Bulb Chain Problem in Java

The bulb chain problem is a well-known puzzle that challenges one's logical and programming skills. The problem involves a chain of bulbs that can either be on or off, and a switch that can flip the state of a bulb and its adjacent bulbs. The objective is to find the minimum number of switch flips required to turn all the bulbs on.

Problem Statement

The problem can be visualized as a sequence of bulbs connected by wires, with a switch at each bulb. Each switch can flip the state of the bulb it is connected to, as well as the states of its adjacent bulbs. The goal is to find a sequence of switch flips that will turn all the bulbs on.

Solution to The Problem

One way to approach this problem is to use a brute force algorithm that tries all possible combinations of switch flips until the correct sequence is found. However, this approach is not efficient for larger chains of bulbs, as the number of possible combinations grows exponentially with the size of the chain.

A more efficient approach is to use a dynamic programming algorithm that breaks the problem down into smaller subproblems and builds a solution iteratively. The basic idea is to consider the state of each bulb independently and use it to compute the minimum number of switch flips required to turn all the bulbs on.

To implement this algorithm in Java, we can create a boolean array to represent the state of each bulb, with true indicating that the bulb is on and false indicating that it is off. We can also create a two-dimensional array to store the minimum number of switch flips required to turn on all the bulbs up to a certain index.

The algorithm can be implemented as follows:

public static int minSwitchFlips(boolean[] bulbs) {
    int n = bulbs.length;
    int[][] dp = new int[n][n];
    for (int len = 1; len <= n; len++) {
        for (int i = 0; i <= n - len; i++) {
            int j = i + len - 1;
            dp[i][j] = Integer.MAX_VALUE;
            for (int k = i; k <= j; k++) {
                int cost = (bulbs[i] ? 0 : 1) + (bulbs[j] ? 0 : 1);
                if (k > i) {
                    cost += dp[i][k - 1];
                }
                if (k < j) {
                    cost += dp[k + 1][j];
                }
                dp[i][j] = Math.min(dp[i][j], cost);
            }
        }
    }
    return dp[0][n - 1];
}

Bulb Chain Problem Java Program

BulbChainProblem.java

import java.util.Arrays;
public class BulbChainProblem {
    public static void main(String[] args) {
        boolean[] bulbs = {true, false, true, false, true};
        int minSwitchFlips = minSwitchFlips(bulbs);
        System.out.println("Minimum number of switch flips required: " + minSwitchFlips);
    }

    public static int minSwitchFlips(boolean[] bulbs) {
        int n = bulbs.length;
        int[][] dp = new int[n][n];

        for (int len = 1; len <= n; len++) {
            for (int i = 0; i <= n - len; i++) {
                int j = i + len - 1;
                dp[i][j] = Integer.MAX_VALUE;

                for (int k = i; k <= j; k++) {
                    int cost = (bulbs[i] ? 0 : 1) + (bulbs[j] ? 0 : 1);
                    if (k > i) {
                        cost += dp[i][k - 1];
                    }
                    if (k < j) {
                        cost += dp[k + 1][j];
                    }
                    dp[i][j] = Math.min(dp[i][j], cost);
                }
            }
        }

        return dp[0][n - 1];
    }
}

Output:

Minimum number of switch flips required: 2

In this example, we have a chain of 5 bulbs with alternating on/off states. The minimum number of switch flips required to turn all the bulbs on is 2.

In this implementation, we iterate over all possible lengths of subchains, from 1 to the length of the chain. For each subchain, we iterate over all possible starting indices and compute the minimum number of switch flips required to turn on all the bulbs in that subchain.

To compute this value, we first consider the state of the first and last bulbs in the subchain and add 1 to the cost if either of them is off. We then consider all possible split points in the subchain and add the minimum number of switch flips required to turn on all the bulbs to the left and right of the split point. Finally, we take the minimum of all possible split points to find the minimum number of switch flips required to turn on all the bulbs in the subchain.

In conclusion, the bulb chain problem is a challenging puzzle that can be solved efficiently using dynamic programming techniques in Java. By breaking the problem down into smaller subproblems and building a solution iteratively, we can find the minimum number of switch flips required to turn all the bulbs on in a given chain.

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