Encoded Tree Sums in C++

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}}" :

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.

Approach

To solve this problem, we need to:

Parse the encoded tree string to construct the tree.
Compute the sum of each node's subtree using a post-order traversal.
Re-encode the tree with the computed sums.

Naive Solution

A naive solution might involve parsing the string multiple times, which is 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 string and compute the sums in a single pass. This approach ensures that each character in the string is processed only once, leading to an O(n) time complexity.

Algorithm

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

Define a recursive function to parse the string and compute the subtree sums.
For each node, recursively compute the sums of its children.
Sum the values of the node and its children to get the subtree sum.
Reconstruct the encoded string with the computed sums.

Code Implementation


#include <iostream>
#include <string>
#include <vector>
#include <stack>

using namespace std;

// Function to parse the tree and compute subtree sums
pair parseAndComputeSum(const string &s, int &index) {
    int num = 0;
    // Parse the number
    while (index < s.size() && isdigit(s[index])) {
        num = num * 10 + (s[index] - '0');
        index++;
    }
    
    int sum = num;
    string result = to_string(num);
    
    if (index < s.size() && s[index] == '{') {
        index++; // Skip '{'
        result += '{';
        bool first = true;
        
        while (index < s.size() && s[index] != '}') {
            if (!first) {
                result += ',';
            }
            first = false;
            
            auto [childSum, childStr] = parseAndComputeSum(s, index);
            sum += childSum;
            result += childStr;
        }
        
        index++; // Skip '}'
        result += '}';
    }
    
    return {sum, to_string(sum) + result.substr(result.find('{'))};
}

// Main function to convert the encoded tree string
string encodedTreeSums(const string &s) {
    int index = 0;
    return parseAndComputeSum(s, index).second;
}

int main() {
    string input1 = "1{2{3,4},5{6}}";
    string input2 = "1{1,1{1,1,1},1}";
    
    cout << "Output 1: " << encodedTreeSums(input1) << endl;
    cout << "Output 2: " << encodedTreeSums(input2) << endl;
    
    return 0;
}

Complexity Analysis

The time complexity of this approach is O(n), where n is the length of the input string. This is because each character in the string is processed exactly once. The space complexity is also O(n) due to the recursion stack and the storage of the result string.

Edge Cases

Potential edge cases include:

Empty string: Should return an empty string.
Single node tree: Should return the same node value.
Deeply nested trees: Ensure the recursion depth is handled properly.

Testing

To test the solution comprehensively, consider the following test cases:

Simple trees with one or two levels.
Complex trees with multiple levels and varying numbers of children.
Edge cases as mentioned above.

Thinking and Problem-Solving Tips

When approaching such problems, it is helpful to:

Break down the problem into smaller, manageable parts.
Use recursion for hierarchical data structures like trees.
Consider edge cases and test thoroughly.

Conclusion

Understanding and solving problems involving tree structures is crucial for many applications. By practicing such problems, you can improve your problem-solving skills and gain a deeper understanding of data structures and algorithms.

Additional Resources

For further reading and practice, consider the following resources:

LeetCode - Practice problems on trees and recursion.
GeeksforGeeks - Tutorials and articles on data structures and algorithms.
cplusplus.com - C++ documentation and tutorials.

Encoded Tree Sums in C++ (Time Complexity: O(n)) /homework