By: Lokman Chan

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Recursive Algorithm
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By Lokman Chan
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Recursion
Definition A function that is define in terms
of itself. Goal Reduce the solution to a
problem with a particular set of
input to the same problem with smaller
input values. Simplify a
complex problem to several sub
problems Examples Looking up a word from a
dictionary (non mathematical)
Solutions to Towers of Hanol problem
(mathematical) Merge
Sort, reversing a list, binary search, etc

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Lookup a phone number in phone book Since the
phone numbers are listed in alphabetical, we can
actually apply the recursive strategy to find a
number Consider ? Search middle page (first
page last page)/2 Open the phone book to
middle page If (name is on middle page)
then done
//this is the base case
else if (name is alphabetically before middle
page) last page middle page
//redefine search area to
front half Search
//recursive call with
reduced number of pages else //name
must be after middle page first page middle
page //redefine
search area to back half Search
//recursive call
with reduced number of pages
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Towers of Hanoi
The recursive algorithm is based on the
observation that moving a tower of height h from
pole A to pole B is equivalent to moving a tower
of height h-1 to pole C, them moving the last
disk from pole A to pole B, and finally moving
the the tower from pole C to B.
So the problem of moving a tower of height h, has
been reduced to the one of moving a tower of
height h-1.
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The idea behind
Objective Move tower of 3 disks from peg a to
peg b, with the help of peg c. Rules Only one
disk can be moved at a time. A larger disk can
never be placed on top of a smaller disk. Assume
the disk1 is smallest, disk2 is larger and disk3
is largest

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2 3 _______
_______ _______ A
B C
2
3 1 _______ _______
_______ A B C
?
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More steps
3 1
2 _______
_______ _______ A
B C


1 3
2 _______ _______ _______ A
B C
?
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Steps

1 3 2
_______ _______ _______ A
B C


1 3
2 _______ _______ _______ A
B C
?
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Finally
2 1 3
_____ _____ _____ A
B C
1
2 3 _______ _______
______ A B C
?
It turns out to be a recurrence relationship and
can be computed using recursive algorithm
For better simulations, please go to
http//www.cut-the-knot.org/recurrence/hanoi.shtml
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Merge Sort
Merge sort can be done nicely with recursive
algorithm but can also be implemented without
using recursion. (For demonstration purpose, we
will discuss only recursive merge sort) Merge
sort is a sorting algorithm using divide and
conquer algorithm.
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Divide and Conquer
General Strategy 1) Split the problem into
subproblems 2) Solve subproblems 3) Combine the
results
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http//www.cs.nott.ac.uk/nza/G5BADS01/slides4.pdf
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http//www.cs.nott.ac.uk/nza/G5BADS01/slides4.pdf
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• Recursive Merge Sort
• Split the array into two halves
• Call merge on each half
• Merge sorted Halves

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Time Complexity Levels of recursion log2N At
each level merging N items Time complexity O(N
log2 N) Space complexity O(N)
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Basic foundation of Recursive Algo

• Base case(s) There must be at least one base
  which can
• be solved without
  recursion
• A recursive call should always making progress,
  heading
• towards the base case. (Simplify the
  problem)
• Dont duplicate work by solving the same instance
  of a
• problem in separate recursive calls (dont
  use recursion
• instead of a simple loop)
• 4) Design rule Assume that all the recursive
  calls work

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Advantages of recursive algorithms
The recursive algorithm gives a more
understandable, code The actaul code can be
expressed with a few lines of code when defined
properly
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Disadvantages of Recursive Algo
Not all mathematically recursive functions can be
efficiently implemented by Javas simulation of
recursion Infinite recursion can lead to stack
overflow if the recursive code run (loops)
forever
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THE END