markdown 查找添加到给定数字的所有组合

Posted 2021-05-05

Given a positive number, find out all combinations of positive numbers that adds upto that number. The program should print only combinations, not permutations. For example, for input 3, either 1, 2 or 2, 1 should be printed.

Examples :
```
Input: N = 3
Output:
1 1 1
1 2
3
```

```
Input: N = 5
Output:
1 1 1 1 1
1 1 1 2
1 1 3
1 2 2
1 4
2 3
5 
```

The idea is to use recursion. We use an array to store combinations and we recursively fill the array and recurse with reduced number. The invariant used in the solution is that each combination will always be stored in increasing order of elements involved. That way we can avoid printing permutations.

Below is implementation of above idea :

```
<?php 
// PHP program to find out all 
// combinations of positive 
// numbers that add upto given number 

/* arr - array to store the combination 
	index - next location in array 
	num - given number 
	reducedNum - reduced number */
function findCombinationsUtil($arr, $index, 
							$num, $reducedNum) 
{ 
	// Base condition 
	if ($reducedNum < 0) 
		return; 

	// If combination is 
	// found, print it 
	if ($reducedNum == 0) 
	{ 
		for ($i = 0; $i < $index; $i++) 
			echo $arr[$i] , " "; 
		echo "\n"; 
		return; 
	} 

	// Find the previous number 
	// stored in arr[] It helps 
	// in maintaining increasing order 
	$prev = ($index == 0) ? 1 : $arr[$index - 1]; 

	// note loop starts from previous 
	// number i.e. at array location 
	// index - 1 
	for ($k = $prev; $k <= $num ; $k++) 
	{ 
		// next element of array is k 
		$arr[$index] = $k; 

		// call recursively with 
		// reduced number 
		findCombinationsUtil($arr, $index + 1, 
							$num, $reducedNum - $k); 
	} 
} 

/* Function to find out all 
combinations of positive numbers 
that add upto given number. 
It uses findCombinationsUtil() */
function findCombinations($n) 
{ 
	// array to store the combinations 
	// It can contain max n elements 
	$arr = array(); 

	//find all combinations 
	findCombinationsUtil($arr, 0, $n, $n); 
}
```

```
// Driver code 
$n = 5; 
findCombinations($n); 

// This code is contributed by ajit 
?> 
```