Given a rod of length n inches and an array of prices that contains prices of all pieces of size smaller than n. Determine the maximum value obtainable by cutting up the rod and selling the pieces. For example, if length of the rod is 8 and the values of different pieces are given as following, then the maximum obtainable value is 22 (by cutting in two pieces of lengths 2 and 6)
length | 1 2 3 4 5 6 7 8 -------------------------------------------- price | 1 5 8 9 10 17 17 20
And if the prices are as following, then the maximum obtainable value is 24 (by cutting in eight pieces of length 1)
length | 1 2 3 4 5 6 7 8 -------------------------------------------- price | 3 5 8 9 10 17 17 20
Following is simple recursive implementation of the Rod Cutting problem. The implementation simply follows the recursive structure mentioned above.
PHP
<?php// A Naive recursive solution for // Rod cutting problem/* Returns the best obtainable price for a rod of length n and price[] as prices of different pieces */function cutRod( $price, $n){ if ($n <= 0) return 0; $max_val = PHP_INT_MIN; // Recursively cut the rod in different // pieces and compare different // configurations for ($i = 0; $i < $n; $i++) $max_val = max($max_val, $price[$i] + cutRod($price, $n - $i - 1)); return $max_val;} // Driver Code $arr = array(1, 5, 8, 9, 10, 17, 17, 20); $size =count($arr); echo "Maximum Obtainable Value is ", cutRod($arr, $size);// This code is contributed anuj_67.?> |
Maximum Obtainable Value is 22
Considering the above implementation, following is recursion tree for a Rod of length 4.
cR() ---> cutRod()
cR(4)
/ /
/ /
cR(3) cR(2) cR(1) cR(0)
/ | / |
/ | / |
cR(2) cR(1) cR(0) cR(1) cR(0) cR(0)
/ | |
/ | |
cR(1) cR(0) cR(0) cR(0)
/
/
CR(0)
In the above partial recursion tree, cR(2) is being solved twice. We can see that there are many subproblems which are solved again and again. Since same subproblems are called again, this problem has Overlapping Subproblems property. So the Rod Cutting problem has both properties (see this and this) of a dynamic programming problem. Like other typical Dynamic Programming(DP) problems, recomputations of same subproblems can be avoided by constructing a temporary array val[] in bottom up manner.
PHP
<?php// A Dynamic Programming solution for// Rod cutting problem/* Returns the best obtainable price for a rod of length n and price[] asprices of different pieces */function cutRod( $price, $n){ $val = array(); $val[0] = 0; $i; $j; // Build the table val[] in bottom // up manner and return the last // entry from the table for ($i = 1; $i <= $n; $i++) { $max_val = PHP_INT_MIN; for ($j = 0; $j < $i; $j++) $max_val = max($max_val, $price[$j] + $val[$i-$j-1]); $val[$i] = $max_val; } return $val[$n];}// Driver program to test above functions$arr = array(1, 5, 8, 9, 10, 17, 17, 20);$size = count($arr);echo "Maximum Obtainable Value is ", cutRod($arr, $size); // This code is contributed by anuj_67.?> |
Maximum Obtainable Value is 22
Please refer complete article on Cutting a Rod | DP-13 for more details!
