Note for Design And Analysis Of Algorithm - DAA by Siddharth Rajput

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UNIT IV: ALGORITHMS Number of questions LAST SIX YEARS' GATE ANALYSIS 9 8 7 6 5 4 3 2 1 0 Marks 1 Marks 2 Total number of questions 2015 2014 2013 2012 2011 2010 Concepts on which questions were asked in the previous six years Year Concept 2015 Algorithms, Unsorted array, BFS, Spanning tree, Recurrence relation, Job sequence, Planar graph, Hash table, Complexity, Asymptotic notations, Polynomial time reducible, SAT Problem, Connected graphs, Algorithm design paradigms, Graphs 2014 Arithmetic expressions, Rooted tree, Graph, Depth-first search, Sorting algorithms and their complexity, Balanced binary, NP-complete problem, Hashing 2013 Sorting algorithms, Time complexity, NP problems, Bellman—Ford algorithm, Graphs, Heap sort, queue 2012 Time complexity, Recurrence relation, Balanced binary search tree, NP problem, Dijkstra’s algorithm, Merge sort 2011 Arrays, Binary heap, Dynamic programming, Time complexity, Hashing, Minimum spanning tree (MST) 2010 Undirected graphs, Spanning tree, Hashing Chapter 4.indd 175 4/9/2015 9:54:02 AM

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Chapter 4.indd 176 4/9/2015 9:54:02 AM

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CHAPTER 4 ALGORITHMS Syllabus: Algorithms: Analysis, Asymptotic notation, Notions of space and time complexity, Worst- and average-case analysis; Design: Greedy approach, Dynamic programming, Divide-and-conquer; Tree and graph traversals, Connected components, Spanning trees, Shortest paths; Hashing, Sorting, Searching. Asymptotic analysis (best, worst, average cases) of time and space, Upper and lower bounds, Basic concepts of complexity classes — P, NP, NP-hard, NP-complete. 4.1  INTRODUCTION 4.2.1  Properties of Algorithm There are certain properties algorithms have: In computer science, an algorithm provides a set of stepby-step instructions for solving various problems. The set of instructions should be unambiguous, finite and provide a clear termination point. An algorithm accepts some input values and produces outputs. The algorithm can be built in simple English language or in programming language. This subject helps in solving and analysing complex problems through logical thinking. 4.2  ALGORITHM An algorithm is a well-defined procedure which takes a set of values as an input and produces a set of values as the output. Chapter 4.indd 177           1.  An algorithm has zero or more inputs. 2.  It should produce at least one output. 3.  It should terminate within the finite time. 4.  It should be deterministic (ambiguous). 5.  It is generic to all programming languages. 4.2.2  Steps to Solve a Problem   1.  Define the problem, which means what is input and what should be the output.   2.  Identify the constraints which are to be satisfied.   3.  Design the algorithm for a given problem.   4.  Draw the flowchart.   5.  Verify the process.   6.  Implement the coding.   7.  Perform time and space analysis. 4/9/2015 9:54:03 AM

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178        Chapter 4:  Algorithms  4.2.3  Algorithm Analysis If a problem has more than one solution, then the technique used to decide which solution is best is known as algorithm analysis. 4.2.3.1 A Posteriori and A Priori Analysis Analysis is done on two parameters: time complexity and space complexity. Time complexity means time taken by an algorithm to solve a problem. Space complexity means memory space required by an algorithm to solve a problem. There are two types of analysis:   1.  A Posteriori Analysis: It is a software- and hardware-dependent analysis. It keeps on changing from one system to other system. This type of analysis tells the exact time and space complexity, meaning how much an algorithm takes time and space for solving a problem on the given system.   2.  A Priori Analysis: It is a software- and hardware-independent analysis. It is constant for all the systems. This type of analysis tells the approximate time and space complexity. It is the determination of the order of magnitude of a statement, meaning how many times a statement is executing. 4.2.3.2 Asymptotic Analysis Suppose there are given two algorithms for a task, how do we find out which one is better? This can be done by actually implementing both the algorithms and running them on the same computer with different input values. Then the comparison is made in terms of time. Algorithm that takes less time is considered better. But there are so many problems with this approach.   1.  It might be possible that for certain inputs first algorithms performs better than second and for other set of inputs second algorithm performs better. So,the decision of choosing best among them becomes difficult.   2. With the change in operating machine, performance of algorithm varies. On one machine first algorithm can work better and on another machine, the second works better. This is another problem associated with this approach. Example 4.1 Let us consider an example of a search problem (searching a given item) in a sorted array. One way to search is linear search (the order of growth is linear) and the other way is binary search. Suppose we have two machines with different processing speeds. We run linear search on the fast machine and binary search on slow machine. For small input size, the linear search will perform better. But as we increase the size of input, binary search will perform better. The reason behind this is,the order of growth of binary search is logarithmic and that of linear search is linear. So, by using this technique the issues of machine dependency are resolved. 4.2.4  Asymptotic Notation Asymptotic notation is based on two assumptions, which hold in most of the cases and have their importance. It  is very much important to understand the assumptions and limitations of asymptotic notations:   1.  Input size: It is interesting to know how the running time of an algorithm grows for a large input size n.   2.  Constant: The running time of an algorithm also depends on various constants. But for a large input n, the constant factors would be ignored. For representing best-case, average-case, and worst-case complexity, there are three different notations as given in the following sub-sections. 4.2.4.1 Theta Notation (Q) Theta notation shows the tightest upper bound and the tightest lower bound to the given function (Fig. 4.1). It is used to define the average-case running time. For a given function f(n), we denote Q(g(n)) as Q(g(n)) = {f(n): there exists positive constants c1 > 0, c2 > 0 and n0 > 0 such that 0 < c1(g(n)) ≤ f (n) ≤ c2 (g(n)) ∀ n ≥ n0} c1g(n) f(n) Above issues can be handled using asymptotic analysis techniques for analysing algorithms. In asymptotic analysis, analysis of algorithm in done in terms of input size. Actual running time is not computed, variation of time with input size is calculated. Now, understand that how the issues with above technique are resolved by asymptotic analysis technique using the below example: Chapter 4.indd 178 f(n) c2g(n) g(n) n n0 Figure 4.1 |   Theta notation (Q) 4/9/2015 9:54:04 AM