INTRODUCTION TO ALOGORITHM DESIGN STRATEGIES SUBJECT DESIGN ANALYSIS

INTRODUCTION TO ALOGORITHM DESIGN STRATEGIES SUBJECT: DESIGN &ANALYSIS OF ALGORITHM, CSE-5 TH SEM Prepared By. Prof. Saquib Ahmed Anjuman College of Engineering & Technology Department of Computer Science & Engineering

SYLLABUS UNIT-I Mathematical foundations, summation of arithmetic and geometric series, n, n 2 , bounding summations using integration, Recursion and Induction: recurrence relations, solutions of recurrence relations using techniques of characteristic equation, generating functions, master method and substitution method. Complexity calculation of various standard functions, principles of designing algorithms. UNIT-II Asymptotic notations of analysis of algorithms, analyzing control structures, worst case and average case analysis, amortized analysis, application of amortized analysis, Sorting networks, comparison networks, bio-tonic sorting network, advanced data structures like Fibonacci heap, disjoint set representation UNIT-III Divide and conquer basic strategy, binary search, quick sort, merge sort, matrix operations, Multiplication Algorithm Greedy method – basic strategy, Knapsack Problem, application to job sequencing with deadlines problem, minimum cost spanning trees, single source shortest path, Optimal Search Patterns. UNIT-IV Dynamic Programming basic strategy, multistage graphs, all pairs shortest path, single source shortest paths, optimal binary search trees, traveling salesman problem, Longest Common Subsequence problem, 0/1 Knapsack Problem, Chained Matrix Multiplication UNIT-V Basic Traversal and Search Techniques, breadth first search and depth first search, connected components. Backtracking basic strategy, 8 -Queen’s problem, graph coloring, Hamiltonian cycles etc, Introduction to Approximation algorithm. UNIT-VI NPhard and NP-complete problems, basic concepts, non-deterministic algorithms, NP-hard and NP-complete, decision and optimization problems, graph based problems on NP Principle

COUURSE OUTCOMES SNO DESCRIPTION CO. 1 Examine the correctness of algorithms using inductive proofs and design the solutions to recursive relations. CO. 2 Explain Asymptotic Analysis and elaborate the methods of Amortized Analysis. Explain different algorithm design techniques like Divide and Conquer & Greedy strategy and make use of algorithms that employ this paradigm. Determine the Dynamic Programming paradigm and solve Dynamic Programming algorithms and simplify them. CO. 3 CO. 4 BLOOM’S TAXONOMY LEVEL 4 2, 5, 6 2, 5, 3 5, 6 CO. 5 Design and illustrate the different traversal techniques and build different graph computations. 2, 3, 6 CO. 6 Explain Polynomial and Non polynomial time complexities and elaborate the deterministic and non deterministic algorithms. 2, 5, 6

CONTENTS Strategies to Algorithm Design: Ø Greedy Method Ø Divide and Conquer Method Ø Dynamic Programming Method

GREEDY ALGORITHMS ØGreedy algorithms are short sighted algorithms ØDecision is based upon incomplete information. ØThe decision are made without bothering about its effect on later part of generated information. ØGreedy algorithms are most efficient algorithms if executed. ØExamples of Greedy Algorithms are Knapsack, Minimum Cost Spanning Trees, Single Source Shortest Path (SSSP) algorithms.

GREEDY ALGORITHMS ARE MOST EFFICIENT ALGORITHMS IF EXECUTED. . JUSTIFY B 5 10 Destination Source A D 10 C 2 Figure 1: Greedy Algorithm Decision

JUSTIFICATION ØAs seen in figure 1, since the information generated is not stored as well as no time is consumed in making the decision. ØTherefore execution speed of algorithm will be much higher. ØDue to this efficiency increases, hence statement is true.

D IVIDE AND ALGORITHMS DIVIDE ANDCONQUER ALGORITHMS ØIn this method, the given problem is divided into sub-problems till the sub-problem is easily solvable ØThe decision is continued until the sub-problems are solvable. ØEach sub-problem is solved using the same principle ØFor every sub-problem there will be a single solution ØHence, in Divide and Conquer (DAC) the size of the input is reduced so as to reduce the size of a problem. p 1 p 3 p 2 p 4

DYNAMIC PROGRAMMING ALGORITHMS ØDynamic Programming operates on principle of Optimality A Shortest Path B Optimal Path ØAssume that the path between A & B is optimal. In this Path, there is another path X-Y. A X Y B Optimal Path ØSince the path between A-B is optimal, therefore path X-Y is also optimal.

DYNAMIC PROGRAMMING ALGORITHMS ØIn Dynamic Programming strategy, the decision making is delayed till all the information is available. ØThis will require larger storage as well as the complexity of decision making process will increase.