product of every difference

Question

Given a sorted array where every element is distinct, we need to evaluate product of every difference, modulo $ 10^9 + 7 $

$$ \prod_{i < j} (arr[j] - arr[i]) \% (10^9 + 7) $$

Best approach I can think of is finding frequency of primes in product by grouping elements with same modular residue, which allows $ O(N * M / \log M) $ if $M$ is maximum number in array.

I have considered mathematical ways by polynomial multiplication and divide and conquer, which seems most promising but can't solve completely.

Is there any way to find this for arrays of length upto $ 10^ 5 $, and numbers upto $ 10 ^ 9 $?

Answer 1

I think a divide and conquer (D&C) approach is valid, and leads to an $O(n \log^3 n)$ solution (maybe the log exponent can be reduced).

The D&C technique reduces the problem to the following: Given two arrays $A$ and $B$ compute

$$\prod_{a \in A, b \in B} a - b$$

Consider the following polynomial:

$$P_A(X) = \prod_{a \in A} a - X$$

If we can evaluate $P_A$ on every element of $B$, then the product of all those results gives the desired solution. To compute the polynomial $\prod_{a \in A} a - X$ you can use a divide and conquer based algorithm and using polynomial multiplication in $O(n \log n)$ you can compute $P_A$ in $O(n \log^2 n)$.

Once $P_A$ has been computed, you can evaluate it on every element in $B$ in $O(n \log^2 n)$ time see this answer.

Since every step of the D&C problem is solvable in $O(n \log^2 n)$, the whole problem is solvable in $O(n \log^3 n)$.

Note that the computation of $P_A$ can be done recursively in the outer D&C as well, so that part should not impart an extra log factor. Therefore the critical part if you want to reduce the complexity is the multipoint evaluation.