2920. Maximum Points After Collecting Coins From All Nodes

Problem Link

Description

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There exists an undirected tree rooted at node 0 with n nodes labeled from 0 to n - 1. You are given a 2D integer array edges of length n - 1, where edges[i] = [ai, bi] indicates that there is an edge between nodes ai and bi in the tree. You are also given a 0-indexed array coins of size n where coins[i] indicates the number of coins in the vertex i, and an integer k.

Starting from the root, you have to collect all the coins such that the coins at a node can only be collected if the coins of its ancestors have been already collected.

Coins at nodei can be collected in one of the following ways:

• Collect all the coins, but you will get coins[i] - k points. If coins[i] - k is negative then you will lose abs(coins[i] - k) points.
• Collect all the coins, but you will get floor(coins[i] / 2) points. If this way is used, then for all the nodej present in the subtree of nodei, coins[j] will get reduced to floor(coins[j] / 2).

Return the maximum points you can get after collecting the coins from all the tree nodes.

 

Example 1:

Input: edges = [[0,1],[1,2],[2,3]], coins = [10,10,3,3], k = 5
Output: 11                        
Explanation: 
Collect all the coins from node 0 using the first way. Total points = 10 - 5 = 5.
Collect all the coins from node 1 using the first way. Total points = 5 + (10 - 5) = 10.
Collect all the coins from node 2 using the second way so coins left at node 3 will be floor(3 / 2) = 1. Total points = 10 + floor(3 / 2) = 11.
Collect all the coins from node 3 using the second way. Total points = 11 + floor(1 / 2) = 11.
It can be shown that the maximum points we can get after collecting coins from all the nodes is 11. 

Example 2:

Input: edges = [[0,1],[0,2]], coins = [8,4,4], k = 0
Output: 16
Explanation: 
Coins will be collected from all the nodes using the first way. Therefore, total points = (8 - 0) + (4 - 0) + (4 - 0) = 16.

 

Constraints:

• n == coins.length
• 2 <= n <= 105
• 0 <= coins[i] <= 104
• edges.length == n - 1
• 0 <= edges[i][0], edges[i][1] < n
• 0 <= k <= 104

Solution

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Python3

class Solution:
    def maximumPoints(self, edges: List[List[int]], coins: List[int], k: int) -> int:
        graph = defaultdict(list)
        
        for a, b in edges:
            graph[a].append(b)
            graph[b].append(a)
        
        cache = {}
        
        def go(node, prev, d):
            if (node, d) in cache:
                return cache[(node, d)]
            
            c = coins[node] // (1 << d)
            o1 = c - k
            o2 = c // 2
            
            for adj in graph[node]:
                if adj != prev:
                    o1 += go(adj, node, d)
                    o2 += go(adj, node, min(d + 1, 32))
            
            res = max(o1, o2)
            cache[(node, d)] = res
            
            return res
        
        return go(0, -1, 0)