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CS 345: Spring, 2014
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1.
a. A distinguishing difference between a depth-first traversal and a breadth-first traversal is the data structure used to hold the nodes to be visited. What data structure is used with a breadth-first traversal? [5 points]
b. If the structure to be traversed is a complete, balanced binary tree (every non-leaf node has two children), does a breadth-first traversal require more or less space to store the nodes to be visited than a depth-first traversal? [5 points]
2.
The textbook presents a solution to the problem of finding a minimum cost spanning tree of a graph by Prim’s algorithm (the Lazy Contractor). What algorithm paradigm does this algorithm use? [10 points]
3.
Write the pseudocode for a function called pre-order(). This is a recursive function which performs a pre-order traversal of a binary tree. The function takes one parameter, n, which is a node of a tree; and it calls a function visit(), which processes the node. [10 points]
CS 345: Spring 2014
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Midterm D
4.
a. Under what conditions might a divide-and-conquer algorithm be useful? [10 points]
b. Describe either a problem that can be solved using a divide-and-conquer algorithm or a well-known algorithm that implements a divide-and-conquer approach. [10 points]
5.
What are the steps necessary to create a dynamic programming solution to a problem? [10 points]
CS 345: Spring 2014
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Midterm D
6.
When a problem can be solved by backtracking, it makes use of a bounding function. Describe the purpose of the bounding function. [10 points]
7.
Graph 1 represents a map of cities. A traveler wants to go from city A to city B.
a. What path would a greedy algorithm take? [10 points]
b. What path would a dynamic programming algorithm take? [10 points]
CS 345: Spring 2014
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Midterm D
8.
Draw a depth-first spanning tree for Graph 3 starting from node A. Process the children of a node in alphabetical order. [10 points]
CS 345: Spring 2014
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Midterm D
9.
Below is a method that calculates a binomial coefficient.
/* int [][] BC = new int[m][n] and int EMPTY = -1 are defined globally. In addition, BC has been initialized to EMPTY before the method is called. */ public int binomialCoefficient( int m, int n ) { if( (m == 0) || (m == 1) ) return 1; else if( (n == 0) || (n == m) ) return 1; else if( (n == 1) || (n == (m-1)) ) return m; else { if( BC[ m – 1 ][ n – 1 ] == EMPTY ) BC[ m – 1 ][ n – 1 ] = binomialCoefficient ( m-1, n-1 ); if( BC[ m – 1 ][ n ] == EMPTY ) BC[ m – 1 ][ n ] = binomialCoefficient ( m-1, n ); return ( BC[ m – 1 ][ n – 1 ] + BC[ m – 1 ][ n ] ); } }
a. What algorithmic paradigm is it using? [5 points]
b. Justify your answer. [5 points]
CS 345: Spring 2014
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Midterm D
10.
a. Describe what it means when an algorithm is partially correct. [5 points]
b. Describe what it means when an algorithm is totally correct. [5 points]
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c. Describe the purpose of a convergent. [5 points]
CS 345: Spring 2014
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Midterm D
This page is scratch paper. It may be removed from the exam, and it does not need to be turned in with the exam.
CS 345: Spring 2014
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Midterm D
Basic Topics
Chapter 4
:
Algorithm Paradigms
•
Trees
•
Traversals and Searches
•
Depth
–
First: pre
–
order, in
–
order, post
–
order
•
Breadth
–
First
•
Divide
–
and
–
Conquer
•
Greedy
•
Dynamic Programming
•
Memoization
•
State Space Search
•
Backtracking
•
Branch and Bound
•
Problems
•
N
–
Queens
•
Knight
’
s Tour
•
Stack Permutation
•
Subset Sum
•
Maximum Subsequence Sum
•
Slobovian Coin
•
Knapsack
•
Minimum Cost Spanning Tree
•
Shortest Path
Chapter 5:
Algorithm Verification
•
Assertions and Invariants
•
Convergence
•
Verification Conditions
•
Partial and Total Correctness
