Collaboration groups will be set up in Blackboard by the end of the week; however, there are no
collaborative problems on this first assignment. All solutions are to be the result of individual effort.
Self-Study Problems
All of the following problems come from the textbook and have solutions posted on the web at
http://mitpress.mit.edu/algorithms.
You are permitted to use this site to examine solutions for these problems as a means of selfchecking your solutions. These problems
will not be graded.
Problems: 2.2-2, 2.3-5, 3.1-2, 12.1-2, 12.3-3, 21.2-6.
Problems for Grading
1. Problem 1 Chapter 2
20 Points Total
Use induction to prove Pn i=1 i3 = n(n2 +1)2.
2. Problem 2 Parts a, b, c, d, e and f
30 Points Total 5 Points Each
Although merge sort runs in Θ(nlgn) worst-case time and insertion sort runs in Θ(n2) worst-case
time, the constant factors in insertion sort can make it faster in practice for small problem sizes
on many machines. Thus, it makes sense to coarsen the leaves of the recursion by using insertion
sort within merge sort when subproblems become sufficiently small. Consider a modification to
merge sort in which n/k sublists of length k are sorted using insertion sort and then merged
using the standard merging mechanism, where k is a value to be determined.
(a) Use insertion sort to sort the unsorted array < 40, 17, 45, 82, 62, 32, 30, 44, 93, 10 >. Make
sure to show the array after every pass.
(b) Use merge sort to sort the unsorted array < 75, 56, 85, 90, 49, 26, 12, 48, 40, 47 >. Make sure
to show the steps of splitting the array then merging the array.
(c) Show that insertion sort can sort the n/k sublists, each of length k, in Θ(nk) worst-case time.
(d) Show how to merge the sublists in Θ(nlg(n/k)) worst-case time.
(e) Given that the modified algorithm runs in Θ(nk + nlg(n/k)) worst-case time, what is the
largest value of k as a function of n for which the modified algorithm has the same running time
as standard merge sort, in terms of Θ-notation?
(f) How should we choose k in practice?
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3. Problem 3
15 Points Total
Write a Θ(m + n) algorithm that prints the in-degree and the out-degree of every vertex in an
m-edge, n-vertex directed graph where the directed graph is represented using adjacency lists.
4. Problem 4 Chapter 12 Binary Search Trees
15 Points Total 3 Points Each
Exercise 12.2-1. Suppose that we have numbers between 1 and 1000 in a binary search tree
and we want to search for the number 363. Which of the following sequences could not be the
sequence of nodes examined?
i. 2, 252, 401, 398, 330, 397, 363.
ii. 924, 220, 911, 244, 898, 258, 362, 363.
iii. 925, 202, 911, 240, 912, 245, 363.
iv. 2, 399, 387, 219, 266, 382, 381, 278, 363.
v. 935, 278, 347, 621, 299, 392, 358, 363.
5. Problem 5 Chapter 12 Binary Search Trees
20 Points
Consider the following algorithm for doing a postorder traversal of a binary tree with root vertex
root.
Algorithm 1 Postorder Traversal
Postorder(root)
1 if root 6= null
2 Postorder(root.left)
3 Postorder(root.right)
4 visit root;
Prove that this algorithm runs in time Θ(n) when the input is an n-vertex binary tree.
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