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Parallel Sorting

Sorting is the canonical parallel-algorithms case study because it is the simplest task where the naive parallelization loses to the sequential library. This page walks the theory (work/span), the two algorithms that actually matter (parallel merge and sample sort), and a measured implementation using shared memory - ending with the honest observation that in CPython, beating sorted() is nearly impossible, and why that fact generalizes to every language with a C-tuned standard library.

Work and Span: The Vocabulary

For a parallel algorithm, count two costs: work T1 (total operations, what one processor does) and span T-infinity (the longest dependency chain, what infinite processors do). Parallelism is T1 / T-infinity; with P processors, runtime is bounded below by both T1/P and T-infinity (this pairing is Brent’s theorem in scheduling terms). A good parallel sort has work close to sequential O(n log n) and span far below it.

Parallel mergesort: splitting is trivial; the merge is the problem. A sequential merge has span O(n) - the entire last level serializes - so total span stays O(n), work/span parallelism is a feeble O(log n), and nothing scales. The fix is parallel merging: to merge sorted halves A and B, take the median position in A, binary-search its insertion point in B, and recurse on both halves independently (two subproblems with disjoint output ranges). This drops merge span to O(log^2 n), giving:

parallel mergesort:  T1 = O(n log n)   T-inf = O(log^2 n)   parallelism = O(n / log^2 n)

For n = 4 million, that predicts parallelism in the tens of thousands - far more than any real core count, meaning mergesort is latency-optimal in theory. Theory’s favorite algorithm, however, is nobody’s production favorite, and the next section explains why.

Sample Sort: What Production Actually Uses

Parallel mergesort’s weakness is memory behavior and load balance on real hardware: merges have poor locality, and fixed splitting assumes uniform costs. Sample sort replaces merging with partitioning:

            input (any order)
                 |
   [p0] [p1] [p2] [p3]   pivots from global samples
   /      |     |     \
 bucket0 bucket1 bucket2 bucket3     every element of bucket i <= every
   |      |      |      |            element of bucket i+1
 local  local  local  local
 sort   sort   sort   sort
   \      |     |     /
        concatenate (already globally ordered)

The mechanics: (1) each worker samples O(p log n) elements from its chunk; (2) a common set of p-1 pivots (splitters) is chosen from the pooled samples; (3) every worker partitions its chunk into p buckets by binary search; (4) bucket i of all workers is concatenated and sorted - possibly in parallel - by whoever owns bucket i. Because buckets are pivot-delimited, concatenating the sorted buckets yields the globally sorted array with no merge step at all.

Balance comes from the sampling: with random samples, bucket sizes concentrate around n/p (Chernoff-style tails), so no worker gets a whale bucket. This is why sample sort, not mergesort, is what PBBS benchmarks, C++ parallel-mode extensions, and Spark’s sortByKey (RangePartitioner - literally sampled range bounds) all use. Radix sort variants compete when keys are fixed-width integers (GPU sorting is almost entirely radix for this reason - no comparisons, perfect coalescing), but comparison-based sample sort is the general-purpose answer.

A Real Implementation: Shared Memory or Bust

The engineering lesson that separates a toy from production code: the data must not move. A naive Python multiprocessing sort that pickles 4M-element chunks to workers and pickles results back spends more time serializing than sorting. The version below uses multiprocessing.shared_memory: workers generate, count, scatter, and sort in place, and only control metadata (samples, counts, offsets) crosses process boundaries:

# 4-process sample sort of 4M floats via multiprocessing.shared_memory.
# All payload data lives in shared memory; IPC carries only samples/counts.
# The sequential "pure-python pipeline" does identical phases in one process.
import multiprocessing as mp, random, statistics, time

pivots = [0.25, 0.5, 0.75]     # replaced after phase 1
_SHM = {}

def _shm_init(name_in, name_out, n):
    from multiprocessing import shared_memory
    _SHM['in'] = shared_memory.SharedMemory(name=name_in)
    _SHM['out'] = shared_memory.SharedMemory(name=name_out)
    _SHM['in_buf'] = _SHM['in'].buf.cast('d')
    _SHM['out_buf'] = _SHM['out'].buf.cast('d')

def _sample_segment(args):
    lo, hi = args
    return _SHM['in_buf'][lo:hi: max(1, (hi - lo) // 256)].tolist()

def _gen_into_shm(args):
    seed, lo, hi = args
    rng = random.Random(seed)
    buf = _SHM['in_buf']
    for i in range(lo, hi):
        buf[i] = rng.random()

def _count_buckets(args):
    lo, hi, pivots = args
    import bisect
    buf = _SHM['in_buf']
    counts = [0] * (len(pivots) + 1)
    for i in range(lo, hi):
        counts[bisect.bisect_left(pivots, buf[i])] += 1
    return counts

def _scatter(args):
    lo, hi, pivots, offsets = args
    import bisect
    pos = list(offsets)
    ib, ob = _SHM['in_buf'], _SHM['out_buf']
    for i in range(lo, hi):
        b = bisect.bisect_left(pivots, ib[i])
        ob[pos[b]] = ib[i]
        pos[b] += 1

def _sort_segment(args):
    import array
    lo, hi = args
    ob = _SHM['out_buf']
    seg = array.array('d', sorted(ob[lo:hi]))
    ob[lo:hi] = seg

def sort_demo():
    from multiprocessing import shared_memory
    n = 4_000_000
    procs = 4
    chunk = n // procs
    shm_in = shared_memory.SharedMemory(create=True, size=n * 8)
    shm_out = shared_memory.SharedMemory(create=True, size=n * 8)
    def pure_python_pipeline(pv):
        import bisect
        data = [0.0] * n
        for s in range(procs):
            rng = random.Random(s)
            data[s * chunk:(s + 1) * chunk] = [rng.random() for _ in range(chunk)]
        counts = [0] * procs
        for x in data:
            counts[bisect.bisect_left(pv, x)] += 1
        pos = [sum(counts[:b]) for b in range(procs)]
        out = [0.0] * n
        for x in data:
            b = bisect.bisect_left(pv, x)
            out[pos[b]] = x
            pos[b] += 1
        for b in range(procs):
            s = pos[b] - counts[b]
            out[s:s + counts[b]] = sorted(out[s:s + counts[b]])
        return out
    try:
        t0 = time.perf_counter()
        with mp.Pool(procs, initializer=_shm_init,
                     initargs=(shm_in.name, shm_out.name, n)) as pool:
            pool.map(_gen_into_shm, [(s, s * chunk, (s + 1) * chunk) for s in range(procs)])
            data = list(shm_in.buf.cast('d'))
            pieces = pool.map(_sample_segment, [(s * chunk, (s + 1) * chunk) for s in range(procs)])
            samples = sorted(x for p in pieces for x in p)
            pivots = [samples[k2 * len(samples) // procs] for k2 in range(1, procs)]
            count_lists = pool.map(_count_buckets,
                                   [(s * chunk, (s + 1) * chunk, pivots) for s in range(procs)])
            bucket_sizes = [sum(count_lists[s][b] for s in range(procs)) for b in range(procs)]
            bucket_start = [sum(bucket_sizes[:b]) for b in range(procs)]
            # worker s writes bucket b at bucket_start[b] + same-bucket counts of
            # EARLIER workers (transposing this into earlier buckets is the bug)
            offs = [[bucket_start[b] + sum(count_lists[s2][b] for s2 in range(s))
                     for b in range(procs)] for s in range(procs)]
            pool.map(_scatter, [(s * chunk, (s + 1) * chunk, pivots, offs[s])
                                for s in range(procs)])
            pool.map(_sort_segment, [(bucket_start[b], bucket_start[b] + bucket_sizes[b])
                                     for b in range(procs)])
            par = list(shm_out.buf.cast('d'))
        t_par = time.perf_counter() - t0
        t1 = time.perf_counter(); seq = sorted(data); t_seq_c = time.perf_counter() - t1
        t1 = time.perf_counter(); seq2 = pure_python_pipeline(pivots); t_seq_py = time.perf_counter() - t1
        print(f"n={n:,} procs={procs} (shared memory, zero payload IPC)")
        print(f"sequential C sorted()           : {t_seq_c:6.3f} s")
        print(f"sequential pure-python pipeline : {t_seq_py:6.3f} s")
        print(f"parallel pipeline (4 workers)   : {t_par:6.3f} s")
        print(f"speedup vs pure-python pipeline : {t_seq_py / t_par:5.2f}x")
        print(f"speedup vs C sorted()           : {t_seq_c / t_par:5.2f}x  <- Amdahl in action")
        print("par==seq:", par == seq, "| seq2==seq:", seq2 == seq)
    finally:
        shm_in.close(); shm_in.unlink()
        shm_out.close(); shm_out.unlink()

if True:
    sort_demo()

Illustrative output (measured on a 4-core container). Unlike the deterministic demos elsewhere in this book, the wall-clock lines below are machine-dependent — absolute times and even the speedup ratios will differ on your hardware; the byte-exact reproducible line is the final correctness check, par==seq: True | seq2==seq: True:

n=4,000,000 procs=4 (shared memory, zero payload IPC)
sequential C sorted()           :  1.197 s   (illustrative)
sequential pure-python pipeline :  2.935 s   (illustrative)
parallel pipeline (4 workers)   :  2.604 s   (illustrative)
speedup vs pure-python pipeline :  1.13x     (illustrative)
speedup vs C sorted()           :  0.46x     (illustrative)  <- Amdahl in action
par==seq: True | seq2==seq: True            <- deterministic, reproducible

Reading the Numbers Honestly

The parallel pipeline is correct and, in the sample run, only 1.1x faster than one process doing identical work, and 0.46x the speed of C sorted(). Three forces produce this, and each one is a permanent fact of parallel programming:

  1. Amdahl’s law on the control path. Sampling, pivot selection, offsets, and pool startup are serial; the per-element Python loops (generate, bisect, scatter) dominate and are parallelized, but sorted() itself is C code - the fastest phase is the one that parallelizes least well from Python. The 0.46x line in the sample run is the whole lesson: parallelism is relative to the implementation quality of the sequential baseline, not to a textbook work count.
  2. Interpreter overhead per element. The scatter loop executes 4M interpreted iterations with attribute lookups; the C sort executes the same work in compiled code. Languages matter: the identical algorithm in C++ with std::sort on 4 cores would beat single-core std::sort by 3-3.8x, which is exactly what PBBS and Intel’s parallel-mode benchmarks report.
  3. Memory bandwidth. Scatter is a random-ish write pattern across 32 MB of buffers; four cores contend for the same DRAM channels. Near n * sizeof(key) = L3 size x several, bandwidth - not cores - becomes the ceiling, which is why sample sort’s cache-friendliness (sequential bucket writes, then independent sorts) is a bigger real-world win than its span.

The transferable takeaways: partition by sampled pivots so no merge phase exists; keep payload in shared memory and ship metadata; expect bandwidth and the sequential baseline’s implementation quality - not core count - to decide the outcome. And the theory still matters: T1 = O(n log n) with T-infinity = O(log n)-ish is what makes the algorithm able to scale once you move to a language where the constants stop drowning the asymptotics.

References