Damaged Roads
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;
import java.util.*;
/*
* ============================================================================================
* PROBLEM: Damaged Roads (Grid-based MST)
* ============================================================================================
*
* ---------------------
* 1. Problem Description
* ---------------------
* The country is a (N+1) x (M+1) grid of cities.
* - Vertical roads connect (i, j) and (i+1, j) with cost A[i]. There are (M+1) such
* identical vertical roads for each row index i.
* - Horizontal roads connect (i, j) and (i, j+1) with cost B[j]. There are (N+1) such
* identical horizontal roads for each column index j.
* * Task: Find the minimum cost to make all cities reachable from (0,0) (i.e., find the MST).
*
* ---------------------
* 2. Logical Breakdown & Mathematical Formulation
* ---------------------
* Total nodes = (N+1) * (M+1). Total edges in an MST = (Nodes - 1).
* In a grid, horizontal edges are repeated (N+1) times and vertical edges are repeated
* (M+1) times. We apply Kruskal's greedy principle:
*
* 1. Combine all costs from A and B into a single list of "Edge Types," keeping
* track of whether they are horizontal (H) or vertical (V).
* 2. Sort all edge types by cost in ascending order.
* 3. Use two variables:
* - h_count: Number of horizontal edge segments used to connect rows.
* - v_count: Number of vertical edge segments used to connect columns.
* Initial values: h_count = 1, v_count = 1.
* 4. For each edge in the sorted list:
* - If it's a Vertical Edge (Cost A[i]):
* It needs to connect (M+1) columns, but some are already connected by
* horizontal edges.
* Contribution = Cost * (M + 1 - (number of horizontal types already picked)).
* Actually, simpler: if we pick a Vertical type, it covers M+1 - h_used edges.
* Update: v_used++.
* - If it's a Horizontal Edge (Cost B[j]):
* Contribution = Cost * (N + 1 - (number of vertical types already picked)).
* Update: h_used++.
*
* ---------------------
* 3. Complexity Analysis
* ---------------------
* Time Complexity: O((N+M) log (N+M)) due to sorting the edge costs.
* Space Complexity: O(N + M) to store the cost pairs.
*
* ---------------------
* 4. Example
* ---------------------
* A = [1], B = [1] (2x2 grid)
* Edges: A[0]=1, B[0]=1.
* Pick A[0]: Cost 1 * (1 horizontal remaining) = 1.
* Pick B[0]: Cost 1 * (1 vertical remaining) = 1.
* Total = 2.
* ============================================================================================
/
public class DamagedRoads {
public int solve(int A) {
// TODO: Implement solution
return 0;
}
public int solve(int[] a3, int[] b3) {
return 1;
}
}