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Chapter 17 Recursion ( 递归 )PowerPoint Presentation

Chapter 17 Recursion ( 递归 )

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### Chapter 17 Recursion (递归)

§17.1 Introduction

§17.2 Methodology of Recursion Design

§17.3 Recursion vs. Iteration

§17.1 Introduction

- Computing Factorial (阶乘)

- Factorial in mathematics:
f(n) = n!

= 1*2*..*n

1 if n=0

=

n*(n-1)! if n>0

- Factorial in C++
int factorial(int n){

int result;

if (n==0)

result =1;

else

result = n *factorial(n-1);

return result;

}

Recursive Call!

Computing Factorial

factorial(3) =

= 3 * factorial(2)

= 3 * (2 * factorial(1))

= 3 * ( 2 * (1 * factorial(0)))

= 3 * ( 2 * ( 1 * 1)))

= 3 * ( 2 * 1)

= 3 * 2

= 6

factorial(0) = 1;

factorial(n) = n*factorial(n-1);

Trace Recursive Factorial

factorial(4)

return 24 to caller

Step 0: execute factorial(4)

Step 9: return 24

Space Required

for factorial(0)

4*factorial(3)

Space Required

Step 1: execute factorial(3)

Step 8: return 6

for factorial(1)

3*factorial(2)

Space Required

Step 2: execute factorial(2)

Step 7: return 2

for factorial(2)

2*factorial(1)

Space Required

Step 3: execute factorial(1)

for factorial(3)

Step 6: return 1

1*factorial(0)

Space Required

for factorial(4)

Step 4: execute factorial(0)

Step 5: return 1

return 1

Recursive Function

- Recursive Function
- A function with recursive call

- Recursive call
- A function calls itself, directly or indirectly

f( ){

……

f( );

……

}

f1( ){ f2( ){

…… ……

f2( ); f1( );

…… ……

} }

Fibonacci Numbers

Finonacci series: 0 1 1 2 3 5 8 13 21 34 55 89 …

indices: 0 1 2 3 4 5 6 7 8 9 10 11 …

0, if i = 0

fib(i) = 1, if i = 1

fib(i -1) + fib(i -2), if i >=2

fib(3) = fib(2) + fib(1)

= (fib(1) + fib(0)) + fib(1)

= (1 + 0) +fib(1)

= 1 + fib(1)

= 1 + 1

= 2

ComputeFibonacci

§17.2 Methodology of Recursion Design

- Characteristics of Recursion
- Different cases using selection statement
- One or more base cases (the simplest case)
- To stop recursion

- Every recursive call reduces the original problem
- To bring it increasingly closer to and eventually to be the base case

Problem Solving Using Recursion

- General – thinking recursively
- Divide and conquer
- Sub-problems resemble the original

void nPrintln(string msg, int times) {

if (times >= 1) {

cout << msg << endl;

nPrintln(msg, times - 1);

}

}

bool isPalindrome(const string s) {

if (strlen(s) <= 1) return true;

else if (s[0] != s[strlen(s) - 1]) return false;

else

return isPalindrome(substring(s, 1, strlen(s) - 2));

}

Recursive Helper Function

- Recursive helper function
- A recursive function with additional parameters to reduce the problem
- Especially useful for functions involving strings/arrays

bool isPalindrome(const char * const s, int low, int high) {

if (high <= low) return true;

else if (s[low] != s[high]) return false;

else return isPalindrome(s, low + 1, high - 1);

}

bool isPalindrome(const char * const s) {

return isPalindrome(s, 0, strlen(s) - 1);

}

Case Studies

- Recursive Selection Sort
- Find the largest number in the list and swaps it with the last number.
- Ignore the last number and sort the remaining smaller list recursively.

RecursiveSelectionSort

Recursive Binary Search

- If the key is less than the middle element, recursively search the key in the first half of the array.
- If the key is equal to the middle element, the search ends with a match.
- If the key is greater than the middle element, recursively search the key in the second half of the array.

RecursiveBinarySearch

Towers of Hanoi

- There are n disks labeled 1, 2, 3, . . ., n, and three towers labeled A, B, and C.
- No disk can be on top of a smaller disk at any time.
- All the disks are initially placed on tower A.
- Only one disk can be moved at a time, and it must be the top disk on the tower.

Solution to Towers of Hanoi

Decompose the problem into three subproblems.

n n-1 n-2 … 1, move it directly!

2

A

B

C

A

B

C

A

B

C

4

5

3

A

B

C

A

B

C

A

B

C

6

7

A

B

C

A

B

C

Towers of HanoiTowersOfHanoi

1. 把n-1个盘从A搬到C，借助B

2. 把A上剩下的最大的一个盘搬到B

3. 再把n-1个盘从C搬到B，借助A

4. 当n==1时，直接从A搬到B即可

Eight Queens

boolisValid(int row, int column)

{

for (int i = 1; i <= row; i++)

if (queens[row - i] == column

|| queens[row - i] == column - i

|| queens[row - i] == column + i)

return false; // There is a conflict

return true; // No conflict

}

EightQueen

§17.3 Recursion vs. Iteration (迭代)

- Negative aspects of recursion
- High cost in both time and memory

- Recursion Iteration
- Any recursive function can be converted to non-recursive (iterative) function

- When to use recursion? Depending on the problem.
- Recursion is suitable for “recursive” problems

int factorial(int n){

int result;

int i;

for(i=1;i<=n;i++)

result *= i;

return result;

}

int factorial(int n){

int result;

if(n==0) result =1;

else result = n*factorial(n-1);

return result;

}

Recursion vs. IterationRecursion vs. Iteration

f(n) = 0 n=0

1 n=1

f(n-1)+f(n-2) n>1

int fib (int n) {

int result, i, pre1, pre2 ;

result = 0; i =2; pre2 = 1 ;

while (i<=n) {

pre1 = pre2 ;

pre2 = result ;

result = pre1 + pre2;

i++;

}

return result;

}

int fib (int n){

int result;

if (n==0) result =0;

else if (n==1) result =1;

else

result = fib (n-1)+ fib (n-2);

return result;

}

Summary

- Concept of recursion
- Design of recursive functions
- Recursion vs. iteration

Homework Questions

1. Please write a recursive function to solve the Sudoku problem.

2. How to find all possible solutions for the Eight Queens problem?

- (Just for exercise by yourself. No need to submitting it.)

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