Download Recursion, Divide and Conquer

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 function
F(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
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
The concept of terminating condition(s) is/are
needed.
It is determined by the parameter(s).
Concept of recursion

Consider the function
F(n)
IF (n=0)
RETURN 1
ELSE
RETURN n*F(n-1)
n
n
n
Stack

When F(2) is called,
F(2)=2*F(1)
=2*(1*F(0))
=2*(1*(1))
=2*(1*1)
=2*1
=2
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
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Factorial
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Permutation
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r
 nn1 Pr
_C
n
r
 (n / r )n1 Cr
Pascal relationship


_P
n
Combination

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n_! n  (n  1)!
_C
n
r
 n1 Cr 1  n1Cr
Integer power

x  x x
_n
n 1
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.

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Denote S(N) as any strings of N bits with no two
consecutive ‘0’s.
When N>=2, N is either
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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).
1
when N  0


f (N )  
2
when N  1
 f ( N  1)  f ( N  2) when N  2

Problem Discussion
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This puzzle is called the Tower of Hanoi.
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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
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Suppose N=4
Problem Discussion
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Suppose procedure
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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
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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=1
F(2)
i=1___i=2___ i=1___i=2___ i=1___i=2___
F(3) i=1____________i=2____________i=3____________
a[ ]
7
9
8
7
8
9
7
9
8
Output:
897, 987, 978, 798, 879, 789
i
1
i
1
2
i
1
3
2
Stack
Complexity of recursion
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First Order Linear Recurrence
h(0)
n0

f ( n)  
 g (n) f (n  1)  h(n) n  0
n
n
i 0
j i 1
f (n)   h(i)  g ( j )

In particular, suppose a>1, c≠-1
f (n)  f (n  1)  (n c )  f (n)  (n c 1 )
f (n)  a  f (n  1)  b  f (n)  (a n )
Complexity of recursion
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_
Recurrence Iteration and Recursion Tree
f (n)  2 f (n  1)  1
f(4)
=1+2f(3)
=1+2+4f(2)
=1+2+4+8f(1)
=1+2+4+8+f(0)
Complexity of recursion
_f (n)  f (n  1)  n
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)

Complexity of recursion
n
_ f ( n)  f ( )  n
2
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)

Complexity of recursion

_ f ( n)  2 f (
n
)n
2
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))
Complexity of recursion
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Master Theorem
T (n)  aT (n / b)  f (n) where a  1, b  1
logb a
logb a 
T ( n )  ( n
) if f (n)  (n
) for some   0
T ( n )  ( n
logb a
log n) if f (n)  (n
logb a
)
T (n)  ( f (n)) if f (n)  (n logb a  ) for some   0

_
_
_
In particular, suppose b>1,
f ( n )  T ( n / b )  n  f ( n )  ( n )
f (n)  bT (n / b)  n  f (n)  (n log n)
Problem Discussion
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Compute Bp mod M
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B,P are integers in [0,2147483647]
M is an integer in [1,46340]
f(p)=B*f(p-1)
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Bp= B*Bp-1
Θ(p)
Problem Discussion

Notice that
p/2 2

(B )
p
B 
 p / 2 2
)
B  ( B

Θ(log p)
when B is even
when B is odd
Properties of recursion
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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
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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
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Divide
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Conquer
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Break a problem into sub problem(s)
Solve all sub problem(s)
Combine

Solve the problem using the results of the sub
problem(s)
Problem Discussion
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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
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Example
Problem Discussion
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Illustration
Recursion outside Computer Science
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Names
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PHP : PHP Hypertext Preprocessor
GNU : GNU is Not Unix
Mathematics
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Definition of the set of non negative numbers N
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0 is in N
If n is in N, then n+1 is in N.
Images
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Fractals
Challenge Problems
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Consider the following recursive function
defined on pairs of nonnegative numbers.
0
when m  0


f (m, n)  
0
when n  0
 f (m  1, n  1)  m  n  1 otherwise

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Find the closed form of the function. Prove
your answer algebraically.
Challenge Problems

Consider the following recursive function defined on
triple of nonnegative numbers.
a
b

f (a, b, c)  c
 f (a  1, b  1, c  1)

 ab  bc  ac  a  b  c  1

when a  0
when b  0
when c  0
otherwise
Find the closed form of the function. Prove your
answer geometrically.
Challenge Problems
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Find a closed form solution for the nth
Fibonacci Number.
Prove the Master Theorem.
Challenge Problems

Consider the following algorithm.

F(x,p)
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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.
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B,P are integers in [0,2147483647]
M is an integer in [1,46340]
State the time and space complexity.
Challenge Problems
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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
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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
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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/