6-Python-Recursion PPT.pptx
- 2. Recursive Functions Recall factorial function: IterativeAlgorithm Loop construct (while) can capture computation in a set of state variables that update on each iteration through loop
- 3. Recursive Functions factorial(n) = n * factorial(n-1) Alternatively: Consider 5! = 5x4x3x2x1 can be re-written as 5!=5x4! In general n! = nx(n-1)!
- 4. Recursive Functions Alternatively: Consider 5! = 5x4x3x2x1 Recursive Algorithm can be re-written as 5!=5x4! function calling itself In general n! = nx(n-1)! factorial(n) = n * factorial(n-1)
- 8. Recursive Functions Execution trace for n = 4 fact (4) 4 * fact (3) 4 * (3 * fact (2)) 4 * (3 * (2 * fact (1))) 4 * (3 * (2 * (1 * fact (0)))) 4 * (3 * (2 * (1 * 1))) 4 * (3 * (2 * 1)) 4 * (3 * 2) 4 * 6 24 • No computation in first phase, only function calls • Deferred/Postponed computation – after function calls terminate, computation starts • Sequence of calls have to be remembered Courtesy Prof PR Panda CSE Department IIT Dehi
- 11. Recursive Functions • Size of the problem reduces at each step • The nature of the problem remains the same • There must be at least one terminating condition • Simpler, more intuitive – For inductively defined computation, recursive algorithm may be natural • close to mathematical specification • Easy from programing point of view • May not efficient computation point of view
- 12. GCD Algorithm gcd (a, b) = b if a mod b = 0 gcd (b, a mod b) otherwise Iterative Algorithm RecursiveAlgorithm
- 13. GCD Algorithm gcd (a, b) = b if a mod b = 0 gcd (b, a mod b) otherwise gcd (6, 10) gcd (10, 6) gcd (6, 4) gcd (4, 2) 2 2 2 2 Courtesy Prof PR Panda CSE Department IIT Dehi
- 14. Fibonacci Numbers fib (n) = 0 1 n = 1 n = 2 fib (n-1) + fib (n-2) n > 2 RecursiveAlgorithm Courtesy Prof PR Panda CSE Department IIT Dehi
- 15. Fibonacci Numbers fib (6) fib (5) fib (4) fib (3) fib (2) fib (2) fib (1) fib (3) fib (2) fib (1) fib (4) fib (3) fib (2) fib (2) fib (1) Courtesy Prof PR Panda CSE Department IIT Dehi
- 16. Power Function • Write a function power (x,n) to compute the nth power of x power(x,n) = 1 if n = 0 x * power(x,n-1) otherwise
- 17. Power Function • Efficient power function • Fast Power – fpower(x,n) = 1 for n = 0 – fpower(x,n) = x * (fpower(x, n/2))2 if n is odd – fpower(x,n) = (fpower(x, n/2))2 if n is even
- 18. Power Function • Efficient power function
- 19. Towers of Hanoi Problem • 64 gold discs with different diameters • Three poles: (Origin, Spare, Final) • Transfer all discs to final pole from origin – one at a time – spare can be used to temporarily store discs – no disk should be placed on a smaller disk • Intial Arrangement: – all on origin pole, largest at bottom, next above it, etc. Origin Courtesy Prof PR Panda CSE Department IIT Dehi Spare Final
- 20. 3-Step Strategy Origin Spare Final Origin Spare Final Origin Spare Final Origin Spare Final Solve for (n-1) disks. Move to Spare. Use Final as Spare. Move bottom disk to Final Move (n-1) disks to Final. Use Origin as Spare. Courtesy Prof PR Panda CSE Department IIT Dehi
- 21. Recursive Solution Courtesy Prof PR Panda CSE Department IIT Dehi • Use algorithm for (n-1) disks to solve n-disk problem • Use algorithm for (n-2) disks to solve (n-1) disk problem • Use algorithm for (n-3) disks to solve (n-2) disk problem • ... • Finally, solve 1-disk problem: – Just move the disk!
- 22. Problem solving with Top Down Design Top down design approach, also called as Stepwise refinement, is essential to develop a well structured program. This approach is a problem solving technique that systematically breaks a complicated problem into smaller, more manageable pieces. If any of these subproblems is not easier to solve, we further divide the subproblem into smaller parts. We repeat the subdividing process until all small parts are not dividable. Then, we can use a few lines of code to solve every trivial problem. In the end, we put all these little pieces together as a solution to the original problem. This approach makes it easy to think about the problem, code the solution read the code and identify bugs.
- 23. Steps to solve a problem using Top Down Design: 1. Start with an initial problem statement. 2. Define subtask at the first level. 3. Divide subtasks at a higher level into more specific tasks. 4. Repeat step (3) until each subtask is trivial. 5. Refine the algorithm into real code.
- 24. Business requirement Analysis Initial Problem statement Read new emails from an email box and save email messages into a database table.
- 27. Case study: Gathering Information from a file system Import os os.mkdir(“d:tempdir”)
- 28. FILES Open( ) Syntax file_object = (“filename.txt”, “Access Mode”) File pointer points to the content of file
- 29. Access Modes r – read mode only for reading the content file pointer points at the beginning of the file file should exist before opening in read mode