Coloring A Cycle Graph

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: Check Bipartite Graph * ============================================================================================ * * --------------------- * 2. Logical Breakdown * --------------------- * A graph is Bipartite if it contains no ODD length cycles. * * Algorithm (2-Coloring via BFS): * 1. Initialize color array with -1 (uncolored). * 2. For each uncolored node: * - Start BFS/DFS, color it 0. * - For every neighbor: * - If uncolored, give it opposite color (1 - current_color). * - If already colored and color is SAME as current, return 0 (Not Bipartite). * 3. Return 1 if all components are successfully colored. * ============================================================================================ / public class ColoringACycleGraph { public int solve(int A) { // TODO: Implement solution return 0; }

public int isBipartite(int i, int[][] edges1) {
    return 1;
}

}