Minimum Time to Collect All Apples in a Tree

Problem Statement

In this task, you are given an undirected tree, represented by n vertices numbered from 0 to n-1. Each vertex may or may not contain an apple. The challenge is to determine the minimum time required to collect all apples in the tree by traversing its edges, starting and ending at vertex 0. Each edge traversal takes exactly one second.

The tree structure is given by an array edges, where each element edges[i] = [ai, bi] represents an undirected edge between vertices ai and bi. To identify which vertices contain apples, a boolean array hasApple is provided where hasApple[i] = true means that vertex i has an apple.

The objective is to calculate and return the total minimum seconds required to traverse from vertex 0, collect all the apples located in various vertices of the tree, and return back to vertex 0.

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

Explanation:

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

Explanation:

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

Constraints

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

Approach and Intuition

To solve the problem, the key is to optimize the path taken to minimize the travel time:

1. Build Tree Connections: Start by constructing the tree using the edges array. This typically involves building an adjacency list to represent the connected nodes.
2. Depth-First Search (DFS) Implementation: Use DFS starting from the root vertex (vertex 0 in this case) to explore all reachable nodes. The DFS should track the cumulative time to reach each vertex and backtrack accordingly.
3. Track Apples with DFS: While performing DFS, the function should keep track of whether any vertex in the subtree of the current vertex has an apple. If it does, the traversal should continue to accumulate time; otherwise, it can cut off exploration, effectively pruning paths that aren't necessary.
4. Calculate Return Time: Each time DFS visits a vertex with an apple, it should calculate the time taken to go to that vertex and double it (as you need to return). This accumulated time will give the total minimum seconds needed.
5. Optimize the Path: Make sure that the path does not include unnecessary vertices—vertices without apples and whose all children also do not have apples should be excluded from the path calculation.

By traversing strategically only through necessary paths where apples are effectively collected, the solution ensures that the time computed is minimal. Use recursion appropriately in DFS to explore each branch and backtrack, adjusting the path time whenever an apple-bearing vertex is found.

Solutions

class Solution {
public:
    int exploreTree(int current, int prev, vector<vector<int>>& graph, vector<bool>& applePresence) {
        int totalCost = 0, timeForChild = 0;
        for (auto childNode : graph[current]) {
            if (childNode == prev) continue;
            timeForChild = exploreTree(childNode, current, graph, applePresence);
            // Only increase total cost if there's an apple in the subtree or at the child node
            if (timeForChild || applePresence[childNode]) totalCost += timeForChild + 2;
        }
    
        return totalCost;
    }
    
    int minTime(int n, vector<vector<int>>& edges, vector<bool>& applePresence) {
        vector<vector<int>> graph(n);
        for (auto& edge : edges) {
            graph[edge[0]].push_back(edge[1]);
            graph[edge[1]].push_back(edge[0]);
        }
        return exploreTree(0, -1, graph, applePresence);
    }
};