. Different Ways to Add Parentheses

241. Different Ways to Add Parentheses

Difficulty: Medium

Topics: Math, String, Dynamic Programming, Recursion, Memoization

Given a string expression of numbers and operators, return all possible results from computing all the different possible ways to group numbers and operators. You may return the answer in any order.

The test cases are generated such that the output values fit in a 32-bit integer and the number of different results does not exceed 10⁴.

Example 1:

• Input: expression = "2-1-1"
• Output: [0,2]
• Explanation:

  ((2-1)-1) = 0
  (2-(1-1)) = 2

Example 2:

• Input: expression = "2*3-4*5"
• Output: [-34,-14,-10,-10,10]
• Explanation:

  (2*(3-(4*5))) = -34
  ((2*3)-(4*5)) = -14
  ((2*(3-4))*5) = -10
  (2*((3-4)*5)) = -10
  (((2*3)-4)*5) = 10

Constraints:

• 1 <= expression.length <= 20
• expression consists of digits and the operator '+', '-', and '*'.
• All the integer values in the input expression are in the range [0, 99].
• The integer values in the input expression do not have a leading '-' or '+' denoting the sign.

Solution:

We can use recursion combined with memoization to store previously computed results for sub-expressions, as this avoids redundant calculations and optimizes the solution.

Approach:

1. Recursion:

   • For each operator in the string (+, -, *), we split the expression at that operator.
   • Recursively compute the left and right parts of the expression.
   • Combine the results of both parts using the operator.

2. Memoization:

   • Store the results of sub-expressions in an associative array to avoid recalculating the same sub-expression multiple times.

3. Base Case:

   • If the expression contains only a number (i.e., no operators), return that number as the result.

Example Walkthrough:

For the input "2*3-4*5":

• Split at * -> 2 and 3-4*5
  • Recursively compute 3-4*5 and 2, then combine.
• Split at - -> 2*3 and 4*5
  • Recursively compute each and combine.
• Similarly for other splits.

Let's implement this solution in PHP: 241. Different Ways to Add Parentheses

<?php
class Solution {

    /**
     * @var array
     */
    private $memo = [];

    /**
     * @param String $expression
     * @return Integer[]
     */
    public function diffWaysToCompute($expression) {
       ...
       ...
       ...
       /**
        * go to ./solution.php
        */
    }

    /**
     * @param $expression
     * @return array|mixed
     */
    private function compute($expression) {
       ...
       ...
       ...
       /**
        * go to ./solution.php
        */
    }
}

// Example usage
$solution = new Solution();
$expression1 = "2-1-1";
$expression2 = "2*3-4*5";
print_r($solution->diffWaysToCompute($expression1)); // Output: [0, 2]
print_r($solution->diffWaysToCompute($expression2)); // Output: [-34, -14, -10, -10, 10]
?>

Explanation:

1. Memoization: The $memo array stores the computed results for each expression to avoid redundant calculations.
2. Base Case: When the expression is numeric, it’s converted to an integer and added to the results.
3. Recursive Splitting: For each operator in the expression, split it into left and right parts, recursively compute the results for both parts, and then combine them based on the operator.
4. Example Usage: The diffWaysToCompute function is tested with example expressions to verify the solution.

This approach ensures that you efficiently compute all possible results by leveraging memoization to avoid redundant computations.

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