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CMSC 132:
Object-Oriented Programming II
Recursive Algorithms
Department of Computer Science
University of Maryland, College Park
Recursion
Recursion is a strategy for solving problems
A procedure that calls itself
Approach
If ( problem instance is simple / trivial ) Solve it directly Else
- Simplify problem instance into smaller instance(s) of the original problem
- Solve smaller instance using same algorithm
- Combine solution(s) to solve original problem
Recursive Algorithm Format
1. Base case
Solve small problem directly
2. Recursive step
Simplify problem into smaller subproblem(s) Recursively apply algorithm to subproblem(s) Calculate overall solution
Example – Find
To find an element in an array
Base case If array is empty, return false Recursive step If 1 st^ element of array is given value, return true Skip 1st^ element and recur on remainder of array
Example – Count
To count # of elements in an array
Base case If array is empty, return 0 Recursive step Skip 1st^ element and recur on remainder of array Add 1 to result
Example – Factorial
Factorial definition
n! = n × n-1 × n-2 × n-3 × … × 3 × 2 × 1 0! = 1
To calculate factorial of n
Base case If n = 0, return 1 Recursive step Calculate the factorial of n- Return n × (the factorial of n-1)
Example – Factorial
Code
int fact ( int n ) { if ( n == 0 ) return 1; // base case return n * fact(n-1); // recursive step }
Properties
Recursion relies on the call stack
State of current procedure is saved when procedure is recursively invoked Every procedure invocation gets own stack space
Any problem solvable with recursion may be
solved with iteration (and vice versa)
Use iteration with explicit stack to store state Algorithm may be simpler for one approach
Recursion vs. Iteration
Recursive algorithm
int fact ( int n ) { if ( n == 0 ) return 1; return n * fact(n-1); }
Recursive algorithm is closer to factorial definition
Iterative algorithm
int fact ( int n ) { int i, res; res = 1; for (i=n; i>0; i--) { res = res * i; } return res; }
Example – Towers of Hanoi
Problem
Move stack of disks between pegs Can only move top disk in stack Only allowed to place disk on top of larger disk
Example – Towers of Hanoi
To move a stack of n disks from peg X to Y
Base case If n = 1, move disk from X to Y Recursive step
- Move top n-1 disks from X to 3 rd^ peg
- Move bottom disk from X to Y
- Move top n-1 disks from 3 rd^ peg to Y
Iterative algorithm would take much longer to describe!
Recursion vs. Iteration
Iterative algorithms
May be more efficient No additional function calls Run faster, use less memory
Possible Problems – Infinite Loop
Infinite recursion
If recursion not applied to simpler problem
int bad ( int n ) { if ( n == 0 ) return 1; return bad(n); }
Will infinite loop Eventually halt when runs out of (stack) memory Stack overflow
Possible Problems – Efficiency
May perform excessive computation
If recomputing solutions for subproblems
Example
Fibonacci numbers fibonacci(0) = 1 fibonacci(1) = 1 fibonacci(n) = fibonacci(n-1) + fibonacci(n-2)
Possible Problems – Efficiency
Recursive algorithm to calculate fibonacci(n)
If n is 0 or 1, return 1 Else compute fibonacci(n-1) and fibonacci(n-2) Return their sum
Simple algorithm ⇒ exponential time O(2n)
Computes fibonacci(1) 2 n^ times
Can solve efficiently using
Iteration Dynamic programming Will examine different algorithm strategies later…
Examples of Recursive Algorithms
Binary search
Quicksort
N-queens
Fractals
N-Queens
Goal
Place queens on a board such that every row and column contains one queen, but no queen can attack another queen
Recursive approach
To place queens on NxN board Assume you’ve already placed K queens
Fractals
Goal
Construct shapes using a simple recursive definition with a natural appearance
Properties
Appears similar at all scales of magnification Therefore “infinitely complex” Not easily described in Euclidean geometry
Mandelbrot Set
