Finite description of steps for solving problem Problem types

Satisfying ⇒ find any legal solution Optimization ⇒ find best solution (vs. cost metric)

Approaches

Iterative Recursive ⇒ execute action in loop ⇒ reapply action to subproblem(s)

Recursive Algorithm

Definition

An algorithm that calls itself

Approach

1. Solve small problem directly 2. Simplify large problem into 1 or more smaller

subproblem(s) & solve recursively 3. Calculate solution from solution(s) for subproblem

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 1st element of array is given value, return true Skip 1st element and recur on remainder of array Example (See ArrayExamples.java)

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-1 Return n × (the factorial of n-1)

Example – Factorial

Code

int fact ( int n ) { if ( n == 0 ) return 1; return n * fact(n-1); }

// base case // recursive step

Requirements

Must have

Small version of problem solvable without recursion Strategy to simplify problem into 1 or more smaller subproblems Ability to calculate overall solution from solution(s) to subproblem(s)

Making Recursion Work

Designing a correct recursive algorithm Verify

1. Base case is

Recognized correctly Solved correctly 2. Recursive case Solves 1 or more simpler subproblems Can calculate solution from solution(s) to subproblems

Uses principle of proof by induction