Edge In MST
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: Edge in MST
* ============================================================================================
*
* ---------------------
* 1. Problem Description
* ---------------------
* Given an undirected weighted graph with A nodes and M edges. For each edge, determine if
* it can belong to at least one Minimum Spanning Tree (MST).
*
* ---------------------
* 2. Logical Breakdown & Mathematical Formulation
* ---------------------
* An edge (u, v) with weight 'W' belongs to some MST if and only if there is no path
* between 'u' and 'v' consisting only of edges with weights strictly less than 'W'.
*
* Step 1: Sorting & Grouping
* Sort all edges by weight. Group edges that have the same weight together.
*
* Step 2: Processing Weight Groups
* We process groups of edges with the same weight W one by one:
* 1. For each edge (u, v) in the current weight group:
* - Find the current roots (using DSU) of 'u' and 'v'.
* - If root(u) == root(v), this edge forms a cycle with edges of weights < W.
* It can NEVER be in an MST.
* - If root(u) != root(v), this edge could be in an MST.
*
* 2. Handling Multiple Edges of same Weight (Bridge Detection):
* - To be precise, within a group of equal weights, an edge belongs to some MST
* if it connects two components that are not already connected by edges of
* smaller weights.
* - However, the problem asks if it can belong to any MST. The condition is:
* an edge (u, v) of weight W belongs to an MST if u and v are in different
* components considering only edges with weight < W.
*
* Step 3: DSU Update
* After processing all edges in a weight group and marking their status, perform
* the DSU union operations for all edges in that group that connect different
* components.
*
* ---------------------
* 3. Constraints & Complexity Analysis
* ---------------------
* Time Complexity: O(M log M)
* - Sorting: O(M log M).
* - DSU operations: O(M α(A)).
* Space Complexity: O(A + M) - Adjacency list for grouping and DSU arrays.
* ============================================================================================
*/
public class EdgeInMST {
public int[] solve(int A, int[][] b3) {
// TODO: Implement solution
return new int[0];
}
}