Maximum Nesting Depth of the Parentheses

Maximum Nesting Depth of the Parentheses

What is this problem about?

The Maximum Nesting Depth of the Parentheses coding problem gives you a valid parentheses string (possibly containing digits and operators as a VPS — valid parentheses string). You need to find the maximum depth of nesting, i.e., the maximum number of open parentheses that are open at any one time. For example, ((())) has depth 3 and (()()) has depth 2.

Why is this asked in interviews?

Microsoft, Meta, Google, Bloomberg, Amazon, and tcs include this as a warm-up problem to verify basic stack or counter reasoning. While trivial for experienced candidates, it tests clean implementation of a counter-based traversal. Interviewers use it to evaluate code quality and ability to handle simple string traversal problems without bugs.

Algorithmic pattern used

Counter-based linear scan (no stack needed): Traverse the string character by character. Maintain a current_depth counter. Increment it on (, decrement it on ). Track the maximum value of current_depth seen during the traversal. This is O(n) time and O(1) space — no stack needed since we only care about the maximum depth, not the structure.

A stack-based approach is also valid but unnecessarily complex for this problem.

Example explanation

String: "(1+(2*3)+((8)/4))+1"

• (: depth=1, max=1
• 1+(2: at ( → depth=2, max=2
• *3): at ) → depth=1
• +((: two ( → depth=2, then depth=3, max=3
• 8)/4): ) → depth=2, /4 no change, ) → depth=1
• ): depth=0
• +1: no parens
• Maximum nesting depth = 3.

Common mistakes candidates make

• Using a stack unnecessarily: A counter is sufficient; pushing/popping to a stack adds code complexity without benefit for this specific problem.
• Counting all parentheses instead of tracking depth: Counting total ( characters gives the total count, not the maximum concurrent depth.
• Not resetting on ): Forgetting to decrement on ) keeps depth growing instead of accurately tracking the current nesting level.

Interview preparation tip

For the String Stack interview pattern, the key distinction is: when do you need a full stack vs just a counter? If you need to remember which parentheses are open (e.g., for matching or balancing), use a stack. If you only need the current count or maximum, a counter is enough and faster to code. Recognizing this distinction saves time in timed interview settings.