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# Source code for ase.geometry.minkowski_reduction

    import itertools
    import numpy as np
    from ase.utils import pbc2pbc

    def reduction_gauss(B, hu, hv):
        """Calculate a Gauss-reduced lattice basis (2D reduction)."""
        u = hu @ B
        v = hv @ B

        max_it = 100000    # in practice this is not exceeded
        for it in range(max_it):

            x = int(round(np.dot(u, v) / np.dot(u, u)))
            hu, hv = hv - x * hu, hu

            u = hu @ B
            v = hv @ B
            if np.dot(u, u) >= np.dot(v, v):
                return hv, hu

        raise RuntimeError("Gaussian basis not found after %d iterations" % max_it)

    def relevant_vectors_2D(u, v):
        cs = np.array([e for e in itertools.product([-1, 0, 1], repeat=2)])
        vs = np.dot(cs, [u, v])
        indices = np.argsort(np.linalg.norm(vs, axis=1))[:7]
        return vs[indices], cs[indices]

    def closest_vector(t0, u, v):
        t = t0
        rs, cs = relevant_vectors_2D(u, v)
        a = np.array([0, 0])

        dprev = float("inf")
        max_it = 100000    # in practice this is not exceeded
        for it in range(max_it):

            ds = np.linalg.norm(rs + t, axis=1)
            index = np.argmin(ds)
            if index == 0 or ds[index] >= dprev:
                return a

            dprev = ds[index]
            r = rs[index]
            kopt = int(round(-np.dot(t, r) / np.dot(r, r)))
            a += kopt * cs[index]
            t = t0 + a[0] * u + a[1] * v

        raise RuntimeError("Closest vector not found after %d iterations" % max_it)

    def reduction_full(B):
        """Calculate a Minkowski-reduced lattice basis (3D reduction)."""
        H = np.eye(3).astype(np.int)
        norms = np.linalg.norm(B, axis=1)

        max_it = 100000    # in practice this is not exceeded
        for it in range(max_it):

            # Sort vectors by norm
            indices = np.argsort(norms)
            H = H[indices]

            # Gauss-reduce smallest two vectors
            hw = H[2]
            hu, hv = reduction_gauss(B, H[0], H[1])

            H = np.array([hu, hv, hw])
            R = H @ B
            u, v, w = R

            X = u / np.linalg.norm(u)
            Y = v - X * np.dot(v, X)
            Y /= np.linalg.norm(Y)

            # Find closest vector to last element of R
            pu, pv, pw = np.dot(R, np.array([X, Y]).T)
            nb = closest_vector(pw, pu, pv)
            hw = np.dot([nb[0], nb[1], 1], H)

            # Update basis
            H = np.array([hu, hv, hw])
            R = H @ B

            norms = np.diag(np.dot(R, R.T))
            if norms[2] >= norms[1] or (nb == 0).all():
                return R, H

        raise RuntimeError("Reduced basis not found after %d iterations" % max_it)

    def minkowski_reduce(cell, pbc=True):
        """Calculate a Minkowski-reduced lattice basis.  The reduced basis
        has the shortest possible vector lengths and has
        norm(a) <= norm(b) <= norm(c).

        Implements the method described in:

        Low-dimensional Lattice Basis Reduction Revisited
        Nguyen, Phong Q. and Stehlé, Damien,
        ACM Trans. Algorithms 5(4) 46:1--46:48, 2009
        https://doi.org/10.1145/1597036.1597050

        Parameters:

        cell: array
            The lattice basis to reduce (in row-vector format).
        pbc: array, optional
            The periodic boundary conditions of the cell (Default `True`).
            If `pbc` is provided, only periodic cell vectors are reduced.

        Returns:

        rcell: array
            The reduced lattice basis.
        op: array
            The unimodular matrix transformation (rcell = op @ cell).
        """
        pbc = pbc2pbc(pbc)
        dim = pbc.sum()

        op = np.eye(3).astype(np.int)
        if dim == 2:
            perm = np.argsort(pbc, kind='merge')[::-1]    # stable sort
            pcell = cell[perm][:, perm]

            norms = np.linalg.norm(pcell, axis=1)
            norms[2] = float("inf")
            indices = np.argsort(norms)
            op = op[indices]

            hu, hv = reduction_gauss(pcell, op[0], op[1])

            op[0] = hu
            op[1] = hv
            invperm = np.argsort(perm)
            op = op[invperm][:, invperm]

        elif dim == 3:
            _, op = reduction_full(cell)

        # maintain cell handedness
        if dim == 3:
            if np.sign(np.linalg.det(cell)) != np.sign(np.linalg.det(op @ cell)):
                op = -op
        elif dim == 2:
            index = np.argmin(pbc)
            _cell = cell.copy()
            _cell[index] = (1, 1, 1)
            _rcell = op @ cell
            _rcell[index] = (1, 1, 1)

            if np.sign(np.linalg.det(_cell)) != np.sign(np.linalg.det(_rcell)):
                index = np.argmax(pbc)
                op[index] *= -1

        norms1 = np.sort(np.linalg.norm(cell, axis=1))
        norms2 = np.sort(np.linalg.norm(op @ cell, axis=1))
        if not (norms2 <= norms1 + 1E-12).all():
            raise RuntimeError("Minkowski reduction failed")
        return op @ cell, op

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