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Design and Analysis of Algorithms Introduction to Divide-and-conquer Haidong Xue Summer 2012, at GSU Knowledge tree Algorithms Analysis Asymptotic notations O(), o(), (), (), () … Sorting Probabilisti c analysis Divide & Conquer … … Shortest path Matrix multiplication … Heap, Hashing, Binary Search Tree, RBT, …. Dynamic Programming Quicksort, Heapsort, Mergesort, … … … … … Greedy Classic data structure Algorithms for classic problems Design … … … … … … … … … … … … What is divide-and-conquer? • An example problem: “eat some cake” • Description: given some amount of cake, eat all of them • When the cake is like: What is divide-and-conquer? • When the cake is like…. • Hint: you need not to eat all of them at the first time What is divide-and-conquer? • Any ideas? • Case 1: a small piece of cake Smaller input size Base case – Eat all of them • Case 2: a lot of cake Recursive case – Cut off a small piece Do some other things – and EatAllTheCake( the small piece) Solve a smaller problem – Put the rest into a refrigerator Do some other things – 24 housrs later, EatAllTheCake( the rest of the Solve a smaller problem cake ) What is divide-and-conquer? • Textbook definition: – Divide the problem into a number of subproblems that are smaller instances of the same problem – Conquer the subproblems by solving them recursively. If the subproblem sizes are small enough, however, just solve the subproblems in a straight forward manner. – Combine the solutions to the subproblems into the solution for the original problem Pros and cons • Pros: – Many problems can be solved by a straightforward divide-and-conquer algorithm • Cons: – Time complexity is not always good An example: calculate factorial of n • • • • Problem: calculate n! What is n!? What is the iterative algorithm? fact1(n) – f=1; – for i = n to 1 – f = f * i; – return f; An example: calculate the factorial of n • What is the divide-and-conquer algorithm? • fact2(n) – if(n<=1) return 1; Base case – return n*fact2(n-1); Recursive case Smaller problem Combination Let’s try it in Java Tail recursion • What happen when fact2(5)? fact2(5) n <- 5; m = fact2(4); n*m; fact2(4) n <- 4; m = fact2(3); n*m; For each call, there is some storage used to save the current state Can we save those space? fact2(3) n <- 3; m = fact2(2); n*m; fact2(n) if(n<=1) return 1; return n*fact2(n-1); fact2(2) n <- 2; m = fact2(1); n*m; fact2(1) return 1; Tail recursion • Tail recursion: there is only one self call, and it is the last operation. • E.g.: • fact3(n, currentFactotial) – if(n<=1) return currentFactotial; – return fact3(n-1, currentFactotial*n); • What is the first call? – fact3(n, 1); • What happened in fact3? Tail recursion • What happen when fact3(5)? fact3(5, currentFactotial) m <- currentFactotial*5; fact3(4,m ); fact3(4, currentFactotial) m <- currentFactotial*4; fact3(3,m ); fact3(3, currentFactotial) m <- currentFactotial*3; fact3(2,m ); fact3(2, currentFactotial) m <- currentFactotial*2; fact3(1,m ); fact3(1, currentFactotial) With a tail recursion, local return currentFactotial; variable need not to be saved fact3(n, currentFactotial) if(n<=1) return currentFactotial; return fact3(n-1, currentFactotial*n); Let’s improve the code More exercises-print all strings • Problem: – Each character could be: a, b, c, d or e. – Print all the possible n character strings • printAllStrings( int n, string head) – if(n==0) • print head; – else • printAllStrings(n-1, head+”a”); More exercises – tower of Hanoi • Tower of Hanoi: move all disks to the third rod • Only one disk may be moved at a time. • No disk may be placed on top of a smaller disk. More exercises – tower of Hanoi • What is the algorithm to solve tower of Hanoi problem? • Hanoi( number of disks n, current rod c, destination rod d, auxiliary rod a ) – if(n==0) //nothing – else • Move top n-1 disks to auxiliary rod: Hanoi(n-1, c, a, d) • Move the last disk to destination rod • Move the n-1 disks from auxiliary rod to destination rod: Hanoi(n-1, a, d, c) Let’s try it in Java More exercises • Have you seen other algorithms? – Quicksort – Mergesort – Heapify – Heapsort …. Analysis of divide-and-conquer algorithms • Recurrence • Solve recurrence