2973. Find Number of Coins to Place in Tree Nodes

Problem Description

In this problem, we are working with an undirected tree (a connected acyclic graph) with n nodes, where each node is assigned a unique label from 0 to n-1, and the tree's root is node 0. We're given a list of edges that define the connections between nodes and an array cost where each element represents the cost associated with the corresponding node.

We need to determine the number of coins to place at each node. The rules for placing coins are as follows:

1. If the subtree of a node contains less than three nodes (either because it's a leaf node or has just one child), we place exactly one coin there.
2. If the subtree contains three or more nodes, we find the maximum product of costs from any three distinct nodes within that subtree. If this product is negative, we place zero coins; otherwise, we place a number of coins equal to this maximum product.

The goal is to return an array coin of size n, where coin[i] represents the number of coins placed at node i.

Intuition

To solve this problem, we need to walk through the tree and calculate the number of coins for each node's subtree. A Depth-First Search (DFS) can be used to traverse the tree, starting at the root node (node 0). When we visit a node, we'll consider the sizes and costs of its subtree nodes to determine the number of coins for the node.

For subtrees with less than three nodes, the situation is straightforward; we simply place one coin. However, for larger subtrees, we seek to optimize the coin count by finding the maximum product of costs using three nodes from the subtree. This involves looking at both the highest and lowest cost values, as negative costs can flip the sign of the product. Thus, we keep track of the highest three and lowest two costs in each subtree.

While any node can have multiple children, for each node, we only need to maintain the information for the smallest two and largest three costs present in its subtree after visiting all its children. These will give us enough information to calculate the maximum product for that node. We then sort those values and decide the number of coins based on the computed products, considering the edge cases where negative costs are involved.

To implement this logic efficiently, our DFS function maintains an array of costs, updating this array as we traverse the children nodes. After visiting all children, we sort this array and calculate the maximum product using either the three largest positive costs or the two smallest (which might be negative) costs and the largest cost.

Finally, after computing these products at each node, the DFS function also prunes the array of costs to contain at most five elements (the lowest two and highest three), as this is all we need for any parent nodes. This array is then returned up the call stack so that the parent nodes can similarly compute their maximum products. We initialize an answer array ans with ones to cover the subtrees of size less than three and then call the DFS function starting from the root to fill in the rest.

Learn more about Tree, Depth-First Search, Dynamic Programming, Sorting and Heap (Priority Queue) patterns.

Solution Approach

The solution approach for this problem makes use of several important algorithmic concepts such as Depth-First Search (DFS) for traversing trees, sorting for maintaining orders of costs, and efficient array manipulation for pruning unnecessary elements.

The DFS algorithm is suitable for problems dealing with trees because it allows us to explore all the nodes starting from the root, going depth-wise until we reach the leaves, and then backtracking. This way, we can explore the size and the costs associated with the subtrees.

To implement the DFS and solve the problem, the following steps are taken:

1. Adjacency List Construction:

   • An adjacency list g is constructed to represent the tree structure, which is a common data structure used to represent graphs in a memory-efficient way. For each node a, g[a] stores the list of its neighbouring nodes (children).

2. DFS Algorithm:

   • A DFS function dfs(a, fa) is defined to return an array of costs associated with the subtree rooted at node a with the parent node fa.
   • The function first includes the cost of the current node a in the results array res.
   • Then, it iterates through all the child nodes (neighbours) b of a:
     • For each child b, if b is not the parent of a, it recursively calls dfs(b, a) to get the costs from the subtree rooted at b and appends them to res.

3. Sorting Subtree Costs:

   • After collecting the subtree costs, res is sorted to easily find the largest and smallest cost values for the maximum product calculation.

4. Calculating Coins:

   • Based on the sorted costs res, the number of coins to be placed at each node is computed and stored in ans[a]:
     • If the subtree size is at least 3, it takes the maximum product of the last three costs or the product of the first two and the last cost, whichever is larger (considering non-negative product), otherwise the number of coins is 1.
     • If the result is negative, 0 coins are placed.

5. Pruning Costs Array:

   • To avoid unnecessary computation for the ancestors, the array res is pruned to keep at most the smallest two and the largest three costs — as these five values are sufficient to calculate the max product for any parent node.

6. Initializing Ans Array:

   • An array ans of size n is initialized with ones, which caters to those nodes whose subtrees are smaller than size three.

7. Starting DFS Traversal:

   • The DFS traversal is started from the root node 0 with an fa (parent node) of -1 (indicating no parent).

8. Returning the Result:

   • After the full DFS traversal, the ans array filled with the number of coins for each node is returned.

By combining these steps, the solution efficiently computes the number of coins to be placed on each node in the tree by taking advantage of DFS traversal to examine each subtree, sorting to organize the costs for quick access, and intelligent array slicing for keeping only the necessary information for further computations.

Example Walkthrough

Let's go through a small example to illustrate the solution approach:

Consider a tree with n = 4 nodes and the following edge list and cost array:

• Edges: [[0, 1], [0, 2], [1, 3]]
• Cost: [1, -2, -3, 4]

First, we’ll create an adjacency list g to represent our tree:

1g = {0: [1, 2], 1: [0, 3], 2: [0], 3: [1]}

The adjacency list describes that node 0 is connected to nodes 1 and 2, node 1 is connected to 0 and 3, and so on.

Let's begin our DFS traversal with the dfs(a, fa) function, starting at the root node 0:

1. We call dfs(0, -1) (since 0 is the root node and has no parent).

2. Initialize res with the cost of the current node, so res = [1] (cost of node 0).

3. We have two children to explore: 1 and 2.

   For child 1:

   • Call dfs(1, 0), since 1 has parent 0.
   • Initialize res for this call as [cost[1]] = [-2].
   • Since node 1 has one child node 3, call dfs(3, 1).
     • Initialize res in this call as [cost[3]] = [4].
     • Node 3 has no more children, so return [4] back to the parent 1.
   • Append 4 to res of node 1 and get res = [-2, 4].
   • Sort res to be [-2, 4] (negative to positive).
   • Since node 1 has a subtree with less than three nodes, it will hold 1 coin (ans[1] = 1).
   • Return [-2, 4] to the root's res.

   For child 2:

   • Node 2 has no children, so res will be [-3].
   • Since node 2 has a subtree with less than three nodes, it will hold 1 coin (ans[2] = 1).

4. The root's res array now holds values [1, -2, 4, -3] from its own cost and its children's subtrees.

5. We sort res to be [-3, -2, 1, 4].

6. To find the maximum product, we consider the largest three numbers or two smallest and one large for negative products, but since the maximum product from the root's perspective is negative (-3 * -2 * 4 which is 24), we set ans[0] to be this positive product.

7. We prune the res array for the root to only hold the necessary costs, which here will be [4] and return this to the "parent" which does not exist as 0 is the root.

After the DFS traversal is completed, the ans array looks like: [24, 1, 1, 1].

This array represents the number of coins at each node, completing our calculation using the major steps of the solution approach.

To summarize, node 0 stores 24 coins (maximum product of its subtree), while all other nodes store 1 coin each because their subtrees consist of less than three nodes.

Solution Implementation

Time and Space Complexity

The time complexity of the code can be analyzed based on the operations performed in the dfs function which is recursively called for each node. Within the dfs function, the sort method is called on the result list. The sort operation has a time complexity of O(k * log k) where k is the length of the list being sorted. As the solution keeps the result list with at most 5 elements after each call, sorting operations persist as O(1) constant time since it does not depend on n. However, the sorting operation is performed once for every node, resulting in a time complexity of O(n).

The algorithm involves a traversal over all edges exactly once, and thus the dfs function is called once per node, leading to an overall time complexity of O(n), where n is the number of nodes. The provided reference claiming a time complexity of O(n * log n) may assume a non-optimized scenario where all subtree outcomes are sorted, which does not hold in this optimized implementation.

Regarding the space complexity, the code utilizes additional data structures such as g which is an adjacency list representation of the graph, and ans which stores the result for each node. Both structures are of size O(n), matching the number of nodes. The recursion stack will at most grow to O(n) in case of a degenerate tree (a tree where each parent has only one child). Thus, the space complexity of the code is O(n), where O(n) is due to the storage of graph structure and results, and the O(n) recursion stack space in the worst case.

Learn more about how to find time and space complexity quickly using problem constraints.