PHP-Programm für minimale Anzahl von Sprüngen zum Erreichen des Endes

What is PHP?

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PHP Program for Minimum number of jumps to reach end

Method 1: Naive Recursive Approach

The naive recursive approach is a basic algorithmic approach where a problem is solved by recursively breaking it down into smaller subproblems. In the context of finding the minimum number of jumps to reach the end of an array, the naive recursive approach involves recursively exploring all possible paths from each position and choosing the minimum number of jumps.

Example

<?php
function minJumpsRecursive($arr, $start, $end) {
   // Base case: If the starting index is the last index, no jumps are needed
   if ($start == $end) {
      return 0;
   }
   // If the current element is 0, it is not possible to make any further jumps
   if ($arr[$start] == 0) {
      return PHP_INT_MAX;
   }
 // Initialize the minimum number of jumps to a large value
   $minJumps = PHP_INT_MAX;
   // Try all possible jumps from the current position
   // and choose the one that requires the minimum number of jumps
   for ($i = $start + 1; $i <= $end && $i <= $start + $arr[$start]; $i++) {
      $jumps = minJumpsRecursive($arr, $i, $end);
      if ($jumps != PHP_INT_MAX && $jumps + 1 < $minJumps) {
         $minJumps = $jumps + 1;
      }
   }
   return $minJumps;
}
// Example usage:
$arr = [1, 3, 5, 8, 9, 2, 6, 7, 6, 8, 9];
$n = count($arr);
$minJumps = minJumpsRecursive($arr, 0, $n - 1);
if ($minJumps != PHP_INT_MAX) {
   echo "Minimum number of jumps required to reach the end: " . $minJumps;
} else {
   echo "It is not possible to reach the end.";
}
?>

Output

Minimum number of jumps required to reach the end: 3

Method 2: Dynamic Programming

Dynamic programming is a technique used in computer programming to solve complex problems by breaking them down into overlapping subproblems and solving each subproblem only once. It stores the solutions of subproblems in a table or array, allowing for efficient lookup and reuse of previously computed results. This approach helps to avoid redundant computations and improve the overall efficiency of the algorithm.

Example

<?php
function minJumpsDynamic($arr, $n) {
   // Create an array to store the minimum number of jumps needed
   $minJumps = array_fill(0, $n, PHP_INT_MAX);
   $minJumps[0] = 0; // Base case: No jumps needed to reach the first element
   // Calculate the minimum number of jumps for each position
   for ($i = 1; $i < $n; $i++) {
      for ($j = 0; $j < $i; $j++) {
         // Check if it is possible to reach position $i from position $j
         if ($j + $arr[$j] >= $i) {
            // Update the minimum number of jumps for position $i
            // by considering the minimum of the current jumps and jumps from position $j plus one
            $minJumps[$i] = min($minJumps[$i], $minJumps[$j] + 1);
         }
      }
   }
   // Return the minimum number of jumps needed to reach the end
   return $minJumps[$n - 1];
}
// Example usage:
$arr = [1, 3, 5, 8, 9, 2, 6, 7, 6, 8, 9];
$n = count($arr);
$minJumps = minJumpsDynamic($arr, $n);
if ($minJumps != PHP_INT_MAX) {
   echo "Minimum number of jumps required to reach the end: " . $minJumps;
} else {
   echo "It is not possible to reach the end.";
}
?>

Output

Minimum number of jumps required to reach the end: 3

Conclusion

In conclusion, the PHP program for finding the minimum number of jumps to reach the end of an array can be implemented using various approaches. The naive recursive approach explores all possible paths, but it suffers from exponential time complexity and is not efficient for large arrays. The dynamic programming approach, on the other hand, optimizes the solution by breaking the problem into overlapping subproblems and storing the solutions in an array. This approach eliminates redundant calculations and significantly improves the efficiency of the algorithm, making it suitable for larger arrays. By leveraging dynamic programming techniques, the PHP program can efficiently determine the minimum number of jumps required to reach the end of the array.

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