sort.h

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// Copyright © 2023-2024 Apple Inc.
 
#define MLX_MTL_CONST static constant constexpr const
#define MLX_MTL_LOOP_UNROLL _Pragma("clang loop unroll(full)")
 
using namespace metal;
 
// Based on GPU merge sort algorithm at
// https://github.com/NVIDIA/cccl/tree/main/cub/cub

// Thread-level sort
 
template <typename T>
METAL_FUNC void thread_swap(thread T& a, thread T& b) {
  T w = a;
  a = b;
  b = w;
}
 
template <typename T>
struct LessThan {
  static constexpr constant T init = Limits<T>::max;
 
  METAL_FUNC bool operator()(T a, T b) {
    return a < b;
  }
};
 
template <
    typename val_t,
    typename idx_t,
    bool ARG_SORT,
    short N_PER_THREAD,
    typename CompareOp>
struct ThreadSort {
  static METAL_FUNC void sort(
      thread val_t (&vals)[N_PER_THREAD],
      thread idx_t (&idxs)[N_PER_THREAD]) {
    CompareOp op;
 
    MLX_MTL_LOOP_UNROLL
    for (short i = 0; i < N_PER_THREAD; ++i) {
      MLX_MTL_LOOP_UNROLL
      for (short j = i & 1; j < N_PER_THREAD - 1; j += 2) {
        if (op(vals[j + 1], vals[j])) {
          thread_swap(vals[j + 1], vals[j]);
          thread_swap(idxs[j + 1], idxs[j]);
        }
      }
    }
  }
};

// Threadgroup-level sort
 
template <
    typename val_t,
    typename idx_t,
    bool ARG_SORT,
    short BLOCK_THREADS,
    short N_PER_THREAD,
    typename CompareOp>
struct BlockMergeSort {
  using thread_sort_t =
      ThreadSort<val_t, idx_t, ARG_SORT, N_PER_THREAD, CompareOp>;
  static METAL_FUNC int merge_partition(
      const threadgroup val_t* As,
      const threadgroup val_t* Bs,
      short A_sz,
      short B_sz,
      short sort_md) {
    CompareOp op;
 
    short A_st = max(0, sort_md - B_sz);
    short A_ed = min(sort_md, A_sz);
 
    while (A_st < A_ed) {
      short md = A_st + (A_ed - A_st) / 2;
      auto a = As[md];
      auto b = Bs[sort_md - 1 - md];
 
      if (op(b, a)) {
        A_ed = md;
      } else {
        A_st = md + 1;
      }
    }
 
    return A_ed;
  }
 
  static METAL_FUNC void merge_step(
      const threadgroup val_t* As,
      const threadgroup val_t* Bs,
      const threadgroup idx_t* As_idx,
      const threadgroup idx_t* Bs_idx,
      short A_sz,
      short B_sz,
      thread val_t (&vals)[N_PER_THREAD],
      thread idx_t (&idxs)[N_PER_THREAD]) {
    CompareOp op;
    short a_idx = 0;
    short b_idx = 0;
 
    for (int i = 0; i < N_PER_THREAD; ++i) {
      auto a = As[a_idx];
      auto b = Bs[b_idx];
      bool pred = (b_idx < B_sz) && (a_idx >= A_sz || op(b, a));
 
      vals[i] = pred ? b : a;
      idxs[i] = pred ? Bs_idx[b_idx] : As_idx[a_idx];
 
      b_idx += short(pred);
      a_idx += short(!pred);
    }
  }
 
  static METAL_FUNC void sort(
      threadgroup val_t* tgp_vals [[threadgroup(0)]],
      threadgroup idx_t* tgp_idxs [[threadgroup(1)]],
      int size_sorted_axis,
      uint3 lid [[thread_position_in_threadgroup]]) {
    // Get thread location
    int idx = lid.x * N_PER_THREAD;
 
    // Load from shared memory
    thread val_t thread_vals[N_PER_THREAD];
    thread idx_t thread_idxs[N_PER_THREAD];
    for (int i = 0; i < N_PER_THREAD; ++i) {
      thread_vals[i] = tgp_vals[idx + i];
      if (ARG_SORT) {
        thread_idxs[i] = tgp_idxs[idx + i];
      }
    }
 
    // Per thread sort
    if (idx < size_sorted_axis) {
      thread_sort_t::sort(thread_vals, thread_idxs);
    }
 
    // Do merges using threadgroup memory
    for (int merge_threads = 2; merge_threads <= BLOCK_THREADS;
         merge_threads *= 2) {
      // Update threadgroup memory
      threadgroup_barrier(mem_flags::mem_threadgroup);
      for (int i = 0; i < N_PER_THREAD; ++i) {
        tgp_vals[idx + i] = thread_vals[i];
        if (ARG_SORT) {
          tgp_idxs[idx + i] = thread_idxs[i];
        }
      }
      threadgroup_barrier(mem_flags::mem_threadgroup);
 
      // Find location in merge step
      int merge_group = lid.x / merge_threads;
      int merge_lane = lid.x % merge_threads;
 
      int sort_sz = N_PER_THREAD * merge_threads;
      int sort_st = N_PER_THREAD * merge_threads * merge_group;
 
      // As = tgp_vals[A_st:A_ed] is sorted
      // Bs = tgp_vals[B_st:B_ed] is sorted
      int A_st = sort_st;
      int A_ed = sort_st + sort_sz / 2;
      int B_st = sort_st + sort_sz / 2;
      int B_ed = sort_st + sort_sz;
 
      const threadgroup val_t* As = tgp_vals + A_st;
      const threadgroup val_t* Bs = tgp_vals + B_st;
      int A_sz = A_ed - A_st;
      int B_sz = B_ed - B_st;
 
      // Find a partition of merge elements
      //  Ci = merge(As[partition:], Bs[sort_md - partition:])
      //       of size N_PER_THREAD for each merge lane i
      //  C = [Ci] is sorted
      int sort_md = N_PER_THREAD * merge_lane;
      int partition = merge_partition(As, Bs, A_sz, B_sz, sort_md);
 
      As += partition;
      Bs += sort_md - partition;
 
      A_sz -= partition;
      B_sz -= sort_md - partition;
 
      const threadgroup idx_t* As_idx =
          ARG_SORT ? tgp_idxs + A_st + partition : nullptr;
      const threadgroup idx_t* Bs_idx =
          ARG_SORT ? tgp_idxs + B_st + sort_md - partition : nullptr;
 
      // Merge starting at the partition and store results in thread registers
      merge_step(As, Bs, As_idx, Bs_idx, A_sz, B_sz, thread_vals, thread_idxs);
    }
 
    // Write out to shared memory
    threadgroup_barrier(mem_flags::mem_threadgroup);
    for (int i = 0; i < N_PER_THREAD; ++i) {
      tgp_vals[idx + i] = thread_vals[i];
      if (ARG_SORT) {
        tgp_idxs[idx + i] = thread_idxs[i];
      }
    }
  }
};

// Kernel sort
 
template <
    typename T,
    typename U,
    bool ARG_SORT,
    short BLOCK_THREADS,
    short N_PER_THREAD,
    typename CompareOp = LessThan<T>>
struct KernelMergeSort {
  using val_t = T;
  using idx_t = uint;
  using block_merge_sort_t = BlockMergeSort<
      val_t,
      idx_t,
      ARG_SORT,
      BLOCK_THREADS,
      N_PER_THREAD,
      CompareOp>;
 
  MLX_MTL_CONST short N_PER_BLOCK = BLOCK_THREADS * N_PER_THREAD;
 
  static METAL_FUNC void block_sort(
      const device T* inp,
      device U* out,
      const constant int& size_sorted_axis,
      const constant int& in_stride_sorted_axis,
      const constant int& out_stride_sorted_axis,
      const constant int& in_stride_segment_axis,
      const constant int& out_stride_segment_axis,
      threadgroup val_t* tgp_vals,
      threadgroup idx_t* tgp_idxs,
      uint3 tid [[threadgroup_position_in_grid]],
      uint3 lid [[thread_position_in_threadgroup]]) {
    // tid.y tells us the segment index
    inp += tid.y * in_stride_segment_axis;
    out += tid.y * out_stride_segment_axis;
 
    // Copy into threadgroup memory
    for (short i = lid.x; i < N_PER_BLOCK; i += BLOCK_THREADS) {
      tgp_vals[i] = i < size_sorted_axis ? inp[i * in_stride_sorted_axis]
                                         : val_t(CompareOp::init);
      if (ARG_SORT) {
        tgp_idxs[i] = i;
      }
    }
 
    // Sort elements within the block
    threadgroup_barrier(mem_flags::mem_threadgroup);
 
    block_merge_sort_t::sort(tgp_vals, tgp_idxs, size_sorted_axis, lid);
 
    threadgroup_barrier(mem_flags::mem_threadgroup);
 
    // Write output
    for (int i = lid.x; i < size_sorted_axis; i += BLOCK_THREADS) {
      if (ARG_SORT) {
        out[i * out_stride_sorted_axis] = tgp_idxs[i];
      } else {
        out[i * out_stride_sorted_axis] = tgp_vals[i];
      }
    }
  }
};
 
template <
    typename T,
    typename U,
    bool ARG_SORT,
    short BLOCK_THREADS,
    short N_PER_THREAD>
[[kernel, max_total_threads_per_threadgroup(BLOCK_THREADS)]] void block_sort(
    const device T* inp [[buffer(0)]],
    device U* out [[buffer(1)]],
    const constant int& size_sorted_axis [[buffer(2)]],
    const constant int& in_stride_sorted_axis [[buffer(3)]],
    const constant int& out_stride_sorted_axis [[buffer(4)]],
    const constant int& in_stride_segment_axis [[buffer(5)]],
    const constant int& out_stride_segment_axis [[buffer(6)]],
    uint3 tid [[threadgroup_position_in_grid]],
    uint3 lid [[thread_position_in_threadgroup]]) {
  using sort_kernel =
      KernelMergeSort<T, U, ARG_SORT, BLOCK_THREADS, N_PER_THREAD>;
  using val_t = typename sort_kernel::val_t;
  using idx_t = typename sort_kernel::idx_t;
 
  if (ARG_SORT) {
    threadgroup val_t tgp_vals[sort_kernel::N_PER_BLOCK];
    threadgroup idx_t tgp_idxs[sort_kernel::N_PER_BLOCK];
    sort_kernel::block_sort(
        inp,
        out,
        size_sorted_axis,
        in_stride_sorted_axis,
        out_stride_sorted_axis,
        in_stride_segment_axis,
        out_stride_segment_axis,
        tgp_vals,
        tgp_idxs,
        tid,
        lid);
  } else {
    threadgroup val_t tgp_vals[sort_kernel::N_PER_BLOCK];
    sort_kernel::block_sort(
        inp,
        out,
        size_sorted_axis,
        in_stride_sorted_axis,
        out_stride_sorted_axis,
        in_stride_segment_axis,
        out_stride_segment_axis,
        tgp_vals,
        nullptr,
        tid,
        lid);
  }
}
 
constant constexpr const int zero_helper = 0;
 
template <
    typename T,
    typename U,
    bool ARG_SORT,
    short BLOCK_THREADS,
    short N_PER_THREAD>
[[kernel, max_total_threads_per_threadgroup(BLOCK_THREADS)]] void block_sort_nc(
    const device T* inp [[buffer(0)]],
    device U* out [[buffer(1)]],
    const constant int& size_sorted_axis [[buffer(2)]],
    const constant int& in_stride_sorted_axis [[buffer(3)]],
    const constant int& out_stride_sorted_axis [[buffer(4)]],
    const constant int& nc_dim [[buffer(5)]],
    const constant int* nc_shape [[buffer(6)]],
    const constant int64_t* in_nc_strides [[buffer(7)]],
    const constant int64_t* out_nc_strides [[buffer(8)]],
    uint3 tid [[threadgroup_position_in_grid]],
    uint3 lid [[thread_position_in_threadgroup]]) {
  using sort_kernel =
      KernelMergeSort<T, U, ARG_SORT, BLOCK_THREADS, N_PER_THREAD>;
  using val_t = typename sort_kernel::val_t;
  using idx_t = typename sort_kernel::idx_t;
 
  auto in_block_idx = elem_to_loc(tid.y, nc_shape, in_nc_strides, nc_dim);
  auto out_block_idx = elem_to_loc(tid.y, nc_shape, out_nc_strides, nc_dim);
  inp += in_block_idx;
  out += out_block_idx;
 
  if (ARG_SORT) {
    threadgroup val_t tgp_vals[sort_kernel::N_PER_BLOCK];
    threadgroup idx_t tgp_idxs[sort_kernel::N_PER_BLOCK];
    sort_kernel::block_sort(
        inp,
        out,
        size_sorted_axis,
        in_stride_sorted_axis,
        out_stride_sorted_axis,
        zero_helper,
        zero_helper,
        tgp_vals,
        tgp_idxs,
        tid,
        lid);
  } else {
    threadgroup val_t tgp_vals[sort_kernel::N_PER_BLOCK];
    sort_kernel::block_sort(
        inp,
        out,
        size_sorted_axis,
        in_stride_sorted_axis,
        out_stride_sorted_axis,
        zero_helper,
        zero_helper,
        tgp_vals,
        nullptr,
        tid,
        lid);
  }
}
 
template <
    typename val_t,
    typename idx_t,
    bool ARG_SORT,
    short BLOCK_THREADS,
    short N_PER_THREAD,
    typename CompareOp = LessThan<val_t>>
struct KernelMultiBlockMergeSort {
  using block_merge_sort_t = BlockMergeSort<
      val_t,
      idx_t,
      ARG_SORT,
      BLOCK_THREADS,
      N_PER_THREAD,
      CompareOp>;
 
  MLX_MTL_CONST short N_PER_BLOCK = BLOCK_THREADS * N_PER_THREAD;
 
  static METAL_FUNC void block_sort(
      const device val_t* inp,
      device val_t* out_vals,
      device idx_t* out_idxs,
      const constant int& size_sorted_axis,
      const constant int& stride_sorted_axis,
      threadgroup val_t* tgp_vals,
      threadgroup idx_t* tgp_idxs,
      uint3 tid [[threadgroup_position_in_grid]],
      uint3 lid [[thread_position_in_threadgroup]]) {
    // tid.y tells us the segment index
    int base_idx = tid.x * N_PER_BLOCK;
 
    // Copy into threadgroup memory
    for (short i = lid.x; i < N_PER_BLOCK; i += BLOCK_THREADS) {
      int idx = base_idx + i;
      tgp_vals[i] = idx < size_sorted_axis ? inp[idx * stride_sorted_axis]
                                           : val_t(CompareOp::init);
      tgp_idxs[i] = idx;
    }
 
    // Sort elements within the block
    threadgroup_barrier(mem_flags::mem_threadgroup);
 
    block_merge_sort_t::sort(tgp_vals, tgp_idxs, size_sorted_axis, lid);
 
    threadgroup_barrier(mem_flags::mem_threadgroup);
 
    // Write output
    for (int i = lid.x; i < N_PER_BLOCK; i += BLOCK_THREADS) {
      int idx = base_idx + i;
      if (idx < size_sorted_axis) {
        out_vals[idx] = tgp_vals[i];
        out_idxs[idx] = tgp_idxs[i];
      }
    }
  }
 
  static METAL_FUNC int merge_partition(
      const device val_t* As,
      const device val_t* Bs,
      int A_sz,
      int B_sz,
      int sort_md) {
    CompareOp op;
 
    int A_st = max(0, sort_md - B_sz);
    int A_ed = min(sort_md, A_sz);
 
    while (A_st < A_ed) {
      int md = A_st + (A_ed - A_st) / 2;
      auto a = As[md];
      auto b = Bs[sort_md - 1 - md];
 
      if (op(b, a)) {
        A_ed = md;
      } else {
        A_st = md + 1;
      }
    }
 
    return A_ed;
  }
};
 
template <
    typename val_t,
    typename idx_t,
    bool ARG_SORT,
    short BLOCK_THREADS,
    short N_PER_THREAD>
[[kernel, max_total_threads_per_threadgroup(BLOCK_THREADS)]] void mb_block_sort(
    const device val_t* inp [[buffer(0)]],
    device val_t* out_vals [[buffer(1)]],
    device idx_t* out_idxs [[buffer(2)]],
    const constant int& size_sorted_axis [[buffer(3)]],
    const constant int& stride_sorted_axis [[buffer(4)]],
    const constant int& nc_dim [[buffer(5)]],
    const constant int* nc_shape [[buffer(6)]],
    const constant int64_t* nc_strides [[buffer(7)]],
    uint3 tid [[threadgroup_position_in_grid]],
    uint3 lid [[thread_position_in_threadgroup]]) {
  using sort_kernel = KernelMultiBlockMergeSort<
      val_t,
      idx_t,
      ARG_SORT,
      BLOCK_THREADS,
      N_PER_THREAD>;
 
  auto block_idx = elem_to_loc(tid.y, nc_shape, nc_strides, nc_dim);
  inp += block_idx;
  out_vals += tid.y * size_sorted_axis;
  out_idxs += tid.y * size_sorted_axis;
 
  threadgroup val_t tgp_vals[sort_kernel::N_PER_BLOCK];
  threadgroup idx_t tgp_idxs[sort_kernel::N_PER_BLOCK];
 
  sort_kernel::block_sort(
      inp,
      out_vals,
      out_idxs,
      size_sorted_axis,
      stride_sorted_axis,
      tgp_vals,
      tgp_idxs,
      tid,
      lid);
}
 
template <
    typename val_t,
    typename idx_t,
    bool ARG_SORT,
    short BLOCK_THREADS,
    short N_PER_THREAD>
[[kernel]] void mb_block_partition(
    device idx_t* block_partitions [[buffer(0)]],
    const device val_t* dev_vals [[buffer(1)]],
    const device idx_t* dev_idxs [[buffer(2)]],
    const constant int& size_sorted_axis [[buffer(3)]],
    const constant int& merge_tiles [[buffer(4)]],
    const constant int& n_blocks [[buffer(5)]],
    uint3 tid [[threadgroup_position_in_grid]],
    uint3 lid [[thread_position_in_threadgroup]],
    uint3 tgp_dims [[threads_per_threadgroup]]) {
  using sort_kernel = KernelMultiBlockMergeSort<
      val_t,
      idx_t,
      ARG_SORT,
      BLOCK_THREADS,
      N_PER_THREAD>;
 
  block_partitions += tid.y * tgp_dims.x;
  dev_vals += tid.y * size_sorted_axis;
  dev_idxs += tid.y * size_sorted_axis;
 
  for (int i = lid.x; i <= n_blocks; i += tgp_dims.x) {
    // Find location in merge step
    int merge_group = i / merge_tiles;
    int merge_lane = i % merge_tiles;
 
    int sort_sz = sort_kernel::N_PER_BLOCK * merge_tiles;
    int sort_st = sort_kernel::N_PER_BLOCK * merge_tiles * merge_group;
 
    int A_st = min(size_sorted_axis, sort_st);
    int A_ed = min(size_sorted_axis, sort_st + sort_sz / 2);
    int B_st = A_ed;
    int B_ed = min(size_sorted_axis, B_st + sort_sz / 2);
 
    int partition_at = min(B_ed - A_st, sort_kernel::N_PER_BLOCK * merge_lane);
    int partition = sort_kernel::merge_partition(
        dev_vals + A_st,
        dev_vals + B_st,
        A_ed - A_st,
        B_ed - B_st,
        partition_at);
 
    block_partitions[i] = A_st + partition;
  }
}
 
template <
    typename val_t,
    typename idx_t,
    bool ARG_SORT,
    short BLOCK_THREADS,
    short N_PER_THREAD,
    typename CompareOp = LessThan<val_t>>
[[kernel, max_total_threads_per_threadgroup(BLOCK_THREADS)]] void
mb_block_merge(
    const device idx_t* block_partitions [[buffer(0)]],
    const device val_t* dev_vals_in [[buffer(1)]],
    const device idx_t* dev_idxs_in [[buffer(2)]],
    device val_t* dev_vals_out [[buffer(3)]],
    device idx_t* dev_idxs_out [[buffer(4)]],
    const constant int& size_sorted_axis [[buffer(5)]],
    const constant int& merge_tiles [[buffer(6)]],
    const constant int& num_tiles [[buffer(7)]],
    uint3 tid [[threadgroup_position_in_grid]],
    uint3 lid [[thread_position_in_threadgroup]]) {
  using sort_kernel = KernelMultiBlockMergeSort<
      val_t,
      idx_t,
      ARG_SORT,
      BLOCK_THREADS,
      N_PER_THREAD,
      CompareOp>;
 
  using block_sort_t = typename sort_kernel::block_merge_sort_t;
 
  block_partitions += tid.y * (num_tiles + 1);
  dev_vals_in += tid.y * size_sorted_axis;
  dev_idxs_in += tid.y * size_sorted_axis;
  dev_vals_out += tid.y * size_sorted_axis;
  dev_idxs_out += tid.y * size_sorted_axis;
 
  int block_idx = tid.x;
  int merge_group = block_idx / merge_tiles;
  int sort_st = sort_kernel::N_PER_BLOCK * merge_tiles * merge_group;
  int sort_sz = sort_kernel::N_PER_BLOCK * merge_tiles;
  int sort_md = sort_kernel::N_PER_BLOCK * block_idx - sort_st;
 
  int A_st = block_partitions[block_idx + 0];
  int A_ed = block_partitions[block_idx + 1];
  int B_st = min(size_sorted_axis, 2 * sort_st + sort_sz / 2 + sort_md - A_st);
  int B_ed = min(
      size_sorted_axis,
      2 * sort_st + sort_sz / 2 + sort_md + sort_kernel::N_PER_BLOCK - A_ed);
 
  if ((block_idx % merge_tiles) == merge_tiles - 1) {
    A_ed = min(size_sorted_axis, sort_st + sort_sz / 2);
    B_ed = min(size_sorted_axis, sort_st + sort_sz);
  }
 
  int A_sz = A_ed - A_st;
  int B_sz = B_ed - B_st;
 
  // Load from global memory
  thread val_t thread_vals[N_PER_THREAD];
  thread idx_t thread_idxs[N_PER_THREAD];
  for (int i = 0; i < N_PER_THREAD; i++) {
    int idx = BLOCK_THREADS * i + lid.x;
    if (idx < (A_sz + B_sz)) {
      thread_vals[i] = (idx < A_sz) ? dev_vals_in[A_st + idx]
                                    : dev_vals_in[B_st + idx - A_sz];
      thread_idxs[i] = (idx < A_sz) ? dev_idxs_in[A_st + idx]
                                    : dev_idxs_in[B_st + idx - A_sz];
    } else {
      thread_vals[i] = CompareOp::init;
      thread_idxs[i] = 0;
    }
  }
 
  // Write to shared memory
  threadgroup val_t tgp_vals[sort_kernel::N_PER_BLOCK];
  threadgroup idx_t tgp_idxs[sort_kernel::N_PER_BLOCK];
  threadgroup_barrier(mem_flags::mem_threadgroup);
  for (int i = 0; i < N_PER_THREAD; i++) {
    int idx = BLOCK_THREADS * i + lid.x;
    tgp_vals[idx] = thread_vals[i];
    tgp_idxs[idx] = thread_idxs[i];
  }
  threadgroup_barrier(mem_flags::mem_threadgroup);
 
  // Merge
  int sort_md_local = min(A_sz + B_sz, N_PER_THREAD * int(lid.x));
 
  int A_st_local = block_sort_t::merge_partition(
      tgp_vals, tgp_vals + A_sz, A_sz, B_sz, sort_md_local);
  int A_ed_local = A_sz;
 
  int B_st_local = sort_md_local - A_st_local;
  int B_ed_local = B_sz;
 
  int A_sz_local = A_ed_local - A_st_local;
  int B_sz_local = B_ed_local - B_st_local;
 
  // Do merge
  block_sort_t::merge_step(
      tgp_vals + A_st_local,
      tgp_vals + A_ed_local + B_st_local,
      tgp_idxs + A_st_local,
      tgp_idxs + A_ed_local + B_st_local,
      A_sz_local,
      B_sz_local,
      thread_vals,
      thread_idxs);
 
  threadgroup_barrier(mem_flags::mem_threadgroup);
  for (int i = 0; i < N_PER_THREAD; ++i) {
    int idx = lid.x * N_PER_THREAD;
    tgp_vals[idx + i] = thread_vals[i];
    tgp_idxs[idx + i] = thread_idxs[i];
  }
 
  threadgroup_barrier(mem_flags::mem_threadgroup);
  // Write output
  int base_idx = tid.x * sort_kernel::N_PER_BLOCK;
  for (int i = lid.x; i < sort_kernel::N_PER_BLOCK; i += BLOCK_THREADS) {
    int idx = base_idx + i;
    if (idx < size_sorted_axis) {
      dev_vals_out[idx] = tgp_vals[i];
      dev_idxs_out[idx] = tgp_idxs[i];
    }
  }
}