How to solve the nearest point pair problem in PHP using divide and conquer method and get the optimal solution?

How to use the divide and conquer method to solve the nearest point pair problem in PHP and obtain the optimal solution?

The closest pair problem refers to finding the two closest point pairs on a given plane. This problem is very common in computational geometry and has many solutions. One of the commonly used methods is divide and conquer.

The divide and conquer method is a method of dividing a problem into smaller sub-problems and solving the original problem by recursively solving the sub-problems. In the nearest point pair problem, we can use the divide-and-conquer method to find the optimal solution efficiently.

The following are the steps to use the divide and conquer method to solve the nearest point pair problem:

1. Input a set of points, where each point is represented by (x, y).
2. Sort the point collection according to the x coordinate.
3. If the number of points is less than or equal to 3, directly use the brute force method to solve the nearest point pair problem. That is, calculate the distance between every two points and find the smallest distance.
4. Divide the point set into two roughly equal sub-sets, called left and right respectively.
5. Recursively call the divide-and-conquer method to find the nearest point pairs in left and right respectively. Denoted as (left_min, left_max) and (right_min, right_max).
6. Take the pair of points with the smallest distance between left_min and right_min, and calculate the distance between them, recorded as min_distance.
7. Find all points in the point collection whose x-coordinate distance from the midline is less than min_distance, and sort them according to their y-coordinate.
8. Among these points, use the linear scan method to calculate the distance between each point and up to 6 subsequent points, and find the minimum distance.
9. Return the pair of points with the smallest distance between left_min and right_min, and the minimum distance obtained by linear scanning.

The following is a code example that uses PHP language to implement the divide-and-conquer method to solve the nearest point-pair problem:

function closestPair($points) {
  $n = count($points);
  
  // 升序排序
  usort($points, function($a, $b){
    return $a['x'] - $b['x'];
  });
  
  // 少于等于3个点直接暴力求解
  if ($n <= 3) {
    return bruteForce($points);
  }
  
  // 分成两个子集合
  $mid = floor($n / 2);
  $left = array_slice($points, 0, $mid);
  $right = array_slice($points, $mid);
  
  // 递归调用分治法
  $leftPair = closestPair($left);
  $rightPair = closestPair($right);
  
  // 找到距离最小的点对
  $delta = min($leftPair['distance'], $rightPair['distance']);
  $minPair = ($leftPair['distance'] < $rightPair['distance']) ? $leftPair : $rightPair;
  
  // 找到中线附近距离小于delta的点
  $strip = [];
  foreach ($points as $point) {
    if (abs($point['x'] - $points[$mid]['x']) < $delta) {
      $strip[] = $point;
    }
  }
  
  // 按照y坐标排序
  usort($strip, function($a, $b){
    return $a['y'] - $b['y'];
  });
  
  // 线性扫描
  $stripPair = stripScan($strip, $delta);
  
  // 返回距离最小的点对
  return ($minPair['distance'] < $stripPair['distance']) ? $minPair : $stripPair;
}

function bruteForce($points) {
  $n = count($points);
  $minDistance = PHP_INT_MAX;
  $minPair = [];
  
  for ($i = 0; $i < $n; $i++) {
    for ($j = $i+1; $j < $n; $j++) {
      $distance = distance($points[$i], $points[$j]);
      if ($distance < $minDistance) {
        $minDistance = $distance;
        $minPair = [$points[$i], $points[$j]];
      }
    }
  }
  
  return [
    'distance' => $minDistance,
    'pair' => $minPair
  ];
}

function stripScan($strip, $delta) {
  $n = count($strip);
  $minDistance = $delta;
  $minPair = [];
  
  for ($i = 0; $i < $n-1; $i++) {
    for ($j = $i+1; $j < $n && ($strip[$j]['y'] - $strip[$i]['y']) < $minDistance; $j++) {
      $distance = distance($strip[$i], $strip[$j]);
      if ($distance < $minDistance) {
        $minDistance = $distance;
        $minPair = [$strip[$i], $strip[$j]];
      }
    }
  }
  
  return [
    'distance' => $minDistance,
    'pair' => $minPair
  ];
}

function distance($a, $b) {
  return sqrt(pow(($b['x'] - $a['x']), 2) + pow(($b['y'] - $a['y']), 2));
}

The above are the detailed steps and details of using the divide-and-conquer method to solve the nearest point-pair problem. Code examples. By dividing the problem into smaller-scale sub-problems, and by solving the sub-problems recursively, we can efficiently solve the nearest point pair problem and obtain the optimal solution. Through reasonable algorithm design and optimization, the efficiency and performance of problem solving can be improved.

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