#1443

Medium

auto_awesomeBinary-tree traversal and state tracking

LeetCode Problem Workspace

Minimum Time to Collect All Apples in a Tree

Minimize the time spent to collect all apples in a tree, considering traversal and state tracking with binary tree techniques.

hash table tree depth first search breadth first search

Problem Statement

You are given an undirected tree with n vertices and n-1 edges. The edges are represented as pairs [ai, bi] where there is a direct connection between vertices ai and bi. Each vertex may or may not contain an apple, as specified by the hasApple array, where hasApple[i] = true means vertex i contains an apple.

Starting at vertex 0, you must return to vertex 0 after collecting all apples in the tree. You spend 1 second for each edge you traverse. Return the minimum total time required to collect all apples, considering that you must traverse the path forward and backward for each apple-containing node.

Examples

Example 1

Input: n = 7, edges = [[0,1],[0,2],[1,4],[1,5],[2,3],[2,6]], hasApple = [false,false,true,false,true,true,false]

Output: 8

The figure above represents the given tree where red vertices have an apple. One optimal path to collect all apples is shown by the green arrows.

Example 2

Input: n = 7, edges = [[0,1],[0,2],[1,4],[1,5],[2,3],[2,6]], hasApple = [false,false,true,false,false,true,false]

Output: 6

The figure above represents the given tree where red vertices have an apple. One optimal path to collect all apples is shown by the green arrows.

Example 3

Input: n = 7, edges = [[0,1],[0,2],[1,4],[1,5],[2,3],[2,6]], hasApple = [false,false,false,false,false,false,false]

Output: 0

Example details omitted.

Constraints

1 <= n <= 105
edges.length == n - 1
edges[i].length == 2
0 <= ai < bi <= n - 1
hasApple.length == n

Solution Approach

Depth-First Search (DFS) Traversal

Use a depth-first search to explore the tree, tracking which nodes contain apples. As you traverse, compute the time needed to visit each apple-containing node, ensuring the traversal path is minimized. Remember, each apple-containing node's path must be traveled both forward and backward.

State Tracking for Apple Nodes

Track which nodes contain apples using a state array. Only the paths that lead to apple-containing nodes should be considered in the traversal. By avoiding unnecessary traversal, you minimize the time spent collecting apples.

Backtracking to Minimize Time

Backtrack to the root only if the current node contains an apple. This ensures that you only traverse the minimum number of edges necessary to collect all apples while returning to the root at the end.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time complexity of this problem is O(n), where n is the number of vertices in the tree. The space complexity is O(n) as well due to the recursion stack and state tracking for each node in the tree.

What Interviewers Usually Probe

The candidate demonstrates familiarity with tree traversal techniques.
The candidate optimizes traversal by focusing only on paths with apples.
The candidate shows ability to backtrack and minimize unnecessary traversal.

Common Pitfalls or Variants

Common pitfalls

Failing to track apple-containing nodes properly, leading to unnecessary traversal.
Not considering the time spent traveling back to the root after collecting apples.
Overcomplicating the solution by adding unnecessary checks or conditions.

Follow-up variants

Handling larger trees with up to 10^5 vertices and ensuring efficiency.
Modifying the problem to return the path taken instead of just the time.
Considering additional constraints such as limits on the number of edges per node.

FAQ

What is the primary algorithm used to solve Minimum Time to Collect All Apples in a Tree?

The solution primarily uses depth-first search (DFS) to traverse the tree while efficiently tracking paths to apple-containing nodes.

How can I optimize my solution for large trees?

Focus on minimizing traversal by only visiting nodes that contain apples, and avoid unnecessary recursion by backtracking efficiently.

What is the time complexity of the Minimum Time to Collect All Apples in a Tree problem?

The time complexity is O(n), where n is the number of vertices in the tree, due to the linear traversal of all nodes.

Why is state tracking necessary in this problem?

State tracking allows you to efficiently determine which paths need to be traversed, minimizing the total time by focusing only on apple-containing nodes.

How does backtracking help in minimizing the time in this problem?

Backtracking ensures that you only return to the root after collecting all apples, preventing unnecessary traversal of non-essential paths.

terminal

Solution

Solution 1

#### Python3

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class Solution:
    def minTime(self, n: int, edges: List[List[int]], hasApple: List[bool]) -> int:
        def dfs(u, cost):
            if vis[u]:
                return 0
            vis[u] = True
            nxt_cost = 0
            for v in g[u]:
                nxt_cost += dfs(v, 2)
            if not hasApple[u] and nxt_cost == 0:
                return 0
            return cost + nxt_cost

        g = defaultdict(list)
        for u, v in edges:
            g[u].append(v)
            g[v].append(u)
        vis = [False] * n
        return dfs(0, 0)

Continue Practicing

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