The Question

Connecting Cities With Minimum Cost

There are n cities labeled from 1 to n. You are given an array connections where connections[i] = [u, v, cost] represents a bidirectional road between city u and city v with a given cost. Your task is to find the minimum cost to connect all cities such that there is at least one path between any two cities. If it is impossible to connect all cities, return -1. Constraints: - 1 <= n <= 10,000 - 1 <= connections.length <= 10,000 - 1 <= cost <= 10^5 - Each connection [u, v, cost] satisfies 1 <= u, v <= n and u != v.

Java

Kruskal's Algorithm

Union-Find

Questions & Insights

Clarifying Questions

What are the constraints on $n$ and the number of connections? (e.g., If

n

is up to

10^4

and connections are up to

10^4

, Kruskal's

O(E \log E)

is efficient).

Can there be multiple connections between the same two cities or self-loops? (Assumed: Multiple connections are possible; the algorithm should naturally pick the cheapest one. Self-loops are ignored).

Does the cost fit within a 32-bit integer? (Assumed: Total cost fits in a signed 32-bit integer based on typical competitive programming limits, but we should be mindful of overflow if costs are very large).

How should we handle the case where $n=1$? (Assumed: A single city is already "connected," so the cost should be 0).

Assumptions

1 \le n \le 10,000

1 \le connections.length \le 10,000

Cities are 1-indexed.

If the graph is disconnected (fewer than

n-1

edges in the MST), return -1.

Thinking Process

Identify the Core Problem: This is a classic Minimum Spanning Tree (MST) problem. We need to find a subset of edges that connects all vertices with the minimum total weight and no cycles.

Choose the Algorithm: Kruskal's algorithm is ideal here because the input is already provided as an edge list. It involves sorting edges by cost and using a Disjoint Set Union (DSU) to avoid cycles.

Detect Connectivity: To ensure all cities are connected, the number of edges added to the MST must exactly be

n - 1

. If we finish processing all edges and have fewer than

n-1

edges, the cities are not fully connected.

DSU Optimization: Implement Union-Find with Path Compression and Union by Rank (or Size) to achieve near-constant time complexity

O(\alpha(n))

for find and union operations.

Implementation Breakdown

Problem Set

Functional Requirement: Find the Minimum Spanning Tree cost.

Connectivity Check: Return -1 if total edges used <

n-1

Input: int n, int[][] connections.

Output: int (Total cost or -1).

Approach

Algorithm: Kruskal's Algorithm.

Data Structure: Disjoint Set Union (DSU).

Complexity:

Time:

O(E \log E)

due to sorting the edges, where

E

is the number of connections. The Union-Find operations take

O(E \alpha(V))

Space:

O(V)

to store the parent and rank arrays in the DSU, where

V

is the number of cities (

n

Implementation

Wrap Up

Advanced Topics

Prim's Algorithm: For dense graphs where

E \approx V^2

, Prim's algorithm using an adjacency list and a Fibonacci heap (or PriorityQueue in Java) might be more efficient

O(E + V \log V)

. However, Kruskal's is generally easier to implement and highly efficient for this problem's likely constraints.

Boruvka's Algorithm: This is a parallelizable alternative for MST. It works by merging components in "steps," which is useful in distributed systems (e.g., finding the MST of a graph stored across multiple machines in a MapReduce framework).

Memory Optimization: If

n

is extremely large, using a primitive array for DSU is much better than an object-oriented approach to avoid overhead and cache misses.