Problem Description

The problem requires finding the minimum degree of a connected trio in an undirected graph. A connected trio is a set of three nodes where there is an edge connecting every pair of nodes within the set. The degree of the connected trio is the total number of edges that are connected to the trio's nodes where the other endpoint of these edges is not within the trio itself.

You are given:

• An integer n indicating the total number of nodes in the graph.
• An array edges where each element is a pair [u, v], representing an undirected edge between nodes u and v.

The task is to calculate the minimum degree among all possible connected trios in the graph. If the graph contains no connected trios, the function should return -1.

Intuition

To solve this problem, we need to find all possible connected trios and determine their degrees. We then find the minimum degree among them.

The intuition behind the solution is to:

1. Create a graph representation that allows us to quickly check if an edge exists between any two nodes.
2. Count the degree of each node, which is the number of edges connected to it.
3. Iterate through all possible combinations of three nodes and check if they form a connected trio.
4. If a connected trio is found, compute its degree by adding up the degrees of the three nodes, then subtracting the six edges that form the trio from the calculation because these internal edges should not count towards the degree of the trio.
5. Keep track of the minimum degree found.

The algorithm uses a 2D list g that acts as an adjacency matrix with a boolean value indicating whether an edge exists between any two nodes (indexed by u-1 and v-1 to convert to zero-based indices). Another list deg holds the degree count for each node.

The solution iterates over three nested loops to consider every possible trio of nodes (i, j, k), checking if g[i][j], g[i][k], and g[j][k] are all True, which would indicate a connected trio. Upon finding a trio, it calculates the degree of the trio and updates the minimum answer (ans) accordingly, ensuring to subtract six to exclude the internal edges of the trio.

The answer (ans) is initially set to infinity (a value representing an unreachable high number) to allow it to be updated to any lower valid degree value found during the iterations. If after all the iterations, the ans value remains infinity, it means that no trios were found, and thus -1 is returned. Otherwise, the minimum degree found is returned.

Learn more about Graph patterns.

Solution Approach

The solution approach consists of several key steps implemented through the use of effective data structures and algorithms:

Graph Representation:

• The solution uses an adjacency matrix g to represent the graph, where g[i][j] holds a boolean value indicating the presence of an edge between nodes i and j.
• The adjacency matrix is chosen for its ease of checking if a pair of nodes is directly connected.
• This matrix is populated in the first loop where we iterate over the edges and mark True for the corresponding positions in the matrix g[u][v] and g[v][u], as well as increment the degree deg[u] and deg[v] for the ends of each edge.

Degree Calculation:

• An array deg of size n is used to keep track of the degree of each node (i.e., the number of edges connected to each node).
• While setting up the adjacency matrix, the degree of each node involved in an edge is incremented, thus populating the deg array.

Finding Connected Trios and Calculating Degrees:

• Three nested loops over the nodes are used to test each possible trio (a combination of three nodes).
• For each potential trio, the solution checks if all three possible edges between them exist. This is done using the adjacency matrix. If g[i][j], g[i][k], and g[j][k] are all True, a trio is found.
• For each connected trio, its degree is calculated by summing up deg[i], deg[j], and deg[k], and then subtracting 6 to exclude the edges within the trio itself.
• The intuition behind subtracting 6 is that each edge within the trio would have been counted twice - once for each node it connects.

Finding the Minimum Degree:

• The variable ans is used to keep the minimum degree found. It is initialized to inf (infinity) to ensure that the first valid degree calculated will be less than this initial value.
• Once a connected trio is found and its degree is calculated, ans is updated if the current trio's degree is lower than the previously stored value.
• After checking all possible trios, if ans is still inf, it means no connected trios exist, and therefore -1 is returned.
• If at least one connected trio is found, ans will hold the minimum degree of a connected trio, which is returned.

Through the use of an adjacency matrix and the precomputation of node degrees, the implementation efficiently finds the connected trios and computes their degrees. This approach is effective for the problem as it minimizes the number of operations needed to determine the existence of edges within possible trios.

Example Walkthrough

Let's walk through the solution approach with a small example:

Suppose we are given 4 nodes (n = 4) and the following edges in a graph: edges = [[1, 2], [2, 3], [3, 1], [2, 4], [3, 4]]. This graph contains multiple connected trios. Now, let's apply the steps of our solution approach to this example:

1. Graph Representation: We first create an adjacency matrix g of size n x n and initialize it with False values. We also create a deg array to keep track of the degrees of each node. Since n = 4, we will have a 4x4 matrix and a degree array with 4 elements.

2. Populate the Matrix and Degree Array:

   • We iterate over the given edges to fill in the adjacency matrix and update the degrees.
   • After iterating through all edges, our g matrix looks like this:

     	1	2	3	4
     1	F	T	T	F
     2	T	F	T	T
     3	T	T	F	T
     4	F	T	T	F

   • Our deg array is [2, 3, 3, 2], because node 1 has 2 edges, node 2 has 3, and so on.

3. Finding Connected Trios and Calculating Degrees:

   • Now, we look for all possible connected trios with 3 nested loops.
   • We check the possible connected trio (1,2,3). Since g[1][2], g[2][3], and g[3][1] are all True, we have found a connected trio.
   • The degree of this trio is the sum of degrees of nodes 1, 2, and 3, which is 2+3+3 = 8, then we subtract 6 (the internal edges of the trio), resulting in 2.
   • Next, we consider the trio (1,2,4), (1,3,4), and (2,3,4). Of these, only (2,3,4) is a connected trio.
   • For the connected trio (2,3,4), we sum up their degrees 3+3+2 = 8 and subtract 6, which gives us a degree of 2.

4. Finding the Minimum Degree:

   • We initialize ans to infinity (or an appropriately large number).
   • As we find connected trios, we update ans with the minimum degree found. With the connected trios we found, ans becomes 2.
   • Since we found at least one connected trio, we don't return -1. The smallest degree among all connected trios is 2, which is our final answer.

By following these steps with our example graph, we determined that the minimum degree of a connected trio is 2.

Solution Implementation

Time and Space Complexity

Time Complexity

The given Python code consists of a triple nested loop. Let's break it down:

• Building the adjacency matrix g and the degree array deg takes O(E) time, where E is the number of edges, because it iterates over all edges once (for u, v in edges) and sets the corresponding elements in g and increments the degrees in deg.

• The outermost loop (for i in range(n)) runs n times, where n is the number of nodes.

• The second loop (for j in range(i + 1, n)) runs at most (n - 1) times, decrementing by 1 with each iteration of the outer loop.

• The innermost loop (for k in range(j + 1, n)) runs at most (n - 2) times, decrementing by 1 for each iteration of the second loop.

• Inside the innermost loop, the code checks if three nodes form a trio (a triangle in the graph) with if g[i][k] and g[j][k]:, which is a constant time operation.

These nested loops lead to a time complexity of O(n^3) in the worst case, as each node is checked with every other pair of nodes to determine if they form a trio.

Space Complexity

The space complexity of the code can also be analyzed:

• The adjacency matrix g takes O(n^2) space as it is a 2D matrix with dimensions n by n.

• The degree array deg takes O(n) space since it stores the degree for each node.

No other significant storage is being used. Therefore, the overall space complexity is O(n^2) due to the adjacency matrix g.

In conclusion, the time complexity of this code is O(n^3) and the space complexity is O(n^2).

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