![Generating 0/1Vectors (3)
9
static void Gen01(int index, int[] vector)
{
if (index == -1)
Print(vector);
else
for (int i=0; i<=1; i++)
{
vector[index] = i;
Gen01(index-1, vector);
}
}
static void Main()
{
int size = 8;
int[] vector = new int[size];
Gen01(size-1, vector);
}](https://image.slidesharecdn.com/13-recursion-120712074623-phpapp02-150420012014-conversion-gate02/85/13-recursion-120712074623-phpapp02-9-320.jpg)
![Find All Paths: Algorithm
Representing the labyrinth as matrix of characters
(in this example 5 rows and 7 columns):
Spaces (' ') are passable cells
Asterisks ('*') are not passable cells
The symbol 'e' is the exit (can be multiple)
14
static char[,] lab =
{
{' ', ' ', ' ', '*', ' ', ' ', ' '},
{'*', '*', ' ', '*', ' ', '*', ' '},
{' ', ' ', ' ', ' ', ' ', ' ', ' '},
{' ', '*', '*', '*', '*', '*', ' '},
{' ', ' ', ' ', ' ', ' ', ' ', 'е'},
};](https://image.slidesharecdn.com/13-recursion-120712074623-phpapp02-150420012014-conversion-gate02/85/13-recursion-120712074623-phpapp02-14-320.jpg)
![Find All Paths: Algorithm (2)
15
static void FindExit(int row, int col)
{
if ((col < 0) || (row < 0) || (col >= lab.GetLength(1))
|| (row >= lab.GetLength(0)))
{
// We are out of the labyrinth -> can't find a path
return;
}
// Check if we have found the exit
if (lab[row, col] == 'е')
{
Console.WriteLine("Found the exit!");
}
if (lab[row, col] != ' ')
{
// The current cell is not free -> can't find a path
return;
}
(example continues)](https://image.slidesharecdn.com/13-recursion-120712074623-phpapp02-150420012014-conversion-gate02/85/13-recursion-120712074623-phpapp02-15-320.jpg)
![Find All Paths: Algorithm (3)
16
// Temporary mark the current cell as visited
lab[row, col] = 's';
// Invoke recursion to explore all possible directions
FindExit(row, col-1); // left
FindExit(row-1, col); // up
FindExit(row, col+1); // right
FindExit(row+1, col); // down
// Mark back the current cell as free
lab[row, col] = ' ';
}
static void Main()
{
FindExit(0, 0);
}](https://image.slidesharecdn.com/13-recursion-120712074623-phpapp02-150420012014-conversion-gate02/85/13-recursion-120712074623-phpapp02-16-320.jpg)
![Find All Paths and PrintThem
How to print all paths found by our recursive
algorithm?
Each move's direction can be stored in array
Need to pass the movement direction at each
recursive call
At the start of each recursive call the current
direction is appended to the array
At the end of each recursive call the last
direction is removed form the array 18
static char[] path =
new char[lab.GetLength(0) * lab.GetLength(1)];
static int position = 0;](https://image.slidesharecdn.com/13-recursion-120712074623-phpapp02-150420012014-conversion-gate02/85/13-recursion-120712074623-phpapp02-18-320.jpg)
![Find All Paths and PrintThem (2)
19
static void FindPathToExit(int row, int col, char direction)
{
...
// Append the current direction to the path
path[position++] = direction;
if (lab[row, col] == 'е')
{
// The exit is found -> print the current path
}
...
// Recursively explore all possible directions
FindPathToExit(row, col - 1, 'L'); // left
FindPathToExit(row - 1, col, 'U'); // up
FindPathToExit(row, col + 1, 'R'); // right
FindPathToExit(row + 1, col, 'D'); // down
...
// Remove the last direction from the path
position--;
}](https://image.slidesharecdn.com/13-recursion-120712074623-phpapp02-150420012014-conversion-gate02/85/13-recursion-120712074623-phpapp02-19-320.jpg)
![Fast Recursive Fibonacci
Each Fibonacci sequence member can be
remembered once it is calculated
Can be returned directly when needed again
25
static decimal[] fib = new decimal[MAX_FIB];
static decimal Fibonacci(int n)
{
if (fib[n] == 0)
{
// The value of fib[n] is still not calculated
if ((n == 1) || (n == 2))
fib[n] = 1;
else
fib[n] = Fibonacci(n - 1) + Fibonacci(n - 2);
}
return fib[n];
}](https://image.slidesharecdn.com/13-recursion-120712074623-phpapp02-150420012014-conversion-gate02/85/13-recursion-120712074623-phpapp02-25-320.jpg)
13 recursion-120712074623-phpapp02
- 1. Recursion The Power of Calling a Method from Itself Svetlin Nakov Telerik Software Academy academy.telerik.com ManagerTechnicalTraining http://nakov.com
- 2. Table of Contents 1. What is Recursion? 2. Calculating Factorial Recursively 3. Generating All 0/1Vectors Recursively 4. Finding All Paths in a Labyrinth Recursively 5. Recursion or Iteration? 2
- 3. What is Recursion? Recursion is when a methods calls itself Very powerful technique for implementing combinatorial and other algorithms Recursion should have Direct or indirect recursive call The method calls itself directly or through other methods Exit criteria (bottom) Prevents infinite recursion 3
- 4. Recursive Factorial – Example Recursive definition of n! (n factorial): 4 n! = n * (n–1)! for n >= 0 0! = 1 5! = 5 * 4! = 5 * 4 * 3 * 2 * 1 * 1 = 120 4! = 4 * 3! = 4 * 3 * 2 * 1 * 1 = 24 3! = 3 * 2! = 3 * 2 * 1 * 1 = 6 2! = 2 * 1! = 2 * 1 * 1 = 2 1! = 1 * 0! = 1 * 1 = 1 0! = 1
- 5. Recursive Factorial – Example Calculating factorial: 0! = 1 n! = n* (n-1)!, n>0 Don't try this at home! Use iteration instead 5 static decimal Factorial(decimal num) { if (num == 0) return 1; else return num * Factorial(num - 1); } The bottom of the recursion Recursive call: the method calls itself
- 7. Generating 0/1Vectors How to generate all 8-bit vectors recursively? 00000000 00000001 ... 01111111 10000000 ... 11111110 11111111 How to generate all n-bit vectors? 7
- 8. Generating 0/1Vectors (2) Algorithm Gen01(n): put 0 and 1 at the last position n and call Gen01(n-1) for the rest: 8 x x x x x x 0 Gen01(6): Gen01(5) x x x x x x 1 Gen01(5) x x x x x 0 y Gen01(5): Gen01(4) x x x x x 1 y Gen01(4) ... Gen01(-1) Stop!
- 9. Generating 0/1Vectors (3) 9 static void Gen01(int index, int[] vector) { if (index == -1) Print(vector); else for (int i=0; i<=1; i++) { vector[index] = i; Gen01(index-1, vector); } } static void Main() { int size = 8; int[] vector = new int[size]; Gen01(size-1, vector); }
- 11. Finding All Paths in a Labyrinth We are given a labyrinth Represented as matrix of cells of size M x N Empty cells are passable, the others (*) are not We start from the top left corner and can move in the all 4 directions: left, right, up, down We need to find all paths to the bottom right corner 11 Start position End position * * * * * * * * * *
- 12. Finding All Paths in a Labyrinth (2) There are 3 different paths from the top left corner to the bottom right corner: 12 0 1 2 * * * 3 * * 6 5 4 7 * * * * * 8 9 10 11 12 13 14 0 1 2 * 8 9 10 * * 3 * 7 * 11 4 5 6 12 * * * * * 13 14 1) 2) 0 1 2 * * * 3 * * 4 5 6 7 8 * * * * * 9 10 3)
- 13. Finding All Paths in a Labyrinth (2) Suppose we have an algorithm FindExit(x,y) that finds and prints all paths to the exit (bottom right corner) starting from position (x,y) If (x,y) is not passable, no paths are found If (x,y) is already visited, no paths are found Otherwise: Mark position (x,y) as visited (to avoid cycles) Find all paths to the exit from all neighbor cells: (x-1,y) , (x+1,y) , (x,y+1) , (x,y-1) Mark position (x,y) as free (can be visited again) 13
- 14. Find All Paths: Algorithm Representing the labyrinth as matrix of characters (in this example 5 rows and 7 columns): Spaces (' ') are passable cells Asterisks ('*') are not passable cells The symbol 'e' is the exit (can be multiple) 14 static char[,] lab = { {' ', ' ', ' ', '*', ' ', ' ', ' '}, {'*', '*', ' ', '*', ' ', '*', ' '}, {' ', ' ', ' ', ' ', ' ', ' ', ' '}, {' ', '*', '*', '*', '*', '*', ' '}, {' ', ' ', ' ', ' ', ' ', ' ', 'е'}, };
- 15. Find All Paths: Algorithm (2) 15 static void FindExit(int row, int col) { if ((col < 0) || (row < 0) || (col >= lab.GetLength(1)) || (row >= lab.GetLength(0))) { // We are out of the labyrinth -> can't find a path return; } // Check if we have found the exit if (lab[row, col] == 'е') { Console.WriteLine("Found the exit!"); } if (lab[row, col] != ' ') { // The current cell is not free -> can't find a path return; } (example continues)
- 16. Find All Paths: Algorithm (3) 16 // Temporary mark the current cell as visited lab[row, col] = 's'; // Invoke recursion to explore all possible directions FindExit(row, col-1); // left FindExit(row-1, col); // up FindExit(row, col+1); // right FindExit(row+1, col); // down // Mark back the current cell as free lab[row, col] = ' '; } static void Main() { FindExit(0, 0); }
- 17. Find All Paths in a Labyrinth Live Demo
- 18. Find All Paths and PrintThem How to print all paths found by our recursive algorithm? Each move's direction can be stored in array Need to pass the movement direction at each recursive call At the start of each recursive call the current direction is appended to the array At the end of each recursive call the last direction is removed form the array 18 static char[] path = new char[lab.GetLength(0) * lab.GetLength(1)]; static int position = 0;
- 19. Find All Paths and PrintThem (2) 19 static void FindPathToExit(int row, int col, char direction) { ... // Append the current direction to the path path[position++] = direction; if (lab[row, col] == 'е') { // The exit is found -> print the current path } ... // Recursively explore all possible directions FindPathToExit(row, col - 1, 'L'); // left FindPathToExit(row - 1, col, 'U'); // up FindPathToExit(row, col + 1, 'R'); // right FindPathToExit(row + 1, col, 'D'); // down ... // Remove the last direction from the path position--; }
- 20. Find and Print All Paths in a Labyrinth Live Demo
- 21. Recursion or Iteration? When to Use andWhen to Avoid Recursion?
- 22. Recursion Can be Harmful! When used incorrectly the recursion could take too much memory and computing power Example: 22 static decimal Fibonacci(int n) { if ((n == 1) || (n == 2)) return 1; else return Fibonacci(n - 1) + Fibonacci(n - 2); } static void Main() { Console.WriteLine(Fibonacci(10)); // 89 Console.WriteLine(Fibonacci(50)); // This will hang! }
- 24. How the Recursive Fibonacci Calculation Works? fib(n) makes about fib(n) recursive calls The same value is calculated many, many times! 24
- 25. Fast Recursive Fibonacci Each Fibonacci sequence member can be remembered once it is calculated Can be returned directly when needed again 25 static decimal[] fib = new decimal[MAX_FIB]; static decimal Fibonacci(int n) { if (fib[n] == 0) { // The value of fib[n] is still not calculated if ((n == 1) || (n == 2)) fib[n] = 1; else fib[n] = Fibonacci(n - 1) + Fibonacci(n - 2); } return fib[n]; }
- 26. Fast Recursive Fibonacci Live Demo
- 27. When to Use Recursion? Avoid recursion when an obvious iterative algorithm exists Examples: factorial, Fibonacci numbers Use recursion for combinatorial algorithm where at each step you need to recursively explore more than one possible continuation Examples: permutations, all paths in labyrinth If you have only one recursive call in the body of a recursive method, it can directly become iterative (like calculating factorial) 27
- 28. Summary Recursion means to call a method from itself It should always have a bottom at which recursive calls stop Very powerful technique for implementing combinatorial algorithms Examples: generating combinatorial configurations like permutations, combinations, variations, etc. Recursion can be harmful when not used correctly 28
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- 30. Exercises 1. Write a recursive program that simulates execution of n nested loops from 1 to n. Examples: 1 1 1 1 1 2 1 1 3 1 1 1 2 1 n=2 -> 1 2 n=3 -> ... 2 1 3 2 3 2 2 3 3 1 3 3 2 3 3 3 30
- 31. Exercises (2) 2. Write a recursive program for generating and printing all the combinations with duplicates of k elements from n-element set. Example: n=3, k=2 (1 1), (1 2), (1 3), (2 2), (2 3), (3 3) 3. Write a recursive program for generating and printing all permutations of the numbers 1, 2, ..., n for given integer number n. Example: n=3 {1, 2, 3}, {1, 3, 2}, {2, 1, 3}, {2, 3, 1}, {3, 1, 2},{3, 2, 1} 31
- 32. Exercises (3) 4. Write a recursive program for generating and printing all ordered k-element subsets from n- element set (variationsVk n). Example: n=3, k=2 (1 1), (1 2), (1 3), (2 1), (2 2), (2 3), (3 1), (3 2), (3 3) 5. Write a program for generating and printing all subsets of k strings from given set of strings. Example: s = {test, rock, fun}, k=2 (test rock), (test fun), (rock fun) 32
- 33. Exercises (4) 6. We are given a matrix of passable and non-passable cells. Write a recursive program for finding all paths between two cells in the matrix. 7. Modify the above program to check whether a path exists between two cells without finding all possible paths.Test it over an empty 100 x 100 matrix. 8. Write a program to find the largest connected area of adjacent empty cells in a matrix. 9. Implement the BFS algorithm to find the shortest path between two cells in a matrix (read about Breath-First Search in Wikipedia). 33
- 34. Exercises (5) 10. We are given a matrix of passable and non-passable cells. Write a recursive program for finding all areas of passable cells in the matrix. 11. Write a recursive program that traverses the entire hard disk drive C: and displays all folders recursively and all files inside each of them. 34
- 35. FreeTrainings @Telerik Academy Fundamentals of C# Programming Course csharpfundamentals.telerik.com Telerik Software Academy academy.telerik.com Telerik Academy @ Facebook facebook.com/TelerikAcademy Telerik Software Academy Forums forums.academy.telerik.com
Editor's Notes
- #3: (c) 2007 National Academy for Software Development - http://academy.devbg.org. All rights reserved. Unauthorized copying or re-distribution is strictly prohibited.*
- #6: (c) 2007 National Academy for Software Development - http://academy.devbg.org. All rights reserved. Unauthorized copying or re-distribution is strictly prohibited.*
- #8: (c) 2007 National Academy for Software Development - http://academy.devbg.org. All rights reserved. Unauthorized copying or re-distribution is strictly prohibited.*
- #10: (c) 2007 National Academy for Software Development - http://academy.devbg.org. All rights reserved. Unauthorized copying or re-distribution is strictly prohibited.*
- #29: (c) 2007 National Academy for Software Development - http://academy.devbg.org. All rights reserved. Unauthorized copying or re-distribution is strictly prohibited.*