Download 3466 - Allama Iqbal Open University

Final: 3-8-2016
ALLAMA IQBAL OPEN UNIVERSITY, ISLAMABAD
(Department of Computer Science)
WARNING
1.
2.
PLAGIARISM OR HIRING OF GHOST WRITER(S) FOR SOLVING
THE ASSIGNMENT(S) WILL DEBAR THE STUDENT FROM AWARD
OF DEGREE/CERTIFICATE, IF FOUND AT ANY STAGE.
SUBMITTING ASSIGNMENT(S) BORROWED OR STOLEN FROM
OTHER(S) AS ONE’S OWN WILL BE PENALIZED AS DEFINED IN
“AIOU PLAGIARISM POLICY”.
Course: Analysis & Design of Algorithms (3466)
Level: BS (CS)
Semester: Autumn 2016
Total Marks: 100
Pass Marks: 50
ASSIGNMENT No. 1
Note: All questions are compulsory. Each question carries equal marks.
Q. 1 (a)
(b)
What is the measure of efficiency in algorithm analysis and design? State
your answer with the help of suitable examples.
prove that:
i.
ii.
iii.
Q. 2 Consider sorting n numbers stored in array A by first finding the smallest element
of A and exchanging it with the element in A[1]. Then find the second smallest
element of A, and exchange it with A [2]. Continue in this manner for the first n-1
elements of A.
(a)
Write pseudo code for this algorithm. What loop invariant does this
algorithm maintain? Why does it need to run for only the first n - 1 element,
rather than for all n elements?
(b)
Give the best-case, average case and worst-case running times of this
algorithm.
Q. 3 (a)
What is recursion tree method? How this method is helpful in solving the
problem?
Solve the following recurrence
T (n) = 2 T (n/2) + cn, n >1
(b)
T (1) = c
Q. 4 (a)
(b)
Q. 5 (a)
(b)
Given an array of 7 elements: [15, 20, 10,5, 17, 19, 22] sort it in ascending
order using heap sort.
What is the running time of heap sort on an array A of length n that is already
sorted in increasing order?
Illustrate the operation of quick sort on the array A = [6, 11, 2, 18, 1, 19, 25,
6, 49, 33, 2, 9, 77, 12, 61].
Show that the running time of QUICKSORT is Θ(n2) when the array A
contains distinct elements and is sorted in decreasing order.
ASSIGNMENT No. 2
Total Marks: 100
Pass Marks: 50
Note: All questions are compulsory. Each question carries equal marks.
Q. 1 (a)
(b)
Write algorithm for counting sort algorithm. Discuss efficiency of algorithm.
Illustrate the operation of counting sort on the array A = [6, 0, 2, 0, 1, 3, 4, 6,
1, 3, 2]
Q. 2 (a)
Which of the following sorting algorithms are stable?
Insertion sort, merge sort, heap sort, and quicksort
Give a simple scheme that makes any sorting algorithm stable. How much
additional time and space does this scheme entail?
(b)
Q. 3 (a)
(b)
Find the optimal sequence and minimum number of operations required for
the following chain matrix multiplication using dynamic programming.
A (30, 40) B (40, 5) C (5, 15) D (15, 6)
Given a Convex Polygon (P= v0, v1, ... ,vn-1) and a eight function
on
triangles, find a triangulation minimizing the total weight.
Q. 4 Trace the following algorithms with the help of an example. Analyze the difference
between the order of nodes or edges visited for the two algorithms.
a)
Prim’s algorithm
b)
Kruskal’s algorithm
Q. 5 Give and explain each step with graph example for the trace of following graph
traversal algorithms.
a)
Breadth first search
b)
Depth first search
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3466-Analysis and Design of Algorithm
Credit Hours: 3 (3 + 0)
Recommended Book:
Introduction to Algorithms by Thomas H. Cormen, Charles E. Leiserson,
Ronald L. Rivest
Course Outline:
Unit–1 Introduction
Introduction to Algorithm Analysis and Design
Growth of Functions, Summations formulas and properties
Unit–2 Recurrences and Sets
Substitution, Iteration and Master Methods
Sets, Relations, Functions, Graph and trees, Counting and Probability
Unit–3 Sorting Algorithms
Heaps, Maintaining the heap property, Heap Sort algorithm,
Quick Sort, Performance and analysis of Quick Sort
Unit–4 Sorting in Linear Time and Order Statistics
Lower bounds for sorting, Counting sort, Radix and Bucket Sort, Medians
and order Statistics
Unit–5 Elementary Data Structures
Analysis of Stack, Queues and Linked List Algorithms, Hash Table and
Functions, Binary Search Trees
Unit–6 Dynamic Programming
Matrix Chain Multiplication, Longest Common Subsequence, Optimal
Polygon Triangulation
Unit–7 Greedy Algorithms
An activity selection problem, Huffman Codes, A task scheduling
problem, Amortized Analysis
Unit–8 Graph Algorithms
Elementary Graph Algorithms, Breadth first search, Depth first search,
Minimum Spanning Trees
Unit–9 Single Source Shortest Paths
Shortest paths and relaxation, Dijkstra’s algorithm, The Bellman-Ford
algorithm, Introduction to NP-Completeness
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