Design and Analysis of Algorithms
Introduction to Divide-and-conquer
Haidong Xue
Summer 2012, at GSU
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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