fn dot_product[
    in_layout: Layout, out_layout: Layout
](
    out: LayoutTensor[mut=True, dtype, out_layout],
    a: LayoutTensor[mut=True, dtype, in_layout],
    b: LayoutTensor[mut=True, dtype, in_layout],
    size: Int,
):
    shared = tb[dtype]().row_major[TPB]().shared().alloc()
    global_i = block_dim.x * block_idx.x + thread_idx.x
    local_i = thread_idx.x

    # Compute element-wise multiplication into shared memory
    if global_i < size:
        shared[local_i] = a[global_i] * b[global_i]

    # Synchronize threads within block
    barrier()

    # Parallel reduction in shared memory
    stride = TPB // 2
    while stride > 0:
        if local_i < stride:
            shared[local_i] += shared[local_i + stride]

        barrier()
        stride //= 2

    # Only thread 0 writes the final result
    if local_i == 0:
        out[0] = shared[0]

The solution implements a parallel reduction for dot product using LayoutTensor. Here’s the detailed breakdown:

Phase 1: Element-wise Multiplication

Each thread performs one multiplication with natural indexing:

shared[local_i] = a[global_i] * b[global_i]

Phase 2: Parallel Reduction

Tree-based reduction with layout-aware operations:

Initial:  [0*0  1*1  2*2  3*3  4*4  5*5  6*6  7*7]
        = [0    1    4    9    16   25   36   49]

Step 1:   [0+16 1+25 4+36 9+49  16   25   36   49]
        = [16   26   40   58   16   25   36   49]

Step 2:   [16+40 26+58 40   58   16   25   36   49]
        = [56   84   40   58   16   25   36   49]

Step 3:   [56+84  84   40   58   16   25   36   49]
        = [140   84   40   58   16   25   36   49]

Key Implementation Features:

1. Memory Management:

   • Clean shared memory allocation with tensor builder
   • Type-safe operations with LayoutTensor
   • Automatic bounds checking
   • Layout-aware indexing

2. Thread Synchronization:

   • barrier() after initial multiplication
   • barrier() between reduction steps
   • Safe thread coordination

3. Reduction Logic:

   stride = TPB // 2
   while stride > 0:
       if local_i < stride:
           shared[local_i] += shared[local_i + stride]
       barrier()
       stride //= 2
   

4. Performance Benefits:

   • \(O(\log n)\) time complexity
   • Coalesced memory access
   • Minimal thread divergence
   • Efficient shared memory usage

The LayoutTensor version maintains the same efficient parallel reduction while providing:

• Better type safety
• Cleaner memory management
• Layout awareness
• Natural indexing syntax

Barrier Synchronization Importance

The barrier() between reduction steps is critical for correctness. Here’s why:

Without barrier(), race conditions occur:

Initial shared memory: [0 1 4 9 16 25 36 49]

Step 1 (stride = 4):
Thread 0 reads: shared[0] = 0, shared[4] = 16
Thread 1 reads: shared[1] = 1, shared[5] = 25
Thread 2 reads: shared[2] = 4, shared[6] = 36
Thread 3 reads: shared[3] = 9, shared[7] = 49

Without barrier:
- Thread 0 writes: shared[0] = 0 + 16 = 16
- Thread 1 starts next step (stride = 2) before Thread 0 finishes
  and reads old value shared[0] = 0 instead of 16!

With barrier():

Step 1 (stride = 4):
All threads write their sums:
[16 26 40 58 16 25 36 49]
barrier() ensures ALL threads see these values

Step 2 (stride = 2):
Now threads safely read the updated values:
Thread 0: shared[0] = 16 + 40 = 56
Thread 1: shared[1] = 26 + 58 = 84

The barrier() ensures:

1. All writes from current step complete
2. All threads see updated values
3. No thread starts next iteration early
4. Consistent shared memory state

Without these synchronization points, we could get:

• Memory race conditions
• Threads reading stale values
• Non-deterministic results
• Incorrect final sum