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Course Outline

Encoded Tree Sums in Python with O(n) Time Complexity


Given a tree encoded as a string, for every node, compute the sum of its subtree values and return the sums tree encoded as a string.

For example, the following tree is given as an encoded string "1{2{3,4},5{6}}" :

    1
   / \
  2   5
 / \   \
3   4   6
After computing the sums, the tree will look like the following: 
    21
   /  \
  9    11
 / \     \
3   4     6
Therefore the output will be "21{9{3,4},11{6}}"

Example 1:

Input:  "1{2{3,4},5{6}}"

Output: "21{9{3,4},11{6}}"

Example 2:

Input:  "1{1,1{1,1,1},1}"

Output: "7{1,4{1,1,1},1}"

Understanding the Problem

The core challenge of this problem is to parse the encoded tree string, compute the sum of each node's subtree, and then re-encode the tree with these sums. This problem is significant in scenarios where hierarchical data needs to be processed and summarized, such as in organizational structures or file systems.

Potential pitfalls include correctly parsing the tree structure and ensuring that the sums are computed accurately for each subtree.

Approach

To solve this problem, we need to:

  1. Parse the encoded tree string into a tree data structure.
  2. Compute the sum of each node's subtree using a post-order traversal.
  3. Re-encode the tree with the computed sums.

Naive Solution

A naive solution might involve parsing the string multiple times, which would be inefficient. Instead, we can use a single traversal to both parse and compute the sums.

Optimized Solution

We can use a recursive approach to parse the tree and compute the sums in a single pass. This approach ensures that we only traverse the tree once, making it more efficient.

Algorithm

Here is a step-by-step breakdown of the algorithm:

  1. Define a recursive function to parse the tree string and compute the sums.
  2. For each node, recursively compute the sums of its children.
  3. Sum the values of the node and its children to get the subtree sum.
  4. Re-encode the tree with the computed sums.

Code Implementation

import re

def parse_tree(s):
    # Helper function to parse the tree string
    def helper(index):
        if index >= len(s):
            return None, index
        
        # Parse the root value
        match = re.match(r'\d+', s[index:])
        if not match:
            return None, index
        
        root_val = int(match.group(0))
        index += len(match.group(0))
        
        # Create the root node
        root = {'val': root_val, 'children': []}
        
        # Check if there are children
        if index < len(s) and s[index] == '{':
            index += 1  # Skip '{'
            while index < len(s) and s[index] != '}':
                child, index = helper(index)
                if child:
                    root['children'].append(child)
                if index < len(s) and s[index] == ',':
                    index += 1  # Skip ','
            index += 1  # Skip '}'
        
        return root, index
    
    tree, _ = helper(0)
    return tree

def compute_sums(node):
    if not node:
        return 0
    
    subtree_sum = node['val']
    for child in node['children']:
        subtree_sum += compute_sums(child)
    
    node['val'] = subtree_sum
    return subtree_sum

def encode_tree(node):
    if not node:
        return ''
    
    result = str(node['val'])
    if node['children']:
        result += '{' + ','.join(encode_tree(child) for child in node['children']) + '}'
    
    return result

def encoded_tree_sums(s):
    tree = parse_tree(s)
    compute_sums(tree)
    return encode_tree(tree)

# Example usage
input_str = "1{2{3,4},5{6}}"
output_str = encoded_tree_sums(input_str)
print(output_str)  # Output: "21{9{3,4},11{6}}"

Complexity Analysis

The time complexity of this approach is O(n), where n is the number of nodes in the tree. This is because we traverse each node exactly once. The space complexity is also O(n) due to the recursion stack and the storage of the tree structure.

Edge Cases

Potential edge cases include:

  • Empty tree: The input string is empty.
  • Single node tree: The tree has only one node.
  • Deeply nested trees: Trees with a large depth.

Each of these cases should be tested to ensure the algorithm handles them correctly.

Testing

To test the solution comprehensively, we can use a variety of test cases:

def test_encoded_tree_sums():
    assert encoded_tree_sums("1{2{3,4},5{6}}") == "21{9{3,4},11{6}}"
    assert encoded_tree_sums("1{1,1{1,1,1},1}") == "7{1,4{1,1,1},1}"
    assert encoded_tree_sums("1") == "1"
    assert encoded_tree_sums("") == ""
    assert encoded_tree_sums("10{20{30,40},50{60}}") == "210{90{30,40},110{60}}"
    print("All test cases passed!")

test_encoded_tree_sums()

Thinking and Problem-Solving Tips

When approaching such problems, it's important to break down the problem into smaller, manageable parts. Focus on understanding the structure of the input and the desired output. Use recursion to handle hierarchical data structures like trees effectively.

Practice solving similar problems and study different tree traversal techniques to improve your problem-solving skills.

Conclusion

In this blog post, we discussed how to solve the problem of computing subtree sums for an encoded tree string. We explored a recursive approach to parse the tree, compute the sums, and re-encode the tree. We also analyzed the complexity and tested the solution with various test cases.

Understanding and solving such problems is crucial for developing strong problem-solving skills, especially when dealing with hierarchical data structures.

Additional Resources

For further reading and practice, consider the following resources: