Ways to Send the Signal

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.dynamic_programming;

import java.util.*;

/ * Problem: Ways to Send the Signal * Group: 11. Dynamic Programming */ / * ============================================================================================ * PROBLEM: Ways to Send the Signal (Binary Strings without Consecutive 1s) * ============================================================================================ * * --------------------- * 1. Problem Description * --------------------- * You need to count the number of binary strings of length A such that there are * no consecutive 1s. * (This is equivalent to: A signal consists of 0s and 1s, where 1 cannot be adjacent to another 1). * * --------------------- * 2. Logical Breakdown & Mathematical Formulation * --------------------- * Let dp[i][0] be the count of valid strings of length i ending in '0'. * Let dp[i][1] be the count of valid strings of length i ending in '1'. * * Transitions: * 1. If we end in '0': * The previous character (at i-1) could have been '0' OR '1'. * dp[i][0] = dp[i-1][0] + dp[i-1][1] * * 2. If we end in '1': * The previous character MUST be '0' (to avoid consecutive 1s). * dp[i][1] = dp[i-1][0] * * Total Ways for Length i = dp[i][0] + dp[i][1]. * * Simplified Recurrence (Fibonacci): * Notice Total[i] = dp[i][0] + dp[i][1] * = (dp[i-1][0] + dp[i-1][1]) + dp[i-1][0] * = Total[i-1] + Total[i-2] * * This problem reduces to the Fibonacci sequence (shifted). * * Base Cases: * Len 1: "0", "1" -> 2 ways. * Len 2: "00", "01", "10" (Not "11") -> 3 ways. * * --------------------- * 3. Constraints & Complexity * --------------------- * Time Complexity: O(A). * Space Complexity: O(1). * * ============================================================================================ */ public class WaysToSendTheSignal { public int solve(int A) { // TODO: Implement solution return 0; }

public int solveSignal(int i) {
    return 0;
}

}