cpp-flashcards/org/study_deck_02/dsa/advanced-graphs/2493-divide-nodes-into-the-maximum-number-of-groups.org

#+ANKI_DECK: study_deck_02
* TODO 2493. Divide Nodes Into the Maximum Number of Groups :hard:
:PROPERTIES:
:NEETCODE: [[file:../../roadmap.org::*2493. Divide Nodes Into the Maximum Number of Groups][2493. Divide Nodes Into the Maximum Number of Groups]]
:END:

You are given a positive integer ~n~ representing the number of nodes in an *undirected* graph. The nodes are labeled from ~1~ to ~n~.

You are also given a 2D integer array ~edges~, where ~edges[i] = [a_{i, }b_{i}]~ indicates that there is a *bidirectional* edge between nodes ~a_{i}~ and ~b_{i}~. *Notice* that the given graph may be disconnected.

Divide the nodes of the graph into ~m~ groups (*1-indexed*) such that:

 - Each node in the graph belongs to exactly one group.

 - For every pair of nodes in the graph that are connected by an edge ~[a_{i, }b_{i}]~, if ~a_{i}~ belongs to the group with index ~x~, and ~b_{i}~ belongs to the group with index ~y~, then ~|y - x| = 1~.

Return /the maximum number of groups (i.e., maximum /~m~/) into which you can divide the nodes/. Return ~-1~ /if it is impossible to group the nodes with the given conditions/.

*Example 1:*


#+begin_src
Input: n = 6, edges = [[1,2],[1,4],[1,5],[2,6],[2,3],[4,6]]
Output: 4
Explanation: As shown in the image we:
- Add node 5 to the first group.
- Add node 1 to the second group.
- Add nodes 2 and 4 to the third group.
- Add nodes 3 and 6 to the fourth group.
We can see that every edge is satisfied.
It can be shown that that if we create a fifth group and move any node from the third or fourth group to it, at least on of the edges will not be satisfied.
#+end_src


*Example 2:*


#+begin_src
Input: n = 3, edges = [[1,2],[2,3],[3,1]]
Output: -1
Explanation: If we add node 1 to the first group, node 2 to the second group, and node 3 to the third group to satisfy the first two edges, we can see that the third edge will not be satisfied.
It can be shown that no grouping is possible.
#+end_src


*Constraints:*

 - ~1 <= n <= 500~

 - ~1 <= edges.length <= 10^{4}~

 - ~edges[i].length == 2~

 - ~1 <= a_{i}, b_{i} <= n~

 - ~a_{i} != b_{i}~

 - There is at most one edge between any pair of vertices.

** TODO Approach
Write your approach here.

** TODO Python
#+begin_src python :lc-problem 2493 :lc-lang python3
class Solution:
    def magnificentSets(self, n: int, edges: List[List[int]]) -> int:
#+end_src

** TODO C++
#+begin_src cpp :lc-problem 2493
class Solution {
public:
    int magnificentSets(int n, vector<vector<int>>& edges) {
        
    }
};
#+end_src