Design and Analysis of AlgorithmsIntroduction to Divide-and-conquer

Haidong XueSummer 2012, at GSU

Knowledge treeAlgorithms

Analysis DesignAlgorithms for

classic problemsClassic data

structure

Asymptotic notations

Probabilistic analysis

Sorting Shortest path

Matrix multiplication …

Divide & Conquer

GreedyDynamic

Programming

O(), o(), (), (), ()

Heap,Hashing,

Binary Search Tree,RBT,….

Quicksort,Heapsort,

Mergesort,…

… … …… … …

… … … … … … … … … … … ……

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– Eat all of them

• Case 2: a lot of cake– Cut off a small piece– and EatAllTheCake( the small piece)– Put the rest into a refrigerator – 24 housrs later, EatAllTheCake( the rest of the cake )

Base case

Recursive case

Solve a smaller problem

Solve a smaller problem

Do some other things

Do some other things

Smaller input size

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;– return n*fact2(n-1); Recursive case

Base 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;

fact2(3) n <- 3; m = fact2(2); n*m;

fact2(2) n <- 2; m = fact2(1); n*m;

fact2(1) return 1;

For each call, there is some storage used to save the current state

Can we save those space?

fact2(n)if(n<=1) return 1;return n*fact2(n-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 );

With a tail recursion, local variable need not to be saved

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;

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