Stages

The conventional wisdom is that the stages of a task should be pipelined, so you don’t need to wait for the completion of one stage before the next is started. It surprises me that it seems you can sometimes do better when performing each stage of a pipeline in a short batch. Useful optimisation opportunities can arise from this phenomenon, with only minor code changes. I recently applied this principle to implement fast batch iterators in RoaringBitmap.

I came across a discussion about shuffling arrays on Twitter, stimulated by a blog post by Daniel Lemire. Imagine you want to randomly shuffle the contents of an array. One approach to take would be to iterate over the array in reverse, at each index i, generate a random index j smaller than i, and swap the elements at i and j. Here’s some benchmark code to measure how long this takes for an assortment of swapping strategies, including one where the swaps are just precomputed and looked up in an array.


  @Benchmark
  public void shuffle(Blackhole bh) {
    for (int i = data.length; i > 1; i--)
      swap(data, i - 1, op.applyAsInt(i));
    bh.consume(data);
  }

  private static void swap(int[] arr, int i, int j) {
    arr[i] ^= arr[j];
    arr[j] ^= arr[i];
    arr[i] ^= arr[j];
  }

There is a large difference between the version where the random swap is precomputed and when the swap is computed on the fly with ThreadLocalRandom.nextInt.

Benchmark Mode Threads Samples Score Score Error (99.9%) Unit Param: mode Param: size
shuffle thrpt 1 10 2198.459182 274.965189 ops/s THREAD_LOCAL_RANDOM 65536
shuffle thrpt 1 10 1015.796005 16.225480 ops/s THREAD_LOCAL_RANDOM 131072
shuffle thrpt 1 10 7300.732038 46.788234 ops/s PRECOMPUTED 65536
shuffle thrpt 1 10 3828.021096 450.874537 ops/s PRECOMPUTED 131072

The difference is large, but a lot more work is being done when the random indices are computed on the fly. A good measure of efficiency per unit of work is cycles per instruction (CPI). Running the benchmark with -prof perfnorm shows that these benchmarks are at parity for cycles per instruction: if throughput is lower when the random numbers are generated on the fly, it’s because there are more instructions to execute.

Benchmark Mode Threads Samples Score Score Error (99.9%) Unit Param: mode Param: size
shuffle:CPI thrpt 1 1 0.427028 NaN #/op THREAD_LOCAL_RANDOM 65536
shuffle:CPI thrpt 1 1 0.447793 NaN #/op THREAD_LOCAL_RANDOM 131072
shuffle:CPI thrpt 1 1 0.477202 NaN #/op PRECOMPUTED 65536
shuffle:CPI thrpt 1 1 0.565153 NaN #/op PRECOMPUTED 131072

Nevertheless, instruction profiling with -prof perfasm shows that the code is qualitatively different when computing the next swapped index is simple. When there is random number generation to do, most of the time is attributed either to mov or just after mov instructions (probably because of profiling skid) during the swap. For example, with the smaller array:


  0.04%    0.00%  │   ││  0x00007fa009c0a8f9: xor    0x10(%rsi,%rdx,4),%r10d  
 15.31%   13.18%  │   ││  0x00007fa009c0a8fe: mov    %r10d,0xc(%rsi,%rcx,4)  
  3.43%    3.05%  │   ││  0x00007fa009c0a903: xor    0x10(%rsi,%rdx,4),%r10d  
  5.37%    5.92%  │   ││  0x00007fa009c0a908: mov    %r10d,0x10(%rsi,%rdx,4)  
  4.15%    4.22%  │   ││  0x00007fa009c0a90d: xor    %r10d,0xc(%rsi,%rcx,4)  
 10.80%    8.80%  │   ││  0x00007fa009c0a912: cmp    $0x1,%r9d ; probably skid

The key difference in the precomputed case is that the loop is unrolled with several isomorphic chains of instructions. None of the loads seem to be quite so expensive according to the sampled frequencies.


  0.08%    0.16%  │      0x00007fda2dc0dfb2: cmp    %r10d,%r9d
                  │      0x00007fda2dc0dfb5: jae    0x00007fda2dc0e264
  0.00%    0.00%  │      0x00007fda2dc0dfbb: xor    0x10(%rdx,%r9,4),%edi
  2.90%    2.89%  │      0x00007fda2dc0dfc0: mov    %edi,0xc(%rdx,%r11,4)
  0.48%    0.33%  │      0x00007fda2dc0dfc5: xor    0x10(%rdx,%r9,4),%edi
  0.45%    0.48%  │      0x00007fda2dc0dfca: mov    %edi,0x10(%rdx,%r9,4)
  0.56%    0.46%  │      0x00007fda2dc0dfcf: xor    %edi,0xc(%rdx,%r11,4)
  4.29%    3.88%  │      0x00007fda2dc0dfd4: mov    0x8(%rdx,%r11,4),%edi
  0.03%    0.01%  │      0x00007fda2dc0dfd9: mov    0x8(%rsi,%r11,4),%r9d
  1.38%    1.46%  │      0x00007fda2dc0dfde: mov    %r11d,%ebx
  0.02%    0.01%  │      0x00007fda2dc0dfe1: add    $0xfffffffe,%ebx   

  0.63%    0.61%  │      0x00007fda2dc0dfe4: cmp    %r10d,%r9d
                  │      0x00007fda2dc0dfe7: jae    0x00007fda2dc0e26f
  0.00%    0.01%  │      0x00007fda2dc0dfed: xor    0x10(%rdx,%r9,4),%edi
  2.60%    2.38%  │      0x00007fda2dc0dff2: mov    %edi,0x8(%rdx,%r11,4)
  0.58%    0.51%  │      0x00007fda2dc0dff7: xor    0x10(%rdx,%r9,4),%edi
  0.90%    0.96%  │      0x00007fda2dc0dffc: mov    %edi,0x10(%rdx,%r9,4)
  0.68%    0.66%  │      0x00007fda2dc0e001: xor    %edi,0x8(%rdx,%r11,4)
  4.85%    4.17%  │      0x00007fda2dc0e006: mov    0x4(%rdx,%r11,4),%edi
  0.01%    0.02%  │      0x00007fda2dc0e00b: mov    0x4(%rsi,%r11,4),%r9d
  1.12%    0.95%  │      0x00007fda2dc0e010: mov    %r11d,%ecx
  0.01%    0.00%  │      0x00007fda2dc0e013: add    $0xfffffffd,%ecx  

  1.02%    1.02%  │      0x00007fda2dc0e016: cmp    %r10d,%r9d
                  │      0x00007fda2dc0e019: jae    0x00007fda2dc0e267
  0.01%    0.01%  │      0x00007fda2dc0e01f: xor    0x10(%rdx,%r9,4),%edi
  2.47%    2.10%  │      0x00007fda2dc0e024: mov    %edi,0x4(%rdx,%r11,4)
  0.69%    0.57%  │      0x00007fda2dc0e029: xor    0x10(%rdx,%r9,4),%edi
  1.37%    1.50%  │      0x00007fda2dc0e02e: mov    %edi,0x10(%rdx,%r9,4)
  0.77%    0.83%  │      0x00007fda2dc0e033: xor    %edi,0x4(%rdx,%r11,4)
  4.28%    3.85%  │      0x00007fda2dc0e038: mov    (%rdx,%r11,4),%edi
  0.03%    0.02%  │      0x00007fda2dc0e03c: mov    (%rsi,%r11,4),%r9d
  1.14%    0.97%  │      0x00007fda2dc0e040: mov    %r11d,%ebx
  0.01%    0.00%  │      0x00007fda2dc0e043: add    $0xfffffffc,%ebx  

With unrolling, some of each chain can take place concurrently, and if there is a cache miss in one chain, it won’t stall the progress of the other chains. Without this capacity for parallelism, a cache miss during the swap will stop all work from progressing. As the probability of a cache miss increases, the cost of the load bottleneck in the swap should increase: this can be stressed by increasing the size of the array. With a large (100M) array, there’s a good chance of a cache miss virtually all the time. CPI increases in both cases, markedly so with the precomputed swaps, but throughput converges: access to main memory has become the bottleneck.

Benchmark Mode Threads Samples Score Score Error (99.9%) Unit Param: mode Param: size
shuffle:CPI thrpt 1 1 1.354325 NaN #/op THREAD_LOCAL_RANDOM 100000000
shuffle:CPI thrpt 1 1 3.854150 NaN #/op PRECOMPUTED 100000000

The perfasm output points to the first load in the generated swap as the bottleneck: notice the large cost on the mov instruction.


  0.15%    0.24%  │      ││  0x00007f8405c0a264: cmp    %r9d,%edx
                  │      ││  0x00007f8405c0a267: jae    0x00007f8405c0a350
  0.10%    0.11%  │      ││  0x00007f8405c0a26d: xor    0x10(%r11,%rdx,4),%eax  
 73.97%   63.58%  │      ││  0x00007f8405c0a272: mov    %eax,0xc(%r11,%rcx,4)  
  2.46%    1.87%  │      ││  0x00007f8405c0a277: xor    0x10(%r11,%rdx,4),%eax 
  1.42%    0.67%  │      ││  0x00007f8405c0a27c: mov    %eax,0x10(%r11,%rdx,4) 
  2.19%    1.44%  │      ││  0x00007f8405c0a281: xor    %eax,0xc(%r11,%rcx,4) 
  2.16%    1.37%  │      ││  0x00007f8405c0a286: cmp    $0x1,%edi

With precomputed swaps, there is no single bottleneck, and my intuition is that there is some concurrency, despite the higher CPI. This is a long way from being proven.


 10.33%   11.23%   ││  0x00007fdb35c09250: mov    %r9d,0xc(%rsi,%r10,4)  
  0.41%    0.45%   ││  0x00007fdb35c09255: xor    0x10(%rsi,%r11,4),%r9d  
  0.36%    0.25%   ││  0x00007fdb35c0925a: mov    %r9d,0x10(%rsi,%r11,4)  
  0.42%    0.42%   ││  0x00007fdb35c0925f: xor    %r9d,0xc(%rsi,%r10,4)  
  0.51%    0.66%   ││  0x00007fdb35c09264: mov    0x8(%rsi,%r10,4),%r9d  
  0.03%    0.09%   ││  0x00007fdb35c09269: mov    0x8(%r13,%r10,4),%r11d 
  0.25%    0.20%   ││  0x00007fdb35c0926e: mov    %r10d,%r8d
  0.03%    0.15%   ││  0x00007fdb35c09271: add    $0xfffffffe,%r8d  
  0.19%    0.17%   ││  0x00007fdb35c09275: cmp    %ebx,%r11d
                   ││  0x00007fdb35c09278: jae    0x00007fdb35c09440
  0.02%    0.06%   ││  0x00007fdb35c0927e: xor    0x10(%rsi,%r11,4),%r9d  
 10.40%   10.66%   ││  0x00007fdb35c09283: mov    %r9d,0x8(%rsi,%r10,4) 
  0.41%    0.35%   ││  0x00007fdb35c09288: xor    0x10(%rsi,%r11,4),%r9d 
  0.41%    0.30%   ││  0x00007fdb35c0928d: mov    %r9d,0x10(%rsi,%r11,4) 
  0.45%    0.39%   ││  0x00007fdb35c09292: xor    %r9d,0x8(%rsi,%r10,4)  
  0.48%    0.60%   ││  0x00007fdb35c09297: mov    0x4(%rsi,%r10,4),%r9d  
  0.03%    0.06%   ││  0x00007fdb35c0929c: mov    0x4(%r13,%r10,4),%r11d 
  0.06%    0.11%   ││  0x00007fdb35c092a1: mov    %r10d,%edi
  0.02%    0.16%   ││  0x00007fdb35c092a4: add    $0xfffffffd,%edi   
  0.25%    0.20%   ││  0x00007fdb35c092a7: cmp    %ebx,%r11d

This can be exploited so the random numbers can be generated on the fly without a single bottleneck by using a hybrid approach. The random swaps can be generated on the fly and written into a small buffer. Once the buffer is full, the swaps are done. This should “decouple” the random number generation from the swapping code, and should allow some of the swaps to be performed independently. Concretely:


  @Benchmark
  public void shuffleBuffered(Blackhole bh) {
    for (int i = data.length; i - unroll > 1; i -= unroll) {
      for (int j = 0; j < buffer.length; ++j) {
        buffer[j] = op.applyAsInt(i - j);
      }
      for (int j = 0; j < buffer.length; ++j) {
        swap(data, i - j - 1, buffer[j]);
      }
    }
    bh.consume(data);
  }

There’s not much to be gained (or lost) from this until the array gets quite large, but it’s a relatively interesting outcome. CPI is on the whole improved, and throughput improves as a function of buffer size, so long as the buffer is small.

Mode Benchmark 16 32 64 128 256
PRECOMPUTED shuffle 0.30639 0.296566 0.309829 0.312449 0.311183
PRECOMPUTED shuffle:CPI 3.004183 3.126903 2.989748 2.987508 3.000369
THREAD_LOCAL_RANDOM shuffle 0.271536 0.266418 0.271858 0.265593 0.264507
THREAD_LOCAL_RANDOM shuffle:CPI 1.303454 1.328127 1.300731 1.32857 1.377559
THREAD_LOCAL_RANDOM shuffleBuffered 0.296098 0.324416 0.346934 0.353246 0.35277
THREAD_LOCAL_RANDOM shuffleBuffered:CPI 0.96738 0.937101 0.893673 0.87786 0.874607

Frankly, I can’t think of anything less interesting than scrambling an array, but this observation made me think there may be something in the idea of decoupling stages of work in choice scenarios. After going through the otherwise pointless exercise above, I decided it might be worthwhile spending some time porting a batch iteration feature implemented in the Go version of RoaringBitmap to Java. Confident in the potential advantages of batching, I decided to try porting the feature to Java. This idea turned out to be hugely profitable, speeding up iteration between 2x and 10x. If you use RoaringBitmap, it might be worth switching to these new batch iterators.

This topic is explored in more depth by Daniel Lemire in Fast Random Integer Generation in an Interval.