Batches
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: Batches (Connected Components with Weights) * ============================================================================================ * * --------------------- * 1. Problem Description * --------------------- * There are A students, each with a strength value B[i]. Relations between students * are given in matrix C. If students x and y are related, and y and z are related, * then x, y, and z form a "batch." * * Strength of a Batch = Sum of strengths of all students in that batch. * * Task: * Find the total number of batches whose total strength is >= D. * * --------------------- * 2. Logical Breakdown & Mathematical Formulation * --------------------- * This problem is a classic application of finding Connected Components* in an * undirected graph. * * Step 1: Graph Representation * - Nodes: Students 1 to A. * - Edges: Relations given in C. * * Step 2: Component Traversal (DFS/BFS) or DSU * - Iterate through all students from 1 to A. * - If a student has not been visited: * a. Start a traversal to find all students in this current batch. * b. During traversal, keep a running sum of the strengths (from array B). * c. Mark all students in this component as visited. * d. After the traversal for one component is finished, check: * If (Component_Sum >= D), increment the count of selected batches. * * Step 3: Result * - Return the final count. * * --------------------- * 3. Constraints & Complexity Analysis * --------------------- * Constraints: * 1 <= A <= 10^5 * 1 <= M <= 2 * 10^5 * 1 <= D <= 10^9 (Total sum can exceed 10^9, use long for sums). * * Time Complexity: * O(A + M) - Every node and edge is processed once. * * Space Complexity: * O(A + M) - Adjacency list and visited array. * * ============================================================================================ / public class Batches { public int solve(int A) { // TODO: Implement solution return 0; }
public int solve(int i, int[] b4, int[][] c4, int i1) {
return 1;
}
}