Minimum Height Trees

 

  1. Clarify the problem:

    • The problem requires us to find the minimum height trees in an undirected graph.
    • We need to identify the root labels of the minimum height trees.
    • The graph is represented as an adjacency list.
    • The number of nodes in the graph can range from 1 to 2 * 10^4.

  2. Analyze the problem:

    • Input: An integer n representing the number of nodes in the graph, and a list of edges representing the connections between nodes.
    • Output: A list of integers representing the root labels of the minimum height trees.
    • Constraints:
      • The input graph is a valid undirected graph, and the given edges form a connected graph.
      • The number of nodes in the graph can range from 1 to 2 * 10^4.

  3. Design an algorithm:

    • We can use a "Topological Sorting" approach to solve this problem.
    • First, we create an adjacency list to represent the graph based on the given edges.
    • We start with a set of leaf nodes, which are nodes with only one neighbor.
    • We iteratively remove the leaf nodes from the graph and update their neighboring nodes.
    • This process continues until we are left with one or two nodes, which will be the minimum height trees.

  4. Explain your approach:

    • We will create an adjacency list to represent the graph.
    • We will initialize a set of leaf nodes.
    • We will iterate through the leaf nodes, remove them from the graph, and update their neighbors.
    • We will continue this process until we are left with one or two nodes.
    • Finally, we will return the remaining nodes as the root labels of the minimum height trees.

  5. Write clean and readable code:

python
def findMinHeightTrees(n, edges):
    if n == 1:
        return [0]

    adjacency_list = {i: [] for i in range(n)}
    for u, v in edges:
        adjacency_list[u].append(v)
        adjacency_list[v].append(u)

    leaves = set()
    for node in adjacency_list:
        if len(adjacency_list[node]) == 1:
            leaves.add(node)

    remaining_nodes = n
    while remaining_nodes > 2:
        remaining_nodes -= len(leaves)
        new_leaves = set()
        for leaf in leaves:
            neighbor = adjacency_list[leaf][0]
            adjacency_list[neighbor].remove(leaf)
            if len(adjacency_list[neighbor]) == 1:
                new_leaves.add(neighbor)
        leaves = new_leaves

    return list(leaves)
  1. Test your code:
    • Let's test the code with the following test case:
      • Test case: n = 6, edges = [[3,0],[3,1],[3,2],[3,4],[5,4]]
      • In this test case, the graph is represented as:
        markdown
      • 0 - 3 - 1
     |
     2
     |
     4 - 5
      • The minimum height trees are rooted at nodes 3 and 4.
python
print(findMinHeightTrees(6, [[3,0],[3,1],[3,2],[3,4],[5,4]]))  # Output: [3, 4]

  1. Optimize if necessary:

    • The current solution has a time complexity of O(n), where n is the number of nodes in the graph.
    • There is no further optimization available for this problem.

  2. Handle error cases:

    • The given code assumes the input is valid and doesn't explicitly handle error cases like invalid graph representation or negative node numbers. Additional error handling logic can be added to validate the input.

  3. Discuss complexity analysis:

    • Time complexity: O(n), where n is the number of nodes in the graph. We iterate through the graph once to build the adjacency list and then remove the leaf nodes iteratively until we are left with one or two nodes.
    • Space complexity: O(n), where n is the number of nodes in the graph. We use the adjacency list to store the graph representation, and the leaves set to store the leaf nodes.
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