1614. Maximum Nesting Depth of the Parentheses

1614. Maximum Nesting Depth of the Parentheses

Difficulty: Easy

Topics: String, Stack, Weekly Contest 210

A string is a valid parentheses string (denoted VPS) if it meets one of the following:

• It is an empty string "", or a single character not equal to "(" or ")",
• It can be written as AB (A concatenated with B), where A and B are VPS's, or
• It can be written as (A), where A is a VPS.

We can similarly define the nesting depth depth(S) of any VPS S as follows:

• depth("") = 0
• depth(C) = 0, where C is a string with a single character not equal to "(" or ")".
• depth(A + B) = max(depth(A), depth(B)), where A and B are VPS's.
• depth("(" + A + ")") = 1 + depth(A), where A is a VPS.

For example, "", "()()", and "()(()())" are VPS's (with nesting depths 0, 1, and 2), and ")(" and "(()" are not VPS's.

Given a VPS represented as string s, return the nesting depth of s.
Given a valid parentheses string s, return the nesting depth of s. The nesting depth is the maximum number of nested parentheses.

Example 1:

• Input: s = "(1+(2*3)+((8)/4))+1"
• Output: 3
• Explanation: Digit 8 is inside of 3 nested parentheses in the string.

Example 2:

• Input: s = "(1)+((2))+(((3)))"
• Output: 3

Constraints:

• 1 <= s.length <= 100
• s consists of digits 0-9 and characters '+', '-', '*', '/', '(', and ')'.
• It is guaranteed that parentheses expression s is a VPS.

Hint:

1. The depth of any character in the VPS is the ( number of left brackets before it ) - ( number of right brackets before it )

Solution:

We need to determine the maximum nesting depth of parentheses in a given valid parentheses string (VPS). The nesting depth is defined as the maximum number of nested parentheses at any point in the string.

Approach

1. Problem Analysis: The input string is a VPS, meaning the parentheses are correctly matched and nested. The string can also contain digits and arithmetic operators, which do not affect the nesting depth.
2. Intuition: As we traverse the string, we keep track of the current nesting depth by incrementing a counter each time we encounter an opening parenthesis '(' and decrementing it when we encounter a closing parenthesis ')'. The maximum value of this counter during the traversal gives us the maximum nesting depth.
3. Algorithm Selection: We iterate through each character in the string. For each character:
   • If it is '(', we increment the current depth and update the maximum depth if the current depth exceeds the previously recorded maximum.
   • If it is ')', we decrement the current depth.
   • All other characters are ignored as they do not affect the nesting depth.
4. Complexity Analysis: The algorithm runs in O(n) time, where n is the length of the string, as we process each character exactly once. The space complexity is O(1) since we only use a few extra variables.

Let's implement this solution in PHP: 1614. Maximum Nesting Depth of the Parentheses

<?php
/**
 * @param String $s
 * @return Integer
 */
function maxDepth($s) {
    ...
    ...
    ...
    /**
     * go to ./solution.php
     */
}

// Test cases
echo maxDepth("(1+(2*3)+((8)/4))+1") . "\n"; // Output: 3
echo maxDepth("(1)+((2))+(((3)))") . "\n";   // Output: 3
?>

Explanation:

• Initialization: We start with maxDepth and currentDepth both set to 0.
• Traversal: For each character in the string:
  • Opening Parenthesis '(': When encountered, we increment currentDepth and check if it exceeds maxDepth. If so, we update maxDepth.
  • Closing Parenthesis ')': When encountered, we decrement currentDepth to indicate that we are exiting a nested level.
  • Other Characters: These are ignored as they do not affect the nesting depth.
• Result: After processing all characters, maxDepth holds the maximum nesting depth encountered, which we return as the result.

This approach efficiently tracks the nesting depth in real-time as we traverse the string, ensuring optimal performance with minimal space usage. The solution leverages the properties of a valid parentheses string to simplify the depth calculation.

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