PHP algorithm analysis: How to use dynamic programming algorithm to solve the longest rising subsequence problem?

PHP algorithm analysis: How to use dynamic programming algorithm to solve the longest rising subsequence problem?

Dynamic Programming (Dynamic Programming) is a commonly used algorithm idea that can be used to solve many practical problems. This article will introduce how to use dynamic programming algorithm to solve the Longest Increasing Subsequence problem and provide specific code examples.

The longest ascending subsequence problem refers to finding a subsequence in a given integer sequence so that the elements in the subsequence are arranged in increasing order and have the longest length. For example, in the sequence [10, 22, 9, 33, 21, 50, 41, 60, 80], the longest ascending subsequence is [10, 22, 33, 50, 60, 80] with a length of 6.

Dynamic programming algorithms usually adopt a bottom-up approach, solving sub-problems first and then gradually solving larger problems. For the longest rising subsequence problem, we can set dp[i] to represent the length of the longest rising subsequence ending with the i-th element. Then the state transition equation is:

dp[i] = max(dp[j]) 1, where 0 ≤ j

First, we define an array dp and initialize all elements to 1, which means that each element itself is an ascending subsequence. Then, the input integer sequence nums is traversed from left to right, and for each element nums[i], all elements nums[j] between 0 and i-1 are traversed. If nums[j]

Next, we only need to traverse the entire dp array and find the largest element, which is the length of the longest ascending subsequence.

The following is a code example implemented using PHP language:

function lengthOfLIS($nums) {
    $n = count($nums);
    $dp = array_fill(0, $n, 1);

    for ($i = 1; $i < $n; $i++) {
        for ($j = 0; $j < $i; $j++) {
            if ($nums[$j] < $nums[$i]) {
                $dp[$i] = max($dp[$i], $dp[$j] + 1);
            }
        }
    }

    $maxLen = 0;
    for ($i = 0; $i < $n; $i++) {
        $maxLen = max($maxLen, $dp[$i]);
    }

    return $maxLen;
}

$nums = array(10, 22, 9, 33, 21, 50, 41, 60, 80);
$result = lengthOfLIS($nums);
echo "最长上升子序列的长度为：" . $result;

In the above code, the function lengthOfLIS accepts an integer sequence nums as a parameter and returns the length of the longest ascending subsequence. In the given example, the output is 6.

Through the dynamic programming algorithm, we can efficiently solve the longest rising subsequence problem. In practical applications, this algorithm is also widely used, such as search engine optimization, data compression and network transmission.

I hope this article can help you understand the dynamic programming algorithm and be able to flexibly apply it to practical problems.

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