PHP algorithm design ideas: How to achieve an efficient solution to the maximum common subsequence problem?

PHP algorithm design ideas: How to achieve an efficient solution to the maximum common subsequence problem?

Longest Common Subsequence (LCS) is the problem of finding the longest identical subsequence in two strings. In practical applications, LCS is widely used in fields such as text similarity comparison, version control, and DNA sequence comparison. This article will introduce an efficient solution to solve this problem and provide specific code examples.

Algorithm idea:

Dynamic programming is a common method to solve LCS problems. The LCS problem has the optimal substructure property, that is, the longest common subsequence of two sequences can be constructed by the longest common subsequence of the subproblem. According to this property, dynamic programming method can be used to solve the LCS problem.

The specific algorithm steps are as follows:

1. Create a two-dimensional array dpm 1, where m and n are the lengths of the two input strings respectively.

   • dpi represents the length of LCS between the first i characters of the first string and the first j characters of the second string.
2. Initialize the first row and column of the dp array to 0, that is, dpi=dp0=0.

3. Traverse each character of the two strings, for the i-th character of the first string and the j-th character of the second string:

   • If the two characters are equal (that is, the i-th character of the first string and the j-th character of the second string are equal), then dpi = dpi-1 1.
   • If the two characters are not equal, then dpi = max(dpi-1, dpi), that is, the larger value of the LCS of the previous character and the next character is taken.
4. After traversing the two strings, the dpm obtained is the length of the longest common subsequence.
5. According to the result of the dp array, the longest common subsequence can be obtained backtracking. Starting from dpm, move to the upper left corner. If dpi is equal to dpi-1 1, it means that the current character belongs to LCS. Add the character to the result sequence and move to the upper left corner.

Code example:

function longestCommonSubsequence($str1, $str2){

$m = strlen($str1);
$n = strlen($str2);
$dp = array();

for($i=0; $i<=$m; $i++){
    $dp[$i] = array_fill(0, $n+1, 0);
}

for($i=1; $i<=$m; $i++){
    for($j=1; $j<=$n; $j++){
        if($str1[$i-1] == $str2[$j-1]){
            $dp[$i][$j] = $dp[$i-1][$j-1] + 1;
        }
        else{
            $dp[$i][$j] = max($dp[$i-1][$j], $dp[$i][$j-1]);
        }
    }
}

$lcs = "";
$i = $m;
$j = $n;

while($i>0 && $j>0){
    if($str1[$i-1] == $str2[$j-1]){
        $lcs = $str1[$i-1] . $lcs;
        $i--;
        $j--;
    }
    else if($dp[$i-1][$j] > $dp[$i][$j-1]){
        $i--;
    }
    else{
        $j--;
    }
}

return $lcs;

}

$ str1 = "ABCBDAB";
$str2 = "BDCAB";
$lcs = longestCommonSubsequence($str1, $str2);
echo "Input string: $str1 and $str2
";
echo "The longest common subsequence is: $lcs
";
?>

The above code will output:

Input string: ABCBDAB and BDCAB
The longest common subsequence is: BCBA

Conclusion:

This article introduces the idea of ​​using dynamic programming algorithm to solve the maximum common subsequence problem and specific PHP code examples. By using dynamic programming, LCS problems can be solved efficiently. The time complexity of this algorithm is O(m*n), where m and n are the lengths of the two input strings respectively. In practical applications, the algorithm can be optimized according to needs, such as using techniques such as rolling arrays to reduce space complexity.

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