# 2493. Divide Nodes Into the Maximum Number of Groups

## Problem Description

This problem presents a scenario in which we have to divide `n` nodes of an undirected graph into groups based on specific rules. The nodes are numbered from 1 to `n`, and the edges are represented as pairs where each pair `[a, b]` signifies a bidirectional connection between node `a` and node `b`. The goal is to create the maximum number of groups, indexed from 1 to `m`, such that each node is in one group, and for every pair of connected nodes, the absolute difference in their group indexes is 1. In case it's not possible to arrange the nodes while satisfying these conditions, we should return -1.

## Intuition

The process of solving this problem can be understood as finding connected components and then verifying if we can arrange them linearly in such a way that neighboring nodes in each component differ in their group indexes by exactly 1. The steps in the solution can be described as follows:

1. We first create a representation of the graph using adjacency lists, which enables us to explore the graph easily.
2. We use depth-first search (DFS) to find all connected components within the graph. This is accomplished by iterating through each node that hasn't been visited and performing DFS to collect all the nodes in the connected component related to the current node.
3. After obtaining a connected component, we use breadth-first search (BFS) to assign distances to each of the nodes from a certain starting node. In this context, the distances represent group indexes. While doing so, we validate whether the assignment is possible following the rules given in the problem. If we enc ounter a situation where the absolute difference between the group indexes of connected nodes isn't equal to 1, we return -1 as the grouping isn't possible.
4. We keep track of the maximum distance which correlates to the largest group index obtained for the particular component. We do this because the answer to the problem is the sum of the maximum group indexes in each component since nodes of different components do not influence each other's grouping.
5. We add the maximum group indexes obtained from each component to generate the final answer which is the maximum number of groups we can divide the nodes into.
6. In case any of the components cannot be arranged as per the given rules, the function will return -1.

By the end of the process, we will either have a maximum number of groups `m` that can be formed under the given constraints or `-1` if the task is impossible.

## Solution Approach

The solution involves several algorithms and data structures, primarily Depth-First Search (DFS), Breadth-First Search (BFS), and the use of a graph representation. Here's a step-by-step explanation:

1. Representation: The graph is represented using a dictionary `g`, which maps each node to a list of its adjacent nodes.

• This is a typical adjacency list representation which allows for efficient traversal of the graph.
2. Depth-First Search (DFS): A recursion-based `dfs` function is used to traverse the graph. It helps identify all nodes belonging to the same connected component.

• `vis` is a list of boolean values indicating whether a node has been visited during DFS.
• `arr` collects all nodes of the current connected component.
• As DFS proceeds, it marks nodes as visited and adds them to `arr`.
3. Breadth-First Search (BFS): Once a connected component is identified by DFS, BFS is used to try to assign each node a group index.

• `bfs` operates on node `i`, starting by assigning distance (group index) 1 to it.
• Then it uses a queue to process nodes level by level, maintaining the invariant that the difference between group indexes of adjacent nodes is 1.
• `dist` is a list where `dist[i]` is the group index of the `i`-th node or `inf` if the node hasn't been assigned a group yet.
• BFS continues until all nodes in `arr` have been assigned a valid group index or a situation is encountered that violates the conditions for such assignment (returning `-1`).
• `ans` within the `bfs` function is the largest group index assigned to any node in the current connected component.
4. Main Logic:

• Iterate over all nodes, if a node hasn't been visited, it means we're encountering a new connected component. Perform `dfs` from this node.
• After identifying all nodes in a connected component via `dfs`, `bfs` is executed for each node `v` in `arr`. This is to ensure that if it's possible to start the assignment of groups from different nodes to maximize the number of groups.
• We keep a variable `t` to store the highest number of groups achievable from the connected component based on the various starting points, and this value is added to the overall `ans`.
5. Final Step:

• After all the components have been processed, the final answer `ans` contains the maximum number of groups that the nodes can be divided into according to the problem's conditions.
• If at any point the grouping conditions cannot be met, the solution function returns `-1`.

In essence, the algorithm is a combination of DFS to identify components and BFS to validate and find the maximal grouping of the nodes within those components. The adjacency list allows for efficient graph traversal, and the queue in BFS ensures that nodes are processed in the correct order for group assignment.

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### Example Walkthrough

Suppose we have an undirected graph with 5 nodes, and edges as follows: `[1, 2]`, `[1, 3]`, `[2, 4]`, `[3, 5]`. The graph can be visualized as:

``````11 - 2 - 4
2|
33 - 5``````

We need to divide these nodes into groups while ensuring connected nodes have group indexes differing by exactly one. Hereโs a step-by-step example using the given approach:

1. Representation: We create an adjacency list for the graph.

``1g = {1: [2, 3], 2: [1, 4], 3: [1, 5], 4: [2], 5: [3]}``
2. Depth-First Search (DFS): Initialize a `vis` list to keep track of visited nodes, starting with all nodes set to `False`.

``1vis = [False, False, False, False, False] // Indexed from 1, so index 0 is ignored.``

We start DFS from node 1, leading to the discovery of all nodes in the connected component.

``1arr = [1, 2, 3, 4, 5]``
3. Breadth-First Search (BFS): We perform BFS starting from node 1.

• Assign group index `1` to node 1 and set `dist` to keep track of group assignments, initializing other distances to `inf`.
``1dist = [inf, 1, inf, inf, inf]``
• Use a queue and begin with node 1. Then, process adjacent nodes. For each node, determine a valid group index based on its neighbors.
• After BFS, we might end up with these group assignments:
``1dist = [inf, 1, 2, 1, 2]``
• The largest group index `ans` within this component is `2`.
4. Main Logic: Since the graph is connected and we start with the first node, all nodes belong to the same connected component and are visited in the DFS step. Hence, there's no need to iterate over other nodes for the DFS step. The BFS step assigns group indexes such that adjacent nodes have indexes differing by 1.

5. Final Step:

• Having traversed all components and found a valid assignment, we add the largest group index from each component (which is 2 in our single-component example) to our overall answer `ans`.
• In this case, the answer is the maximum group index, which is `2`.

Therefore, the nodes can be divided into a maximum of 2 groups under the given constraints, and all nodes in this graph follow the rule that every connected pair of nodes has group indexes differing by 1.

## Python Solution

``````1from collections import defaultdict, deque
2
3class Solution:
4    def magnificentSets(self, n: int, edges: List[List[int]]) -> int:
5
6        # Depth-first search function to explore all the nodes within a connected component
7        def dfs(node):
8            component_nodes.append(node)
9            visited[node] = True
10            for neighbor in graph[node]:
11                if not visited[neighbor]:
12                    dfs(neighbor)
13
14        # Breadth-first search function to calculate the maximum distance from the start node
15        def bfs(start_node):
16            maximum_distance = 1
17            distances = [float('inf')] * (n + 1)
18            distances[start_node] = 1
19            queue = deque([start_node])
20
21            while queue:
22                current_node = queue.popleft()
23                for neighbor in graph[current_node]:
24                    if distances[neighbor] == float('inf'):
25                        maximum_distance = distances[neighbor] = distances[current_node] + 1
26                        queue.append(neighbor)
27
28            # Assign an incremental distance value for disconnected nodes in the component
29            for node in component_nodes:
30                if distances[node] == float('inf'):
31                    maximum_distance += 1
32                    distances[node] = maximum_distance
33
34            # Verify the resultant distances conform to the problem constraints
35            for node in component_nodes:
36                for neighbor in graph[node]:
37                    if abs(distances[node] - distances[neighbor]) != 1:
38                        return -1
39            return maximum_distance
40
41        # Initialize an adjacency list graph and visited list
42        graph = defaultdict(list)
43        for edge in edges:
44            a, b = edge
45            graph[a].append(b)
46            graph[b].append(a)
47
48        visited = [False] * (n + 1)
49        total_sets = 0
50
51        # Traverse all nodes to find disconnected components
52        for i in range(1, n + 1):
53            if not visited[i]:
54                component_nodes = []
55                dfs(i)  # DFS to populate component_nodes
56
57                # Perform BFS on all nodes in the component and find the maximum distance
58                max_dist = max(bfs(node) for node in component_nodes)
59
60                if max_dist == -1:
61                    return -1  # Early return if any component doesn't fulfill the requirements
62
63                total_sets += max_dist
64
65        # Return the total potential magnificent sets across all components
67``````

## Java Solution

``````1class Solution {
2    private List<Integer>[] graph; // The graph represented as an adjacency list
3    private List<Integer> componentNodes = new ArrayList<>(); // List to store nodes of the currently visited component
4    private boolean[] visited; // Array to track visited nodes
5    private int totalNodes; // Total number of nodes in the graph
6
7    // Method to compute the magnificent sets value for the given graph
8    public int magnificentSets(int n, int[][] edges) {
9        totalNodes = n;
10        graph = new List[n + 1];
11        Arrays.setAll(graph, k -> new ArrayList<>()); // Initialize adjacency list for each node
12        for (int[] edge : edges) { // Build the graph
13            int nodeA = edge[0], nodeB = edge[1];
16        }
17
18        visited = new boolean[n + 1];
19        int totalMagnificentSets = 0;
20        for (int i = 1; i <= n; ++i) {
21            if (!visited[i]) {
22                dfs(i); // Perform Depth-First Search to find all nodes in the component
23                int largestDepth = -1;
24                // For each node in the component, use BFS to find the largest depth
25                for (int node : componentNodes) {
26                    largestDepth = Math.max(largestDepth, bfs(node));
27                }
28                if (largestDepth == -1) {
29                    return -1; // If it's not a magnificent set, return -1
30                }
31                totalMagnificentSets += largestDepth; // Add the largest depth to the total
32                componentNodes.clear(); // Clear nodes list for next component
33            }
34        }
35        return totalMagnificentSets; // Return the total magnificent sets of all components
36    }
37
38    // Helper method for BFS to calculate the largest depth of the BFS tree from the starting node
39    private int bfs(int startNode) {
40        int[] depth = new int[totalNodes + 1];
41        Arrays.fill(depth, Integer.MAX_VALUE); // Set initial depth to a high value
42        depth[startNode] = 1; // Depth of start node is 1
43        Deque<Integer> queue = new ArrayDeque<>();
44        queue.offer(startNode); // Initialize the queue with the starting node
45
46        int maxDepth = 1; // Track the maximum depth
47        while (!queue.isEmpty()) {
48            int currentNode = queue.poll();
49            for (int neighbor : graph[currentNode]) {
50                if (depth[neighbor] == Integer.MAX_VALUE) {
51                    depth[neighbor] = depth[currentNode] + 1; // Update the depth of the neighbor
52                    maxDepth = depth[neighbor]; // Update the max depth
53                    queue.offer(neighbor); // Add the neighbor node to the queue
54                }
55            }
56        }
57
58        // Update the depth for any node that hasn't been reached by BFS
59        for (int node : componentNodes) {
60            if (depth[node] == Integer.MAX_VALUE) {
61                depth[node] = ++maxDepth;
62            }
63        }
64
65        // Verify that all edges in the component have depths differing by 1
66        for (int node : componentNodes) {
67            for (int neighbor : graph[node]) {
68                if (Math.abs(depth[node] - depth[neighbor]) != 1) {
69                    return -1; // This component isn't magnificent
70                }
71            }
72        }
73
74        return maxDepth; // Return the largest depth in the BFS tree
75    }
76
77    // Helper method for DFS to traverse all nodes in a connected component
78    private void dfs(int currentNode) {
80        visited[currentNode] = true; // Mark current node as visited
81        for (int neighbor : graph[currentNode]) { // Visit all unvisited neighbors
82            if (!visited[neighbor]) {
83                dfs(neighbor); // Recursively visit neighbors
84            }
85        }
86    }
87}
88``````

## C++ Solution

``````1class Solution {
2public:
3    // Function to find the number of magnificent sets in a graph
4    int magnificentSets(int n, vector<vector<int>>& edges) {
5        // Graph represented as an adjacency list
6        vector<vector<int>> graph(n + 1);
7        // Building the graph
8        for (auto& edge : edges) {
9            int from = edge[0], to = edge[1];
10            graph[from].emplace_back(to);
11            graph[to].emplace_back(from);
12        }
13
14        vector<int> nodes;
15        bool visited[n + 1] = {false}; // Array to keep track of visited nodes
16        int totalCount = 0; // To store the sum of magnificent sets found
17
18        // Depth First Search (DFS) to traverse components of the graph
19        function<void(int)> depthFirstSearch = [&](int node) {
20            nodes.emplace_back(node);
21            visited[node] = true;
22            for (int neighbor : graph[node]) {
23                if (!visited[neighbor]) {
24                    depthFirstSearch(neighbor);
25                }
26            }
27        };
28
29        // Breadth First Search (BFS) to find the largest distance in a single component
30        auto breadthFirstSearch = [&](int startNode) {
31            int localCount = 1;
32            int distances[n + 1];
33            memset(distances, 0x3f, sizeof distances);
34            distances[startNode] = 1;
35            queue<int> queue;
36            queue.push(startNode);
37
38            while (!queue.empty()) {
39                int currentNode = queue.front();
40                queue.pop();
41                for (int neighbor : graph[currentNode]) {
42                    if (distances[neighbor] == 0x3f3f3f3f) {
43                        localCount = distances[neighbor] = distances[currentNode] + 1;
44                        queue.push(neighbor);
45                    }
46                }
47            }
48
49            // Update distances for all nodes to ensure they are reachable
50            for (int& node : nodes) {
51                if (distances[node] == 0x3f3f3f3f) {
52                    distances[node] = ++localCount;
53                }
54            }
55
56            // Check if there is a violation in magnificent set property
57            for (int& node : nodes) {
58                for (int& neighbor : graph[node]) {
59                    if (abs(distances[node] - distances[neighbor]) != 1) {
60                        return -1;
61                    }
62                }
63            }
64            return localCount;
65        };
66
67        // Loop over all nodes to explore every component of the graph
68        for (int i = 1; i <= n; ++i) {
69            if (!visited[i]) {
70                depthFirstSearch(i);
71                int maxDistance = -1;
72                for (int& node : nodes) {
74                }
75                if (maxDistance == -1) return -1; // If impossible to find a magnificent set
76                totalCount += maxDistance; // Add the maximum distance found in this component
77                nodes.clear();
78            }
79        }
81    }
82};
83``````

## Typescript Solution

``````1// Define the magnificentSets function with 'n' of type number and
2// 'edges' as a tuple array of type number.
3const magnificentSets = (n: number, edges: [number, number][]): number => {
4
5    // Initialize graph as an array of Sets to hold edges for each vertex.
6    const graph: Set<number>[] = Array.from({ length: n + 1 }, () => new Set<number>());
7
8    // Populate the graph with the edges from the edges list.
9    for (const [u, v] of edges) {
12    }
13
14    // Create a hash map to keep track of the greatest distances.
15    const hash: Map<number, number> = new Map();
16
17    // Perform BFS to find the shortest paths.
18    for (let i = 1; i <= n; i++) {
19        let queue: number[] = [i];
20        const distances: number[] = Array(n + 1).fill(0);
21        distances[i] = 1;
22        let maxDistance: number = 1,
23            minVertex: number = n;
24
25        while (queue.length > 0) {
26            let next: number[] = [];
27            for (let u of queue) {
28                minVertex = Math.min(minVertex, u);
29                for (const v of graph[u]) {
30                    if (!distances[v]) {
31                        distances[v] = distances[u] + 1;
32                        maxDistance = Math.max(maxDistance, distances[v]);
33                        next.push(v);
34                    }
35                    if (Math.abs(distances[u] - distances[v]) !== 1) {
36                        return -1;
37                    }
38                }
39            }
40            queue = next;
41        }
42
43        // Update the hash for each node with the maximum distance.
44        hash.set(minVertex, Math.max(maxDistance, hash.get(minVertex) || 0));
45    }
46
47    // Calculate the sum of maximum distances.
48    let result: number = 0;
49    hash.forEach((value) => {
50        result += value;
51    });
52
53    // Return the total sum.
54    return result;
55};
56``````

## Time and Space Complexity

### Time Complexity

The given code consists of two Depth First Search (DFS) and Breadth First Search (BFS) traversals of a graph. Here, `n` is the number of nodes, and let's assume `e` is the number of edges.

1. DFS Part:

• The DFS is done once for each component of the graph. In the worst case, if the graph is fully connected, the DFS will visit all `n` nodes and it would traverse all edges twice (since it is an undirected graph and each edge will be considered from both ends), which gives us `O(n + 2e)`.
• However, in a DFS, each node is visited exactly once, so the complexity boils down to `O(n + e)` regardless of how many times each edge is considered.
2. BFS Part:

• For BFS in the `bfs` function, in the worst case, it will visit all nodes and traverse all edges in the connected component it starts from. This again is `O(n + e)`.
• The BFS is called once for each node in the connected component found by DFS. Therefore, in the worst case, its complexity is `O(n*(n + e))`.
3. Overall Time Complexity:

• Combining DFS and BFS, and since BFS is nested within the scope of DFS results, the overall time complexity is therefore `O(n*(n + e))`.

Note that `max(bfs(v) for v in arr)` is a potentially costly operation since it will execute BFS for each vertex `v` in `arr`. The above analysis assumes that the graph is sparse and that `e` is much smaller than `n^2`. If the graph were dense with `e` close to `n^2`, the time complexity would be `O(n^3)` because performing BFS from each vertex would amount to checking all edges for all vertices.

### Space Complexity

The space complexity is determined by the storage required for the graph representation, visited array, `arr`, and the queue used in BFS.

1. Graph Representation (`g`): The adjacency list will have at most `2e` entries, so it requires `O(e)` space.
2. Visited Array (`vis`): An array of length `n + 1`, requiring `O(n)` space.
3. Array (`arr`): In the worst case, it can hold all the nodes in the graph, so `O(n)`.
4. Queue for BFS: At most, it can hold all nodes in a connected component. Again, this is `O(n)` in the worst case.

The auxiliary space for BFS and DFS is not considered since the queue and call stack would at most hold `O(n)` entries at any point.

Thus, Overall Space Complexity: `O(n + e)`.

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