There are n stairs, a person standing at the bottom wants to reach the top. The person can climb either 1 stair or 2 stairs at a time. Count the number of ways, the person can reach the top.
Consider the example shown in diagram. The value of n is 3. There are 3 ways to reach the top. The diagram is taken from Easier Fibonacci puzzles
PHP
<?php// A PHP program to count // number of ways to reach // n'th stair when a person// can climb 1, 2, ..m stairs // at a time. // A simple recursive// function to find n'th // fibonacci numberfunction fib($n){if ($n <= 1) return $n;return fib($n - 1) + fib($n - 2);}// Returns number of // ways to reach s'th stairfunction countWays($s){ return fib($s + 1);}// Driver Code$s = 4;echo "Number of ways = ", countWays($s);// This code is contributed// by m_kit?> |
Output:
Number of ways = 5
The time complexity of the above implementation is exponential (golden ratio raised to power n). It can be optimized to work in O(Logn) time using the previously discussed Fibonacci function optimizations.
PHP
<?php// A PHP program to count // number of ways to reach // n'th stair when a person // can climb either 1 or 2 // stairs at a time // A recursive function// used by countWaysfunction countWaysUtil($n, $m){ if ($n <= 1) return $n; $res = 0; for ($i = 1; $i <= $m && $i <= $n; $i++) $res += countWaysUtil($n - $i, $m); return $res;}// Returns number of ways// to reach s'th stairfunction countWays($s, $m){ return countWaysUtil($s + 1, $m);}// Driver Code$s = 4; $m = 2;echo "Number of ways = ", countWays($s, $m); // This code is contributed// by akt_mit?> |
Output
Number of ways = 5
Time complexity: O(2n)
Auxiliary Space: O(n)
PHP
<?php// A PHP program to count number // of ways to reach n't stair when// a person can climb 1, 2, ..m // stairs at a time// A recursive function used by countWaysfunction countWaysUtil($n, $m){ $res[0] = 1; $res[1] = 1; for ($i = 2; $i < $n; $i++) { $res[$i] = 0; for ($j = 1; $j <= $m && $j <= $i; $j++) $res[$i] += $res[$i - $j]; } return $res[$n - 1];}// Returns number of ways// to reach s'th stairfunction countWays($s, $m){ return countWaysUtil($s + 1, $m);} // Driver Code $s = 4; $m = 2; echo "Number of ways = ", countWays($s, $m);// This code is contributed by m_kit ?> |
Output:
of ways = 5
Time Complexity: O(m*n)
Auxiliary Space: O(n)
Please refer complete article on Count ways to reach the n’th stair for more details!
