utils.cpp

/**
 * Copyright (c) 2015-present, Facebook, Inc.
 * All rights reserved.
 *
 * This source code is licensed under the BSD+Patents license found in the
 * LICENSE file in the root directory of this source tree.
 */

// -*- c++ -*-

#include "utils.h"

#include <cstdio>
#include <cassert>
#include <cstring>
#include <cmath>

#include <immintrin.h>


#include <sys/time.h>
#include <sys/types.h>
#include <unistd.h>

#include <omp.h>


#include <algorithm>
#include <vector>

#include "AuxIndexStructures.h"
#include "FaissAssert.h"



#ifndef FINTEGER
#define FINTEGER long
#endif


extern "C" {

/* declare BLAS functions, see http://www.netlib.org/clapack/cblas/ */

int sgemm_ (const char *transa, const char *transb, FINTEGER *m, FINTEGER *
            n, FINTEGER *k, const float *alpha, const float *a,
            FINTEGER *lda, const float *b, FINTEGER *
            ldb, float *beta, float *c, FINTEGER *ldc);

/* Lapack functions, see http://www.netlib.org/clapack/old/single/sgeqrf.c */

int sgeqrf_ (FINTEGER *m, FINTEGER *n, float *a, FINTEGER *lda,
                 float *tau, float *work, FINTEGER *lwork, FINTEGER *info);

int sorgqr_(FINTEGER *m, FINTEGER *n, FINTEGER *k, float *a,
            FINTEGER *lda, float *tau, float *work,
            FINTEGER *lwork, FINTEGER *info);


}


/**************************************************
 * Get some stats about the system
 **************************************************/

namespace faiss {

#ifdef __AVX__
#define USE_AVX
#endif

double getmillisecs () {
    struct timeval tv;
    gettimeofday (&tv, nullptr);
    return tv.tv_sec * 1e3 + tv.tv_usec * 1e-3;
}


#ifdef __linux__

size_t get_mem_usage_kb ()
{
    int pid = getpid ();
    char fname[256];
    snprintf (fname, 256, "/proc/%d/status", pid);
    FILE * f = fopen (fname, "r");
    FAISS_THROW_IF_NOT_MSG (f, "cannot open proc status file");
    size_t sz = 0;
    for (;;) {
        char buf [256];
        if (!fgets (buf, 256, f)) break;
        if (sscanf (buf, "VmRSS: %ld kB", &sz) == 1) break;
    }
    fclose (f);
    return sz;
}

#elif __APPLE__

size_t get_mem_usage_kb ()
{
    fprintf(stderr, "WARN: get_mem_usage_kb not implemented on the mac\n");
    return 0;
}

#endif



/**************************************************
 * Random data generation functions
 **************************************************/

/**
 * The definition of random functions depends on the architecture:
 *
 * - for Linux, we rely on re-entrant functions (random_r). This
 *   provides good quality reproducible random sequences.
 *
 * - for Apple, we use rand_r. Apple is trying so hard to deprecate
 *   this function that it removed its definition form stdlib.h, so we
 *   re-declare it below. Fortunately, since it is deprecated, its
 *   prototype should not change much in the forerseeable future.
 *
 * Unfortunately, system designers are more concerned with making the
 * most unpredictable random sequences for cryptographic use, when in
 * scientific contexts what acutally matters is having reproducible
 * squences in multi-threaded contexts.
 */


#ifdef __linux__




int RandomGenerator::rand_int ()
{
    int32_t a;
    random_r (&rand_data, &a);
    return a;
}

long RandomGenerator::rand_long ()
{
    int32_t a, b;
    random_r (&rand_data, &a);
    random_r (&rand_data, &b);
    return long(a) | long(b) << 31;
}


RandomGenerator::RandomGenerator (long seed)
{
    memset (&rand_data, 0, sizeof (rand_data));
    initstate_r (seed, rand_state, sizeof (rand_state), &rand_data);
}


RandomGenerator::RandomGenerator (const RandomGenerator & other)
{
    memcpy (rand_state, other.rand_state, sizeof(rand_state));
    rand_data = other.rand_data;
    setstate_r (rand_state, &rand_data);
}


#elif __APPLE__

extern "C" {
int rand_r(unsigned *seed);
}

RandomGenerator::RandomGenerator (long seed)
{
    rand_state = seed;
}


RandomGenerator::RandomGenerator (const RandomGenerator & other)
{
    rand_state = other.rand_state;
}


int RandomGenerator::rand_int ()
{
    // RAND_MAX is 31 bits
    // try to add more randomness in the lower bits
    int lowbits = rand_r(&rand_state) >> 15;
    return rand_r(&rand_state) ^ lowbits;
}

long RandomGenerator::rand_long ()
{
    return long(random()) | long(random()) << 31;
}



#endif

int RandomGenerator::rand_int (int max)
{   // this suffers form non-uniform probabilities when max is not a
    // power of 2, but if RAND_MAX >> max the bias is limited.
    return rand_int () % max;
}

float RandomGenerator::rand_float ()
{
    return rand_int() / float(1L << 31);
}

double RandomGenerator::rand_double ()
{
    return rand_long() / double(1L << 62);
}


/***********************************************************************
 * Random functions in this C file only exist because Torch
 *  counterparts are slow and not multi-threaded.  Typical use is for
 *  more than 1-100 billion values. */


/* Generate a set of random floating point values such that x[i] in [0,1]
   multi-threading. For this reason, we rely on re-entreant functions.  */
void float_rand (float * x, size_t n, long seed)
{
    // only try to parallelize on large enough arrays
    const size_t nblock = n < 1024 ? 1 : 1024;

    RandomGenerator rng0 (seed);
    int a0 = rng0.rand_int (), b0 = rng0.rand_int ();

#pragma omp parallel for
    for (size_t j = 0; j < nblock; j++) {

        RandomGenerator rng (a0 + j * b0);

        const size_t istart = j * n / nblock;
        const size_t iend = (j + 1) * n / nblock;

        for (size_t i = istart; i < iend; i++)
            x[i] = rng.rand_float ();
    }
}


void float_randn (float * x, size_t n, long seed)
{
    // only try to parallelize on large enough arrays
    const size_t nblock = n < 1024 ? 1 : 1024;

    RandomGenerator rng0 (seed);
    int a0 = rng0.rand_int (), b0 = rng0.rand_int ();

#pragma omp parallel for
    for (size_t j = 0; j < nblock; j++) {
        RandomGenerator rng (a0 + j * b0);

        double a = 0, b = 0, s = 0;
        int state = 0;  /* generate two number per "do-while" loop */

        const size_t istart = j * n / nblock;
        const size_t iend = (j + 1) * n / nblock;

        for (size_t i = istart; i < iend; i++) {
            /* Marsaglia's method (see Knuth) */
            if (state == 0) {
                do {
                    a = 2.0 * rng.rand_double () - 1;
                    b = 2.0 * rng.rand_double () - 1;
                    s = a * a + b * b;
                } while (s >= 1.0);
                x[i] = a * sqrt(-2.0 * log(s) / s);
            }
            else
                x[i] = b * sqrt(-2.0 * log(s) / s);
            state = 1 - state;
        }
    }
}


/* Integer versions */
void long_rand (long * x, size_t n, long seed)
{
    // only try to parallelize on large enough arrays
    const size_t nblock = n < 1024 ? 1 : 1024;

    RandomGenerator rng0 (seed);
    int a0 = rng0.rand_int (), b0 = rng0.rand_int ();

#pragma omp parallel for
    for (size_t j = 0; j < nblock; j++) {

        RandomGenerator rng (a0 + j * b0);

        const size_t istart = j * n / nblock;
        const size_t iend = (j + 1) * n / nblock;
        for (size_t i = istart; i < iend; i++)
            x[i] = rng.rand_long ();
    }
}



void rand_perm (int *perm, size_t n, long seed)
{
    for (size_t i = 0; i < n; i++) perm[i] = i;

    RandomGenerator rng (seed);

    for (size_t i = 0; i + 1 < n; i++) {
        int i2 = i + rng.rand_int (n - i);
        std::swap(perm[i], perm[i2]);
    }
}




void byte_rand (uint8_t * x, size_t n, long seed)
{
    // only try to parallelize on large enough arrays
    const size_t nblock = n < 1024 ? 1 : 1024;

    RandomGenerator rng0 (seed);
    int a0 = rng0.rand_int (), b0 = rng0.rand_int ();

#pragma omp parallel for
    for (size_t j = 0; j < nblock; j++) {

        RandomGenerator rng (a0 + j * b0);

        const size_t istart = j * n / nblock;
        const size_t iend = (j + 1) * n / nblock;

        size_t i;
        for (i = istart; i < iend; i++)
            x[i] = rng.rand_long ();
    }
}



void reflection (const float * __restrict u,
                 float * __restrict x,
                 size_t n, size_t d, size_t nu)
{
    size_t i, j, l;
    for (i = 0; i < n; i++) {
        const float * up = u;
        for (l = 0; l < nu; l++) {
            float ip1 = 0, ip2 = 0;

            for (j = 0; j < d; j+=2) {
                ip1 += up[j] * x[j];
                ip2 += up[j+1] * x[j+1];
            }
            float ip = 2 * (ip1 + ip2);

            for (j = 0; j < d; j++)
                x[j] -= ip * up[j];
            up += d;
        }
        x += d;
    }
}


/* Reference implementation (slower) */
void reflection_ref (const float * u, float * x, size_t n, size_t d, size_t nu)
{
    size_t i, j, l;
    for (i = 0; i < n; i++) {
        const float * up = u;
        for (l = 0; l < nu; l++) {
            double ip = 0;

            for (j = 0; j < d; j++)
                ip += up[j] * x[j];
            ip *= 2;

            for (j = 0; j < d; j++)
                x[j] -= ip * up[j];

            up += d;
        }
        x += d;
    }
}

/*********************************************************
 * Optimized distance computations
 *********************************************************/



/* Functions to compute:
   - L2 distance between 2 vectors
   - inner product between 2 vectors
   - L2 norm of a vector

   The functions should probably not be invoked when a large number of
   vectors are be processed in batch (in which case Matrix multiply
   is faster), but may be useful for comparing vectors isolated in
   memory.

   Works with any vectors of any dimension, even unaligned (in which
   case they are slower).

*/


/*********************************************************
 * Reference implementations
 */



/* same without SSE */
float fvec_L2sqr_ref (const float * x,
                     const float * y,
                     size_t d)
{
    size_t i;
    float res = 0;
    for (i = 0; i < d; i++) {
        const float tmp = x[i] - y[i];
       res += tmp * tmp;
    }
    return res;
}

float fvec_inner_product_ref (const float * x,
                             const float * y,
                             size_t d)
{
    size_t i;
    float res = 0;
    for (i = 0; i < d; i++)
       res += x[i] * y[i];
    return res;
}

float fvec_norm_L2sqr_ref (const float * __restrict x,
                          size_t d)
{
    size_t i;
    double res = 0;
    for (i = 0; i < d; i++)
       res += x[i] * x[i];
    return res;
}


/*********************************************************
 * SSE and AVX implementations
 */

// reads 0 <= d < 4 floats as __m128
static inline __m128 masked_read (int d, const float *x)
{
    assert (0 <= d && d < 4);
    __attribute__((__aligned__(16))) float buf[4] = {0, 0, 0, 0};
    switch (d) {
      case 3:
        buf[2] = x[2];
      case 2:
        buf[1] = x[1];
      case 1:
        buf[0] = x[0];
    }
    return _mm_load_ps (buf);
    // cannot use AVX2 _mm_mask_set1_epi32
}

#ifdef USE_AVX

// reads 0 <= d < 8 floats as __m256
static inline __m256 masked_read_8 (int d, const float *x)
{
    assert (0 <= d && d < 8);
    if (d < 4) {
        __m256 res = _mm256_setzero_ps ();
        res = _mm256_insertf128_ps (res, masked_read (d, x), 0);
        return res;
    } else {
        __m256 res = _mm256_setzero_ps ();
        res = _mm256_insertf128_ps (res, _mm_loadu_ps (x), 0);
        res = _mm256_insertf128_ps (res, masked_read (d - 4, x + 4), 1);
        return res;
    }
}

float fvec_inner_product (const float * x,
                          const float * y,
                          size_t d)
{
    __m256 msum1 = _mm256_setzero_ps();

    while (d >= 8) {
        __m256 mx = _mm256_loadu_ps (x); x += 8;
        __m256 my = _mm256_loadu_ps (y); y += 8;
        msum1 = _mm256_add_ps (msum1, _mm256_mul_ps (mx, my));
        d -= 8;
    }

    __m128 msum2 = _mm256_extractf128_ps(msum1, 1);
    msum2 +=       _mm256_extractf128_ps(msum1, 0);

    if (d >= 4) {
        __m128 mx = _mm_loadu_ps (x); x += 4;
        __m128 my = _mm_loadu_ps (y); y += 4;
        msum2 = _mm_add_ps (msum2, _mm_mul_ps (mx, my));
        d -= 4;
    }

    if (d > 0) {
        __m128 mx = masked_read (d, x);
        __m128 my = masked_read (d, y);
        msum2 = _mm_add_ps (msum2, _mm_mul_ps (mx, my));
    }

    msum2 = _mm_hadd_ps (msum2, msum2);
    msum2 = _mm_hadd_ps (msum2, msum2);
    return  _mm_cvtss_f32 (msum2);
}

float fvec_L2sqr (const float * x,
                 const float * y,
                 size_t d)
{
    __m256 msum1 = _mm256_setzero_ps();

    while (d >= 8) {
        __m256 mx = _mm256_loadu_ps (x); x += 8;
        __m256 my = _mm256_loadu_ps (y); y += 8;
        const __m256 a_m_b1 = mx - my;
        msum1 += a_m_b1 * a_m_b1;
        d -= 8;
    }

    __m128 msum2 = _mm256_extractf128_ps(msum1, 1);
    msum2 +=       _mm256_extractf128_ps(msum1, 0);

    if (d >= 4) {
        __m128 mx = _mm_loadu_ps (x); x += 4;
        __m128 my = _mm_loadu_ps (y); y += 4;
        const __m128 a_m_b1 = mx - my;
        msum2 += a_m_b1 * a_m_b1;
        d -= 4;
    }

    if (d > 0) {
        __m128 mx = masked_read (d, x);
        __m128 my = masked_read (d, y);
        __m128 a_m_b1 = mx - my;
        msum2 += a_m_b1 * a_m_b1;
    }

    msum2 = _mm_hadd_ps (msum2, msum2);
    msum2 = _mm_hadd_ps (msum2, msum2);
    return  _mm_cvtss_f32 (msum2);
}

#else

/* SSE-implementation of L2 distance */
float fvec_L2sqr (const float * x,
                 const float * y,
                 size_t d)
{
    __m128 msum1 = _mm_setzero_ps();

    while (d >= 4) {
        __m128 mx = _mm_loadu_ps (x); x += 4;
        __m128 my = _mm_loadu_ps (y); y += 4;
        const __m128 a_m_b1 = mx - my;
        msum1 += a_m_b1 * a_m_b1;
        d -= 4;
    }

    if (d > 0) {
        // add the last 1, 2 or 3 values
        __m128 mx = masked_read (d, x);
        __m128 my = masked_read (d, y);
        __m128 a_m_b1 = mx - my;
        msum1 += a_m_b1 * a_m_b1;
    }

    msum1 = _mm_hadd_ps (msum1, msum1);
    msum1 = _mm_hadd_ps (msum1, msum1);
    return  _mm_cvtss_f32 (msum1);
}


float fvec_inner_product (const float * x,
                         const float * y,
                         size_t d)
{
    __m128 mx, my;
    __m128 msum1 = _mm_setzero_ps();

    while (d >= 4) {
        mx = _mm_loadu_ps (x); x += 4;
        my = _mm_loadu_ps (y); y += 4;
        msum1 = _mm_add_ps (msum1, _mm_mul_ps (mx, my));
        d -= 4;
    }

    // add the last 1, 2, or 3 values
    mx = masked_read (d, x);
    my = masked_read (d, y);
    __m128 prod = _mm_mul_ps (mx, my);

    msum1 = _mm_add_ps (msum1, prod);

    msum1 = _mm_hadd_ps (msum1, msum1);
    msum1 = _mm_hadd_ps (msum1, msum1);
    return  _mm_cvtss_f32 (msum1);
}



#endif

float fvec_norm_L2sqr (const float *  x,
                      size_t d)
{
    __m128 mx;
    __m128 msum1 = _mm_setzero_ps();

    while (d >= 4) {
        mx = _mm_loadu_ps (x); x += 4;
        msum1 = _mm_add_ps (msum1, _mm_mul_ps (mx, mx));
        d -= 4;
    }

    mx = masked_read (d, x);
    msum1 = _mm_add_ps (msum1, _mm_mul_ps (mx, mx));

    msum1 = _mm_hadd_ps (msum1, msum1);
    msum1 = _mm_hadd_ps (msum1, msum1);
    return  _mm_cvtss_f32 (msum1);
}




/***************************************************************************
 * Matrix/vector ops
 ***************************************************************************/



/* Compute the inner product between a vector x and
   a set of ny vectors y.
   These functions are not intended to replace BLAS matrix-matrix, as they
   would be significantly less efficient in this case. */
void fvec_inner_products_ny (float * __restrict ip,
                             const float * x,
                             const float * y,
                             size_t d, size_t ny)
{
    for (size_t i = 0; i < ny; i++) {
        ip[i] = fvec_inner_product (x, y, d);
        y += d;
    }
}




/* compute ny L2 distances between x and a set of vectors y */
void fvec_L2sqr_ny (float * __restrict dis,
                    const float * x,
                    const float * y,
                    size_t d, size_t ny)
{
    for (size_t i = 0; i < ny; i++) {
        dis[i] = fvec_L2sqr (x, y, d);
        y += d;
    }
}




/* Compute the L2 norm of a set of nx vectors */
void fvec_norms_L2 (float * __restrict nr,
                    const float * __restrict x,
                    size_t d, size_t nx)
{

#pragma omp parallel for
    for (size_t i = 0; i < nx; i++) {
        nr[i] = sqrtf (fvec_norm_L2sqr (x + i * d, d));
    }
}

void fvec_norms_L2sqr (float * __restrict nr,
                       const float * __restrict x,
                       size_t d, size_t nx)
{
#pragma omp parallel for
    for (size_t i = 0; i < nx; i++)
        nr[i] = fvec_norm_L2sqr (x + i * d, d);
}



void fvec_renorm_L2 (size_t d, size_t nx, float * __restrict x)
{
#pragma omp parallel for
    for (size_t i = 0; i < nx; i++) {
        float * __restrict xi = x + i * d;

        float nr = fvec_norm_L2sqr (xi, d);

        if (nr > 0) {
            size_t j;
            const float inv_nr = 1.0 / sqrtf (nr);
            for (j = 0; j < d; j++)
                xi[j] *= inv_nr;
        }
    }
}

















/***************************************************************************
 * KNN functions
 ***************************************************************************/



/* Find the nearest neighbors for nx queries in a set of ny vectors */
static void knn_inner_product_sse (const float * x,
                        const float * y,
                        size_t d, size_t nx, size_t ny,
                        float_minheap_array_t * res)
{
    size_t k = res->k;

#pragma omp parallel for
    for (size_t i = 0; i < nx; i++) {
        const float * x_i = x + i * d;
        const float * y_j = y;

        float * __restrict simi = res->get_val(i);
        long * __restrict idxi = res->get_ids (i);

        minheap_heapify (k, simi, idxi);

        for (size_t j = 0; j < ny; j++) {
            float ip = fvec_inner_product (x_i, y_j, d);

            if (ip > simi[0]) {
                minheap_pop (k, simi, idxi);
                minheap_push (k, simi, idxi, ip, j);
            }
            y_j += d;
        }
        minheap_reorder (k, simi, idxi);
    }

}

static void knn_L2sqr_sse (
                const float * x,
                const float * y,
                size_t d, size_t nx, size_t ny,
                float_maxheap_array_t * res)
{
    size_t k = res->k;

#pragma omp parallel for
    for (size_t i = 0; i < nx; i++) {
        const float * x_i = x + i * d;
        const float * y_j = y;
        size_t j;
        float * __restrict simi = res->get_val(i);
        long * __restrict idxi = res->get_ids (i);

        maxheap_heapify (k, simi, idxi);
        for (j = 0; j < ny; j++) {
            float disij = fvec_L2sqr (x_i, y_j, d);

            if (disij < simi[0]) {
                maxheap_pop (k, simi, idxi);
                maxheap_push (k, simi, idxi, disij, j);
            }
            y_j += d;
        }
        maxheap_reorder (k, simi, idxi);
    }

}


/** Find the nearest neighbors for nx queries in a set of ny vectors */
static void knn_inner_product_blas (
        const float * x,
        const float * y,
        size_t d, size_t nx, size_t ny,
        float_minheap_array_t * res)
{
    res->heapify ();

    // BLAS does not like empty matrices
    if (nx == 0 || ny == 0) return;

    /* block sizes */
    const size_t bs_x = 4096, bs_y = 1024;
    // const size_t bs_x = 16, bs_y = 16;
    float *ip_block = new float[bs_x * bs_y];

    for (size_t i0 = 0; i0 < nx; i0 += bs_x) {
        size_t i1 = i0 + bs_x;
        if(i1 > nx) i1 = nx;

        for (size_t j0 = 0; j0 < ny; j0 += bs_y) {
            size_t j1 = j0 + bs_y;
            if (j1 > ny) j1 = ny;
            /* compute the actual dot products */
            {
                float one = 1, zero = 0;
                FINTEGER nyi = j1 - j0, nxi = i1 - i0, di = d;
                sgemm_ ("Transpose", "Not transpose", &nyi, &nxi, &di, &one,
                        y + j0 * d, &di,
                        x + i0 * d, &di, &zero,
                        ip_block, &nyi);
            }

            /* collect maxima */
            res->addn (j1 - j0, ip_block, j0, i0, i1 - i0);
        }
    }
    delete [] ip_block;
    res->reorder ();
}

// distance correction is an operator that can be applied to transform
// the distances
template<class DistanceCorrection>
static void knn_L2sqr_blas (const float * x,
        const float * y,
        size_t d, size_t nx, size_t ny,
        float_maxheap_array_t * res,
        const DistanceCorrection &corr)
{
    res->heapify ();

    // BLAS does not like empty matrices
    if (nx == 0 || ny == 0) return;

    size_t k = res->k;

    /* block sizes */
    const size_t bs_x = 4096, bs_y = 1024;
    // const size_t bs_x = 16, bs_y = 16;
    float *ip_block = new float[bs_x * bs_y];

    float *x_norms = new float[nx];
    fvec_norms_L2sqr (x_norms, x, d, nx);

    float *y_norms = new float[ny];
    fvec_norms_L2sqr (y_norms, y, d, ny);

    for (size_t i0 = 0; i0 < nx; i0 += bs_x) {
        size_t i1 = i0 + bs_x;
        if(i1 > nx) i1 = nx;

        for (size_t j0 = 0; j0 < ny; j0 += bs_y) {
            size_t j1 = j0 + bs_y;
            if (j1 > ny) j1 = ny;
            /* compute the actual dot products */
            {
                float one = 1, zero = 0;
                FINTEGER nyi = j1 - j0, nxi = i1 - i0, di = d;
                sgemm_ ("Transpose", "Not transpose", &nyi, &nxi, &di, &one,
                        y + j0 * d, &di,
                        x + i0 * d, &di, &zero,
                        ip_block, &nyi);
            }

            /* collect minima */
#pragma omp parallel for
            for (size_t i = i0; i < i1; i++) {
                float * __restrict simi = res->get_val(i);
                long * __restrict idxi = res->get_ids (i);
                const float *ip_line = ip_block + (i - i0) * (j1 - j0);

                for (size_t j = j0; j < j1; j++) {
                    float ip = *ip_line++;
                    float dis = x_norms[i] + y_norms[j] - 2 * ip;

                    dis = corr (dis, i, j);

                    if (dis < simi[0]) {
                        maxheap_pop (k, simi, idxi);
                        maxheap_push (k, simi, idxi, dis, j);
                    }
                }
            }
        }
    }
    res->reorder ();

    delete [] ip_block;
    delete [] x_norms;
    delete [] y_norms;
}









/*******************************************************
 * KNN driver functions
 *******************************************************/

int distance_compute_blas_threshold = 20;

void knn_inner_product (const float * x,
        const float * y,
        size_t d, size_t nx, size_t ny,
        float_minheap_array_t * res)
{
    if (d % 4 == 0 && nx < distance_compute_blas_threshold) {
        knn_inner_product_sse (x, y, d, nx, ny, res);
    } else {
        knn_inner_product_blas (x, y, d, nx, ny, res);
    }
}



struct NopDistanceCorrection {
  float operator()(float dis, size_t /*qno*/, size_t /*bno*/) const {
    return dis;
    }
};

void knn_L2sqr (const float * x,
                const float * y,
                size_t d, size_t nx, size_t ny,
                float_maxheap_array_t * res)
{
    if (d % 4 == 0 && nx < distance_compute_blas_threshold) {
        knn_L2sqr_sse (x, y, d, nx, ny, res);
    } else {
        NopDistanceCorrection nop;
        knn_L2sqr_blas (x, y, d, nx, ny, res, nop);
    }
}

struct BaseShiftDistanceCorrection {
    const float *base_shift;
    float operator()(float dis, size_t /*qno*/, size_t bno) const {
      return dis - base_shift[bno];
    }
};

void knn_L2sqr_base_shift (
         const float * x,
         const float * y,
         size_t d, size_t nx, size_t ny,
         float_maxheap_array_t * res,
         const float *base_shift)
{
    BaseShiftDistanceCorrection corr = {base_shift};
    knn_L2sqr_blas (x, y, d, nx, ny, res, corr);
}



/***************************************************************************
 * compute a subset of  distances
 ***************************************************************************/

/* compute the inner product between x and a subset y of ny vectors,
   whose indices are given by idy.  */
void fvec_inner_products_by_idx (float * __restrict ip,
                                 const float * x,
                                 const float * y,
                                 const long * __restrict ids, /* for y vecs */
                                 size_t d, size_t nx, size_t ny)
{
#pragma omp parallel for
    for (size_t j = 0; j < nx; j++) {
        const long * __restrict idsj = ids + j * ny;
        const float * xj = x + j * d;
        float * __restrict ipj = ip + j * ny;
        for (size_t i = 0; i < ny; i++) {
            if (idsj[i] < 0)
                continue;
            ipj[i] = fvec_inner_product (xj, y + d * idsj[i], d);
        }
    }
}

/* compute the inner product between x and a subset y of ny vectors,
   whose indices are given by idy.  */
void fvec_L2sqr_by_idx (float * __restrict dis,
                        const float * x,
                        const float * y,
                        const long * __restrict ids, /* ids of y vecs */
                        size_t d, size_t nx, size_t ny)
{
#pragma omp parallel for
    for (size_t j = 0; j < nx; j++) {
        const long * __restrict idsj = ids + j * ny;
        const float * xj = x + j * d;
        float * __restrict disj = dis + j * ny;
        for (size_t i = 0; i < ny; i++) {
            if (idsj[i] < 0)
                continue;
            disj[i] = fvec_L2sqr (xj, y + d * idsj[i], d);
        }
    }
}





/* Find the nearest neighbors for nx queries in a set of ny vectors
   indexed by ids. May be useful for re-ranking a pre-selected vector list */
void knn_inner_products_by_idx (const float * x,
                                const float * y,
                                const long * ids,
                                size_t d, size_t nx, size_t ny,
                                float_minheap_array_t * res)
{
    size_t k = res->k;

#pragma omp parallel for
    for (size_t i = 0; i < nx; i++) {
        const float * x_ = x + i * d;
        const long * idsi = ids + i * ny;
        size_t j;
        float * __restrict simi = res->get_val(i);
        long * __restrict idxi = res->get_ids (i);
        minheap_heapify (k, simi, idxi);

        for (j = 0; j < ny; j++) {
            if (idsi[j] < 0) break;
            float ip = fvec_inner_product (x_, y + d * idsi[j], d);

            if (ip > simi[0]) {
                minheap_pop (k, simi, idxi);
                minheap_push (k, simi, idxi, ip, idsi[j]);
            }
        }
        minheap_reorder (k, simi, idxi);
    }

}

void knn_L2sqr_by_idx (const float * x,
                       const float * y,
                       const long * __restrict ids,
                       size_t d, size_t nx, size_t ny,
                       float_maxheap_array_t * res)
{
    size_t k = res->k;

#pragma omp parallel for
    for (size_t i = 0; i < nx; i++) {
        const float * x_ = x + i * d;
        const long * __restrict idsi = ids + i * ny;
        float * __restrict simi = res->get_val(i);
        long * __restrict idxi = res->get_ids (i);
        maxheap_heapify (res->k, simi, idxi);
        for (size_t j = 0; j < ny; j++) {
            float disij = fvec_L2sqr (x_, y + d * idsi[j], d);

            if (disij < simi[0]) {
                maxheap_pop (k, simi, idxi);
                maxheap_push (k, simi, idxi, disij, idsi[j]);
            }
        }
        maxheap_reorder (res->k, simi, idxi);
    }

}





/***************************************************************************
 * Range search
 ***************************************************************************/

/** Find the nearest neighbors for nx queries in a set of ny vectors
 * compute_l2 = compute pairwise squared L2 distance rather than inner prod
 */
template <bool compute_l2>
static void range_search_blas (
        const float * x,
        const float * y,
        size_t d, size_t nx, size_t ny,
        float radius,
        RangeSearchResult *result)
{

    // BLAS does not like empty matrices
    if (nx == 0 || ny == 0) return;

    /* block sizes */
    const size_t bs_x = 4096, bs_y = 1024;
    // const size_t bs_x = 16, bs_y = 16;
    float *ip_block = new float[bs_x * bs_y];

    float *x_norms = nullptr, *y_norms = nullptr;

    if (compute_l2) {
        x_norms = new float[nx];
        fvec_norms_L2sqr (x_norms, x, d, nx);
        y_norms = new float[ny];
        fvec_norms_L2sqr (y_norms, y, d, ny);
    }

    std::vector <RangeSearchPartialResult *> partial_results;

    for (size_t j0 = 0; j0 < ny; j0 += bs_y) {
        size_t j1 = j0 + bs_y;
        if (j1 > ny) j1 = ny;
        RangeSearchPartialResult * pres = new RangeSearchPartialResult (result);
        partial_results.push_back (pres);

        for (size_t i0 = 0; i0 < nx; i0 += bs_x) {
            size_t i1 = i0 + bs_x;
            if(i1 > nx) i1 = nx;

            /* compute the actual dot products */
            {
                float one = 1, zero = 0;
                FINTEGER nyi = j1 - j0, nxi = i1 - i0, di = d;
                sgemm_ ("Transpose", "Not transpose", &nyi, &nxi, &di, &one,
                        y + j0 * d, &di,
                        x + i0 * d, &di, &zero,
                        ip_block, &nyi);
            }


            for (size_t i = i0; i < i1; i++) {
                const float *ip_line = ip_block + (i - i0) * (j1 - j0);

                RangeSearchPartialResult::QueryResult & qres =
                    pres->new_result (i);

                for (size_t j = j0; j < j1; j++) {
                    float ip = *ip_line++;
                    if (compute_l2) {
                        float dis =  x_norms[i] + y_norms[j] - 2 * ip;
                        if (dis < radius) {
                            qres.add (dis, j);
                        }
                    } else {
                        if (ip > radius) {
                            qres.add (ip, j);
                        }
                    }
                }
            }
        }

    }
    delete [] ip_block;
    delete [] x_norms;
    delete [] y_norms;

    { // merge the partial results
        int npres = partial_results.size();
        // count
        for (size_t i = 0; i < nx; i++) {
            for (int j = 0; j < npres; j++)
                result->lims[i] += partial_results[j]->queries[i].nres;
        }
        result->do_allocation ();
        for (int j = 0; j < npres; j++) {
            partial_results[j]->set_result (true);
            delete partial_results[j];
        }

        // reset the limits
        for (size_t i = nx; i > 0; i--) {
            result->lims [i] = result->lims [i - 1];
        }
        result->lims [0] = 0;
    }
}


template <bool compute_l2>
static void range_search_sse (const float * x,
                const float * y,
                size_t d, size_t nx, size_t ny,
                float radius,
                RangeSearchResult *res)
{
    FAISS_THROW_IF_NOT (d % 4 == 0);

#pragma omp parallel
    {
        RangeSearchPartialResult pres (res);

#pragma omp for
        for (size_t i = 0; i < nx; i++) {
            const float * x_ = x + i * d;
            const float * y_ = y;
            size_t j;

            RangeSearchPartialResult::QueryResult & qres =
                pres.new_result (i);

            for (j = 0; j < ny; j++) {
                if (compute_l2) {
                    float disij = fvec_L2sqr (x_, y_, d);
                    if (disij < radius) {
                        qres.add (disij, j);
                    }
                } else {
                    float ip = fvec_inner_product (x_, y_, d);
                    if (ip > radius) {
                        qres.add (ip, j);
                    }
                }
                y_ += d;
            }

        }
        pres.finalize ();
    }
}





void range_search_L2sqr (
        const float * x,
        const float * y,
        size_t d, size_t nx, size_t ny,
        float radius,
        RangeSearchResult *res)
{

    if (d % 4 == 0 && nx < distance_compute_blas_threshold) {
        range_search_sse<true> (x, y, d, nx, ny, radius, res);
    } else {
        range_search_blas<true> (x, y, d, nx, ny, radius, res);
    }
}

void range_search_inner_product (
        const float * x,
        const float * y,
        size_t d, size_t nx, size_t ny,
        float radius,
        RangeSearchResult *res)
{

    if (d % 4 == 0 && nx < distance_compute_blas_threshold) {
        range_search_sse<false> (x, y, d, nx, ny, radius, res);
    } else {
        range_search_blas<false> (x, y, d, nx, ny, radius, res);
    }
}



/***************************************************************************
 * Some matrix manipulation functions
 ***************************************************************************/


/* This function exists because the Torch counterpart is extremly slow
   (not multi-threaded + unexpected overhead even in single thread).
   It is here to implement the usual property |x-y|^2=|x|^2+|y|^2-2<x|y>  */
void inner_product_to_L2sqr (float * __restrict dis,
                             const float * nr1,
                             const float * nr2,
                             size_t n1, size_t n2)
{

#pragma omp parallel for
    for (size_t j = 0 ; j < n1 ; j++) {
        float * disj = dis + j * n2;
        for (size_t i = 0 ; i < n2 ; i++)
            disj[i] = nr1[j] + nr2[i] - 2 * disj[i];
    }
}


void matrix_qr (int m, int n, float *a)
{
    FAISS_THROW_IF_NOT (m >= n);
    FINTEGER mi = m, ni = n, ki = mi < ni ? mi : ni;
    std::vector<float> tau (ki);
    FINTEGER lwork = -1, info;
    float work_size;

    sgeqrf_ (&mi, &ni, a, &mi, tau.data(),
             &work_size, &lwork, &info);
    lwork = size_t(work_size);
    std::vector<float> work (lwork);

    sgeqrf_ (&mi, &ni, a, &mi,
             tau.data(), work.data(), &lwork, &info);

    sorgqr_ (&mi, &ni, &ki, a, &mi, tau.data(),
             work.data(), &lwork, &info);

}


void pairwise_L2sqr (long d,
                     long nq, const float *xq,
                     long nb, const float *xb,
                     float *dis,
                     long ldq, long ldb, long ldd)
{
    if (nq == 0 || nb == 0) return;
    if (ldq == -1) ldq = d;
    if (ldb == -1) ldb = d;
    if (ldd == -1) ldd = nb;

    // store in beginning of distance matrix to avoid malloc
    float *b_norms = dis;

#pragma omp parallel for
    for (long i = 0; i < nb; i++)
        b_norms [i] = fvec_norm_L2sqr (xb + i * ldb, d);

#pragma omp parallel for
    for (long i = 1; i < nq; i++) {
        float q_norm = fvec_norm_L2sqr (xq + i * ldq, d);
        for (long j = 0; j < nb; j++)
            dis[i * ldd + j] = q_norm + b_norms [j];
    }

    {
        float q_norm = fvec_norm_L2sqr (xq, d);
        for (long j = 0; j < nb; j++)
            dis[j] += q_norm;
    }

    {
        FINTEGER nbi = nb, nqi = nq, di = d, ldqi = ldq, ldbi = ldb, lddi = ldd;
        float one = 1.0, minus_2 = -2.0;

        sgemm_ ("Transposed", "Not transposed",
                &nbi, &nqi, &di,
                &minus_2,
                xb, &ldbi,
                xq, &ldqi,
                &one, dis, &lddi);
    }


}



/***************************************************************************
 * Kmeans subroutine
 ***************************************************************************/

// a bit above machine epsilon for float16

#define EPS (1 / 1024.)

/* For k-means, compute centroids given assignment of vectors to centroids */
int km_update_centroids (const float * x,
                         float * centroids,
                         long * assign,
                         size_t d, size_t k, size_t n,
                         size_t k_frozen)
{
    k -= k_frozen;
    centroids += k_frozen * d;

    std::vector<size_t> hassign(k);
    memset (centroids, 0, sizeof(*centroids) * d * k);

#pragma omp parallel
    {
        int nt = omp_get_num_threads();
        int rank = omp_get_thread_num();
        // this thread is taking care of centroids c0:c1
        size_t c0 = (k * rank) / nt;
        size_t c1 = (k * (rank + 1)) / nt;
        const float *xi = x;
        size_t nacc = 0;

        for (size_t i = 0; i < n; i++) {
            long ci = assign[i];
            assert (ci >= 0 && ci < k + k_frozen);
            ci -= k_frozen;
            if (ci >= c0 && ci < c1)  {
                float * c = centroids + ci * d;
                hassign[ci]++;
                for (size_t j = 0; j < d; j++)
                    c[j] += xi[j];
                nacc++;
            }
            xi += d;
        }

    }

#pragma omp parallel for
    for (size_t ci = 0; ci < k; ci++) {
        float * c = centroids + ci * d;
        float ni = (float) hassign[ci];
        if (ni != 0) {
            for (size_t j = 0; j < d; j++)
                c[j] /= ni;
        }
    }

    /* Take care of void clusters */
    size_t nsplit = 0;
    RandomGenerator rng (1234);
    for (size_t ci = 0; ci < k; ci++) {
        if (hassign[ci] == 0) { /* need to redefine a centroid */
            size_t cj;
            for (cj = 0; 1; cj = (cj + 1) % k) {
                /* probability to pick this cluster for split */
                float p = (hassign[cj] - 1.0) / (float) (n - k);
                float r = rng.rand_float ();
                if (r < p) {
                    break; /* found our cluster to be split */
                }
            }
            memcpy (centroids+ci*d, centroids+cj*d, sizeof(*centroids) * d);

            /* small symmetric pertubation. Much better than  */
            for (size_t j = 0; j < d; j++) {
                if (j % 2 == 0) {
                    centroids[ci * d + j] *= 1 + EPS;
                    centroids[cj * d + j] *= 1 - EPS;
                } else {
                    centroids[ci * d + j] *= 1 - EPS;
                    centroids[cj * d + j] *= 1 + EPS;
                }
            }

            /* assume even split of the cluster */
            hassign[ci] = hassign[cj] / 2;
            hassign[cj] -= hassign[ci];
            nsplit++;
        }
    }

    return nsplit;
}

#undef EPS



/***************************************************************************
 * Result list routines
 ***************************************************************************/


void ranklist_handle_ties (int k, long *idx, const float *dis)
{
    float prev_dis = -1e38;
    int prev_i = -1;
    for (int i = 0; i < k; i++) {
        if (dis[i] != prev_dis) {
            if (i > prev_i + 1) {
                // sort between prev_i and i - 1
                std::sort (idx + prev_i, idx + i);
            }
            prev_i = i;
            prev_dis = dis[i];
        }
    }
}

size_t merge_result_table_with (size_t n, size_t k,
                                long *I0, float *D0,
                                const long *I1, const float *D1,
                                bool keep_min,
                                long translation)
{
    size_t n1 = 0;

#pragma omp parallel reduction(+:n1)
    {
        std::vector<long> tmpI (k);
        std::vector<float> tmpD (k);

#pragma omp for
        for (size_t i = 0; i < n; i++) {
            long *lI0 = I0 + i * k;
            float *lD0 = D0 + i * k;
            const long *lI1 = I1 + i * k;
            const float *lD1 = D1 + i * k;
            size_t r0 = 0;
            size_t r1 = 0;

            if (keep_min) {
                for (size_t j = 0; j < k; j++) {

                    if (lI0[r0] >= 0 && lD0[r0] < lD1[r1]) {
                        tmpD[j] = lD0[r0];
                        tmpI[j] = lI0[r0];
                        r0++;
                    } else if (lD1[r1] >= 0) {
                        tmpD[j] = lD1[r1];
                        tmpI[j] = lI1[r1] + translation;
                        r1++;
                    } else { // both are NaNs
                        tmpD[j] = NAN;
                        tmpI[j] = -1;
                    }
                }
            } else {
                for (size_t j = 0; j < k; j++) {
                    if (lI0[r0] >= 0 && lD0[r0] > lD1[r1]) {
                        tmpD[j] = lD0[r0];
                        tmpI[j] = lI0[r0];
                        r0++;
                    } else if (lD1[r1] >= 0) {
                        tmpD[j] = lD1[r1];
                        tmpI[j] = lI1[r1] + translation;
                        r1++;
                    } else { // both are NaNs
                        tmpD[j] = NAN;
                        tmpI[j] = -1;
                    }
                }
            }
            n1 += r1;
            memcpy (lD0, tmpD.data(), sizeof (lD0[0]) * k);
            memcpy (lI0, tmpI.data(), sizeof (lI0[0]) * k);
        }
    }

    return n1;
}



size_t ranklist_intersection_size (size_t k1, const long *v1,
                                   size_t k2, const long *v2_in)
{
    if (k2 > k1) return ranklist_intersection_size (k2, v2_in, k1, v1);
    long *v2 = new long [k2];
    memcpy (v2, v2_in, sizeof (long) * k2);
    std::sort (v2, v2 + k2);
    { // de-dup v2
        long prev = -1;
        size_t wp = 0;
        for (size_t i = 0; i < k2; i++) {
            if (v2 [i] != prev) {
                v2[wp++] = prev = v2 [i];
            }
        }
        k2 = wp;
    }
    const long seen_flag = 1L << 60;
    size_t count = 0;
    for (size_t i = 0; i < k1; i++) {
        long q = v1 [i];
        size_t i0 = 0, i1 = k2;
        while (i0 + 1 < i1) {
            size_t imed = (i1 + i0) / 2;
            long piv = v2 [imed] & ~seen_flag;
            if (piv <= q) i0 = imed;
            else          i1 = imed;
        }
        if (v2 [i0] == q) {
            count++;
            v2 [i0] |= seen_flag;
        }
    }
    delete [] v2;

    return count;
}

double imbalance_factor (int k, const int *hist) {
    double tot = 0, uf = 0;

    for (int i = 0 ; i < k ; i++) {
        tot += hist[i];
        uf += hist[i] * (double) hist[i];
    }
    uf = uf * k / (tot * tot);

    return uf;
}


double imbalance_factor (int n, int k, const long *assign) {
    std::vector<int> hist(k, 0);
    for (int i = 0; i < n; i++) {
        hist[assign[i]]++;
    }

    return imbalance_factor (k, hist.data());
}



int ivec_hist (size_t n, const int * v, int vmax, int *hist) {
    memset (hist, 0, sizeof(hist[0]) * vmax);
    int nout = 0;
    while (n--) {
        if (v[n] < 0 || v[n] >= vmax) nout++;
        else hist[v[n]]++;
    }
    return nout;
}


void bincode_hist(size_t n, size_t nbits, const uint8_t *codes, int *hist)
{
    FAISS_THROW_IF_NOT (nbits % 8 == 0);
    size_t d = nbits / 8;
    std::vector<int> accu(d * 256);
    const uint8_t *c = codes;
    for (size_t i = 0; i < n; i++)
        for(int j = 0; j < d; j++)
            accu[j * 256 + *c++]++;
    memset (hist, 0, sizeof(*hist) * nbits);
    for (int i = 0; i < d; i++) {
        const int *ai = accu.data() + i * 256;
        int * hi = hist + i * 8;
        for (int j = 0; j < 256; j++)
            for (int k = 0; k < 8; k++)
                if ((j >> k) & 1)
                    hi[k] += ai[j];
    }

}



size_t ivec_checksum (size_t n, const int *a)
{
    size_t cs = 112909;
    while (n--) cs = cs * 65713 + a[n] * 1686049;
    return cs;
}


namespace {
    struct ArgsortComparator {
        const float *vals;
        bool operator() (const size_t a, const size_t b) const {
            return vals[a] < vals[b];
        }
    };

    struct SegmentS {
        size_t i0; // begin pointer in the permutation array
        size_t i1; // end
        size_t len() const {
            return i1 - i0;
        }
    };

    // see https://en.wikipedia.org/wiki/Merge_algorithm#Parallel_merge
    // extended to > 1 merge thread

    // merges 2 ranges that should be consecutive on the source into
    // the union of the two on the destination
    template<typename T>
    void parallel_merge (const T *src, T *dst,
                         SegmentS &s1, SegmentS & s2, int nt,
                         const ArgsortComparator & comp) {
        if (s2.len() > s1.len()) { // make sure that s1 larger than s2
            std::swap(s1, s2);
        }

        // compute sub-ranges for each thread
        SegmentS s1s[nt], s2s[nt], sws[nt];
        s2s[0].i0 = s2.i0;
        s2s[nt - 1].i1 = s2.i1;

        // not sure parallel actually helps here
#pragma omp parallel for num_threads(nt)
        for (int t = 0; t < nt; t++) {
            s1s[t].i0 = s1.i0 + s1.len() * t / nt;
            s1s[t].i1 = s1.i0 + s1.len() * (t + 1) / nt;

            if (t + 1 < nt) {
                T pivot = src[s1s[t].i1];
                size_t i0 = s2.i0, i1 = s2.i1;
                while (i0 + 1 < i1) {
                    size_t imed = (i1 + i0) / 2;
                    if (comp (pivot, src[imed])) {i1 = imed; }
                    else                         {i0 = imed; }
                }
                s2s[t].i1 = s2s[t + 1].i0 = i1;
            }
        }
        s1.i0 = std::min(s1.i0, s2.i0);
        s1.i1 = std::max(s1.i1, s2.i1);
        s2 = s1;
        sws[0].i0 = s1.i0;
        for (int t = 0; t < nt; t++) {
            sws[t].i1 = sws[t].i0 + s1s[t].len() + s2s[t].len();
            if (t + 1 < nt) {
                sws[t + 1].i0 = sws[t].i1;
            }
        }
        assert(sws[nt - 1].i1 == s1.i1);

        // do the actual merging
#pragma omp parallel for num_threads(nt)
        for (int t = 0; t < nt; t++) {
            SegmentS sw = sws[t];
            SegmentS s1t = s1s[t];
            SegmentS s2t = s2s[t];
            if (s1t.i0 < s1t.i1 && s2t.i0 < s2t.i1) {
                for (;;) {
                    // assert (sw.len() == s1t.len() + s2t.len());
                    if (comp(src[s1t.i0], src[s2t.i0])) {
                        dst[sw.i0++] = src[s1t.i0++];
                        if (s1t.i0 == s1t.i1) break;
                    } else {
                        dst[sw.i0++] = src[s2t.i0++];
                        if (s2t.i0 == s2t.i1) break;
                    }
                }
            }
            if (s1t.len() > 0) {
                assert(s1t.len() == sw.len());
                memcpy(dst + sw.i0, src + s1t.i0, s1t.len() * sizeof(dst[0]));
            } else if (s2t.len() > 0) {
                assert(s2t.len() == sw.len());
                memcpy(dst + sw.i0, src + s2t.i0, s2t.len() * sizeof(dst[0]));
            }
        }
    }

};

void fvec_argsort (size_t n, const float *vals,
                    size_t *perm)
{
    for (size_t i = 0; i < n; i++) perm[i] = i;
    ArgsortComparator comp = {vals};
    std::sort (perm, perm + n, comp);
}

void fvec_argsort_parallel (size_t n, const float *vals,
                            size_t *perm)
{
    size_t * perm2 = new size_t[n];
    // 2 result tables, during merging, flip between them
    size_t *permB = perm2, *permA = perm;

    int nt = omp_get_max_threads();
    { // prepare correct permutation so that the result ends in perm
      // at final iteration
        int nseg = nt;
        while (nseg > 1) {
            nseg = (nseg + 1) / 2;
            std::swap (permA, permB);
        }
    }

#pragma omp parallel
    for (size_t i = 0; i < n; i++) permA[i] = i;

    ArgsortComparator comp = {vals};

    SegmentS segs[nt];

    // independent sorts
#pragma omp parallel for
    for (int t = 0; t < nt; t++) {
        size_t i0 = t * n / nt;
        size_t i1 = (t + 1) * n / nt;
        SegmentS seg = {i0, i1};
        std::sort (permA + seg.i0, permA + seg.i1, comp);
        segs[t] = seg;
    }
    int prev_nested = omp_get_nested();
    omp_set_nested(1);

    int nseg = nt;
    while (nseg > 1) {
        int nseg1 = (nseg + 1) / 2;
        int sub_nt = nseg % 2 == 0 ? nt : nt - 1;
        int sub_nseg1 = nseg / 2;

#pragma omp parallel for num_threads(nseg1)
        for (int s = 0; s < nseg; s += 2) {
            if (s + 1 == nseg) { // otherwise isolated segment
                memcpy(permB + segs[s].i0, permA + segs[s].i0,
                       segs[s].len() * sizeof(size_t));
            } else {
                int t0 = s * sub_nt / sub_nseg1;
                int t1 = (s + 1) * sub_nt / sub_nseg1;
                printf("merge %d %d, %d threads\n", s, s + 1, t1 - t0);
                parallel_merge(permA, permB, segs[s], segs[s + 1],
                               t1 - t0, comp);
            }
        }
        for (int s = 0; s < nseg; s += 2)
            segs[s / 2] = segs[s];
        nseg = nseg1;
        std::swap (permA, permB);
    }
    assert (permA == perm);
    omp_set_nested(prev_nested);
    delete [] perm2;
}
















/***************************************************************************
 * heavily optimized table computations
 ***************************************************************************/


static inline void fvec_madd_ref (size_t n, const float *a,
                           float bf, const float *b, float *c) {
    for (size_t i = 0; i < n; i++)
        c[i] = a[i] + bf * b[i];
}


static inline void fvec_madd_sse (size_t n, const float *a,
                                  float bf, const float *b, float *c) {
    n >>= 2;
    __m128 bf4 = _mm_set_ps1 (bf);
    __m128 * a4 = (__m128*)a;
    __m128 * b4 = (__m128*)b;
    __m128 * c4 = (__m128*)c;

    while (n--) {
        *c4 = _mm_add_ps (*a4, _mm_mul_ps (bf4, *b4));
        b4++;
        a4++;
        c4++;
    }
}

void fvec_madd (size_t n, const float *a,
                       float bf, const float *b, float *c)
{
    if ((n & 3) == 0 &&
        ((((long)a) | ((long)b) | ((long)c)) & 15) == 0)
        fvec_madd_sse (n, a, bf, b, c);
    else
        fvec_madd_ref (n, a, bf, b, c);
}

static inline int fvec_madd_and_argmin_ref (size_t n, const float *a,
                                         float bf, const float *b, float *c) {
    float vmin = 1e20;
    int imin = -1;

    for (size_t i = 0; i < n; i++) {
        c[i] = a[i] + bf * b[i];
        if (c[i] < vmin) {
            vmin = c[i];
            imin = i;
        }
    }
    return imin;
}

static inline int fvec_madd_and_argmin_sse (size_t n, const float *a,
                                         float bf, const float *b, float *c) {
    n >>= 2;
    __m128 bf4 = _mm_set_ps1 (bf);
    __m128 vmin4 = _mm_set_ps1 (1e20);
    __m128i imin4 = _mm_set1_epi32 (-1);
    __m128i idx4 = _mm_set_epi32 (3, 2, 1, 0);
    __m128i inc4 = _mm_set1_epi32 (4);
    __m128 * a4 = (__m128*)a;
    __m128 * b4 = (__m128*)b;
    __m128 * c4 = (__m128*)c;

    while (n--) {
        __m128 vc4 = _mm_add_ps (*a4, _mm_mul_ps (bf4, *b4));
        *c4 = vc4;
        __m128i mask = (__m128i)_mm_cmpgt_ps (vmin4, vc4);
        // imin4 = _mm_blendv_epi8 (imin4, idx4, mask); // slower!

        imin4 = _mm_or_si128 (_mm_and_si128 (mask, idx4),
                              _mm_andnot_si128 (mask, imin4));
        vmin4 = _mm_min_ps (vmin4, vc4);
        b4++;
        a4++;
        c4++;
        idx4 = _mm_add_epi32 (idx4, inc4);
    }

    // 4 values -> 2
    {
        idx4 = _mm_shuffle_epi32 (imin4, 3 << 2 | 2);
        __m128 vc4 = _mm_shuffle_ps (vmin4, vmin4, 3 << 2 | 2);
        __m128i mask = (__m128i)_mm_cmpgt_ps (vmin4, vc4);
        imin4 = _mm_or_si128 (_mm_and_si128 (mask, idx4),
                              _mm_andnot_si128 (mask, imin4));
        vmin4 = _mm_min_ps (vmin4, vc4);
    }
    // 2 values -> 1
    {
        idx4 = _mm_shuffle_epi32 (imin4, 1);
        __m128 vc4 = _mm_shuffle_ps (vmin4, vmin4, 1);
        __m128i mask = (__m128i)_mm_cmpgt_ps (vmin4, vc4);
        imin4 = _mm_or_si128 (_mm_and_si128 (mask, idx4),
                              _mm_andnot_si128 (mask, imin4));
        // vmin4 = _mm_min_ps (vmin4, vc4);
    }
    return  _mm_extract_epi32 (imin4, 0);
}


int fvec_madd_and_argmin (size_t n, const float *a,
                                 float bf, const float *b, float *c)
{
    if ((n & 3) == 0 &&
        ((((long)a) | ((long)b) | ((long)c)) & 15) == 0)
        return fvec_madd_and_argmin_sse (n, a, bf, b, c);
    else
        return fvec_madd_and_argmin_ref (n, a, bf, b, c);
}



const float *fvecs_maybe_subsample (
          size_t d, size_t *n, size_t nmax, const float *x,
          bool verbose, long seed)
{

    if (*n <= nmax) return x; // nothing to do

    size_t n2 = nmax;
    if (verbose) {
        printf ("  Input training set too big (max size is %ld), sampling "
                "%ld / %ld vectors\n", nmax, n2, *n);
    }
    std::vector<int> subset (*n);
    rand_perm (subset.data (), *n, seed);
    float *x_subset = new float[n2 * d];
    for (long i = 0; i < n2; i++)
        memcpy (&x_subset[i * d],
                &x[subset[i] * size_t(d)],
                sizeof (x[0]) * d);
    *n = n2;
    return x_subset;
}


} // namespace faiss