Detailed explanation of dynamic programming algorithm in PHP

Detailed explanation of dynamic programming algorithm in PHP

Dynamic Programming (Dynamic Programming) is an algorithmic idea for solving problems by decomposing the problem into smaller sub-problems and utilizing the solved sub-problems. to solve the overall problem. In PHP, dynamic programming algorithms can be widely used in many fields of computer science and mathematics, such as shortest paths, string matching, and knapsack problems. This article will introduce the principles of dynamic programming algorithm in PHP in detail and provide code examples to illustrate.

1. Principle of dynamic programming algorithm

Dynamic programming algorithm usually includes the following steps:

1. Define the state of the problem: divide the problem into smaller sub-divisions problem and determine the status of each sub-problem.
2. Determine the state transition equation: According to the state of the sub-problems, find the recursive relationship between the sub-problems, that is, the state transition equation.
3. Set boundary conditions: Determine the boundary conditions of the problem, that is, the solution to the minimum subproblem.
4. Recursive solution: Starting from the smallest subproblem, the solution to the final problem is solved recursively according to the state transition equation.

2. Dynamic programming algorithm example

The following uses the Fibonacci sequence as an example to demonstrate the dynamic programming algorithm in PHP in detail.

The Fibonacci sequence starts from 0, the 0th item is 0, the 1st item is 1, and starting from the 2nd item, each item is equal to the sum of the previous two items. That is, the recurrence relationship of the sequence is F(n) = F(n-1) F(n-2), and the boundary conditions are F(0) = 0, F(1) = 1.

First, define the state of the problem, that is, take the nth term of the Fibonacci sequence as the state of the sub-problem:

function fibonacci($n) {

// 定义状态数组
$dp = array();

// 设置边界条件
$dp[0] = 0;
$dp[1] = 1;

// 递推求解
for ($i = 2; $i <= $n; $i++) {
    $dp[$i] = $dp[$i-1] + $dp[$i-2];
}

// 返回结果
return $dp[$n];

}

In the above code, the $dp array is used to save the value of each Fibonacci sequence. First set the boundary conditions $dp[0] = 0, $dp[1] = 1. Then, recurse from item 2 through the for loop, and solve the final problem according to the state transition equation $dp[$i] = $dp[$i-1] $dp[$i-2].

By calling the fibonacci function, you can get the value of the nth term of the Fibonacci sequence. For example:

$n = 10;
$result = fibonacci($n);

echo "The value of the ". $n." item in the Fibonacci sequence is: " . $result;

Run the above code, the output result is:

The value of the 10th item in the Fibonacci sequence is: 55

3. Summary

Dynamic programming is an important algorithmic idea that can provide efficient solutions when solving some complex problems. This article uses the Fibonacci sequence as an example to introduce the principle of dynamic programming algorithm in PHP in detail, and provides code examples for explanation. By understanding the principles and examples of dynamic programming algorithms, you can better apply them in solving practical problems.

The above is the detailed content of Detailed explanation of dynamic programming algorithm in PHP. For more information, please follow other related articles on the PHP Chinese website!