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Data Structures Using C++ 2
Chapter Objectives
• 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
Data Structures Using C++ 3
Recursive Definitions
• Recursion– Process of solving a problem by reducing it to
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
Data Structures Using C++ 4
Recursive 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– Implemented using recursive functions
• Recursive function– function that calls itself
• Base case– Case in recursive definition in which the solution is
obtained directly – Stops the recursion
Data Structures Using C++ 5
Recursive 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
Data Structures Using C++ 6
Tracing a Recursive Function
Recursive function– Has unlimited copies of itself– Every recursive call has
• its own code• own set of parameters• own set of local variables
Data Structures Using C++ 7
Tracing a Recursive Function
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
Data Structures Using C++ 8
Recursive 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
Data Structures Using C++ 9
Designing Recursive Functions
• Understand problem requirements
• Determine limiting conditions
• Identify base cases
Data Structures Using C++ 10
Designing Recursive Functions
• Provide direct solution to each base case
• Identify general case(s)
• Provide solutions to general cases in terms of smaller versions of itself
Data Structures Using C++ 11
Recursive Factorial Function
int fact(int num)
{
if(num == 0)
return 1;
else
return num * fact(num – 1);
}
Data Structures Using C++ 13
Recursive Implementation: Largest Value in Array
int largest(const int list[], int lowerIndex, int upperIndex){ int max; if(lowerIndex == upperIndex) //the size of the sublist is 1 return list[lowerIndex]; else { max = largest(list, lowerIndex + 1, upperIndex); if(list[lowerIndex] >= max) return list[lowerIndex]; else return max; }}
Data Structures Using C++ 15
Recursive Fibonacci
int rFibNum(int a, int b, int n){ if(n == 1) return a; else if(n == 2) return b; else return rFibNum(a, b, n - 1) + rFibNum(a, b, n - 2);}
Data Structures Using C++ 20
Towers of Hanoi: Recursive Algorithm
void moveDisks(int count, int needle1, int needle3, int needle2)
{ if(count > 0) { moveDisks(count - 1, needle1, needle2,
needle3); cout<<"Move disk "<<count<<“ from "<<needle1 <<“ to "<<needle3<<"."<<endl; moveDisks(count - 1, needle2, needle3,
needle1); }}
Data Structures Using C++ 21
Decimal to Binary: Recursive Algorithm
void decToBin(int num, int base)
{
if(num > 0)
{
decToBin(num/base, base);
cout<<num % base;
}
}
Data Structures Using C++ 23
Recursion 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
Data Structures Using C++ 24
Backtracking 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
Data Structures Using C++ 25
Backtracking 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
Data Structures Using C++ 30
Chapter Summary
• Recursive definitions
• Recursive algorithms
• Recursive functions
• Base cases
• General cases