Minimum Cost to Change the Final Value of Expression

Start by immediately considering the boolean expression and the final result it evaluates to. Use a state transition dynamic programming approach to track minimal changes needed for each subexpression. Focus on how operators and parentheses influence the cost, ensuring the computation efficiently handles nested structures and all valid states.

Problem Statement

You are given a valid boolean expression as a string containing '1','0','&','|','(', and ')'. Each operator affects the expression value, and parentheses determine order of evaluation. Your goal is to compute the minimal number of operations required to flip the final evaluated value of the entire expression.

An operation consists of changing an operator from '&' to '|' or vice versa. The expression is guaranteed to be well-formed with matched parentheses and non-empty subexpressions. Determine the least number of such operations needed so that the expression evaluates to the opposite boolean value of its original result.

Examples

Example 1

Input: expression = "1&(0|1)"

Output: 1

We can turn "1&(0|1)" into "1&(0&1)" by changing the '|' to a '&' using 1 operation. The new expression evaluates to 0.

Example 2

Input: expression = "(0&0)&(0&0&0)"

Output: 3

We can turn "(0&0)&(0&0&0)" into "(0|1)|(0&0&0)" using 3 operations. The new expression evaluates to 1.

Example 3

Input: expression = "(0|(1|0&1))"

Output: 1

We can turn "(0|(1|0&1))" into "(0|(0|0&1))" using 1 operation. The new expression evaluates to 0.

Constraints

• 1 <= expression.length <= 105
• expression only contains '1','0','&','|','(', and ')'
• All parentheses are properly matched.
• There will be no empty parentheses (i.e: "()" is not a substring of expression).

Solution Approach

Recursive State DP with Memoization

For each subexpression, calculate the minimal cost to make it evaluate to 0 or 1. Use recursion with memoization to avoid recomputation. Each state represents the subexpression bounds and target value.

Combine Results via Operators

For each operator, compute the minimal cost by combining left and right subexpression costs. Account for flipping the operator itself if it reduces total cost. Track all combinations to ensure global minimality.

Handle Parentheses Efficiently

Treat each pair of parentheses as a subexpression boundary. Recursively apply DP inside parentheses and propagate results outward. This avoids recalculating overlapping subexpressions and respects expression hierarchy.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

Time and space depend on the number of subexpressions and states tracked. With memoization, the complexity is O(n^3) worst-case for all subexpression ranges, but practical pruning reduces it. Space stores DP tables per subexpression and target value.

What Interviewers Usually Probe

• Ask how to represent subexpression states and minimal costs clearly.
• Probe whether memoization prevents redundant recalculation of nested parentheses.
• Check if candidate considers flipping operators versus changing subexpression values.

Common Pitfalls or Variants

Common pitfalls

• Ignoring that changing an operator might not always reduce total cost.
• Failing to correctly track all possible states for subexpressions inside parentheses.
• Assuming linear traversal works without considering operator precedence and nested structures.

Follow-up variants

• Compute minimal changes to achieve a specific target value other than flipping.
• Allow changing operands '0' to '1' and vice versa, not just operators.
• Handle expressions with additional operators like '^' while maintaining minimal cost.

FAQ

What pattern does this problem rely on?

It relies on state transition dynamic programming, tracking minimal changes for each subexpression target value.

Can parentheses nesting affect the solution?

Yes, each pair defines a subexpression, and DP must respect nested boundaries to compute minimal costs correctly.

Is flipping operators always optimal?

Not always; sometimes combining subexpression results without flipping yields a lower cost.

What is the maximum expression length supported?

Expressions can be up to 105 characters, including '1','0','&','|','(', and ')'.

How does GhostInterview handle 'Minimum Cost to Change the Final Value of Expression'?

It automatically computes minimal operator flips for all subexpressions using state transition DP and memoization.