CS 415 Algorithm Analysis
Spring 2006
Practice Problems for the final examination
1) An algorithm solves a problem of size n (in the worst-case) in 20 n2 steps,
while another algorithm takes n3 steps in the worst-case for the same
problem. What is the smallest value of n for which the former algorithm
would outperform the latter?
2) Use iterative substitution to solve the recurrence equation T(n) = 2 T(n – 1)
+ 1, with the initial condition T(1) = 1.
3) In practical implementations of recursive divide and conquer algorithms,
the splitting stops when the size of the sub-problem reaches a small
enough value at which point a simple straight-forward algorithm is used to
solve it. Suppose we use this idea to implement Merge-sorting and switch
to Insertion-sorting when the size of the sub-problem is 16. Write a
recurrence formula for the worst-case number of comparisons performed
by the resulting algorithm. (Use the fact that the worst-case number of key
comparisons performed by Insertion-sorting is n(n – 1)/2 for n keys.) Also
assume that the number of input keys is a power of 2.
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4) Describe an algorithm that takes as input a heap A, and an index j
size,
and removes the key A[j] from the heap. Your algorithm should be time
complexity O(log n) where n is the size of the heap. Use Heapify-up and
Heapify-Down, with minimal additional code.
5) Consider the following knapsack problem:
Object
Weight
Profit
1 2 3 4
7 4 8 6
21 11 23 15
Volume of the knapsack is 20.
(a) Let T[j, k] denote the optimal solution of the problem with the first j
objects, where the profit is k. What values does T[4, 15] depend on?
What values does T[3, 10] depend on?
(b) If we are allowed to choose the same object more than once, what is
the optimum solution to the above problem?
6) Consider a version of the knapsack problem in which each object has a
weight and a profit, and a specified knapsack capacity. Each object can
be taken 0 or 1 time. Will any of the following ((greedy) algorithms always
produce an optimal solution? In each case, an order of the objects is
specified. The algorithm includes the objects one at a time in that order
and skips over the objects that will cause the weight of the knapsack to
exceed the capacity. Prove or give a counter-example:
(a) increasing order of weight.
(b) decreasing order of profit
(c) decreasing order of profit to weight ratio
7) You bring a t-foot log of wood to your favorite sawmill. You want it cut in k
specific places: t1 , t 2 ,..., t k feet from the left end. The sawmill charges x
dollars to cut an x-foot log any place you want. Give a dynamic
programming based algorithm that determines the order in which they
should cut in order to minimize the cost. Estimate the time complexity of
your algorithm. (It should be a polynomial in k.)
8) Exercise 3.12, page 112.
9) Exercise 4.22, page 200.
10) Exercise 5.2, page 246.
11) Exercise 7.4, page 416.
12) Exercise 7.5, page 416.