Valid Path

Logical Breakdown

• [ ] Subproblem Identification:
• [ ] Optimal Substructure:
• [ ] Constraint Handling: (e.g., Modulo \(10^9 + 7\))
• [ ] Optimization: (Matrix Exponentiation if 'A' is huge)

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Mathematical Rigor

State Definition

Let \(dp[i]\) be the state for the \(i\)-th subproblem.

Recurrence Relation

\[ dp[i] = ... \]

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Visualization

graph TD A["Current State"] --> B["Previous State 1"] A --> C["Previous State 2"]

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Complexity Analysis

Approach	Time Complexity	Space Complexity
Iterative DP	\(O(N)\)	\(O(1)\)

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Code Reference

package com.dsa.graphs;

import java.util.*;

/* * ============================================================================================ * PROBLEM: Valid Path * ============================================================================================ * * --------------------- * 1. Problem Description * --------------------- * Given a rectangle defined by (0,0) and (x,y), and N circles of radius R centered at * (A[i], B[i]). Determine if a path exists from (0,0) to (x,y) that avoids all circular * zones (including boundaries). * * Rules: * - Movement: 8-directional (including diagonals). * - Boundaries: Cannot step outside [0, x] or [0, y]. * - Obstacles: Any cell (i, j) that lies within or on the boundary of any circle is blocked. * * --------------------- * 2. Logical Breakdown & Mathematical Formulation * --------------------- * This problem is solved using a combination of Geometry and Graph Traversal. * * Step 1: Geometry Pruning (Grid Preprocessing) * We treat every integer coordinate (i, j) within the rectangle as a node in a graph. * A cell (i, j) is "unsafe" if the Euclidean distance to any circle center (Cx, Cy) * is <= R. * Formula: (i - Cx)^2 + (j - Cy)^2 <= R^2 * * Step 2: Pathfinding (BFS) * 1. Initialize a 2D boolean 'visited' array or reuse the grid array to mark unsafe cells. * 2. If start (0,0) or end (x,y) is unsafe, return "NO". * 3. Use Breadth-First Search starting from (0,0): * - For each cell, explore all 8 adjacent neighbors. * - If a neighbor is within bounds, not unsafe, and not visited: * - Mark as visited and enqueue. * - If we reach (x, y), return "YES". * 4. If queue becomes empty, return "NO". * * --------------------- * 3. Constraints & Complexity Analysis * --------------------- * Constraints: * 1 <= x, y <= 100 * 1 <= N <= 100 * 1 <= R <= 100 * * Time Complexity: * O(x * y * N) - To mark unsafe cells (each cell checked against N circles). * O(x * y) - For BFS traversal. * Total: O(x * y * N) * * Space Complexity: * O(x * y) - To store the grid/visited status. * * ============================================================================================ / public class ValidPath { public int solve(int A) { // TODO: Implement solution return 0; }

public String solve(int i, int i1, int i2, int i3, int[] circlesX4, int[] circlesY4) {
    return "";
}

}