Algorithms

CS3401    

ALGORITHMS

COURSE OBJECTIVES:

•        To understand and apply the algorithm analysis techniques on searching  and sorting algorithms

•        To critically analyze the efficiency of graph algorithms

•        To understand different algorithm design techniques

•        To solve programming problems using state space tree

•        To understand the concepts behind NP Completeness, Approximation algorithms and randomized algorithms.

UNIT I       INTRODUCTION     

Algorithm analysis: Time and space complexity - Asymptotic Notations and its properties Best case, Worst case and average case analysis – Recurrence relation: substitution method - Lower bounds – searching: linear search, binary search and Interpolation Search, Pattern search: The naïve string-matching algorithm - Rabin-Karp algorithm - Knuth-Morris-Pratt algorithm. Sorting: Insertion sort – heap sort

UNIT II     GRAPH ALGORITHMS  

Graph algorithms: Representations of graphs - Graph traversal: DFS – BFS - applications - Connectivity, strong connectivity, bi-connectivity - Minimum spanning tree: Kruskal’s and Prim’s algorithm- Shortest path: Bellman-Ford algorithm - Dijkstra’s algorithm - Floyd-Warshall algorithm Network flow: Flow networks - Ford-Fulkerson method – Matching: Maximum bipartite matching

UNIT III    ALGORITHM DESIGN TECHNIQUES

Divide and Conquer methodology: Finding maximum and minimum - Merge sort - Quick sort Dynamic programming: Elements of dynamic programming — Matrix-chain multiplication - Multi stage graph — Optimal Binary Search Trees. Greedy Technique: Elements of the greedy strategy-     Activity-selection problem –- Optimal Merge pattern — Huffman Trees.

UNIT IV    STATE SPACE SEARCH ALGORITHMS

Backtracking: n-Queens problem - Hamiltonian Circuit Problem - Subset Sum Problem – Graph colouring problem Branch and Bound: Solving 15-Puzzle problem - Assignment problem - Knapsack Problem - Travelling Salesman Problem

UNIT V     NP-COMPLETE AND APPROXIMATION ALGORITHM

Tractable and intractable problems: Polynomial time algorithms – Venn diagram representation - NP-algorithms - NP-hardness and NP-completeness – Bin Packing problem - Problem reduction: TSP – 3-CNF problem. Approximation Algorithms: TSP - Randomized Algorithms: concept and application - primality testing - randomized quick sort - Finding k^th smallest number

PRACTICAL EXERCISES:

Searching and Sorting Algorithms

1.       Implement Linear Search. Determine the time required to search for an element. Repeat the experiment for different values of n, the number of elements in the list to be searched and plot a graph of the time taken versus n.

2.       Implement recursive Binary Search. Determine the time required to search an element. Repeat the experiment for different values of n, the number of elements in the list to be searched and plot a graph of the time taken versus n.

3.       Given a text txt [0...n-1] and a pattern pat [0...m-1], write a function search (char pat [ ], char txt [ ]) that prints all occurrences of pat [ ] in txt [ ]. You may assume that n > m.

4.       Sort a given set of elements using the Insertion sort and Heap sort methods and determine the time required to sort the elements. Repeat the experiment for different values of n, the number of elements in the list to be sorted and plot a graph of the time taken versus n.

Graph Algorithms

1.       Develop a program to implement graph traversal using Breadth First Search

2.       Develop a program to implement graph traversal using Depth First Search

3.       From a given vertex in a weighted connected graph, develop a program to find the shortest paths to other vertices using Dijkstra’s algorithm.

4.       Find the minimum cost spanning tree of a given undirected graph using Prim’s algorithm.

5.       Implement Floyd’s algorithm for the All-Pairs- Shortest-Paths problem.

6.       Compute the transitive closure of a given directed graph using Warshall's algorithm.

Algorithm Design Techniques

1.       Develop a program to find out the maximum and minimum numbers in a given list of n numbers using the divide and conquer technique.

2.       Implement Merge sort and Quick sort methods to sort an array of elements and determine the time required to sort. Repeat the experiment for different values of n, the number of elements in the list to be sorted and plot a graph of the time taken versus n.

State Space Search Algorithms

1.       Implement N Queens problem using Backtracking.

Approximation Algorithms Randomized Algorithms

1. Implement any scheme to find the optimal solution for the Traveling Salesperson problem and then solve the same problem instance using any approximation algorithm and determine the error in the approximation.

2. Implement randomized algorithms for finding the kth smallest number. The programs can be implemented in C/C++/JAVA/ Python.

COURSE OUTCOMES:

At the end of this course, the students will be able to:

CO1: Analyze the efficiency of algorithms using various frameworks

CO2: Apply graph algorithms to solve problems and analyze their efficiency.

CO3: Make use of algorithm design techniques like divide and conquer, dynamic programming and greedy techniques to solve problems

CO4: Use the state space tree method for solving problems.

CO5: Solve problems using approximation algorithms and randomized algorithms

TEXT BOOKS:

1.       Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest and Clifford Stein, "Introduction to Algorithms", 3rd Edition, Prentice Hall of India, 2009.

2.       Ellis Horowitz, Sartaj Sahni, Sanguthevar Rajasekaran “Computer Algorithms/C++” Orient Blackswan, 2nd Edition, 2019.

REFERENCES:

1.       Anany Levitin, “Introduction to the Design and Analysis of Algorithms”, 3rd Edition, Pearson Education, 2012.

2.       Alfred V. Aho, John E. Hopcroft and Jeffrey D. Ullman, "Data Structures and Algorithms", Reprint Edition, Pearson Education, 2006.

3.       S. Sridhar, “Design and Analysis of Algorithms”, Oxford university press, 2014.