Lecture 01-2.ppt
- 1. CS-311 Design and Analysis of Algorithms Muhammad Naman
- 2. Objectives Design algorithms using different algorithms design techniques i.e. Divide and Conquer, Dynamic Programming, Greedy Algorithms & Backtracking etc for different problem areas (sorting, pattern matching, graphs, compression, encryption etc.) Analyse Algorithms (estimate upper & lower bounds without coding and running the algorithms) and compare the efficiency of more than one algorithm for a problem. Logically think and develop problem solving skills Read and understand research papers in this area
- 3. What is an Algorithm? Although there is no universally agreed-on wording to describe this notion, there is general agreement about what the concept means: An algorithm is a sequence of unambiguous instructions for solving a problem, i.e., for obtaining a required output for any legitimate input in a finite amount of time.
- 4. What is an Algorithm?
- 5. What is an Algorithm? The reference to "instructions" in the definition implies that there is something or someone capable of understanding and following the instructions given. We call this a "computer," keeping in mind that before the electronic computer was invented, the word "computer" meant a human being involved in performing numeric calculations. Nowadays, of course, "computers" are those ubiquitous electronic devices that have become indispensable in almost everything we do.
- 6. illustrate several important points As examples illustrating the notion of algorithm, we consider three methods for solving the same problem (detail in Lecture 02): Computing the greatest common divisor of two integers. These examples will help us to illustrate several important points: The non-ambiguity requirement for each step of an algorithm cannot be compromised.
- 7. illustrate several important points The range of inputs for which an algorithm works has to be specified carefully. The same algorithm can be represented in several different ways. Several algorithms for solving the same problem may exist. Algorithms for the same problem can be based on very different ideas and can solve the problem with dramatically different speeds.
- 8. Process to Solve a Problem Understand the problem Formulate a solution / algorithm Design a program Implement the program Execute the code Measure the performance See if the solution is ok
- 9. Sorting Problem Consider the problem of sorting numbers. INPUT: Sequence of n numbers <a1,a2,a3, ….an> OUTPUT: Permutation (reordering) <a1`,a2`,a3`,….an`> of the input sequence such that a1`<a2`<a3`<…..<an` Many algorithms are available.
- 10. Selection Sort 1. Get a list of unsorted numbers 2. Set a marker for the unsorted section at the front of the list 3. Repeat steps 4 - 6 until one number remains in the unsorted section 4. Compare all unsorted numbers in order to select the smallest one 5. Swap this number with the first number in the unsorted section 6. Advance the marker to the right one position 7. Stop
- 11. Merge Sort Algorithm Divide sequence of m elements into two sequences of m/2 elements Conquer both sub-sequences using Merge Sort (recursively) Combine two sorted sub-sequences using merge
- 12. Sorting Problem….. Which sorting algorithm is best. Implement both algorithms See the result of simulation
- 13. Sorting Problem….. Sorting Analysis, processor speed 500MHz Data Size Selection Sort Merge Sort 10 K 1 0 20 K 4 0 30 K 11 0 50 K 31 0 75 K 72 0 1 Lac 132 0 2 Lac 586 1 1 Million 10 2 Million 21 3 Million 33 5 Million 60 7.5 Million 90 10 Million 136 Execution Time in Seconds
- 14. Sorting Problem….. Sorting Analysis 0 100 200 300 400 500 600 700 1 0 K 3 0 K 7 5 K 2 L a c 2 M i l l i o n 5 M i l l i o n 1 0 M i l l i o n Data Size Execution time in seconds Data Size Selection Sort Merge Sort
- 15. Process to Solve a Problem Understand the problem Formulate a solution / algorithm Analyze the algorithm Design a program Implement the program Execute the code Measure the performance See if the solution is ok
- 16. Algorithm Analysis How to analyze an algorithm? Predict the resources that the algorithm requires Memory Communications Bandwidth Logic gates etc Most important is Computational Time
- 17. Algorithm Analysis Important thing before analysis Model of the machine upon which the algorithms is executed. Random Access Machine (RAM) (Sequential) Running Time: No. of primitive operations or “steps executed”.
- 18. Algorithm Analysis How do we write algorithms? Pseudo Code: Similar construct / keywords as in a high level programming languages, e.g. in C, Pascal etc. Structured semantics of the high level languages without caring about the syntactic errors / grammatical rules
- 19. Algorithm Analysis How much time each construct / keyword of a pseudo code takes to execute. Assume it takes ti (the ith construct) Sum / Add up the execution time of all the constructs / keywords. if there are m constructs then m i i t T 1
- 20. Algorithm Analysis That will be the execution time for a given input size (say n) m i i n t T 1 Running time as the function of the input size T(n)
- 21. Algorithm Analysis What are the constructs / Keywords. Time for each construct Total Time Total time as a function of input size
- 22. Algorithm Analysis Construct: Sequence Selection Iterations Recursion
- 23. Algorithm Analysis Sequence Statements: Just add the running time of the statements If-Then-Else: if (condition) S1 else S2 Running time of the test plus the larger of the running times of S1 and S2. Iteration is at most the running time of the statements inside the loop, (including tests) times the number of iterations. Nested Loops: Analyze these inside out. The total Running time of a statement inside a group of nested loops is the running time of the statement multiplied by the product of the size of all the loops. Function Calls: Analyzing from inside to out. If there are function calls, these must be analyzed first.
- 24. Home Work Read Mathematics Revision Handouts available at photocopy shop Solve Exercises Given at the end Assignment # 1 Due date Monday (12 September 2022) at Commencement of Class.