Problem Description

You need to implement a parser that can deserialize a string representation of a nested list structure. The input string s represents a serialization of nested lists where each element can be either:

• An integer
• A list that may contain integers or other nested lists

The goal is to parse this string and return a NestedInteger object that represents the deserialized structure.

The string follows these rules:

• An integer is represented directly as its numeric value (e.g., "123" represents the integer 123)
• A list is represented with square brackets containing comma-separated elements (e.g., "[123,456]" represents a list with two integers)
• Lists can be nested within other lists (e.g., "[123,[456,789]]" represents a list containing an integer 123 and another list with integers 456 and 789)

The solution uses a recursive approach:

1. Base cases:

   • If the string is empty or "[]", return an empty NestedInteger list
   • If the string doesn't start with "[", it must be a single integer, so parse it and return a NestedInteger containing that integer

2. Recursive case (for lists):

   • Create a new NestedInteger to hold the list
   • Track the depth of nested brackets using a counter
   • Iterate through the string starting from index 1 (after the opening bracket)
   • When at depth 0 and encountering a comma or reaching the end of the list (closing bracket), extract the substring between commas as an element
   • Recursively deserialize each element and add it to the current list
   • Adjust the depth counter when encountering brackets: increment for "[" and decrement for "]"

The algorithm correctly handles nested structures by tracking bracket depth to identify where each element at the current level begins and ends, ensuring that nested lists are treated as single elements when parsing the outer list.

Flowchart Walkthrough

First, let's pin down the algorithm using the Flowchart. Here's a step-by-step walkthrough:

Is it a graph?

• Yes: While this problem doesn't involve a traditional graph with nodes and edges, the nested list structure forms a tree-like graph where each list can contain integers or other lists as children. The parsing process needs to traverse this hierarchical structure.

Is it a tree?

• Yes: The nested list structure is essentially a tree where each NestedInteger can either be a leaf node (integer) or an internal node (list containing other NestedIntegers). The serialized string represents this tree structure with brackets denoting parent-child relationships.

DFS

• Yes: The flowchart leads us to DFS as the appropriate algorithm.

Conclusion: The flowchart suggests using DFS (Depth-First Search) for this problem. This aligns perfectly with the recursive solution approach, where:

• We recursively parse nested structures by going deep into each nested list before moving to the next element at the same level
• The recursive calls in the solution (self.deserialize(s[j:i])) represent the DFS traversal, diving deep into nested lists
• Each recursive call processes a complete subtree (either an integer or a nested list) before returning to process siblings
• The depth tracking with the depth variable helps identify when we've completed parsing a nested structure and can move to the next element at the current level

The DFS pattern is ideal here because we need to fully parse each nested element before continuing, which is exactly what the recursive solution accomplishes.

Intuition

When we look at the nested list structure like "[123,[456,789]]", we can observe that it has a recursive nature - a list can contain integers or other lists, which themselves follow the same pattern. This immediately suggests that a recursive solution would be natural.

The key insight is that we need to identify what we're currently parsing at each step:

1. If we see a number (no opening bracket), it's just an integer - parse it directly
2. If we see an opening bracket [, we're dealing with a list that needs to be broken down into its elements

For parsing a list, the challenge is determining where each element begins and ends. Consider "[123,[456,789],10]" - we can't simply split by commas because commas inside nested lists aren't delimiters for the outer list.

This leads us to the depth-tracking approach:

• When we encounter [, we're going deeper into a nested structure (depth increases)
• When we encounter ], we're coming out of a nested structure (depth decreases)
• Only when depth is 0 should we consider a comma as a delimiter for the current list

Think of it like matching parentheses - we need to find complete, balanced segments. When we're at depth 0 and hit a comma, everything from the last position to here forms one complete element (could be a number or a nested list). We can then recursively parse that substring.

The recursion naturally handles the tree-like structure: at each level, we identify the immediate children (elements separated by commas at depth 0), and for each child that's a list, we recursively apply the same logic. This is essentially a DFS traversal where we fully process each branch before moving to the next sibling.

The base cases emerge naturally: an empty string or "[]" represents an empty list, and a string without brackets must be a single integer. These stop the recursion and provide the building blocks for constructing more complex nested structures.

Solution Approach

The implementation uses recursion to parse the nested list structure, treating each element as either a base case (integer or empty list) or a recursive case (non-empty list).

Base Cases:

1. If s is empty or equals "[]", return an empty NestedInteger() object
2. If s doesn't start with "[", it must be a single integer, so parse it with int(s) and return NestedInteger(int(s))

Recursive Case (parsing a list): The main challenge is identifying where each element at the current level begins and ends. The solution uses a depth-tracking mechanism:

1. Initialize ans = NestedInteger() to hold the result list
2. Set depth = 0 to track nesting level and j = 1 to mark the start of the current element (skip the opening bracket)
3. Iterate through the string from index 1 to len(s):

   • If depth == 0 and we encounter either:

     • A comma "," - marks the end of current element
     • The last character (which would be "]") - marks the end of the last element

     Then extract the substring s[j:i] as a complete element, recursively deserialize it, and add to ans using ans.add(self.deserialize(s[j:i])). Update j = i + 1 to mark the start of the next element.

   • If we encounter "[", increment depth += 1 (entering a nested structure)

   • If we encounter "]", decrement depth -= 1 (exiting a nested structure)

The depth tracking ensures that commas inside nested lists (where depth > 0) are ignored as delimiters for the current level. Only commas at depth = 0 separate elements of the current list.

Example walkthrough for "[123,[456,789]]":

• Start with depth = 0, j = 1
• At index 4 (comma after "123"), depth = 0, so extract "123", recursively parse it (base case returns integer 123), add to result
• At index 5 ("["), depth becomes 1
• At index 9 (comma inside nested list), depth = 1, so ignore it
• At index 13 ("]"), depth becomes 0
• At index 14 (final "]"), depth = 0 and it's the last character, so extract "[456,789]", recursively parse it, add to result

This approach elegantly handles arbitrary nesting levels while maintaining O(n) time complexity for parsing the string once, though the recursion can go as deep as the nesting level.

Example Walkthrough

Let's trace through the algorithm with the input string "[12,[3,4],5]":

Initial call: deserialize("[12,[3,4],5]")

• String starts with "[", so we enter the recursive case
• Initialize: ans = NestedInteger() (empty list), depth = 0, j = 1
• Iterate from index 1 to 11:

Index 1-2: Characters are "1" and "2", no special handling needed

Index 3: Character is "," and depth = 0

• Extract substring s[1:3] = "12"
• Recursively call deserialize("12")
  • Doesn't start with "[", so it's an integer
  • Return NestedInteger(12)
• Add this to ans
• Update j = 4

Index 4: Character is "[", increment depth to 1

Index 5: Character is "3", no special handling (depth > 0)

Index 6: Character is ",", but depth = 1, so ignore as delimiter

Index 7: Character is "4", no special handling (depth > 0)

Index 8: Character is "]", decrement depth to 0

Index 9: Character is "," and depth = 0

• Extract substring s[4:9] = "[3,4]"
• Recursively call deserialize("[3,4]")
  • Starts with "[", enter recursive case
  • Parse and find two elements: "3" and "4"
  • Return NestedInteger containing list [3,4]
• Add this nested list to ans
• Update j = 10

Index 10: Character is "5", no special handling

Index 11: Last character "]" and depth = 0

• Extract substring s[10:11] = "5"
• Recursively call deserialize("5")
  • Return NestedInteger(5)
• Add this to ans

Final result: NestedInteger containing the list [12, [3,4], 5]

The key insight is how the depth counter helps us identify which commas are delimiters at the current level (when depth = 0) versus commas inside nested structures (when depth > 0). This allows us to correctly parse each element and recursively handle nested lists.

Solution Implementation

Time and Space Complexity

Time Complexity: O(n²)

The algorithm processes the string recursively. In the worst case, for a deeply nested structure like [[[[...]]]], each recursive call processes a substring and makes another recursive call. At each level, we iterate through the substring once. Since we can have up to n levels of nesting and each level processes up to n characters, the worst-case time complexity is O(n²).

However, if we consider that each character is only visited a constant number of times across all recursive calls (once in each ancestor call's loop), the time complexity can be argued to be O(n). The reference answer suggests O(n), which assumes this more optimistic analysis where despite the recursive nature, each character contributes O(1) work to the total computation.

Space Complexity: O(n)

The space complexity comes from two sources:

1. Recursion stack: In the worst case of maximum nesting depth, we can have O(n) recursive calls on the stack simultaneously.
2. Substring creation: Each recursive call creates substrings using slicing s[j:i], which takes O(k) space where k is the substring length. The total space used by all substrings across all recursive calls is bounded by O(n).
3. NestedInteger objects: The created NestedInteger structure itself takes O(n) space to represent all the elements.

Therefore, the overall space complexity is O(n).

Common Pitfalls

1. Incorrect Handling of Negative Numbers

The current implementation fails to handle negative integers correctly. When parsing a negative number like "-123", the int(s) conversion works fine for the base case, but in the recursive case for lists, the element extraction logic doesn't account for the minus sign.

Problem Example:

• Input: "[-1,2,-3]"
• The parser would incorrectly extract elements because it doesn't treat the minus sign as part of the number.

Solution: No changes needed to the core logic since int() handles negative numbers correctly. The current implementation actually works for negative numbers because the element extraction is based on bracket depth and commas, not individual characters.

2. Off-by-One Error in Element Extraction

The most critical issue is in the element extraction logic. When reaching the last element (at current_index == len(s) - 1), the substring extraction s[start_index:current_index] excludes the last character before the closing bracket.

Problem Example:

• Input: "[123,456]"
• When current_index reaches len(s) - 1 (the closing bracket), extracting s[start_index:current_index] would give "45" instead of "456".

Solution:

# Corrected element extraction
for current_index in range(1, len(s)):
    if bracket_depth == 0 and (s[current_index] == ',' or current_index == len(s) - 1):
        # For the last element, include everything up to (but not including) the closing bracket
        if current_index == len(s) - 1:
            element_string = s[start_index:current_index]  # Excludes the ']'
        else:
            element_string = s[start_index:current_index]
      
        # Only add non-empty elements
        if element_string:
            result.add(self.deserialize(element_string))
        start_index = current_index + 1

3. Empty Elements Between Consecutive Commas

If the input contains consecutive commas or trailing commas like "[1,,2]" or "[1,2,]", the current implementation would try to deserialize an empty string, causing an error.

Problem Example:

• Input: "[1,,3]" would extract an empty string between the commas
• int("") would raise a ValueError

Solution: Add a check to skip empty elements:

if bracket_depth == 0 and (s[current_index] == ',' or current_index == len(s) - 1):
    element_string = s[start_index:current_index]
    # Only process non-empty elements
    if element_string:
        result.add(self.deserialize(element_string))
    start_index = current_index + 1

4. Whitespace Handling

The implementation doesn't handle whitespace in the input string. Real-world serialized data might contain spaces for readability.

Problem Example:

• Input: "[123, 456, [789, 10]]" (with spaces after commas)
• The parser would try to convert " 456" to an integer, which works in Python but might not be intended behavior

Solution: Strip whitespace from extracted elements:

element_string = s[start_index:current_index].strip()
if element_string:
    result.add(self.deserialize(element_string))

5. Stack Overflow for Deeply Nested Structures

The recursive approach could cause stack overflow for extremely deep nesting levels (typically around 1000 levels in Python).

Problem Example:

• Input with 10,000 levels of nesting: "[[[[...]]]]"

Solution: Implement an iterative approach using an explicit stack:

def deserialize(self, s: str) -> NestedInteger:
    stack = []
    current = None
    num_str = ""
  
    for char in s:
        if char == '[':
            if current is not None:
                stack.append(current)
            current = NestedInteger()
        elif char == ']':
            if num_str:
                current.add(NestedInteger(int(num_str)))
                num_str = ""
            if stack:
                prev = stack.pop()
                prev.add(current)
                current = prev
        elif char == ',':
            if num_str:
                current.add(NestedInteger(int(num_str)))
                num_str = ""
        else:
            num_str += char
  
    return current if current else NestedInteger(int(s))