Transcript Lecture 7
CIT 590 Recursion Agenda • Style ‘rules’ • More on recursion • Divide and conquer • Some thoughts on running time Style ‘rules’ • http://www.cis.upenn.edu/~matuszek/cit590- 2013/Pages/style-rules.html • http://www.cis.upenn.edu/~matuszek/cit5902013/Pages/programming-hints.html • Before submitting your assignments, go through a code review process with your partner. • Ideally, look at the same large screen • Skype/Facetime/other communication Recursion • Ask yourself one question ‘Does solving a smaller problem help me solve the bigger one’ • Lists are naturally recursive • A list of n elements contains in it a list of n-1 elements • Python makes extraction of portions of the list super easy List recursion examples • Sum of elements of a list • Finding max of a list • Replicate - I want to initialize a list with repeats of a certain number Divide and Conquer • Divide the problem into smaller problem with (usually) similar structure • Solve the smaller problems • Now aggregate/rollup the smaller solutions to get the solution to the original problem Classic divide and conquer = mergesort • Sort a list • If I divide the list into 2 halves and sort each half does that help? • 2 sorted lists can be merged mergesort 1.#sorting example 2.def mergeSort(a): 3. #base case/ simple case 4. if len(a)== 0 or len(a) == 1: 5. return a 6. else: 7. #divide 8. firstHalf =mergeSort(a[:len(a)/2]) 9. secondHalf=mergeSort(a[len(a)/2:]) 10. #and conquer! 11. return merge(firstHalf,secondHalf) Merge • Making a larger sorted array from 2 smaller sorted ones Towers of Hanoi (and complexity of recursion) • Classic recursion problem involving 3 towers and discs of different sizes • http://www.dynamicdrive.com/dynamicindex12/towerhanoi .htm • The recursive solution involves recursing on the largest disc’s movement • Show how this problem is exponential in nature How to solve recurrences • This is an informal method. Real proofs will be done by induction or using some theorems. • But, you can usually get a good ‘feel’ for the complexity Sum of elements of a list T(n) = T(n-1) + k, because we can assume it takes some small constant unit of time to add 2 numbers Solving the recurrence for Towers of Hanoi M(n) = Move n-1 disc to the extra tower, move the biggest disc to the destination tower, move the n-1 discs from the extra tower to the destination tower. M(n) = M(n-1) + 1 + M(n-1) = 2M(n-1) + 1 = 2(2M(n-2) + 1) + 1 = 4M(n-2) + 2 + 1 = 4 (2M(n-3) + 1) + 2 + 1 = 8M(n-3) + 4 + 2 + 1 So the number of moves is following the series 1 + 2 + 4 + 8 + …+ which is bounded by 2n Getting all permutations of a string • Take out the first character of the string • Make permutations from all the other characters • Now put the first character in all possible places for each of the permutations of the ‘n-1’ characters Caution • Recursive growth with things like Fibonacci, towers of Hanoi, permutation printing • Write base cases first • In fact if you are doing TDD, why not write these as your first test cases!