One optimization I would propose is over the brute force search:
$$
\begin{align*}
d(\mathbf{x}_i, \mathbf{x}_j) &= \lVert (\mathbf{x}_i-\mathbf{x}_j) - \mathbf{v} \rVert^2\\
&= \sum\limits_{k=1}^N (x_i^k - x_j^k-v^k)^2\\
&= \sum\limits_{k=1}^N ((x_i^k-x_j^k)^2+(v^k)^2-2v^k(x_i^k-x_j^k))\\
&= \sum\limits_{k=1}^N ((x_i^k)^2+(x_j^k)^2-2x_i^kx_j^k+(v^k)^2-2v^k(x_i^k-x_j^k))\\
\end{align*}
$$
as $(v^k)^2$ is the same for all pairs, we could simply drop it - doesn't effect minimization.
\begin{align*}
d(\mathbf{x}_i, \mathbf{x}_j) &= \sum\limits_{k=1}^N ((x_i^k)^2+(x_j^k)^2-2x_i^kx_j^k-2v^kx_i^k+2v^kx_j^k)\\
&= \sum\limits_{k=1}^N (x_i^k)^2 + \sum\limits_{k=1}^N(x_j^k)^2 - 2\sum\limits_{k=1}^N x_i^kx_j^k - 2\sum\limits_{k=1}^N v^kx_i^k + 2\sum\limits_{k=1}^N v^kx_j^k\\
\end{align*}
Let's go back to matrix notation:
\begin{align*}
d(\mathbf{x}_i, \mathbf{x}_j) &= \lVert \mathbf{x}_i \rVert+\lVert \mathbf{x}_j \rVert - 2(\mathbf{x}_i \cdot \mathbf{x}_j)- 2(\mathbf{x}_i \cdot \mathbf{v}) + 2(\mathbf{x}_j \cdot \mathbf{v})\\
\end{align*}
Note that all the terms, except the middle one is free of the pairwise computations and can be computed in $O(N)$ time and stored. To compute $(\mathbf{x}_i \cdot \mathbf{x}_j)$, one can assemble matrix $X$, which contains $\mathbf{x}_i^T$ at each row and compute $D=XX^T$. Each element in this huge symmetric matrix, would then give you the dot product per pair: $D(i,j)=(\mathbf{x}_i \cdot \mathbf{x}_j)$. If memory is of concern, you can simply revert to iterative computation and not store the intermediate dot products. All in all, this would save a lot of time in pairwise comptutations, speeding up the entire search. I assume that you could couple this easy to implement approach with any other optimization to further boost the performance. In all the calculations I omitted $sqrt$ because it doesn't influence the relative comparison of distances.
If the assumption is that $\mathbf{v}=\mathbf{0}$ ($\mathbf{v}$ is null), the entire procedure boils down to a fast computation of distance matrix - this view might benefit certain applications.
