Recursion

2Overview

Learn about recursive definitions Explore the base case and the general case of a recursive

definition Discover recursive algorithms Learn about recursive functions Explore how to use recursive functions to implement

recursive algorithms Learn about recursion and backtracking

3Recursive Definitions

Recursion The process of solving a problem by reducing it to a

smaller versions of itself Recursive definition

Definition in which a problem is expressed in terms of a smaller version of itself

Has one or more base cases

4Recursive Definitions

Recursive algorithm Algorithm that finds the solution to a given problem by reducing

the problem to smaller versions of itself Has one or more base cases Can be implemented using recursive functions

Recursive function function that calls itself

Base case Case in recursive definition (or function) in which the solution is

obtained directly (without making a recursive function call) Halts the recursive function calling Without a base case, recursion would continue endlessly

5Recursive Definitions

General solution Breaks problem into smaller versions of itself

General case Case in recursive definition in which a smaller version

of itself is called Must eventually be reduced to a base case

6Tracing a Recursive Function

Every recursive call has its own code own set of parameters own set of local variables

After completing recursive call: Control goes back to calling environment Recursive call must execute completely before control goes back

to previous call Execution in previous call begins from point immediately

following recursive call

7Recursive Definitions

Directly recursive: a function that calls itself Indirectly recursive: a function that calls another function

and eventually results in the original function call Tail recursive function: recursive function in which the

last statement executed is the recursive call Infinite recursion: the case where every recursive call

results in another recursive call

8Designing Recursive Functions

Understand problem requirements Determine limiting conditions Identify base cases Provide direct solution to each base case Identify general case(s) Provide solutions to general cases in terms of

smaller versions of itself

9Recursive Factorial Function

int fact(int num){ if(num <= 0) return 1; else return num * fact(num – 1);}

10Largest Value in Array

int largest(const int list[], //in int lowerIndex, //in int upperIndex //in){ int max; if(lowerIndex == upperIndex) return list[lowerIndex]; else{ max = largest(list, lowerIndex + 1, upperIndex); if(list[lowerIndex] >= max) return list[lowerIndex]; else return max; }}

11Towers of Hanoi Problem with 3 Disks

12Towers of Hanoi: Three Disk Solution

13Towers of Hanoi: Three Disk Solution

14Towers of Hanoi: Recursive Algorithmvoid moveDisks(int count, int source, int target,

int spare){ if(count > 0) { moveDisks(count - 1, source, spare, target); cout<<"Move disk "<<count<<“ from "<<source <<“ to "<<target<<"."<<endl; moveDisks(count - 1, spare, target, source); }}

15Decimal to Binary

void decToBin(int num, int base){ if(num > 0) { decToBin(num/base, base); cout<<num % base; }}

16Recursion or Iteration?

Two ways to solve particular problem Iteration Recursion

Iterative control structures: uses looping to repeat a set of statements

Tradeoffs between two options Sometimes recursive solution is easier Recursive solution is often slower

17Backtracking Algorithm

Attempts to find solutions to a problem by constructing partial solutions

Makes sure that any partial solution does not violate the problem requirements

Tries to extend partial solution towards completion

18Backtracking Algorithm

If it is determined that partial solution would not lead to solution partial solution would end in dead end algorithm backs up by removing the most recently

added part and then tries other possibilities

19Solution to 8-Queens Puzzle

204-Queens Puzzle

214-Queens Tree