The most efficient way rather than using np.setdiff1d and np.in1d, to remove common values of 1D arrays with unique values

I need a much faster code to remove values of an 1D array (array length ~ 10-15) that are common with another 1D array (array length ~ 1e5-5e5 –> rarely up to 7e5), which are index arrays contain integers. There is no duplicate in the arrays, and they are not sorted and the order of the values must be kept in the main array after modification. I know that can be achieved using such np.setdiff1d or np.in1d (which both are not supported for numba jitted in no-python mode), and other similar posts (e.g. this) have not much more efficient way to do so, but performance is important here because all the values in the main index array will be gradually be removed in loops.

import numpy as np
import numba as nb

n = 500000
r = 10
arr1 = np.random.permutation(n)
arr2 = np.random.randint(0, n, r)

# @nb.jit
def setdif1d_np(a, b):
    return np.setdiff1d(a, b, assume_unique=True)


# @nb.jit
def setdif1d_in1d_np(a, b):
    return a[~np.in1d(a, b)]

There is another related post that proposed by norok2 for 2D arrays, that is ~15 times faster solution (hashing-like way using numba) than usual methods described there. This solution may be the best if it could be prepared for 1D arrays:

@nb.njit
def mul_xor_hash(arr, init=65537, k=37):
    result = init
    for x in arr.view(np.uint64):
        result = (result * k) ^ x
    return result


@nb.njit
def setdiff2d_nb(arr1, arr2):
    # : build `delta` set using hashes
    delta = {mul_xor_hash(arr2[0])}
    for i in range(1, arr2.shape[0]):
        delta.add(mul_xor_hash(arr2[i]))
    # : compute the size of the result
    n = 0
    for i in range(arr1.shape[0]):
        if mul_xor_hash(arr1[i]) not in delta:
            n += 1
    # : build the result
    result = np.empty((n, arr1.shape[-1]), dtype=arr1.dtype)
    j = 0
    for i in range(arr1.shape[0]):
        if mul_xor_hash(arr1[i]) not in delta:
            result[j] = arr1[i]
            j += 1
    return result

I tried to prepare that for 1D arrays, but I have some problems/question with that.

• At first, IDU what does mul_xor_hash exactly do, and if init and k are arbitrary selected or not
• Why mul_xor_hash will not work without nb.njit:

  File "C:/Users/Ali/Desktop/test - Copy - Copy.py", line 21, in mul_xor_hash
    result = (result * k) ^ x
TypeError: ufunc 'bitwise_xor' not supported for the input types, and the inputs could not be safely coerced to any supported types according to the casting rule ''safe''

• IDK how to implement mul_xor_hash on 1D arrays (if it could), which I guess may make it faster more than for 2Ds, so I broadcast the input arrays to 2D by [None, :], which get the following error just for arr2:

    print(mul_xor_hash(arr2[0]))
ValueError: new type not compatible with array

• and what does delta do

I am searching the most efficient way in this regard. In the absence of better method than norok2 solution, how to prepare this solution for 1D arrays?

Answer

Understanding the hash-based solution

At first, IDU what does mul_xor_hash exactly do, and if init and k are arbitrary selected or not

mul_xor_hash is a custom hash function. Functions mixing xor and multiply (possibly with shifts) are known to be relatively fast to compute the hash of a raw data buffer. The multiplication tends to shuffle bits and the xor is used to somehow combine/accumulate the result in a fixed size small value (ie. the final hash). There are many different hashing functions. Some are faster than others, some cause more collisions than other in a given context. A fast hashing function causing too many collisions can be useless in practice as it would result in a pathological situation where all conflicting values needs to be compared. This is why fast hash functions are hard to implement.

init and k are parameter certainly causing the hash to be pretty balance. This is pretty common in such a hash function. k needs to be sufficiently big for the multiplication to shuffle bits and it should typically also be a prime number (values like power of two tends to increase collisions due to modular arithmetic behaviours). init plays a significant role only for very small arrays (eg. with 1 item): it helps to reduce collisions by xoring the final hash by a non-trivial constant. Indeed, if arr.size = 1, then result = (init * k) ^ arr[0] where init * k is a constant. Having an identity hash function equal to arr[0] is known to be bad since it tends to result in many collisions (this is a complex topic, but put it shortly, arr[0] can be divided by the number of buckets in the hash table for example). Thus, init should be a relatively big number and init * k should also be a big non-trivial value (a prime number is a good target value).

Why mul_xor_hash will not work without nb.njit

It depends of the input. The input needs to be a 1D array and have a raw size in byte divisible by 8 (eg. 64-bit items, 2n x 32-bit ones, 4n x 16-bit one or 8n 8-bit ones). Here is some examples:

mul_xor_hash(np.random.rand(10))
mul_xor_hash(np.arange(10)) # Do not work with 9

and what does delta do

It is a set containing the hash of the arr2 row so to find matching lines faster than comparing them without hashes.

how to prepare this solution for 1D arrays?

AFAIK, hashes are only use to avoid comparisons of rows but this is because the input is the 2D array. In 1D, there is no such a problem.

There is big catch with this method: it only works if there is no hash collisions. Otherwise, the implementation wrongly assumes that values are equal even if they are not! @norok explicitly mentioned it in the comments though:

Note that the collision handling for the hashings should also be implemented

---

Faster implementation

Using the 2D solution of @norok2 for 1D is not a good idea since hashes will not make it faster the way they are used. In fact, a set already use a hash function internally anyway. Not to mention collisions needs to be properly implemented (which is done by a set).

Using a set is a relatively good idea since it causes the complexity to be O(n + m) where n = len(arr1) and m = len(arr2). That being said, if arr1 is converted to a set, then it will be too big to fit in L1 cache (due to the size of arr1 in your case) resulting in slow cache misses. Additionally, the growing size of the set will cause values to be re-hashed which is not efficient. If arr2 is converted to a set, then the many hash table fetches will not be very efficient since arr2 is very small in your case. This is why this solution is sub-optimal.

One solution is to split arr1 in chunks and then build a set based on the target chunk. You can then check if a value is in the set or not efficiently. Building the set is still not very efficient due to the growing size. This problem is due to Python itself which do not provide a way to reserve some space for the data structure like other languages do (eg. C++). One solution to avoid this issue is simply to reimplement an hash-table which is not trivial and cumbersome. Actually, Bloom filters can be used to speed up this process since they can quickly find if there is no collision between the two sets arr1 and arr2 in average (though they are not trivial to implement).

Another optimization is to use multiple threads to compute the chunks in parallel since they are independent. That being said, the appending to the final array is not easy to do efficiently in parallel, especially since you do not want the order to be modified. One solution is to move away the copy from the parallel loop and do it serially but this is slow and AFAIK there is no simple way to do that in Numba currently (since the parallelism layer is very limited). Consider using native languages like C/C++ for an efficient parallel implementation.

In the end, hashing can be pretty complex and the speed up can be quite small compared to a naive implementation with two nested loops since arr2 only have few items and modern processors can compare values quickly using SIMD instructions (while hash-based method can hardly benefit from them on mainstream processors). Unrolling can help to write a pretty simple and fast implementation. Again, unfortunately, Numba use LLVM-Jit internally which appear to fail to vectorize such a simple code (certainly due to missing optimizations in either LLVM-Jit or even LLVM itself). As a result, the non vectorized code is finally a bit slower (rather than 4~10 times faster on a modern mainstream processor). One solution is to use a C/C++ code instead to do that (or possibly Cython).

Here is a serial implementation using basic Bloom filters:

@nb.njit('uint32(int32)')
def hash_32bit_4k(value):
    return (np.uint32(value) * np.uint32(27_644_437)) & np.uint32(0x0FFF)

@nb.njit(['int32[:](int32[:], int32[:])', 'int32[:](int32[::1], int32[::1])'])
def setdiff1d_nb_faster(arr1, arr2):
    out = np.empty_like(arr1)
    bloomFilter = np.zeros(4096, dtype=np.uint8)
    for j in range(arr2.size):
        bloomFilter[hash_32bit_4k(arr2[j])] = True
    cur = 0
    for i in range(arr1.size):
        # If the bloom-filter value is true, we know arr1[i] is not in arr2.
        # Otherwise, there is maybe a false positive (conflict) and we need to check to be sure.
        if bloomFilter[hash_32bit_4k(arr1[i])] and arr1[i] in arr2:
            continue
        out[cur] = arr1[i]
        cur += 1
    return out[:cur]

Here is an untested variant that should work for 64-bit integers (floating point numbers need memory views and possibly a prime constant too):

@nb.njit('uint64(int64)')
def hash_32bit_4k(value):
    return (np.uint64(value) * np.uint64(67_280_421_310_721)) & np.uint64(0x0FFF)

Note that if all the values in the small array are contained in the main array in each loop, then we can speed up the arr1[i] in arr2 part by removing values from arr2 when we find them. That being said, collisions and findings should be very rare so I do not expect this to be significantly faster (not to mention it adds some overhead and complexity). If items are computed in chunks, then the last chunks can be directly copied without any check but the benefit should still be relatively small. Note that this strategy can be effective for the naive (C/C++) SIMD implementation previously mentioned though (it can be about 2x faster).

---

Generalization and parallel implementation

This section focus on the algorithm to use regarding the input size. It particularly details an SIMD-based implementation and discuss about the use of multiple threads.

First of all, regarding the value r, the best algorithm to use can be different. More specifically:

• when r is 0, the best thing to do is to return the input array arr1 unmodified (possibly a copy to avoid issue with in-place algorithms);
• when r is 1, we can use one basic loop iterating over the array, but the best implementation is likely to use np.where of Numpy which is highly optimized for that
• when r is small like <10, then using a SIMD-based implementation should be particularly efficient, especially if the iteration range of the arr2-based loop is known at compile-time and is unrolled
• for bigger r values that are still relatively small (eg. r < 1000 and r << n), the provided hash-based solution should be one of the best;
• for larger r values with r << n, the hash-based solution can be optimized by packing boolean values as bits in bloomFilter and by using multiple hash-functions instead of one so to better handle collisions while being more cache-friendly (in fact, this is what actual bloom filters does); note that multi-threading can be used so speed up the lookups when r is huge and r << n;
• when r is big and not much smaller than n, then the problem is pretty hard to solve efficiently and the best solution is certainly to sort both arrays (typically with a radix sort) and use a merge-based method to remove the duplicates, possibly with multiple threads when both r and n are huge (hard to implement).

Let’s start with the SIMD-based solution. Here is an implementation:

@nb.njit('int32[:](int32[::1], int32[::1])')
def setdiff1d_nb_simd(arr1, arr2):
    out = np.empty_like(arr1)
    limit = arr1.size // 4 * 4
    limit2 = arr2.size // 2 * 2
    cur = 0
    z32 = np.int32(0)

    # Tile (x4) based computation
    for i in range(0, limit, 4):
        f0, f1, f2, f3 = z32, z32, z32, z32
        v0, v1, v2, v3 = arr1[i], arr1[i+1], arr1[i+2], arr1[i+3]
        # Unrolled (x2) loop searching for a match in `arr2`
        for j in range(0, limit2, 2):
            val1 = arr2[j]
            val2 = arr2[j+1]
            f0 += (v0 == val1) + (v0 == val2)
            f1 += (v1 == val1) + (v1 == val2)
            f2 += (v2 == val1) + (v2 == val2)
            f3 += (v3 == val1) + (v3 == val2)
        # Remainder of the previous loop
        if limit2 != arr2.size:
            val = arr2[arr2.size-1]
            f0 += v0 == val
            f1 += v1 == val
            f2 += v2 == val
            f3 += v3 == val
        if f0 == 0: out[cur] = arr1[i+0]; cur += 1
        if f1 == 0: out[cur] = arr1[i+1]; cur += 1
        if f2 == 0: out[cur] = arr1[i+2]; cur += 1
        if f3 == 0: out[cur] = arr1[i+3]; cur += 1

    # Remainder
    for i in range(limit, arr1.size):
        if arr1[i] not in arr2:
            out[cur] = arr1[i]
            cur += 1

    return out[:cur]

It turns out this implementation is always slower than the hash-based one on my machine since Numba clearly generate an inefficient for the inner arr2-based loop and this appears to come from broken optimizations related to the ==: Numba simply fail use SIMD instructions for this operation (for no apparent reasons). This prevent many alternative SIMD-related codes to be fast as long as they are using Numba.

Another issue with Numba is that np.where is slow since it use a naive implementation while the one of Numpy has been heavily optimized. The optimization done in Numpy can hardly be applied to the Numba implementation due to the previous issue. This prevent any speed up using np.where in a Numba code.

In practice, the hash-based implementation is pretty fast and the copy takes a significant time on my machine already. The computing part can be speed up using multiple thread. This is not easy since the parallelism model of Numba is very limited. The copy cannot be easily optimized with Numba (one can use non-temporal store but this is not yet supported by Numba) unless the computation is possibly done in-place.

To use multiple threads, one strategy is to first split the range in chunk and then:

• build a boolean array determining, for each item of arr1, whether the item is found in arr2 or not (fully parallel)
• count the number of item found by chunk (fully parallel)
• compute the offset of the destination chunk (hard to parallelize, especially with Numba, but fast thanks to chunks)
• copy the chunk to the target location without copying found items (fully parallel)

Here is an efficient parallel hash-based implementation:

@nb.njit('int32[:](int32[:], int32[:])', parallel=True)
def setdiff1d_nb_faster_par(arr1, arr2):
    # Pre-computation of the bloom-filter
    bloomFilter = np.zeros(4096, dtype=np.uint8)
    for j in range(arr2.size):
        bloomFilter[hash_32bit_4k(arr2[j])] = True

    chunkSize = 1024 # To tune regarding the kind of input
    chunkCount = (arr1.size + chunkSize - 1) // chunkSize

    # Find for each item of `arr1` if the value is in `arr2` (parallel)
    # and count the number of item found for each chunk on the fly.
    # Note: thanks to page fault, big parts of `found` are not even written in memory if `arr2` is small
    found = np.zeros(arr1.size, dtype=nb.bool_)
    foundCountByChunk = np.empty(chunkCount, dtype=nb.uint16)
    for i in nb.prange(chunkCount):
        start, end = i * chunkSize, min((i + 1) * chunkSize, arr1.size)
        foundCountInChunk = 0
        for j in range(start, end):
            val = arr1[j]
            if bloomFilter[hash_32bit_4k(val)] and val in arr2:
                found[j] = True
                foundCountInChunk += 1
        foundCountByChunk[i] = foundCountInChunk

    # Compute the location of the destination chunks (sequential)
    outChunkOffsets = np.empty(chunkCount, dtype=nb.uint32)
    foundCount = 0
    for i in range(chunkCount):
        outChunkOffsets[i] = i * chunkSize - foundCount
        foundCount += foundCountByChunk[i]

    # Parallel chunk-based copy
    out = np.empty(arr1.size-foundCount, dtype=arr1.dtype)
    for i in nb.prange(chunkCount):
        srcStart, srcEnd = i * chunkSize, min((i + 1) * chunkSize, arr1.size)
        cur = outChunkOffsets[i]
        # Optimization: we can copy the whole chunk if there is nothing found in it 
        if foundCountByChunk[i] == 0:
            out[cur:cur+(srcEnd-srcStart)] = arr1[srcStart:srcEnd]
        else:
            for j in range(srcStart, srcEnd):
                if not found[j]:
                    out[cur] = arr1[j]
                    cur += 1

    return out

This implementation is the fastest for the target input on my machine. It is generally fast when n is quite big and the overhead to create threads is relatively small on the target platform (eg. on PCs but typically not computing servers with many cores). The overhead of the parallel implementation is significant so the number of core on the target machine needs to be at least 4 so the implementation can be significantly faster than the sequential implementation.

It may be useful to tune the chunkSize variable for the target inputs. If r << n, it is better to use a pretty big chunkSize. That being said, the number of chunk needs to be sufficiently big for multiple thread to operate on many chunks. Thus, chunkSize should be significantly smaller than n / numberOfThreads.

On my machine most of the time (65-70%) is spent in the final copy which is mostly memory-bound and can hardly be optimized further with Numba.

---

Results

Here are results on my i5-9600KF-based machine (with 6 cores):

setdif1d_np:               2.65 ms
setdif1d_in1d_np:          2.61 ms
setdiff1d_nb:              2.33 ms
setdiff1d_nb_simd:         1.85 ms
setdiff1d_nb_faster:       0.73 ms
setdiff1d_nb_faster_par:   0.49 ms

The best provided implementation is about 4~5 time faster than the other ones.