Solution Explanation

This problem requires designing a data structure that efficiently handles two operations: inserting key-value pairs and calculating the sum of values associated with keys sharing a common prefix. A Trie (prefix tree) is an ideal choice for this because it's optimized for prefix-based searches.

Approach:

The solution uses a Trie to store the keys and their associated values. Each node in the Trie represents a character in a key. Each node also stores a val which represents the sum of all values associated with keys passing through that node.

1. insert(key, val): When inserting a key-value pair, the algorithm traverses the Trie, adding nodes as needed. For each node visited, the val is incremented by the difference between the new value and the previous value associated with that key (if it existed). This efficiently handles updates.

2. sum(prefix): To find the sum of values for keys starting with a given prefix, the algorithm traverses the Trie down to the node representing the last character of the prefix. The val of this node is returned, which represents the sum of all values associated with keys that share this prefix.

Time Complexity Analysis:

• insert(key, val): The time complexity is O(m), where 'm' is the length of the key. This is because we traverse at most 'm' nodes in the Trie to insert a key.

• sum(prefix): The time complexity is O(n), where 'n' is the length of the prefix. This is because we traverse at most 'n' nodes to find the sum of values for keys with that prefix.

Space Complexity Analysis:

The space complexity is O(NM), where N is the number of keys inserted and M is the average length of the keys. In the worst-case scenario where all keys share the same prefix, space complexity can be O(NM).

Code Explanation (Python3 as an Example):

The Python code implements the Trie data structure within the MapSum class. The insert method efficiently updates the Trie with the new key-value pair, adjusting node values accordingly. The sum method quickly calculates the prefix sum by traversing the Trie to the prefix's end node.

class Trie:
    def __init__(self):
        self.children = [None] * 26  # Array to store children nodes (a-z)
        self.val = 0  # Sum of values for keys passing through this node
 
    def insert(self, w: str, x: int):
        node = self
        for c in w:
            idx = ord(c) - ord('a')  # Get index (0-25) for character
            if node.children[idx] is None:
                node.children[idx] = Trie()
            node = node.children[idx]
            node.val += x
 
    def search(self, w: str) -> int:
        node = self
        for c in w:
            idx = ord(c) - ord('a')
            if node.children[idx] is None:
                return 0
            node = node.children[idx]
        return node.val
 
class MapSum:
    def __init__(self):
        self.d = defaultdict(int) # To store key-value pairs for efficient updates
        self.tree = Trie() # Trie for prefix sum calculations
 
    def insert(self, key: str, val: int) -> None:
        x = val - self.d[key] #Difference for efficient update
        self.d[key] = val
        self.tree.insert(key, x)
 
    def sum(self, prefix: str) -> int:
        return self.tree.search(prefix)
 

The other language implementations follow a similar structure, adapting the Trie and MapSum class implementations to their respective syntax and data structures. The core logic and time/space complexities remain consistent across all languages.