使用分而治之算法的最近点对

给定平面中包含 n 个点的数组，问题是找出数组中最近的点对。这个问题出现在许多应用程序中。例如，在空中交通管制中，您可能希望监视靠得太近的飞机，因为这可能表明可能发生碰撞。回想以下关于两点 p 和 q 之间距离的公式。
$\left \|pq \right \| = \sqrt{(p_{x}-q_{x})^{2}+ (p_{y}-q_{y})^{2}}$
蛮力解决方案是 O(n^2)，计算每对之间的距离并返回最小的。我们可以使用分而治之策略在 O(nLogn) 时间内计算最小距离。在这篇文章中，讨论了一种 O(nx (Logn)^2) 方法。我们将在另一篇文章中讨论 O(nLogn) 方法。

算法
以下是 O(n (Logn)^2) 算法的详细步骤。
输入： n 个点的数组P[]
输出：给定数组中两点之间的最小距离。
作为预处理步骤，输入数组根据 x 坐标进行排序。
1)在排序后的数组中找到中点，我们可以取P[n/2]作为中点。
2)将给定的数组分成两半。第一个子数组包含从 P[0] 到 P[n/2] 的点。第二个子数组包含从 P[n/2+1] 到 P[n-1] 的点。
3)递归地找出两个子数组中的最小距离。让距离为 dl 和 dr。找出 dl 和 dr 的最小值。让最小值为 d。

4)从上面的3个步骤，我们有一个最小距离的上限d。现在我们需要考虑成对的一个点来自左半部分，另一个来自右半部分。考虑通过 P[n/2] 的垂直线并找到所有 x 坐标比 d 更靠近中间垂直线的点。构建所有这些点的数组 strip[]。

5)根据y坐标对数组strip[]进行排序。这一步是 O(nLogn)。它可以通过递归排序和合并优化为 O(n)。
6)在strip[]中找到最小的距离。这很棘手。乍一看，这似乎是一个 O(n^2) 的步骤，但实际上是 O(n)。可以从几何上证明，对于条带中的每个点，我们最多只需要检查它后面的 7 个点(注意条带是根据 Y 坐标排序的)。更多分析请看这里。
7)最后返回上一步计算出的d和距离的最小值(步骤6)

执行
下面是上述算法的实现。

C++

// A divide and conquer program in C++
// to find the smallest distance from a
// given set of points.
 
#include 
using namespace std;
 
// A structure to represent a Point in 2D plane
class Point
{
    public:
    int x, y;
};
 
/* Following two functions are needed for library function qsort().
Refer: http://www.cplusplus.com/reference/clibrary/cstdlib/qsort/ */
 
// Needed to sort array of points
// according to X coordinate
int compareX(const void* a, const void* b)
{
    Point *p1 = (Point *)a, *p2 = (Point *)b;
    return (p1->x - p2->x);
}
 
// Needed to sort array of points according to Y coordinate
int compareY(const void* a, const void* b)
{
    Point *p1 = (Point *)a, *p2 = (Point *)b;
    return (p1->y - p2->y);
}
 
// A utility function to find the
// distance between two points
float dist(Point p1, Point p2)
{
    return sqrt( (p1.x - p2.x)*(p1.x - p2.x) +
                (p1.y - p2.y)*(p1.y - p2.y)
            );
}
 
// A Brute Force method to return the
// smallest distance between two points
// in P[] of size n
float bruteForce(Point P[], int n)
{
    float min = FLT_MAX;
    for (int i = 0; i < n; ++i)
        for (int j = i+1; j < n; ++j)
            if (dist(P[i], P[j]) < min)
                min = dist(P[i], P[j]);
    return min;
}
 
// A utility function to find
// minimum of two float values
float min(float x, float y)
{
    return (x < y)? x : y;
}
 
 
// A utility function to find the
// distance between the closest points of
// strip of given size. All points in
// strip[] are sorted according to
// y coordinate. They all have an upper
// bound on minimum distance as d.
// Note that this method seems to be
// a O(n^2) method, but it's a O(n)
// method as the inner loop runs at most 6 times
float stripClosest(Point strip[], int size, float d)
{
    float min = d; // Initialize the minimum distance as d
 
    qsort(strip, size, sizeof(Point), compareY);
 
    // Pick all points one by one and try the next points till the difference
    // between y coordinates is smaller than d.
    // This is a proven fact that this loop runs at most 6 times
    for (int i = 0; i < size; ++i)
        for (int j = i+1; j < size && (strip[j].y - strip[i].y) < min; ++j)
            if (dist(strip[i],strip[j]) < min)
                min = dist(strip[i], strip[j]);
 
    return min;
}
 
// A recursive function to find the
// smallest distance. The array P contains
// all points sorted according to x coordinate
float closestUtil(Point P[], int n)
{
    // If there are 2 or 3 points, then use brute force
    if (n <= 3)
        return bruteForce(P, n);
 
    // Find the middle point
    int mid = n/2;
    Point midPoint = P[mid];
 
    // Consider the vertical line passing
    // through the middle point calculate
    // the smallest distance dl on left
    // of middle point and dr on right side
    float dl = closestUtil(P, mid);
    float dr = closestUtil(P + mid, n - mid);
 
    // Find the smaller of two distances
    float d = min(dl, dr);
 
    // Build an array strip[] that contains
    // points close (closer than d)
    // to the line passing through the middle point
    Point strip[n];
    int j = 0;
    for (int i = 0; i < n; i++)
        if (abs(P[i].x - midPoint.x) < d)
            strip[j] = P[i], j++;
 
    // Find the closest points in strip.
    // Return the minimum of d and closest
    // distance is strip[]
    return min(d, stripClosest(strip, j, d) );
}
 
// The main function that finds the smallest distance
// This method mainly uses closestUtil()
float closest(Point P[], int n)
{
    qsort(P, n, sizeof(Point), compareX);
 
    // Use recursive function closestUtil()
    // to find the smallest distance
    return closestUtil(P, n);
}
 
// Driver code
int main()
{
    Point P[] = {{2, 3}, {12, 30}, {40, 50}, {5, 1}, {12, 10}, {3, 4}};
    int n = sizeof(P) / sizeof(P[0]);
    cout << "The smallest distance is " << closest(P, n);
    return 0;
}
 
// This is code is contributed by rathbhupendra

C

// A divide and conquer program in C/C++ to find the smallest distance from a
// given set of points.
 
#include 
#include 
#include 
#include 
 
// A structure to represent a Point in 2D plane
struct Point
{
    int x, y;
};
 
/* Following two functions are needed for library function qsort().
   Refer: http://www.cplusplus.com/reference/clibrary/cstdlib/qsort/ */
 
// Needed to sort array of points according to X coordinate
int compareX(const void* a, const void* b)
{
    Point *p1 = (Point *)a,  *p2 = (Point *)b;
    return (p1->x - p2->x);
}
// Needed to sort array of points according to Y coordinate
int compareY(const void* a, const void* b)
{
    Point *p1 = (Point *)a,   *p2 = (Point *)b;
    return (p1->y - p2->y);
}
 
// A utility function to find the distance between two points
float dist(Point p1, Point p2)
{
    return sqrt( (p1.x - p2.x)*(p1.x - p2.x) +
                 (p1.y - p2.y)*(p1.y - p2.y)
               );
}
 
// A Brute Force method to return the smallest distance between two points
// in P[] of size n
float bruteForce(Point P[], int n)
{
    float min = FLT_MAX;
    for (int i = 0; i < n; ++i)
        for (int j = i+1; j < n; ++j)
            if (dist(P[i], P[j]) < min)
                min = dist(P[i], P[j]);
    return min;
}
 
// A utility function to find a minimum of two float values
float min(float x, float y)
{
    return (x < y)? x : y;
}
 
 
// A utility function to find the distance between the closest points of
// strip of a given size. All points in strip[] are sorted according to
// y coordinate. They all have an upper bound on minimum distance as d.
// Note that this method seems to be a O(n^2) method, but it's a O(n)
// method as the inner loop runs at most 6 times
float stripClosest(Point strip[], int size, float d)
{
    float min = d;  // Initialize the minimum distance as d
 
    qsort(strip, size, sizeof(Point), compareY);
 
    // Pick all points one by one and try the next points till the difference
    // between y coordinates is smaller than d.
    // This is a proven fact that this loop runs at most 6 times
    for (int i = 0; i < size; ++i)
        for (int j = i+1; j < size && (strip[j].y - strip[i].y) < min; ++j)
            if (dist(strip[i],strip[j]) < min)
                min = dist(strip[i], strip[j]);
 
    return min;
}
 
// A recursive function to find the smallest distance. The array P contains
// all points sorted according to x coordinate
float closestUtil(Point P[], int n)
{
    // If there are 2 or 3 points, then use brute force
    if (n <= 3)
        return bruteForce(P, n);
 
    // Find the middle point
    int mid = n/2;
    Point midPoint = P[mid];
 
    // Consider the vertical line passing through the middle point
    // calculate the smallest distance dl on left of middle point and
    // dr on right side
    float dl = closestUtil(P, mid);
    float dr = closestUtil(P + mid, n-mid);
 
    // Find the smaller of two distances
    float d = min(dl, dr);
 
    // Build an array strip[] that contains points close (closer than d)
    // to the line passing through the middle point
    Point strip[n];
    int j = 0;
    for (int i = 0; i < n; i++)
        if (abs(P[i].x - midPoint.x) < d)
            strip[j] = P[i], j++;
 
    // Find the closest points in strip.  Return the minimum of d and closest
    // distance is strip[]
    return min(d, stripClosest(strip, j, d) );
}
 
// The main function that finds the smallest distance
// This method mainly uses closestUtil()
float closest(Point P[], int n)
{
    qsort(P, n, sizeof(Point), compareX);
 
    // Use recursive function closestUtil() to find the smallest distance
    return closestUtil(P, n);
}
 
// Driver program to test above functions
int main()
{
    Point P[] = {{2, 3}, {12, 30}, {40, 50}, {5, 1}, {12, 10}, {3, 4}};
    int n = sizeof(P) / sizeof(P[0]);
    printf("The smallest distance is %f ", closest(P, n));
    return 0;
}

Python3

# A divide and conquer program in Python3
# to find the smallest distance from a
# given set of points.
import math
import copy
# A class to represent a Point in 2D plane
class Point():
    def __init__(self, x, y):
        self.x = x
        self.y = y
 
# A utility function to find the
# distance between two points
def dist(p1, p2):
    return math.sqrt((p1.x - p2.x) *
                     (p1.x - p2.x) +
                     (p1.y - p2.y) *
                     (p1.y - p2.y))
 
# A Brute Force method to return the
# smallest distance between two points
# in P[] of size n
def bruteForce(P, n):
    min_val = float('inf')
    for i in range(n):
        for j in range(i + 1, n):
            if dist(P[i], P[j]) < min_val:
                min_val = dist(P[i], P[j])
 
    return min_val
 
# A utility function to find the
# distance between the closest points of
# strip of given size. All points in
# strip[] are sorted according to
# y coordinate. They all have an upper
# bound on minimum distance as d.
# Note that this method seems to be
# a O(n^2) method, but it's a O(n)
# method as the inner loop runs at most 6 times
def stripClosest(strip, size, d):
     
    # Initialize the minimum distance as d
    min_val = d
 
    
    # Pick all points one by one and
    # try the next points till the difference
    # between y coordinates is smaller than d.
    # This is a proven fact that this loop
    # runs at most 6 times
    for i in range(size):
        j = i + 1
        while j < size and (strip[j].y -
                            strip[i].y) < min_val:
            min_val = dist(strip[i], strip[j])
            j += 1
 
    return min_val
 
# A recursive function to find the
# smallest distance. The array P contains
# all points sorted according to x coordinate
def closestUtil(P, Q, n):
     
    # If there are 2 or 3 points,
    # then use brute force
    if n <= 3:
        return bruteForce(P, n)
 
    # Find the middle point
    mid = n // 2
    midPoint = P[mid]
 
    #keep a copy of left and right branch
    Pl = P[:mid]
    Pr = P[mid:]
 
    # Consider the vertical line passing
    # through the middle point calculate
    # the smallest distance dl on left
    # of middle point and dr on right side
    dl = closestUtil(Pl, Q, mid)
    dr = closestUtil(Pr, Q, n - mid)
 
    # Find the smaller of two distances
    d = min(dl, dr)
 
    # Build an array strip[] that contains
    # points close (closer than d)
    # to the line passing through the middle point
    stripP = []
    stripQ = []
    lr = Pl + Pr
    for i in range(n):
        if abs(lr[i].x - midPoint.x) < d:
            stripP.append(lr[i])
        if abs(Q[i].x - midPoint.x) < d:
            stripQ.append(Q[i])
 
    stripP.sort(key = lambda point: point.y) #<-- REQUIRED
    min_a = min(d, stripClosest(stripP, len(stripP), d))
    min_b = min(d, stripClosest(stripQ, len(stripQ), d))
     
     
    # Find the self.closest points in strip.
    # Return the minimum of d and self.closest
    # distance is strip[]
    return min(min_a,min_b)
 
# The main function that finds
# the smallest distance.
# This method mainly uses closestUtil()
def closest(P, n):
    P.sort(key = lambda point: point.x)
    Q = copy.deepcopy(P)
    Q.sort(key = lambda point: point.y)   
 
    # Use recursive function closestUtil()
    # to find the smallest distance
    return closestUtil(P, Q, n)
 
# Driver code
P = [Point(2, 3), Point(12, 30),
     Point(40, 50), Point(5, 1),
     Point(12, 10), Point(3, 4)]
n = len(P)
print("The smallest distance is",
                   closest(P, n))
 
# This code is contributed
# by Prateek Gupta (@prateekgupta10)

输出：

The smallest distance is 1.414214

时间复杂度令上述算法的时间复杂度为 T(n)。让我们假设我们使用 O(nLogn) 排序算法。上述算法将所有点分成两组，递归调用两组。划分后，在 O(n) 时间内找到条带，在 O(nLogn) 时间内对条带进行排序，最后在 O(n) 时间内找到条带中最近的点。所以T(n)可以表示如下
T(n) = 2T(n/2) + O(n) + O(nLogn) + O(n)
T(n) = 2T(n/2) + O(nLogn)
T(n) = T(nx Logn x Logn)

笔记
1)通过优化上述算法的第5步，时间复杂度可以提高到O(nLogn)。我们很快将在单独的帖子中讨论优化的解决方案。
2)代码找到最小距离。可以很容易地修改它以找到距离最小的点。
3)代码使用快速排序，在最坏的情况下可以是 O(n^2)。要将上限设为 O(n (Logn)^2)，可以使用 O(nLogn) 排序算法，如归并排序或堆排序

参考：
http://www.cs.umd.edu/class/fall2013/cmsc451/Lects/lect10.pdf
http://www.youtube.com/watch?v=vS4Zn1a9KUc
http://www.youtube.com/watch?v=T3T7T8Ym20M
http://en.wikipedia.org/wiki/Closest_pair_of_points_problem

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