PHP Program to Count Number of Binary Strings without Consecutive 1’s

In PHP, counting binary strings without consecutive 1's is a classic dynamic programming problem. A binary string contains only 0's and 1's, and we need to count valid strings where no two 1's appear next to each other.

Understanding the Problem

Let's analyze binary strings of length 3 to understand the pattern

All possible binary strings of length 3: 000, 001, 010, 011, 100, 101, 110, 111

Valid strings (without consecutive 1's): 000, 001, 010, 100

Invalid strings (with consecutive 1's): 011, 110, 111

So for length 3, we have 5 valid binary strings without consecutive 1's.

Method 1: Using Dynamic Programming

This approach uses the Fibonacci pattern where dp[i] = dp[i-1] + dp[i-2] ?

<?php
function countBinaryStrings($n) {
    if ($n == 0) return 1;
    if ($n == 1) return 2;
    
    $dp = array();
    $dp[0] = 1; // Empty string
    $dp[1] = 2; // "0", "1"
    
    for ($i = 2; $i <= $n; $i++) {
        $dp[$i] = $dp[$i - 1] + $dp[$i - 2];
    }
    
    return $dp[$n];
}

$n = 5;
$count = countBinaryStrings($n);
echo "Number of binary strings of length $n without consecutive 1's: " . $count;
?>

Number of binary strings of length 5 without consecutive 1's: 13

Method 2: State-based Dynamic Programming

This method tracks strings ending with 0 and strings ending with 1 separately ?

<?php
function countStrings($n) {
    if ($n == 0) return 1;
    
    // a[i] = count of strings ending with 0
    // b[i] = count of strings ending with 1
    $a = array_fill(0, $n, 0);
    $b = array_fill(0, $n, 0);
    
    $a[0] = 1; // "0"
    $b[0] = 1; // "1"
    
    for ($i = 1; $i < $n; $i++) {
        $a[$i] = $a[$i - 1] + $b[$i - 1]; // Can append 0 after both 0 and 1
        $b[$i] = $a[$i - 1];              // Can append 1 only after 0
    }
    
    return $a[$n - 1] + $b[$n - 1];
}

$n = 5;
echo "Number of binary strings of length $n without consecutive 1's: " . countStrings($n);
?>

Number of binary strings of length 5 without consecutive 1's: 13

Pattern Analysis

Conclusion

Both methods solve the problem using dynamic programming principles. Method 1 follows the Fibonacci sequence pattern, while Method 2 uses state-based tracking. The result follows the Fibonacci sequence starting with 2, 3, 5, 8, 13... for lengths 1, 2, 3, 4, 5 respectively.