Recursion,Divide and Conquer

Lam Chi Kit (George)

HKOI2007

Meaning of recursion

To recur means to happen again. In computer science, we say that a

subroutine (function or procedure) is recursive when it calls itself one or more times.

We call the programming technique by using recursive subroutine(s) as recursion.

The correctness, time complexity and space complexity can be proved by mathematical induction.

Concept of recursion

Consider the function.F()

temp←2*F()RETURN temp

When F() is called,F()=temp=2*F()It will not solve the equation. It continues.

=2*(2*F()) =2*(2*(2*F())) ...

temp

temp

temp

Stack

Concept of recursion

The concept of parameter(s) is needed. This can be implemented by:

Pass by value Pass by pointer Pass by reference Use of global variable

Concept of recursion

Consider the functionF(n)

temp←n*F(n-1)RETURN temp

When F(1) is called,F(1)=temp=1*F(0) =1*(0*F(-1))It will not simplify the expression. It continues.

=1*(0*(-1*F(-2))) ...

n, temp

n, temp

n, temp

Stack

Concept of recursion

The concept of terminating condition(s) is/are needed.

It is determined by the parameter(s).

Concept of recursion

Consider the functionF(n)

IF (n=0)RETURN 1

ELSERETURN n*F(n-1)

When F(2) is called,F(2)=2*F(1) =2*(1*F(0)) =2*(1*(1)) =2*(1*1) =2*1 =2

n

n

n

Stack

Structure of recursion

Similar to mathematic induction, recursion requires the following two components Recurrence relation(s) Base case(s)

Example

F(n)

IF (n=0) RETURN 1 //Base Case

ELSE RETURN n*F(n-1)

//Recurrence Relation

Recurrence in mathematical functions Factorial

_ Permutation

_ Combination

_ Pascal relationship

_ Integer power

_

)!1(! nnn

rnrn CrnC 1)/(

rnrnrn CCC 111

rnrn PnP 1

1 nn xxx

Problem Discussion

S is a string of N bits (either ‘0’ or ‘1’). Furthermore, there is no two consecutive ‘0’s in S. Find the number of solutions to S. Denote S(N) as any strings of N bits with no two

consecutive ‘0’s. When N>=2, N is either

Beginning with ‘1’, followed by S(N-1) Beginning with ’01’, followed by S(N-2)

S(1) must either be ‘0’ or ‘1’. S(0) is the null string ‘’.

Problem Discussion

Let f(N) be the number of solutions to S(N).

2when )2()1(

1when 2

0when 1

)(

NNfNf

N

N

Nf

Problem Discussion

This puzzle is called the Tower of Hanoi. There are three pegs and N disks of different

sizes. Initially all disks are stacked on the same peg,

with the smaller disk on top of the larger disk. In each step, you can move a disk from the top of

a stack to the top of another peg, provided that this will not result in a larger disk stacked on top of a smaller disk.

Your goal is to move all disks to another peg.

Problem Discussion

Suppose N=4

Problem Discussion

Suppose procedure MOVE_DISK(N,A,B) moves a single disk N from

A to B within one step. F(N,X,Y,Z) can move a stack of N disks from peg

X to peg Z via peg Y F(N,X,Y,Z) where N≥1

F(N-1,X,Z,Y) MOVE_DISK(N,X,Z) F(N-1,Y,Z,X)

Problem Discussion

Given a sequence of n distinct integers. Output all permutations of the sequence.

Let F(n) be a procedure that permute a[1..n]. When n>0, for all 1≤i ≤ n, swap a[i] to a[n] and then permute

a[1..n-1] by calling F(n-1)F(n) IF(n=0) OUTPUT ELSE FOR i = 1 to n SWAP(a[i],a[n]) F(n-1) SWAP(a[i],a[n])

Problem Discussion

Suppose a[ ]=(7,8,9)F(0) * * * * * *F(1) i=1 i=1 i=1 i=1 i=1 i=1F(2) i=1___i=2___ i=1___i=2___ i=1___i=2___F(3) i=1____________i=2____________i=3____________

i

i

i

Stack

9

23

a[ ] 87

1

987 987

21

1

Output: 897, 987, 978, 798, 879, 789

Complexity of recursion

First Order Linear Recurrence

In particular, suppose a>1, c≠-1

0)()1()(

0)0()(

nnhnfng

nhnf

n

i

n

ij

jgihnf0 1

)()()(

)()()1()(

)()()()1()( 1

n

cc

anfbnfanf

nnfnnfnf

Complexity of recursion

Recurrence Iteration and Recursion Tree

_f(4)=1+2f(3)=1+2+4f(2)=1+2+4+8f(1)=1+2+4+8+f(0)

1)1(2)( nfnf

Complexity of recursion

_f(8)=8+f(7)=8+7+f(6)=8+7+6+f(5)=8+7+6+5+f(4)=8+7+6+5+4+f(3)=8+7+6+5+4+3+f(2)=8+7+6+5+4+3+2+f(1)= 8+7+6+5+4+3+2+1+f(0)

nnfnf )1()(

Complexity of recursion

_f(256)=256+f(128)=256+128+f(64)=256+128+64+f(32)=256+128+64+32+f(16)=256+128+64+32+16+f(8)=256+128+64+32+16+8+f(4)=256+128+64+32+16+8+4+f(2)=256+128+64+32+16+8+4+2+f(1)

nn

fnf )2

()(

Complexity of recursion

_

f(32)=32+2(f(16))=32+2(16)+4(f(8))=32+2(16)+4(8)+8(f(4))=32+2(16)+4(8)+8(4)+16(f(2))=32+2(16)+4(8)+8(4)+16(2)+32(f(1))

nn

fnf )2

(2)(

Complexity of recursion

Master Theorem

In particular, suppose b>1,___

1,1 where)()/()( banfbnaTnT

0 somefor )()( if ))(()(

)()( if )log()(

0 somefor )()( if )()(

log

loglog

loglog

a

aa

aa

b

bb

bb

nnfnfnT

nnfnnnT

nnfnnT

)log()()/()(

)()()/()(

nnnfnbnbTnf

nnfnbnTnf

Problem Discussion

Compute Bp mod M B,P are integers in [0,2147483647] M is an integer in [1,46340]

f(p)=B*f(p-1) Bp= B*Bp-1

Θ(p)

Problem Discussion

Notice that

Θ(log p)

odd is when )(

even is when )(22/

22/

BBB

BBB

p

pp

Properties of recursion

NP solutions, including exponential, factorial and combinatorial, are usually easy to be implemented by recursion.

It requires more memory than iteration. Every recursive function can be transformed

into iterative function. Every iterative function can be transformed

into recursive function.

Meaning of Divide and Conquer In military, “divide and conquer” is a kind of

strategy. In computer science, “divide and conquer” is

a programming technique by breaking down a problem into one or more sub problems.

“Divide and conquer” is usually implemented by recursion.

Structure of Divide and Conquer Divide

Break a problem into sub problem(s) Conquer

Solve all sub problem(s) Combine

Solve the problem using the results of the sub problem(s)

Problem Discussion

You are given a 2k* 2k square board with one square taken away.

An “L piece” is made up of 3 squares sharing a common corner. You can freely rotate the any of these pieces.

You are required to cover the board with these pieces without overlapping.

Problem Discussion

Example

Problem Discussion

Illustration

Recursion outside Computer Science Names

PHP : PHP Hypertext Preprocessor GNU : GNU is Not Unix

Mathematics Definition of the set of non negative numbers N

0 is in N If n is in N, then n+1 is in N.

Images Fractals

Challenge Problems

Consider the following recursive function defined on pairs of nonnegative numbers.

Find the closed form of the function. Prove your answer algebraically.

otherwise1)1,1(

0when 0

0when 0

),(

nmnmf

n

m

nmf

Challenge Problems

Consider the following recursive function defined on triple of nonnegative numbers.

Find the closed form of the function. Prove your answer geometrically.

otherwise1

)1,1,1(0when

0when

0when

),,(

cbaacbcab

cbafcc

bb

aa

cbaf

Challenge Problems

Find a closed form solution for the nth Fibonacci Number.

Prove the Master Theorem.

Challenge Problems

Consider the following algorithm. F(x,p)

IF (p=0) RETURN 1 IF (p=1) RETURN x BINARY_SEARCH for the largest int q s.t. q2≤p RETURN F(F(x,q),q)*F(x,p-q2)

What is the purpose of F(x,p)? Prove your answer.

Challenge Problems

Describe an efficient non recursive algorithm to compute Bp mod M. B,P are integers in [0,2147483647] M is an integer in [1,46340]

State the time and space complexity.

Challenge Problems

Consider the problem “Tower of Hanoi”. Describe an algorithm which requires the

minimum number of moves. Prove that it requires the minimum number of moves.

Show that this sequence of moves is unique. Calculate the minimum number of moves

required in terms of n.

Challenge Problems

Consider the problem “Tower of Hanoi”. Suppose you have to move a stack from Peg 0 to

Peg 2 via Peg 1. Suppose the ith largest disk is on Peg pi(t) after t

moves. Denote hi(t) be the number of disks under the ith

largest disk after t moves. Consider the algorithm that requires the minimum

number of moves. Show that pi(t)+hi(t)+i is always odd.

Challenge Problems

Consider the problem “L pieces”. Explain whether it is possible to have a faster

algorithm in terms of complexity. Calculate the worst case minimum of colours

required such that no two pieces sharing a common edge has the same colour.

Reference

http://en.wikipedia.org/ http://www.hkoi.org/