这个n体问题是O（n ^ 2）还是O（n log n）？

Question

我正在写一篇关于n体问题的文章，并且我希望在技术上是准确的。

代码为here。这是注释和循环：

/**
 * Given N bodies with mass, in a 3d space, calculate the forces of gravity to be applied to each body.  
 * 
 * This function is exported to JavaScript, so only takes/returns numbers and arrays.
 * For N bodies, pass and array of 4N values (x,y,z,mass) and expect a 3N array of forces (x,y,z)
 * Those forcess can be applied to the bodies mass to update the its position in the simulation.
 * Calculate the 3-vector each unique pair of bodies applies to each other.
 * 
 *   0 1 2 3 4 5
 * 0   x x x x x
 * 1     x x x x
 * 2       x x x
 * 3         x x
 * 4           x
 * 5
 * 
 * Sum those forces together into an array of 3-vector x,y,z forces
 * 
 * Return 0 on success
 */

 // For all bodies:

  for (let i: i32 = 0; i < numBodies; i++) {                   // TypeScript.  i32 is type 32bit int
    // Given body i: pair with every body[j] where j > i
    for (let j: i32 = i + 1; j < numBodies; j++) {             // is this "n" or "log n"?
      // Calculate the force the bodies apply to one another
      stuff = stuff
    }
  }
  return stuff

我相当确定算法是> O（n）和<= O（n * n）。

通过process of elimination将O（n log n）保留为另一个选项。

看着网格，我认为O（1/2 n ^ 2）= O（n ^ 2）

看着循环，我认为内部循环是< n，但是我不确定是否一直到log n。

如果我遍历n，添加一个log n内部循环是什么样的？如果不是内循环，外循环？

Answer 1

假设Calculate the force the bodies apply to one another是O（1）运算，则下面的总和。