Problem Description

In this problem, we are given an undirected tree consisting of n nodes, where each node is labeled from 0 to n - 1. Each node has an associated integer value given in an array nums[]. Alongside, we are provided a list of edges edges[], which represent the links between the trees' nodes.

Our objective is to figure out the maximum number of edges we can remove from this tree in such a way that the resultant connected components all have the same value. The value of a component is defined as the sum of the values of nodes within that component.

To summarize the problem: With a given set of nodes and their connections, we must strategically delete edges so as to partition the tree into equal-valued components, and we want to maximize the number of deletions we can make.

Flowchart Walkthrough

First, let's pin down the algorithm using the Flowchart. Here's a step-by-step walkthrough:

Is it a graph?

• Yes: The problem involves analyzing edges and nodes in a graph to split nodes into components.

Is it a tree?

• Yes: The problem description indicates that we are working with a tree (no cycles and exactly n-1 edges for n nodes).

Perform a DFS:

• Since it's a tree, we can use Depth-First Search (DFS) to solve the problem efficiently. DFS allows traversal of all nodes while checking and modifying values as needed to create components of equal values.

Conclusion: The flowchart navigates through the criteria of graph and tree being true, suggesting the use of DFS for this problem.

Intuition

To solve this problem, we need to determine if the tree can be partitioned into k equal-valued components. If the sum of all node values is s, then each component's total value must be t = s / k. It is only possible to have components of equal value if s is divisible by k.

The solution approach uses the following insights:

1. We iterate over potential component sizes by checking whether the sum of all node values s is divisible by k.
2. For a given k, we compute the target sum t that each component should have (s/k) and then use depth-first search (DFS) to compute the sum of values in each component, starting from the root node.
3. During DFS, if we encounter a subtree whose sum is exactly t, we consider this subtree as a potential component. The edge connecting this subtree to the rest of the tree can be one of the deletable edges.
4. If a subtree's sum exceeds t or if after considering a subtree as component we end up with a partial sum exceeding t, then this partitioning is not possible, and we continue to check the next possible value of k.
5. If the dfs function returns 0 for a given k, it means we have successfully partitioned the tree into k components of equal sum t, and thus can delete k-1 edges to create these k components.
6. We try dividing s from the smallest k possible (s divided by the maximum value in nums, which is the theoretically highest possible k that makes t larger than any single node value) to 2, and return k-1 for the first successful partitioning.

This approach ensures that we test the maximum number of partitions first, which inherently means we would be removing the maximum number of edges possible to achieve equal-valued components.

Learn more about Tree, Depth-First Search and Math patterns.

Solution Approach

To implement the approach described above, the solution script defines a dfs function and iteratively seeks the right k that can divide the total sum of all node values s into equal parts. Here's how the implementation unfolds:

• First, we prepare a graph g in the form of an adjacency list with the help of a defaultdict of lists. This graph g represents the tree structure defined by the edges array. Each edge connects two nodes, and for every edge [a, b], we add b to the list of adjacent nodes of a, and vice versa.

• The variable s is the sum of all node values and represents the total value that we seek to divide into equal parts.

• The dfs function performs a Depth-First Search starting from the root node (which is node 0) of the tree. It calculates the sum of values of each subtree. The critical points are:

  • If the subtree's value sum equals the target value t (meaning this subtree can constitute a standalone component), the dfs returns 0, indicating the potential edge to cut.
  • If the subtree's sum exceeds the target value t, or if collecting this subtree's sum would exceed t for any component, dfs returns -1, which indicates that this partitioning configuration is invalid.

• The main part of the solution involves the loop for k in range(min(n, s // mx), 1, -1):. This loop seeks a k that is a divisor of s. It attempts to partition the tree into k components with equal values starting from the largest possible k to 2. The reason behind this is that we wish to maximize the number of edges we can delete (which corresponds to k - 1). The maximum possible value of k is dictated by the total sum s and the maximum node value mx, as s divided by mx gives an upper bound on k.

• Every iteration checks if s is divisible by k by checking if s % k == 0:. If the condition holds, it sets the target value t to s // k.

• Then, it calls dfs(0, -1). If dfs traverses the entire tree and still returns 0, meaning it successfully identified components with value equal to t, then we found a successful partition count k.

• The solution returns k - 1 because if we can partition the tree into k equal components, then there are k - 1 edges that we can remove to separate each component.

• If no value for k results in a successful partition, the solution returns 0.

The algorithm effectively balances between the depth-first traversal to calculate subcomponent sums and the strategy of checking from the highest possible partition count. By combining these two, it successfully finds the maximum number of deletable edges to ensure all connected components of the tree have the same value.

Example Walkthrough

Let's illustrate the solution approach using a small example:

Suppose we have n = 4 nodes with values nums = [1, 2, 2, 1] and edges edges = [[0, 1], [1, 2], [1, 3]]. The tree can be visualized as follows:

    0(1)
     |
    1(2)
   /   \
2(2)  3(1)

The values in parentheses represent the nums values for each node.

We are looking to maximize the number of removable edges while ensuring that each resulting connected component has the same sum of values.

Step 1 (Initialize the graph):

We construct an adjacency list from the edges:

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

Step 2 (Calculate the total sum s of all nodes and determine possible k):

Total s is 1 + 2 + 2 + 1 = 6. We can partition the tree into components with equal sums only if s is divisible by k. So we could try k=2 or k=3, since 6 is not divisible by 4.

Step 3 (Depth-First Search to find possible partitions):

We start with the maximum possible k, which is s divided by the maximum value in nums. Since s = 6 and the maximum value is 2, we initially try for k = 3.

We set our target sum t = s / k, which is 6 / 3 = 2. We'd need to find three components where the sum is 2 in each, but since our individual subtree sums are either 1 or greater than 2, this isn't possible. We move onto k = 2.

With k = 2, the target sum t becomes 6 / 2 = 3. We perform DFS to find sum of subtrees:

• Starting from root (0), we have a value 1.
• We move to node 1 with value 2, the running sum is now 3.
  • Check child node 2, which would add 2, making the running sum 5.
  • Check child node 3, which would add 1, making the running sum 4.

In both cases for node 1, the running sum exceeds t=3, thus we cannot form a component from this subtree that equals exactly t. However, if we consider the subtree rooted at node 1 as a separate part, its sum is 2 + 2 + 1 = 5. The remaining component (node 0) has a sum of 1. This does not satisfy our conditions, so this partition attempt fails.

Since no value of k yielded a successful partition in this example, the answer is 0: we cannot remove any edges that would result in equal-sum components.

In a case where we would find a subtree that matches our target t, the DFS would effectively "cut" the edge connecting that subtree to the larger tree and return a count that enabled us to determine if that "cut" was valid. With a valid cut, and a subsequent valid partition scheme, we would report the number of edges we can remove to fulfill the problem's conditions.

Solution Implementation

Time and Space Complexity

Time Complexity

The time complexity of the given code can be broken down into two main parts: the construction of the graph and the depth-first search (DFS) operation.

• Graph Construction: The loop over the list of edges which constructs the adjacency list representation of the graph runs in O(E), where E is the number of edges. Since we're working with a tree (as implied by the nature of the problem), the number of edges is E = N - 1, where N is the number of nodes. Hence, constructing the graph takes O(N) time.

• Depth-First Search (DFS): The main for-loop runs from s // mx down to 1. In the worst case, this would be O(N). The DFS function is called once for each node in the worst case, and in each call, it iterates over all the adjacent nodes. Because it's a tree, there are no repeated edges for each node, so total iterations of all DFS calls amount to O(N) as well. Therefore, the DFS complexity is O(N^2) in the worst case.

Overall, the loop that iterates from s // mx to 1 combines with the DFS to form a nested loop, so the total time complexity is O(N^2).

Space Complexity

The space complexity of the code is dominated by the recursion stack from the DFS and the storage of graph g.

• Graph Storage: The graph g is stored as an adjacency list, which takes O(V + E) space where V is the number of vertices and E is the number of edges. For a tree, since E = V - 1, the space required for the graph is O(N).

• DFS Recursion Stack: In the worst case, DFS could go as deep as the height of the tree, which in the case of a skewed tree (worst case) is O(N). Thus, the worst-case space complexity due to the recursion stack is also O(N).

Combining these factors, the total space complexity of the algorithm is O(N).

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