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Design and Analysis of Algorithms Introduction to Divide-and-conquer

Design and Analysis of Algorithms Introduction to Divide-and-conquer. Haidong Xue Summer 2012, at GSU. Knowledge tree. Algorithms. Classic data s tructure. Analysis. Algorithms for classic problems. Design. …. Sorting. Shortest path. Matrix multiplication. Asymptotic notations.

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Design and Analysis of Algorithms Introduction to Divide-and-conquer

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  1. Design and Analysis of AlgorithmsIntroduction to Divide-and-conquer HaidongXue Summer 2012, at GSU

  2. Knowledge tree Algorithms Classic data structure Analysis Algorithms for classic problems Design … Sorting Shortest path Matrix multiplication Asymptotic notations Probabilistic analysis Heap, Hashing, Binary Search Tree, RBT, …. Dynamic Programming Greedy Divide & Conquer O(), o(), (), (), () Quicksort, Heapsort, Mergesort, … … … … … … … … … … … … … … … … … … … …

  3. What is divide-and-conquer? • An example problem: “eat somecake” • Description: given some amount of cake, eat all of them • When the cake is like:

  4. What is divide-and-conquer? • When the cake is like…. • Hint: you need not to eat all of them at the first time

  5. What is divide-and-conquer? Smaller input size • Any ideas? • Case 1: a small piece of cake • Eat all of them • Case 2: a lot of cake • Cut off a small piece • andEatAllTheCake( the small piece) • Put the rest into a refrigerator • 24 housrs later,EatAllTheCake( the rest of the cake ) Base case Recursive case Do some other things Solve a smaller problem Do some other things Solve a smaller problem

  6. 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

  7. Pros and cons • Pros: • Many problems can be solved by a straightforward divide-and-conquer algorithm • Cons: • Time complexity is not always good

  8. 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;

  9. An example: calculate the factorial of n • What is the divide-and-conquer algorithm? • fact2(n) • if(n<=1) return 1; • return n*fact2(n-1); Base case Recursive case Smaller problem Combination Let’s try it in Java

  10. Tail recursion • fact2(n) • if(n<=1) return 1; • return n*fact2(n-1); • What happen when fact2(5)? fact2(5) n <- 5; m = fact2(4); n*m; fact2(4) n <- 4; m = fact2(3); n*m; fact2(3) n <- 3; m = fact2(2); n*m; fact2(2) n <- 2; m = fact2(1); n*m; For each call, there is some storage used to save the current state fact2(1) return 1; Can we save those space?

  11. 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?

  12. 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) return currentFactotial; With a tail recursion, local 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

  13. 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”);

  14. 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.

  15. 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

  16. More exercises • Have you seen other algorithms? • Quicksort • Mergesort • Heapify • Heapsort ….

  17. Analysis of divide-and-conquer algorithms • Recurrence • Solve recurrence

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