Different Ways to Add Parentheses

This problem explores the different possible results you can obtain when evaluating an arithmetic expression containing only non-negative integers, '+', '-', and '*' operators, and parentheses. The goal is to generate all possible values that can be computed by inserting parentheses at different locations in the expression. This is a classic dynamic programming problem with a recursive structure, useful for understanding expression evaluation and combinatorial possibilities.

Problem Description

Given a string expression representing an arithmetic expression consisting of non-negative integers, '+', '-', and '' operators, and potentially parentheses, your task is to compute all possible values that can be obtained by inserting parentheses at different locations. The expression will only contain digits, '+', '-', '', and '('. The expression is guaranteed to be valid, meaning it can be evaluated.

You need to return a list of all possible integer values that can be computed by evaluating the expression with different parenthesizations. The order of the values in the returned list does not matter.

Key Requirements:

• The input expression is a string.
• The expression contains only digits, '+', '-', '*', and '('.
• The expression is guaranteed to be valid.
• Return a list of integers representing all possible evaluation results.
• Handle cases with no operators (just a number).
• Handle cases with nested parentheses.

Expected Behavior:

The function should parse the expression, identify the numbers and operators, and recursively explore all possible ways to group the numbers and operators using parentheses. For each possible grouping, it should evaluate the expression and add the result to the list of possible values.

Edge Cases to Consider:

• Empty expression (should not happen given the problem constraints, but good to consider).
• Expression containing only a single number.
• Expressions with nested parentheses.
• Expressions with multiple operators of the same precedence.
• Expressions with only '+' or '*' operators (results will always be positive).
• Expressions with only '-' operators (results can be negative).

Examples

Example 1:

Input: "2-1-1"
Output: [0, 2]
Explanation:
((2-1)-1) = 0
(2-(1-1)) = 2

Example 2:

Input: "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

Example 3:

Input: "1+2*3"
Output: [7, 5]
Explanation:
(1+(2*3)) = 7
(1+2)*3 = 5

Constraints

• The length of the expression string is between 1 and 20.
• The integers in the expression are non-negative and less than 100.
• The expression contains only digits, '+', '-', '*', and '('.
• The expression is guaranteed to be valid.
• The number of operators is less than or equal to the number of numbers.

Notes

A recursive approach with memoization (dynamic programming) is highly recommended for efficiency. Consider breaking down the problem into smaller subproblems by splitting the expression at each operator. The base case is when the expression contains only a single number. Think about how to represent the possible results of sub-expressions to avoid redundant calculations. The order of operations (PEMDAS/BODMAS) is implicitly handled by the recursive exploration of different parenthesizations.