and best-case) (d) What is the efficiency class of this algorithm? (worst-case and best-case) (e) Provide an improvement that improves its worst-case efficiency class and indicate the new worst-case efficiency class.
- (20 pts.) This is a variation of Exercise 2.4.7.
(a) Provide pseudocode (in the book’s style) for two recursive algorithms that com- pute 2n^ for any integer n ≥ 0 based on the formula 2n^ = 2n−^1 + 2n−^1. One algorithm should have two recursive calls in the code and the other only one. (b) Draw a tree of recursive calls for each algorithm and use the tree to count the number of additions. (c) For both algorithms, set up and solve a recurrence relation for the number of additions made by the algorithm. (d) For each algorithm, comment on whether it is a good algorithm for solving this problem.
- (20 pts.) This is a variation of Exercise 2.5.3.
Find the smallest n for which the nth Fibonacci number does not fit in varables of type int, long, float, and double. For float and double, I want to know when the number cannot be represented exactly, i.e., when you start losing last digit.
- (20 pts.) Do Exercise 2.5.4. For further exercise, perform all the possible ways to climb a staircase on one of the stairways between the 3rd and 4th floor of the Science Building.
