对3d点的查询

Question

I have an array of 3d points. Each point has an x,y and z coordinate. The maximum size of the array can be 777777. I have Q queries each providing me with four numbers A, B, C, D. for each query I have to output the following sum.

Q <=77
1 <= A, B, C <=77
1 <= Xi, Yi, Zi <=77
1 <= Di <= 777
N <= 777777

What I have done: For each query calculate the given sum using two nested loops giving me a complexity of O(Q*N^2). Is there a better way to calculate it ?

Edit: What I know for sure is this is not a geometric problem. The maximum value of xi-xj is 76 and min is -76. This applies to yi and zi too. so total possible combinations are 153*153*153. Now in a query we have to count how many times a particular combination occurs in the array and solve the sum for that combination only once. The problem reduces to finding how many times a particular combination of ( xi-xj , yi-yj, zi-zj). Can someone take it ahead from here? I suspect we can use fast Fourier transform here. I have seen them getting used in these kind of problems before. But I have no clue how to start

Answer 1

The following is a mistake, because it neglects the absolute value function. I don't see a way to get past this.

You can rewrite the query as A * SUM_i!=j (Xi-Xj)/sqrt(...) + B * SUM_i!=j(Yi-Yj)/sqrt(...) + ...

Once you have done this you can see that if you have calculated the values for A=1, B=C=D=0, and B=1,A=C=D=0 and C=1,A=B=D=0 and D=1, A=B=C=0 as eg Qa, Qb, Qc, Qd, then you can find the values for any other values of A, B, C, and D as A * Qa + B * Qb + C * Qc + D * Qd.

This takes the complexity down to O(N^2)