06-recursivealgorithms

8/2/2019 06-RecursiveAlgorithms

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B.B. Karki, LSU0.1CSC 3102

Recursive Algorithms


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Definition and Examples

Recursive algorithm invokes (makes reference to) itself repeatedly

until a certain condition matches

Examples:

Computing factorial function

Tower of Hanoi puzzle

Digits in binary representation

Non-recursive algorithm is executed only once to solve the problem.


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Analyzing Efficiency of Recursive Algorithms

Steps in mathematical analysis of recursive algorithms:

Decide on parameter n indicating input size.

Identify algorithms basic operation

Determine worst, average, and bestcase for input of size n

If the basic operation count also depends on other conditions.

Set up a recurrence relation and initial condition(s) for C(n)-the number oftimes the basic operation will be executed for an input of size n

Alternatively count recursive calls.

Solve the recurrence to obtain a closed form or estimate the order ofmagnitude of the solution (see Appendix B)


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Example 1: Recursive Evaluation ofn !

Definition of factorial functionF(n)= n!: n ! = 1 2 (n-1) n

0! = 1

Recurrence relation for the number of multiplications:

M(n) =M(n-1) + 1 forn > 0

M(0) = 0 initial condition

Solve the recurrence relation using the method of backward substitution:

M(n) =M(n-1) + 1 =M(n-2) + 2 = M(n-3) +3 = .. =M(n-i) + i=.

=M(n-n) + n

M(n) = n

Algorithm F(n) //Compute n!recursively

//Input: A nonnegative integer n

//Output: The value of n!if n= 0 return 1else return F(n-1) * n


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Example 2: Tower of Hanoi Puzzle

Given:n disks of different sizes andthree pegs. Initially all disks are on thefirst peg in order of size, the largestbeing on the bottom

Problem: move all the disks to the thirdpeg, using the second one as an

auxiliary. One can move only one disk at a time,

and it is forbidden to place a larger diskon the top of a smaller one.

Recursive solution: Three stepsinvolved are

First move recursively n - 1 disks frompeg 1 to peg 2 (with peg 3 as auxiliary)

Move the largest disk directly from peg1 to peg 3

Move recursively n - 1 disks from peg 2and peg 3 (using peg 1 as auxiliary)

Peg 1 Peg 2 Peg 3

http://www.mazeworks.com/hanoi/


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Tower of Hanoi: Recurrence Relation

Recurrence for the number of moves

M(n) =M(n-1) + 1 +M(n-1) forn > 1

= 2M(n-1) + 1

M(1) = 1 initial condition

Solve the recurrence relation using themethod of backward substitution:

M(n) = 2M(n-1) + 1 = 22M(n-2) + 22 - 1

= 23M(n-3) + 23 - 1 == 2iM(n-i) +2i- 1 =.

= 2n-1M(n - (n-1)) + 2n-1 - 1= 2n-1 + 2n-1 - 1

M(n) = 2n - 1

n

n-1n-1

n-2n-2 n-2n-2

22 22

11 11 11 11

.. .. ..

Tree of recursive calls made bythe recursive algorithm


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Example 3: Binary Recursive

Problem: Investigate a recursive version of binary algorithm, which findsthe number of binary digits in the binary representation of a positivedecimal integer n.

Recursive relation for the number of additions made by the algorithm.

A(n) =An/2 + 1 forn > 1A(1) = 0 initial condition

For n = 1, no addition is made because no recursive call is executed.

Algorithm BinRec(n)//Input: A positive decimal integer n//Output: The number of binary digits in ns binary representationif n= 1 return 1else return BinRec(n/2) +1


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Binary Recursive: Solution

Use backward substitution method to solve the problem only forn = 2k. Smoothness rule implies that the observed order of growth is valid for all values of n.

A(2k) =A(2k-1) + 1

= [A(2k-2) + 1] + 1 =A(2k-2) + 2

= [A(2k-3) + 1] + 2 =A(2k-3) + 3

=A(2k-i) + i

=A(2k-k) + k

= k A(20) = A(1) = 0

= log2n n = 2k

A(n) = log2 n (log n)Exact solution is A(n) = log2 n

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