Recursive Algorithm - Advance Data Structures | CSC 3102, Study notes of Computer Science

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Recursive Algorithms

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.

Example 1: Recursive Evaluation of n!

 Definition of factorial function F ( n )= n !:  n****! = 1 • 2 • … • ( n- 1) • n  0! = 1  Recurrence relation for the number of multiplications : M ( n ) = M ( n -1) + 1 for n > 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 1 else return F( n-1) * n

Example 2: Tower of Hanoi Puzzle

 Given: n disks of different sizes and three pegs. Initially all disks are on the first peg in order of size, the largest being on the bottom  Problem: move all the disks to the third peg, using the second one as an auxiliary.  One can move only one disk at a time, and it is forbidden to place a larger disk on the top of a smaller one.  Recursive solution: Three steps involved are  First move recursively n - 1 disks from peg 1 to peg 2 (with peg 3 as auxiliary)  Move the largest disk directly from peg 1 to peg 3  Move recursively n - 1 disks from peg 2 and peg 3 (using peg 1 as auxiliary) Peg (^1) Peg 2 Peg 3 http://www.mazeworks.com/hanoi/

Example 3: Binary Recursive

 Problem: Investigate a recursive version of binary algorithm, which finds the number of binary digits in the binary representation of a positive decimal integer n.  Recursive relation for the number of additions made by the algorithm. A ( n ) = A  n /2  + 1 for n > 1 A (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 n ’s binary representation if n = 1 return 1 else return BinRec (  n/2  ) +

Binary Recursive: Solution

 Use backward substitution method to solve the problem only for n = 2 k****.  Smoothness rule implies that the observed order of growth is valid for all values of n.  A (2 k ) = A (2 k-^1 ) + 1 = [ A (2 k-^2 ) + 1] + 1 = A (2 k-^2 ) + 2 = [ A (2 k-^3 ) + 1] + 2 = A (2 k-^3 ) + 3 …… = A (2 k-i ) + i …… = A (2 k-k ) + k = k A (2^0 ) = A (1) = 0 = log 2 n n = 2 k A ( n ) = log 2 n ∈ Θ (log n ) Exact solution is A ( n ) =  log 2 n 