2872. Maximum Number of K-Divisible Components - Detailed Explanation

Problem Statement

Formulation:
You are given a tree with n nodes (labeled from 0 to n–1). Each node has an integer value given in an array nums (where nums[i] is the value of node i). You are also given a list of edges representing the undirected tree (with n–1 edges) and an integer k.

You are allowed to remove some edges from the tree. Removing an edge splits the tree into two connected components. A connected component is considered k-divisible if the sum of its node values is divisible by k.

Your task is to remove as many edges as possible (each removal increases the number of components by one) so that every resulting connected component has a sum of node values that is divisible by k.

Return the maximum number of connected components you can obtain. (Since the number of components is one plus the number of removed edges, you may also think of maximizing the removals.) If it is not possible to partition the tree into k-divisible components, return -1.

Example 1

Input:

nums = [5, 10, 5, 5]
edges = [[0,1], [1,2], [1,3]]
k = 5

Output:

4

Explanation:

• Total sum = 5 + 10 + 5 + 5 = 25, which is divisible by 5.
• By rooting the tree (say at node 1) we can perform a DFS:
  • The subtree at node 0 has sum = 5, divisible by 5 → remove edge (1,0).
  • The subtree at node 2 has sum = 5, divisible by 5 → remove edge (1,2).
  • The subtree at node 3 has sum = 5, divisible by 5 → remove edge (1,3).
• We can remove 3 edges; thus, the number of components is 3 + 1 = 4.

Example 2

Input:

nums = [4, 3, 6, 8]
edges = [[0,1], [1,2], [1,3]]
k = 5

Output:

-1

Explanation:

• Total sum = 4 + 3 + 6 + 8 = 21, which is not divisible by 5.
• Since the entire tree’s sum is not divisible by 5, it is impossible to split the tree into components that are all k-divisible.
• Hence, the answer is -1.

Constraints

• n (number of nodes): Can be moderately large.
• nums[i]: May be positive, negative, or zero.
• Edges: Represent a valid tree (i.e. n–1 edges with no cycles).
• k: A positive integer.

Hints

1. Total Sum Check:
   Before any removals, verify that the sum of all node values is divisible by k. If not, it is impossible to split the tree as required; return -1.

2. Subtree DFS:
   Use a depth-first search (DFS) to compute the sum of the subtree for every node.

   • For every edge between a node and one of its children, if the child’s subtree sum is divisible by k, then you can remove that edge.
   • Removing that edge isolates a valid component (with sum divisible by k) and does not affect the divisibility of the remaining part.

3. Count Valid Removals:
   Each valid removal increases the component count by one. Finally, the maximum number of k-divisible components is the number of removals plus one (for the remaining component).

Approach

Approach 1: DFS on the Tree (Optimal)

• Step 1:
  Compute the total sum of all node values. If the total sum is not divisible by k, return -1 immediately.

• Step 2:
  Pick an arbitrary node as the root (commonly node 0) and perform a DFS.
  For each node:

  • Compute the sum of its subtree.
  • For every child subtree, if the subtree sum is divisible by k, mark that edge as removable (i.e. increment the count of valid removals).
  • When returning to the parent, add the child’s subtree sum to the parent’s running sum (unless that subtree was removed—but note that since its sum is divisible by k, subtracting a multiple of k does not affect divisibility).

• Step 3:
  After the DFS, the answer is the count of removable edges plus one. (Because removing an edge increases the number of components by one.)

• Intuition:
  Removing an edge where the child subtree’s sum is divisible by k ensures that both the removed subtree and the remaining tree still satisfy the divisibility condition (since the total sum is divisible by k).

Step-by-Step Walkthrough with Visual Example

Consider the tree in Example 1:

    1 (value 10)
   / | \
  /  |  \
 0   2   3
(value 5) (value 5) (value 5)

1. Total Sum:
   Total = 5 + 10 + 5 + 5 = 25 → Divisible by 5? Yes.

2. DFS Starting at Node 1 (Root):

   • Node 0:
     Subtree sum = 5. Since 5 % 5 == 0, edge (1,0) can be removed.

   • Node 2:
     Subtree sum = 5. Valid removal (edge (1,2)).

   • Node 3:
     Subtree sum = 5. Valid removal (edge (1,3)).

   • Back at Node 1:
     After removing these valid subtrees, node 1’s value remains 10.
     (Note: Removing an edge does not require us to subtract the subtree sum because subtracting a multiple of k keeps the remainder unchanged.)

3. Result:
   Valid removals = 3 → Components = 3 + 1 = 4.

Code Implementation

Python Code