Partha Sarathi Mandal

Transcription

1 MA 252: Data Structures and Algorithms Lecture 1 Partha Sarathi Mandal Dept. of Mathematics, IIT Guwahati

2 Time Table D / T 8-8:55 9-9: : :55 2-4:55 Sun: Mon: MA252 Tue: MA252 MA253 Wed: Thu: Fri: MA252 MA252 Class Room MA252: 1102, MA253: In Lab (Tutorial: 2-3PM + Comp. Lab) TA: Mr. Barun Garain

3 Tests and Marks distribution MA252: Data Structures and Algorithms Mid semester 30% Quizzes 20% End Semester 50% MA253: Data Structures Lab Quiz + Assignments + attendance 40% Mid semester 30% End semester 30%

4 Laboratory Laboratory attendance is mandatory. You must be punctual. There may be pop quizzes in the labs. (A pop quiz is a quiz or test given to the class without prior warning or announcement)

5 Cheating Gets You equivalent -ve marks of the Question Specific Examples Of Cheating: If you help somebody for cheating or providing your login passwd. If you're going to copy someone's assignment or copy code during test.

6 Reference Books Cormen, Leiserson and Rivest, Introduction to Algorithms, Prentice Hall-India. Horowitz, Sahni, Mehta, Fundamentals of Data Structures in C++, Galgotia. A. H. Aho, J. E. Hopcroft and J. Ullman, Data Structures and Algorithms, Addison-Wesley. Bjarne Stroustrup, The C++ Programming Language, Addison-Wesley (Third Edition & Special Edition).

7 Outline of the course MA252 Asymptotic notation; Sorting - merge sort, heap sort, priority queue, quick sort, sorting in linear time, order statistics; Data structures - heap, hash tables, binary search tree, balanced trees (red-black tree, AVL tree); Algorithm design techniques - divide and conquer, dynamic programming, greedy algorithm, amortized analysis; Elementary graph algorithms, minimum spanning tree, shortest path algorithms.

8 Data Structures and Algorithms Algorithms: Sequence of finite number of steps to solve a given problem. Programs: Implementation of an algorithm in a programming language. Data Structures: Organization of data needed to solve the problem.

9 Algorithm Input instance ALGORITHM Output related to input

10 What is a good algorithm? Efficient algorithm with respect to run time time complexity with respect to memory requirement space complexity Efficiency of an algorithm function of input size.

11 Running Time Running Time Most algorithms transform input objects into output objects. The running time of an algorithm typically grows with the input size. Average case time is often difficult to determine. We focus on the worst case running time. Easier to analyze. Crucial to applications such as games, finance and robotics best case average case worst case Input Size

12 Measure of the running time of an Experiential study algorithm Implement using some programming language Clock the CPU time for an algorithm In C++, you may measure time as follows: #include <ctime> clock_t start, finish; double time; start = clock(); sort something finish = clock(); time =(double(finish)-double(start))/clocks_per_sec;

13 Experimental study and limitation It is necessary to implement the algorithm, which may be difficult and time consuming. Limited set of inputs You have to check with an input instance for which execution time is maximum In order to compare two algorithms, the same hardware and software environments must be used.

14 Beyond experimental studies It is possible to develop a general methodology for analyzing running time of algorithms with a high-level description Without implementations Takes into account of all possible inputs Independent of environment

15 Pseudo-Code It is high-level programming concepts used to write in natural language which describe the main ideas of an algorithm. Example: Algorithm arraymax(a,n): Input: An array A storing n integers: Output: maximum element in A currentmax A[0] for I 1 to n-1 do If currentmax < A[i] then cuttentmax A[i] return currentmax

16 Analysis of algorithms Primitive operations: Data movement (assignments) Control (loop, branch, function call, return) Arithmetic and logical operations (add, subtract, comparison, etc.) We have to count numbers of primitive operations in a pseudo-code for an algorithm

17 The Random Access Machine (RAM) Model Machine-independent algorithm design depends upon a hypothetical computer called the Random Access Machine or RAM. Each simple operation (+, *, -, =, if, call) takes exactly 1 time step. Loops and subroutines are not considered simple operations. Instead, they are the composition of many single-step operations. Each memory access takes exactly one time step, and we have as much memory as we need. The RAM model takes no notice of whether an item is in cache or on the disk, which simplifies the analysis.

18 Pseudo-Code By inspecting the pseudo code, we can determine the maximum number of primitive operations executed by an algorithm, as a function of the input size. Example: Algorithm arraymax(a,n): Input: An array A storing n integers: Output: maximum element in A currentmax A[0] for i 1 to n-1 do If currentmax < A[i] then cuttentmax A[i] return currentmax # operation 2 2n 2(n-1) 2(n-1) 1 Total 6n 1

19 Estimating Running Time Algorithm arraymax executes 6n-1 primitive operations in the worst case. Define: a = Time taken by the fastest primitive operation b = Time taken by the slowest primitive operation Let T(n) be worst-case time of arraymax. Then a (6n-1) <= T(n) <= b (6n-1) Hence, the running time T(n) is bounded by two linear functions.

20 Growth Rate of Running Time Changing the hardware/ software environment Affects T(n) by a constant factor, but Does not alter the growth rate of T(n) The linear growth rate of the running time T(n) is an intrinsic/basic property of algorithm arraymax

21 The Growth Rate of the Six Popular functions n logn n nlogn n 2 n 3 2 n ,096 65, ,024 32,768 4,294,967, , , * ,384 2,097, * ,048 65,536 16,777, * , , ,217, * ,024 10,240 P. S. Mandal, 1,048,576 IITG 1,073,741, *