Recursion

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Recursion
Thinking about Algorithms Abstractly

• Credits Jeff Edmonds, Ping Xuan

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Code Representation of an Algorithm
MULT(X,Y) If X Y 1 then RETURN XY
Break X into ab and Y into cd e MULT(a,c)
and f MULT(b,d) RETURN e 10n (MULT(ab,
cd) e - f) 10n/2 f
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Tree of Frames
MULT(X,Y) If X Y 1 then RETURN XY
Break X into ab and Y into cd e MULT(a,c)
and f MULT(b,d) RETURN e 10n (MULT(ab,
cd) e - f) 10n/2 f
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Friends
MULT(X,Y) If X Y 1 then RETURN XY
Break X into ab and Y into cd e MULT(a,c)
and f MULT(b,d) RETURN e 10n (MULT(ab,
cd) e - f) 10n/2 f
One Friend for each stack frame. Each worries
only about his job.
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Friends
MULT(X,Y) If X Y 1 then RETURN XY
Break X into ab and Y into cd e MULT(a,c)
and f MULT(b,d) RETURN e 10n (MULT(ab,
cd) e - f) 10n/2 f
Worry about one step at a time. Imagine that you
are one specific friend.
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Friends
MULT(X,Y) If X Y 1 then RETURN XY
Break X into ab and Y into cd e MULT(a,c)
and f MULT(b,d) RETURN e 10n (MULT(ab,
cd) e - f) 10n/2 f

• Consider your input instance

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Friends
MULT(X,Y) If X Y 1 then RETURN XY
Break X into ab and Y into cd e MULT(a,c)
and f MULT(b,d) RETURN e 10n (MULT(ab,
cd) e - f) 10n/2 f

• Consider your input instance
• Allocate work
• Construct one or more subinstances

X 3 Y 2 XY?
X 3 Y 1 XY?
X 6 Y 3 XY?
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Friends
MULT(X,Y) If X Y 1 then RETURN XY
Break X into ab and Y into cd e MULT(a,c)
and f MULT(b,d) RETURN e 10n (MULT(ab,
cd) e - f) 10n/2 f

• Consider your input instance
• Allocate work
• Construct one or more subinstances
• Assume by magic your friends give you the answer
  for these.

X 3 Y 2 XY6
X 3 Y 1 XY3
X 6 Y 3 XY18
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Friends
MULT(X,Y) If X Y 1 then RETURN XY
Break X into ab and Y into cd e MULT(a,c)
and f MULT(b,d) RETURN e 10n (MULT(ab,
cd) e - f) 10n/2 f

• Consider your input instance
• Allocate work
• Construct one or more subinstances
• Assume by magic your friends give you the answer
  for these.
• Use this help to solve your own instance.

X 3 Y 2 XY6
X 3 Y 1 XY3
X 6 Y 3 XY18
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Friends
MULT(X,Y) If X Y 1 then RETURN XY
Break X into ab and Y into cd e MULT(a,c)
and f MULT(b,d) RETURN e 10n (MULT(ab,
cd) e - f) 10n/2 f

• Consider your input instance
• Allocate work
• Construct one or more subinstances
• Assume by magic your friends give you the answer
  for these.
• Use this help to solve your own instance.
• Do not worry about anything else.
• Who your boss is.
• How your friends solve their instance.

X 3 Y 2 XY6
X 3 Y 1 XY3
X 6 Y 3 XY18
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Friends
MULT(X,Y) If X Y 1 then RETURN XY
Break X into ab and Y into cd e MULT(a,c)
and f MULT(b,d) RETURN e 10n (MULT(ab,
cd) e - f) 10n/2 f

• This technique is often
• referred to as
• Divide and Conquer

X 3 Y 2 XY6
X 3 Y 1 XY3
X 6 Y 3 XY18
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Friends
MULT(X,Y) If X Y 1 then RETURN XY
Break X into ab and Y into cd e MULT(a,c)
and f MULT(b,d) RETURN e 10n (MULT(ab,
cd) e - f) 10n/2 f

• Consider generic instances.

MULT(X,Y) Break X into ab and Y into cd e
MULT(a,c) and f MULT(b,d) RETURN e 10n
(MULT(ab, cd) e - f) 10n/2 f
MULT(X,Y) If X Y 1 then RETURN XY

• Remember!
• Do not worry about
• Who your boss is.
• How your friends solve their instance.

bd
(ab)(cd)
ac
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Friends
MULT(X,Y) If X Y 1 then RETURN XY
Break X into ab and Y into cd e MULT(a,c)
and f MULT(b,d) RETURN e 10n (MULT(ab,
cd) e - f) 10n/2 f
This solves the problem for every possible
instance.
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Friends

• Recursive Algorithm
• Assume you have an algorithm that works.
• Use it to write an algorithm that works.

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Friends

• Recursive Algorithm
• Assume you have an algorithm that works.
• Use it to write an algorithm that works.

If I could get in, I could get the key. Then I
could unlock the door so that I can get in.
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Friends Strong Induction

• Recursive Algorithm
• Assume you have an algorithm that works.
• Use it to write an algorithm that works.

To get into my house I must get the key from a
smaller house
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Friends Strong Induction
MULT(X,Y) If X Y 1 then RETURN XY
Break X into ab and Y into cd e MULT(a,c)
and f MULT(b,d) RETURN e 10n (MULT(ab,
cd) e - f) 10n/2 f

• Allocate work
• Construct one or more subinstances

Each subinstance must be a smaller instance to
the same problem.
X 3 Y 2 XY?
X 3 Y 1 XY?
X 6 Y 3 XY?
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Friends Strong Induction

• Recursive Algorithm
• Assume you have an algorithm that works.
• Use it to write an algorithm that works.

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Friends
MULT(X,Y) If X Y 1 then RETURN XY
Break X into ab and Y into cd e MULT(a,c)
and f MULT(b,d) RETURN e 10n (MULT(ab,
cd) e - f) 10n/2 f
MULT(X,Y) If X Y 1 then RETURN XY
Use brute force to solve the base case instances.
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Towers of Hanoi
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Towers of Hanoi
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Towers of Hanoi
Get a job at the Towers of Hanoi company at
level n1,2,3,4,..
Or get transferred in as the president you
decide what your vice president does.
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Towers of Hanoi
(Homework 1. Assume n 2k)
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Tower of Hanoi

• http//www.mazeworks.com/hanoi/index.htm
• http//www.cut-the-knot.com/recurrence/hanoi.shtml
  
• http//www.cs.yorku.ca/andy/courses/3101/lecture-
  notes/LN0.html

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Sorting Problem Specification

• Precondition The input is a list of n values
  with the same value possibly repeated.
• Post condition The output is a list consisting
  of the same n values in non-decreasing order.

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Recursive Sorts

• Given list of objects to be sorted
• Split the list into two sublists.
• Recursively have a friend sort the two sublists.
• Combine the two sorted sublists into one entirely
  sorted list.

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Merge Sort
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Merge Sort
Split Set into Two
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Merge Sort
Merge two sorted lists into one
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Merge Sort
merge_sort ( array a ) if (length of a gt
1) then merge_sort ( left half of a )
merge_sort ( right half of a) merge the
above two sorted half arrays
Time T(n)
Q(n log(n))
2T(n/2) n
(Homework 2. Assume n 2k)
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Recursive Images
if n1
if n0, draw
n0
else draw
And recursively Draw here with n-1
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Recursive Images
if n2
if n0, draw
n1
else draw
And recursively Draw here with n-1
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Recursive Images
if n3
if n0, draw
n2
else draw
And recursively Draw here with n-1
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Recursive Images
if n30
if n0, draw
else draw
And recursively Draw here with n-1
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Recursive Images
if n0
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Recursive Images
if n0
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Recursive Images
Þ
(4/3)n
L(n) 4/3 L(n-1)
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Homework 3 Derive the recurrence relation for
the time complexity T( N ) and solve it. Assume
that N 2k .
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