给定二维平面中的n个点,后跟Xi,Yi描述n个点。任务是计算n点的锤击距离。
注意:锤击距离是每对点之间最短距离的平方之和。
例子:
Input: n = 3
0 1
0 0
1 0
Output: 4
Input: n = 4
1 0
2 0
3 0
4 0
Output: 20
基本方法:由于我们必须找出所有对中最短距离的平方和,因此,我们可以取每个可能的对并计算距离的平方和。
// Pseudo code to find hammered-distance using above approach.
//this will store hammered distance
Distance=0
for(int i=0;i
它的时间复杂度将是O(n ^ 2)。
高效的方法:这个问题可以通过O(N)的时间复杂度来解决。
下面是上述方法的实现:
C++
// C++ implementation of above approach
#include
#define ll long long int
using namespace std;
// Function calculate cummalative sum
// of x, y, x^2, y^2 coordinates.
void cumm(vector& x, vector& y,
vector& cummx, vector& cummy,
vector& cummx2, vector& cummy2, ll n)
{
for (int i = 1; i <= n; i++) {
cummx[i] = cummx[i - 1] + x[i];
cummy[i] = cummy[i - 1] + y[i];
cummx2[i] = cummx2[i - 1] + x[i] * x[i];
cummy2[i] = cummy2[i - 1] + y[i] * y[i];
}
}
// Function ot calculate the hammered distance
int calHammeredDistance(int n, vector& x, vector& y)
{
// cummx conatins cummulative sum of x
// cummy conatins cummulative sum of y
vector cummx(n + 1, 0), cummy(n + 1, 0);
// cummx2 conatins cummulative sum of x^2
// cummy2 conatins cummulative sum of y^2
vector cummx2(n + 1, 0), cummy2(n + 1, 0);
// calculate cummalative of x
//, y, x^2, y^2, because these terms
// required in formula to reduce complexity.
// this function calculate all required terms.
cumm(x, y, cummx, cummy, cummx2, cummy2, n);
// hdx calculate hammer distance for x coordinate
// hdy calculate hammer distance for y coordinate
ll hdx = 0, hdy = 0;
for (int i = 1; i <= n; i++) {
// came from formula describe in explanation
hdx += (i - 1) * x[i] * x[i] + cummx2[i - 1]
- 2 * x[i] * cummx[i - 1];
// came from formula describe in explanation
hdy += (i - 1) * y[i] * y[i] + cummy2[i - 1]
- 2 * y[i] * cummy[i - 1];
}
// total is the sum of both x and y.
ll total = hdx + hdy;
return total;
}
// Driver code
int main()
{
// number of points
int n = 3;
// x contains the x coordinates
// y conatins the y coordinates
//and converting the size to n+1
vector x = {0, 0, 1, 0};
vector y = {1, 0, 0, 0};
cout << calHammeredDistance(n, x, y);
return 0;
}
Java
// Java implementation of above approach
class GFG{
// Function calculate cummalative sum
// of x, y, x^2, y^2 coordinates.
static void cumm(int [] x, int [] y,
int [] cummx, int [] cummy,
int [] cummx2, int [] cummy2, int n)
{
for (int i = 1; i <= n; i++) {
cummx[i] = cummx[i - 1] + x[i];
cummy[i] = cummy[i - 1] + y[i];
cummx2[i] = cummx2[i - 1] + x[i] * x[i];
cummy2[i] = cummy2[i - 1] + y[i] * y[i];
}
}
// Function ot calculate the hammered distance
static int calHammeredDistance(int n, int [] x, int [] y)
{
// cummx conatins cummulative sum of x
// cummy conatins cummulative sum of y
int []cummx = new int[n + 1];
int []cummy = new int[n + 1];
// cummx2 conatins cummulative sum of x^2
// cummy2 conatins cummulative sum of y^2
int []cummx2 = new int[n + 1];
int []cummy2 = new int[n + 1];
// calculate cummalative of x
//, y, x^2, y^2, because these terms
// required in formula to reduce complexity.
// this function calculate all required terms.
cumm(x, y, cummx, cummy, cummx2, cummy2, n);
// hdx calculate hammer distance for x coordinate
// hdy calculate hammer distance for y coordinate
int hdx = 0, hdy = 0;
for (int i = 1; i <= n; i++) {
// came from formula describe in explanation
hdx += (i - 1) * x[i] * x[i] + cummx2[i - 1]
- 2 * x[i] * cummx[i - 1];
// came from formula describe in explanation
hdy += (i - 1) * y[i] * y[i] + cummy2[i - 1]
- 2 * y[i] * cummy[i - 1];
}
// total is the sum of both x and y.
int total = hdx + hdy;
return total;
}
// Driver code
public static void main(String[] args)
{
// number of points
int n = 3;
// x contains the x coordinates
// y conatins the y coordinates
int []x = new int[n + 1];
int []y = new int[n + 1];
x[2] = 1;
y[0] = 1;
System.out.print(calHammeredDistance(n, x, y));
}
}
// This code contributed by Rajput-Ji
Python3
# Python3 implementation of the
# above approach
# Function calculate cummalative sum
# of x, y, x^2, y^2 coordinates.
def cumm(x, y, cummx, cummy,
cummx2, cummy2, n):
for i in range(1, n+1):
cummx[i] = cummx[i - 1] + x[i]
cummy[i] = cummy[i - 1] + y[i]
cummx2[i] = cummx2[i - 1] + x[i] * x[i]
cummy2[i] = cummy2[i - 1] + y[i] * y[i]
# Function ot calculate the
# hammered distance
def calHammeredDistance(n, x, y):
# cummx conatins cummulative sum of x
# cummy conatins cummulative sum of y
cummx = [0] * (n + 1)
cummy = [0] * (n + 1)
# cummx2 conatins cummulative sum of x^2
# cummy2 conatins cummulative sum of y^2
cummx2 = [0] * (n + 1)
cummy2 = [0] * (n + 1)
# calculate cumulative of x , y, x^2, y^2,
# because these terms are required in the
# formula to reduce complexity.
# This function calculate all required terms.
cumm(x, y, cummx, cummy, cummx2, cummy2, n)
# hdx calculate hammer distance for x coordinate
# hdy calculate hammer distance for y coordinate
hdx, hdy = 0, 0
for i in range(1, n + 1):
# came from formula describe in explanation
hdx += ((i - 1) * x[i] * x[i] + cummx2[i - 1] -
2 * x[i] * cummx[i - 1])
# came from formula describe in explanation
hdy += ((i - 1) * y[i] * y[i] + cummy2[i - 1] -
2 * y[i] * cummy[i - 1])
# total is the sum of both x and y.
total = hdx + hdy
return total
# Driver Code
if __name__ == "__main__":
# number of points
n = 3
# x contains the x coordinates
# y conatins the y coordinates
x = [0, 0, 1, 0]
y = [1, 0, 0, 0]
print(calHammeredDistance(n, x, y))
# This code is contributed by Rituraj Jain
C#
// C# implementation of above approach
using System;
class GFG{
// Function calculate cummalative sum
// of x, y, x^2, y^2 coordinates.
static void cumm(int [] x, int [] y,
int [] cummx, int [] cummy,
int [] cummx2, int [] cummy2, int n)
{
for (int i = 1; i <= n; i++) {
cummx[i] = cummx[i - 1] + x[i];
cummy[i] = cummy[i - 1] + y[i];
cummx2[i] = cummx2[i - 1] + x[i] * x[i];
cummy2[i] = cummy2[i - 1] + y[i] * y[i];
}
}
// Function ot calculate the hammered distance
static int calHammeredDistance(int n, int [] x, int [] y)
{
// cummx conatins cummulative sum of x
// cummy conatins cummulative sum of y
int []cummx = new int[n + 1];
int []cummy = new int[n + 1];
// cummx2 conatins cummulative sum of x^2
// cummy2 conatins cummulative sum of y^2
int []cummx2 = new int[n + 1];
int []cummy2 = new int[n + 1];
// calculate cummalative of x
//, y, x^2, y^2, because these terms
// required in formula to reduce complexity.
// this function calculate all required terms.
cumm(x, y, cummx, cummy, cummx2, cummy2, n);
// hdx calculate hammer distance for x coordinate
// hdy calculate hammer distance for y coordinate
int hdx = 0, hdy = 0;
for (int i = 1; i <= n; i++) {
// came from formula describe in explanation
hdx += (i - 1) * x[i] * x[i] + cummx2[i - 1]
- 2 * x[i] * cummx[i - 1];
// came from formula describe in explanation
hdy += (i - 1) * y[i] * y[i] + cummy2[i - 1]
- 2 * y[i] * cummy[i - 1];
}
// total is the sum of both x and y.
int total = hdx + hdy;
return total;
}
// Driver code
public static void Main(String[] args)
{
// number of points
int n = 3;
// x contains the x coordinates
// y conatins the y coordinates
int []x = new int[n + 1];
int []y = new int[n + 1];
x[2] = 1;
y[0] = 1;
Console.Write(calHammeredDistance(n, x, y));
}
}
// This code is contributed by PrinciRaj1992
输出
2