Commutable Islands
Logical Breakdown
- [ ] Subproblem Identification:
- [ ] Optimal Substructure:
- [ ] Constraint Handling: (e.g., Modulo \(10^9 + 7\))
- [ ] Optimization: (Matrix Exponentiation if 'A' is huge)
Mathematical Rigor
State Definition
Let \(dp[i]\) be the state for the \(i\)-th subproblem.
Recurrence Relation
Visualization
Complexity Analysis
| Approach | Time Complexity | Space Complexity |
|---|---|---|
| Iterative DP | \(O(N)\) | \(O(1)\) |
Code Reference
package com.dsa.graphs;
/*
* ============================================================================================
* PROBLEM: Commutable Islands (Minimum Spanning Tree)
* ============================================================================================
*
* ---------------------
* 1. Problem Description
* ---------------------
* Given A islands and M bridges with associated costs. Find the minimum total cost
* to connect all islands such that every island is reachable from every other island.
* This is the classic definition of finding the Minimum Spanning Tree (MST) of a graph.
*
* ---------------------
* 2. Logical Breakdown & Mathematical Formulation
* ---------------------
* We use Kruskal's Algorithm to solve this, which greedily picks the cheapest edges.
*
* Step 1: Sorting
* Sort all M bridges in non-descending order based on their cost.
*
* Step 2: Disjoint Set Union (DSU) Initialization
* Initialize a DSU structure for A nodes. Each node is initially its own parent
* (its own component).
*
* Step 3: Edge Selection (Greedy)
* Iterate through the sorted bridges. For each bridge connecting islands u and v:
* 1. Find the absolute root of u and v using the find operation.
* 2. If find(u) != find(v):
* - The islands are currently in different components.
* - Adding this bridge will NOT form a cycle.
* - Add the bridge's cost to the total MST cost.
* - Perform union(u, v) to merge the two components.
* 3. If find(u) == find(v), skip the bridge to avoid a cycle.
*
* Step 4: Termination
* Stop once we have successfully added (A - 1) bridges or processed all edges.
*
* ---------------------
* 3. Constraints & Complexity Analysis
* ---------------------
* Constraints:
* 1 <= A, M <= 6 * 10^4
* 1 <= Cost <= 10^3
*
* Time Complexity:
* O(M log M + M α(A))
* - Sorting edges: O(M log M).
* - DSU operations: O(M α(A)), where α is the inverse Ackermann function (near-constant).
*
* Space Complexity:
* O(A + M) - To store the parent/rank arrays for DSU and the edge list.
*
* ============================================================================================
/
public class CommutableIslands {
public int solve(int A, int[][] b3) {
// TODO: Implement solution
return 0;
}
}