Problem Description

The given problem scenario presents a set of cities along with possible connections between them. Each city is uniquely identified by a numeric label ranging from 1 to n, and the goal is to find the most cost-effective way to ensure that there is at least one path between every pair of cities.

We are provided with the total number of cities (n) and an array connections, where each element of connections is a triplet consisting of two city labels and the cost of connecting those two cities (for example, [xi, yi, costi]). The connections are bidirectional, meaning the cost is the same regardless of the direction of travel between the two cities.

The task is to determine the minimum total cost required to connect all cities in such a way that every city is reachable from every other city. If this task is impossible with the given connections, the function should return -1.

Intuition

To solve this problem, we need to connect all cities while minimizing the total cost. This is a classic example of a Minimum Spanning Tree (MST) problem which can be solved using algorithms like Kruskal's or Prim's algorithm.

Here, the Kruskal's algorithm is suitable because we start with edges (connections) and aim to create the MST by picking the shortest edges while avoiding cycles.

The intuition behind the solution is as follows:

1. Sorting Edges: We begin by sorting the connections by cost. This allows us to consider the cheapest connection available at any step, which is a core part of Kruskal's algorithm.

2. Union-Find Data Structure: We utilize a union-find data structure to keep track of which cities are already connected. This data structure helps us to efficiently find the root (representative) of each city and to unite two separate trees (cities that aren't yet connected).

3. Processing Connections: Iterate through each connection. Use the union-find data struture to determine whether the current connection will form a cycle. We only consider a connection if it connects two different trees (does not form a cycle).

4. Connecting Cities: When a valid connection is found (it doesn't form a cycle), we join the cities. The cost of this connection is added to the total minimum cost, and the number of disconnected components (n) is reduced by 1 because two components are now connected.

5. Checking Early Termination: If at any moment the number of disconnected components becomes 1, it means we have connected all cities, and we can return the total minimum cost at this stage to improve efficiency.

6. Return Result: If we exit the loop and multiple disconnected components (n) are still present, this indicates it's impossible to connect all cities; hence, we return -1.

The provided solution implements Kruskal's algorithm with a union-find data structure to connect cities in the least costly manner.

Learn more about Union Find, Graph, Minimum Spanning Tree and Heap (Priority Queue) patterns.

Solution Approach

The solution provided follows the Kruskal's algorithm approach for finding the minimum spanning tree (MST). Let's walk through the implementation details:

1. Union-Find Initialization: We initiate an array p which stands for the parent array, mostly used by union-find data structures. It's initialized such that each city is its own parent:

   p = list(range(n))

   Here, range(n) is used instead of range(n+1) because the cities are labeled from 1 to n, so an adjustment is made in the index while processing.

2. Sorting the Connections: The connections are sorted in ascending order of their cost using the sort method with a lambda function specifying the sorting key:

   connections.sort(key=lambda x: x[2])

3. Find Function: This is a part of the union-find data structure to find the root parent of a city. The function is recursive, and it also performs path compression, updating the parent reference to the root parent, which saves time in subsequent find operations:

   def find(x):
       if p[x] != x:
           p[x] = find(p[x])
       return p[x]

4. Iterating Over Connections: For each connection [x, y, cost], we first adjust the city labels to zero-based indexing (since the initial array p was created with zero-based index):

   x, y = x - 1, y - 1

5. Checking and Union: For each connection, we check if both cities x and y have the same root parent. If they do, we skip the connection as it would form a cycle:

   if find(x) == find(y):
       continue

   Otherwise, we proceed to union (merge) the two components by setting one city's root parent to the other's, effectively connecting the two components:

   p[find(x)] = find(y)

   The cost of the connection is then added to the total cost ans:

   ans += cost

   Following the union, we reduce n by 1. In this context, n is being used to count the number of disconnected components, with the goal of having exactly 1 component once all cities are connected:

   n -= 1

6. Early Termination: If n becomes 1, that means all cities have been connected, and we can return the total cost at that point:

   if n == 1:
       return ans

7. Return -1 if Impossible: If the loop completes and there are still multiple disconnected components remaining (i.e., n is not 1), we return -1, indicating that it is not possible to connect all the cities with the given connections:

   return -1

This algorithm ensures all cities are connected with the minimum cost, or it returns -1 if they can't be connected with the given connections.

Example Walkthrough

Let's consider a small example scenario. Suppose we have 4 cities with the following possible connections between them:

n = 4
connections = [[1, 2, 3], [3, 4, 4], [2, 3, 2], [1, 4, 5]]

Here's a step-by-step implementation of the provided solution approach:

1. Union-Find Initialization: We initiate the parent array with the cities being their own parent:

   p = [0, 1, 2, 3]  # Adjusted for zero-index based (subtracting 1 from each label)

2. Sorting the Connections: We sort the connections by their cost in ascending order:

   connections.sort(key=lambda x: x[2])
   # connections becomes [[2, 3, 2], [1, 2, 3], [3, 4, 4], [1, 4, 5]]

3. Processing Each Connection:

   • Look at the first connection [2, 3, 2] (zero-based indexing used: [1, 2, 2]):

     • Both do not have the same root, apply union:

     p[find(1)] = find(2)
     # p becomes [0, 2, 2, 3]
     ans += 2
     n = 4 - 1 = 3

   • Next, connection [1, 2, 3] (zero-based indexing used: [0, 1, 3]):

     • Roots are different, apply union:

     p[find(0)] = find(1)
     # p becomes [2, 2, 2, 3]
     ans += 3
     n = 3 - 1 = 2

   • Then, connection [3, 4, 4] (zero-based indexing: [2, 3, 4]):

     • Roots are different, apply union:

     p[find(2)] = find(3)
     # p becomes [2, 2, 3, 3]
     ans += 4
     n = 2 - 1 = 1

     After this operation, we have only one disconnected component because n becomes 1, so we don't need to process any more connections. The current total cost is 2 + 3 + 4 = 9.

4. Early Termination: Since n is now 1, we can terminate the process early and return the total cost ans, which is 9.

In this example, the minimum total cost required to connect all the cities with the given connections is 9, ensuring a path exists between every pair of cities.

Solution Implementation

Time and Space Complexity

The code implements a variation of Kruskal's algorithm with path compression for the union-find operation.

Time Complexity

The time complexity of the code is as follows:

1. Sorting the connections: The sort() function has a time complexity of O(E log E), where E is the number of connections.
2. Iterating through sorted connections: In the worst case, each connection can involve a union operation. Each find operation takes O(log N) in practice due to path compression, although the exact complexity can be effectively considered as O(α(N)), where α(N) is the inverse Ackermann function, which grows very slowly and is less than 5 for any practical input size.
3. The union operation takes O(1) after finding the roots.

Combining all the above steps, the total time complexity is O(E log E + E α(N)). Since the sorting term is dominant for larger input sizes, we can approximate the time complexity to O(E log E).

Space Complexity

The space complexity of the code is as follows:

1. The p array of size O(N), where N is the number of vertices.
2. Minimal space for local variables.

Thus, the space complexity is O(N).

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