Introduction

In this blog post, we will explore how to solve the LeetCode problem "1617" using AI. LeetCode is a popular platform for preparing for coding interviews, and with the help of AI tools like Chatmagic, we can generate solutions quickly and efficiently - helping you pass the interviews and get the job offer without having to study for months.

Problem Statement

There are n cities numbered from 1 to n. You are given an array edges of size n-1, where edges[i] = [ui, vi] represents a bidirectional edge between cities ui and vi. There exists a unique path between each pair of cities. In other words, the cities form a tree. A subtree is a subset of cities where every city is reachable from every other city in the subset, where the path between each pair passes through only the cities from the subset. Two subtrees are different if there is a city in one subtree that is not present in the other. For each d from 1 to n-1, find the number of subtrees in which the maximum distance between any two cities in the subtree is equal to d. Return an array of size n-1 where the dth element (1-indexed) is the number of subtrees in which the maximum distance between any two cities is equal to d. Notice that the distance between the two cities is the number of edges in the path between them. Example 1:
Input: n = 4, edges = [[1,2],[2,3],[2,4]] Output: [3,4,0] Explanation: The subtrees with subsets {1,2}, {2,3} and {2,4} have a max distance of 1. The subtrees with subsets {1,2,3}, {1,2,4}, {2,3,4} and {1,2,3,4} have a max distance of 2. No subtree has two nodes where the max distance between them is 3. Example 2: Input: n = 2, edges = [[1,2]] Output: [1] Example 3: Input: n = 3, edges = [[1,2],[2,3]] Output: [2,1] Constraints: 2 <= n <= 15 edges.length == n-1 edges[i].length == 2 1 <= ui, vi <= n All pairs (ui, vi) are distinct.

Explanation

Here's a breakdown of the solution approach, complexity, and the Python code:

• High-Level Approach:

  • Iterate through all possible subtrees using bit masking. Each bit in the mask represents whether a city is included in the subtree.
  • For each subtree, check if it's connected. If connected, calculate the maximum distance between any two nodes in the subtree using a modified Floyd-Warshall algorithm (or simply BFS starting from all nodes in subtree).
  • Count the number of subtrees for each maximum distance d.

• Complexity:

  • Runtime: O(n³ * 2ⁿ) - dominated by the Floyd-Warshall distance calculation within each subtree iteration. The outer loop iterates through 2ⁿ subtrees. Checking connectivity and distance calculation involves up to O(n³) operations (where n is the maximum possible node number).
  • Storage: O(n²) - used for the adjacency matrix and distance matrix.

Code

    def count_subtrees(n: int, edges: list[list[int]]) -> list[int]:
    """
    Counts the number of subtrees with a given maximum distance between any two cities.

    Args:
        n: The number of cities.
        edges: A list of edges representing the tree structure.

    Returns:
        A list of size n-1 where the dth element is the number of subtrees with maximum distance d.
    """

    adj_matrix = [[float('inf')] * n for _ in range(n)]
    for i in range(n):
        adj_matrix[i][i] = 0
    for u, v in edges:
        adj_matrix[u - 1][v - 1] = 1
        adj_matrix[v - 1][u - 1] = 1

    result = [0] * (n - 1)

    for mask in range(1, 2**n):
        subtree_nodes = []
        for i in range(n):
            if (mask >> i) & 1:
                subtree_nodes.append(i)

        if not subtree_nodes:
            continue

        # Check if the subtree is connected
        is_connected = False
        if len(subtree_nodes) == 1:
            is_connected = True
        else:
            start_node = subtree_nodes[0]
            visited = {start_node}
            q = [start_node]
            while q:
                curr = q.pop(0)
                for neighbor in subtree_nodes:
                    if adj_matrix[curr][neighbor] == 1 and neighbor not in visited:
                        visited.add(neighbor)
                        q.append(neighbor)

            if len(visited) == len(subtree_nodes):
                is_connected = True
            else:
                is_connected = False

        if not is_connected:
            continue

        # Calculate the maximum distance within the subtree (Floyd-Warshall)
        dist_matrix = [[float('inf')] * len(subtree_nodes) for _ in range(len(subtree_nodes))]
        for i in range(len(subtree_nodes)):
            dist_matrix[i][i] = 0

        for i in range(len(subtree_nodes)):
            for j in range(len(subtree_nodes)):
                node1 = subtree_nodes[i]
                node2 = subtree_nodes[j]
                if adj_matrix[node1][node2] == 1:
                    dist_matrix[i][j] = 1

        for k in range(len(subtree_nodes)):
            for i in range(len(subtree_nodes)):
                for j in range(len(subtree_nodes)):
                    dist_matrix[i][j] = min(dist_matrix[i][j], dist_matrix[i][k] + dist_matrix[k][j])

        max_dist = 0
        for i in range(len(subtree_nodes)):
            for j in range(i + 1, len(subtree_nodes)):
                max_dist = max(max_dist, dist_matrix[i][j])

        if max_dist > 0 and max_dist < n:
            result[max_dist - 1] += 1

    return result