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CS173 Discrete Mathematical Structures
Fall 2006
Homework
due Tuesday, October 31, 2006, 8:00 a.m.
(36 points + 4 bonus points)
- Recursive Definition (6 points) Early members of the Pythagorean Society defined figurate numbers to be the number of dots in certain geometrical configurations. a. The first four triangular numbers are 1, 3, 6, and 10. (3 points)
Find a recursive expression for the nth triangular number. What is the value of the 10th triangular number?
b. The first four pentagonal numbers are 1, 5, 12, and 22. (3 points)
Find a recursive expression for the nth pentagonal number. What is the value of the 7th pentagonal number?
- Recursive Algorithm (8 points) ========= choose a format for your pseudocode ==========
Consider following two formats to write a recursive algorithm for computing an^ where a is a
nonzero real number and n is a nonnegative integer. Format 1:
Format 2:
Choose one of these formats to write pseudocode for the following questions.
a. Devise a recursive algorithm to find xn^ mod m when n, x, and m are positive integers
based on the fact that xn^ mod m = (xnโ^1 mod m โ
x mod m) mod m. (4 points)
b. Devise a recursive algorithm to evaluate the nth term of the following sequence S:
p, p โ q, p + q, p โ 2 q, p + 2 q, p โ 3 q, p + 3 q,... (4 points)
procedure power(a: nonzero real number, n: nonnegative integer)
if n = 0 then power(a, n) := 1
else power(a,n) := a โ
power(a, n โ 1)
algorithm power(nonzero real number a; nonnegative integer n) if n = 0 then
power(a, n) := 1
else
power(a, n) := a โ
power(a, n โ 1)
end if end algorithm power
c. Let A be an n ร n matrix, and suppose A[i, j] contains the value in row i column j of the
matrix. Assume that the value of A[i, j] can be accessed in constant time. What does the following algorithm do? Give a tight upper bound for the running time of this algorithm. (3 points)
d. (Bonus points for this one!! Your total homework score will not be hurt if you skip this problem.) S is an array of integers {1, 2, 3, โฆ, n} (each number appears in S exactly once). Subroutine swap(S[a], S[b]) exchanges array items indexed by a and b. Each swap takes constant time. What does the following algorithm do? What is the worst case time complexity? Why? (4 points)
algorithm X(array S, nonnegative integer n) for i = 1 to n -1 do
while S[i] โ i do
swap(S[i], S[S[i]]) end while end for end algorithm X
algorithm MatrixSum(matrix A) sum := 0 for k = 1 to n do for j = k to n do sum := sum + A[j, k] end for end for return sum end algorithm MatrixSum
