Bit-Sliced Signatures and Bloom Filters

While the inverted index is a familiar indexing structure for many with a casual interest in information retrieval, the concept of a document signature may not be. The idea seems to have a long history, having been popular until it lost favour in the late 90s, owing to file sizes and performance empirically inferior to inverted indices. However, there has been new research into the use of signatures for search problems in the last decade, for instance, TopSig, and they are an important building block of the BitFunnel data structure.

Information Retrieval with Signatures

An m bit signature of a document consists of a bit array of fixed length m containing the (not guaranteed to be disjoint) union of the signatures of the terms found in the document. The signature of a term t is the m bit array with each jth bit set where j = hash(k, t) mod m, for each of k hash functions. This is obviously a Bloom filter by another name. In a recent post I noted that a Bloom filter’s behaviour can be implemented efficiently in less than ten lines of Java code.

In an information retrieval setting, a document signature would be computed for each document in a corpus by tokenising each document and building a Bloom filter from the extracted terms. To retrieve the documents matching a given query, the query signature (which is the union of the signatures of the query’s terms) is computed. In what I will call the signature scan phase of the retrieval, the document signatures are iterated over; each document being included if its signature contains all the bits in the query signature. The result set will contain false positives but should also be quite small, so a cheap filter for correctness is executed after the signature scan is complete.

While the hash function and ordering of the documents could be controlled creatively to cut down the number of signatures to be inspected for a given query, this approach is obviously inefficient, despite the appeal of low level bit-wise instructions. What happens when there are several billion documents in a corpus?

Parallelism and Bit-Slicing

Threads and SIMD instructions could be thrown at the signature scan to speed it up, but one approach to parallelising the retrieval is to increase the number of documents represented by each processed word. If there are n documents with m bit signatures, the signature list is an n * m bit matrix. Its transpose, an m * n bit matrix, is referred to as a bit sliced signature. When querying the bit sliced signature, only the rows specified in the query need be accessed and intersected, and each word represents the presence of a term in up to 64 documents. This is a very old technique – the the earliest formulation of a variant of this data structure I could find, where it was referred to as superimposed coding, was published in 1984, but references implementations from the 60s. An accessible evaluation was published in 1994.

Java Implementation

An implementation of such a structure for an immutable corpus is trivial but informative. Typically terms will be strings but needn’t be, whereas documents can be reduced to a set of terms. Using ToIntFunction to abstract hash functions again, we just need to map all term-document tuples into the bits of a long[][]. When querying the structure, we need to map the query’s terms into a sorted sequence of integers, determining which rows of the bit matrix to access.

On the way in, rather than using the hash functions to compute the bit to set (this is constant for each document), the row index is calculated. For each term, for each hash function, the appropriate row index is calculated and the document’s bit is set in the corresponding array. Clean Java would do this outside of the constructor, of course.


public class BitSlicedSignature<D extends Supplier<Set<T>>, T, Q extends Set<T>> {

    private final long[][] bitMatrix;
    private final int width;
    private final int height;
    private final List<ToIntFunction<T>> hashFunctions;

    public BitSlicedSignature(List<D> documents,
                              List<ToIntFunction<T>> hashFunctions,
                              int height) {
        this.hashFunctions = hashFunctions;
        this.width = (documents.size() + 63) / 64;
        this.height = height;
        this.bitMatrix = new long[height][width];
        int docIndex = 0;
        for (D doc : documents) {
            int docWordIndex = docIndex >> 6;
            long docWord = 1L << docIndex;
            for (T term : doc.get()) {
                for (ToIntFunction<T> hash : hashFunctions) {
                    int row = mapHash(hash.applyAsInt(term));
                    bitMatrix[row][docWordIndex] |= docWord;
                }
            }
            ++docIndex;
        }
    }

    private int mapHash(int hash) {
        return Math.abs(hash % height);
    }
}

To query the structure, the query is mapped into row indices and the corresponding rows are intersected word by word, matching document IDs are emitted lazily as an IntStream. The appeal of doing this lazily is that we should expect there to be a lot of documents, this way the bit-wise intersections can be done in chunks as and when the caller wants more documents. This can be achieved with the help of two utility methods:


    public IntStream query(Q query) {
        int[] rows = query.stream()
                          .flatMapToInt(t -> hashFunctions.stream().mapToInt(h -> mapHash(h.applyAsInt(t))))
                          .distinct()
                          .toArray();
        return IntStream.range(0, width).flatMap(i -> bitsOf(intersection(rows, i), i));
    }

    private long intersection(int[] rows, int offset) {
        long word = -1L;
        for (int i = 0; i < rows.length && word != 0; ++i) {
            word &= bitMatrix[rows[i]][offset];
        }
        return word;
    }

    private static IntStream bitsOf(long word, int offset) {
        return IntStream.range(0, Long.SIZE)
                        .filter(i -> (1L << i & word) != 0)
                        .map(i -> Long.SIZE * offset + i);
    }

As you can probably see, you can leave vast swathes of the long[][] untouched, assuming the query is for a small number of terms. A more sophisticated implementation might partition the documents into several bit matrices.

Shortcomings

There are some obvious problems with this data structure. Firstly, a long[][] uses the same amount of memory whether its bits are set or not. What happens when you have some small documents and lots of terms? You have a column in the bit matrix where most of the bits are zero – it’s likely that a compressed bit set would be preferable. Similarly with very common terms you will have long horizontal runs of set bits.

Even worse, what happens when a term is very rare? If you are providing a search service, it’s likely you only ever need to provide a page of results at a time. If the term is rare enough, you may need to scan the entire row to fill a page, which could take a long time. To get around that, BitFunnel uses bit-sliced block signatures, which I will write about in the next post.

Building a Bloom Filter from Scratch

The Bloom filter is an interesting data structure for modelling approximate sets of objects. It can tell you with certainty that an object is not in a collection, but may give false positives. If you write Java for a living, I would not suggest you implement your own, because there is a perfectly good implementation in Guava. It can be illuminating to write one from scratch, however.

Interface

You can do two things with a Bloom filter: put things in it, and check if the filter probably contains items. This leads to the following interface:


public class BloomFilter<T> {
    void add(T value);

    boolean mightContain(T value);
}

Java Bloom Filter Implementation

A Bloom filter represents a set of objects quite succinctly as a bit array of finite length. Unlike a bitset, the bit array stores hashes of the objects, rather than object identities. In practice, you need multiple hash functions, which, modulo the capacity of the bit array, will collide only with low probability. The more hashes you have, the slower insertions and look ups will be. There is also a space trade off for improved accuracy: the larger the array, the less likely collisions of hashes modulo the capacity will be.

Since you probably want to be able to control the hash functions you use, the interface ToIntFunction fits in nicely as the perfect abstraction. You can set this up simply with a builder.


    public static <T> Builder<T> builder() {
        return new Builder<>();
    }

    public static class Builder<T> {
        private int size;
        private List<ToIntFunction<T>> hashFunctions;

        public Builder<T> withSize(int size) {
            assert Integer.bitCount(size) == 1; 
            this.size = size;
            return this;
        }

        public Builder<T> withHashFunctions(List<ToIntFunction<T>> hashFunctions) {
            this.hashFunctions = hashFunctions;
            return this;
        }

        public BloomFilter<T> build() {
            return new BloomFilter<>(new long[size], size, hashFunctions);
        }
    }

    private final long[] array;
    private final int size;
    private final List<ToIntFunction<T>> hashFunctions;

    public BloomFilter(long[] array, int logicalSize, List<ToIntFunction<T>> hashFunctions) {
        this.array = array;
        this.size = logicalSize;
        this.hashFunctions = hashFunctions;
    }

    private int mapHash(int hash) {
        return hash & (size - 1);
    }

Adding Values

To add a value, you need to take an object, and for each hash function hash it modulo the capacity of the bit array. Using a long[] as the substrate of the bit set you must locate the word each hash belongs to, and set the bit corresponding to the remainder of the hash modulo 64. You can do this quickly as follows:


   public void add(T value) {
        for (ToIntFunction<T> function : hashFunctions) {
            int hash = mapHash(function.applyAsInt(value));
            array[hash >>> 6] |= 1L << hash;
        }
    }

Note that each bit may already be set.

Querying the bit set

To check if an object is contained in the bitset, you need to evaluate the hashes again, and find the corresponding words. You can return false if the intersection of the appropriate word and the bit corresponding to the remainder modulo 64 is zero. That looks like this:


    public boolean mightContain(T value) {
        for (ToIntFunction<T> function : hashFunctions) {
            int hash = mapHash(function.applyAsInt(value));
            if ((array[hash >>> 6] & (1L << hash)) == 0) {
                return false;
            }
        }
        return true;
    }

Note that this absolutely does not mean the object is contained within the set, but you can get more precise results if you are willing to perform more hash evaluations, and if you are willing to use a larger bit array. Modeling the precise false positive rate is not clear cut.

BitFunnel

Just as is the case for bitmap indices, bit slicing is a useful enhancement for Bloom filters, forming the basis of BitFunnel. In a follow up post I will share a simplified implementation of a bit sliced Bloom filter.

Confusing Sets and Lists

I have often seen the roles of lists and sets confused. An application can be brought to its knees – that is, cease to deliver commercial value – if List.contains is called frequently enough on big enough lists. And then there is the workaround… When I moved over to Java from C++ several years ago, it seemed utterly crazy that there was even a Collection interface – exactly what Scott Meier’s Effective STL said not to do. I still think it’s crazy. Sets and lists cannot be substituted, and when you add up the marginal costs, as well as the costs of any compensatory workarounds, confusing them is responsible for a lot of performance bugs. As an application developer, it is part of your job to choose. Here are a few simple examples of when to use each collection type.

Contains

Is an element in the collection?

Never ever do this with a List. This operation is often referred to as being O(n). Which means in your worst case will touch every element in the list (technically, at least once). You have a choice between HashSet and a TreeSet, and both have costs and benefits.

If you choose a HashSet, your best case is O(1): you evaluate a hash code, take its modulus with respect to the size of an array, and look up a bucket containing only one element. The worst case occurs with a degenerate hash code which maps all elements to the same bucket. This is again O(n): you probe a linked list testing each element for equality. On average you get something between these two cases and it depends on the uniformity of your hash code implementation.

If you choose a TreeSet you get a worst case O(log n): this is effectively just a binary search through the nodes in a red black tree. Performance is limited by the cost of the comparator, and suffers systematically from cache misses for large sets (like any kind of pointer chasing, branch prediction and prefetching is difficult if not impossible).

If you’re working with numbers, and small to medium collections, a sorted primitive array may be the best approach, so long as it fits in cache. If you’re working with integers, you can do this in constant time in the worst case by using a BitSet.

Select

What is the value of the element at a given index with respect to a sort order?

This is an obvious use case for a List: it’s O(1) – this is just a lookup at an array offset.

You couldn’t even write the code with a HashSet without copying the data into an intermediate ordered structure, at which point you would probably think again. You see this sort of thing done in code written by inexpensive programmers at large outsourcing consultancies, who were perhaps just under unreasonable pressure to deliver to arbitrary deadlines.

SortedSet, and anything conceptually similar, is the wrong data structure for this operation. The only way to compute this is O(n): you iterate through the set incrementing a counter until you reach the index, and then return the element you’ve iterated to. If you reach the end of the set, you throw. If you do this a lot, you’ll notice.

Rank

How many predecessors does an element have with respect to a sort order?

Another classic operation for List, so long as you keep it sorted. Use Collections.binarySearch to find the index of the element in the collection with complexity O(log n). This is its rank. If you can get away with it, primitive arrays will be much better here, especially if they are small enough to fit in cache.

Once again, there are creativity points available for the solution involving a HashSet, and they do exist in the wild, but a clean looking solution is at least possible with a SortedSet. However, it involves an iteration with another check against an incrementing counter. It’s O(n) and if you do it a lot, you’ll blow your performance profile, so use a sorted list instead.

What if you had the source code?

Is this fundamental or just a limitation of the Collections framework? Maybe if you had the source code you could just make these data structures optimal for every use case, without having to choose the right one? Not without creating a Frankenstein, and not without a lot of memory. Optimisation isn’t free.

Sorting Unsigned Integers Faster in Java

I discovered a curious resource for audio-visualising sort algorithms, which is exciting for two reasons. The first is that I finally feel like I understand Alexander Scriabin: he was not a composer. He discovered Tim Sort 80 years before Tim Peters and called it Black Mass. (If you aren’t familiar with the piece, fast-forward to 1:40 to hear the congruence.)

The second reason was that I noticed Radix Sort (LSD). While it was an affront to my senses, it used a mere 800 array accesses and no comparisons! I was unaware of this algorithm so delved deeper and implemented it for integers, and benchmarked my code against Arrays.sort.

Radix Sort

It is taken as given by many (myself included, or am I just projecting my thoughts on to others?) that O(n \log n) is the best you can do in a sort algorithm. But this is actually only true for sort algorithms which depend on comparison. If you can afford to restrict the data types your sort algorithm supports to types with a positional interpretation (java.util can’t because it needs to be ubiquitous and maintainable), you can get away with a linear time algorithm.

Radix sort, along with the closely related counting sort, does not use comparisons. Instead, the data is interpreted as a fixed length string of symbols. For each position, the cumulative histogram of symbols is computed to calculate sort indices. While the data needs to be scanned several times, the algorithm scales linearly and the overhead of the multiple scans is amortised for large arrays.

As you can see on Wikipedia, there are two kinds of radix sort: Least Significant Digit and Most Significant Digit. This dichotomy relates to the order the (representational) string of symbols is traversed in. I implemented and benchmarked the LSD version for integers.

Implementation

The implementation interprets an integer as the concatenation of n bit string symbols of fixed size size 32/n. It performs n passes over the array, starting with the least significant bits, which it modifies in place. For each pass the data is scanned three times, in order to:

  1. Compute the cumulative histogram over the symbols in their natural sort order
  2. Copy the value with symbol k to the mth position in a buffer, where m is defined by the cumulative density of k.
  3. Copy the buffer back into the original array

The implementation, which won’t work unless the chunks are proper divisors of 32, is below. The bonus (or caveat) is that it automatically supports unsigned integers. The code could be modified slightly to work with signed integers at a performance cost.


import java.util.Arrays;

public class RadixSort {

    private final int radix;

    public RadixSort() {
        this(Byte.SIZE);
    }

    public RadixSort(int radix) {
        assert 32 % radix== 0;
        this.radix= radix;
    }

    public void sort(int[] data) {
        int[] histogram = new int[(1 << radix) + 1];
        int shift = 0;
        int mask = (1 << radix) - 1;
        int[] copy = new int[data.length];
        while (shift < Integer.SIZE) {
            Arrays.fill(histogram, 0);
            for (int i = 0; i < data.length; ++i) {
                ++histogram[((data[i] & mask) >> shift) + 1];
            }
            for (int i = 0; i < 1 << radix; ++i) {
                histogram[i + 1] += histogram[i];
            }
            for (int i = 0; i < data.length; ++i) {
                copy[histogram[(data[i] & mask) >> shift]++] = data[i];
            }
            for (int i = 0; i < data.length; ++i) {
                data[i] = copy[i];
            }
            shift += radix;
            mask <<= radix;
        }
    }
}

The time complexity is obviously linear, a temporary buffer is allocated, but in comparison to Arrays.sort it looks fairly spartan. Instinctively, cache locality looks fairly poor because the second inner loop of the three jumps all over the place. Will this implementation beat Arrays.sort (for integers)?

Benchmark

The algorithm is measured using arrays of random positive integers, for which both algorithms are equivalent, from a range of sizes. This isn’t always the best idea (the Tim Sort algorithm comes into its own on nearly sorted data), so take the result below with a pinch of salt. Care must be taken to copy the array in the benchmark since both algorithms are in-place.


public void launchBenchmark(String... jvmArgs) throws Exception {
        Options opt = new OptionsBuilder()
                .include(this.getClass().getName() + ".*")
                .mode(Mode.SampleTime)
                .mode(Mode.Throughput)
                .timeUnit(TimeUnit.MILLISECONDS)
                .measurementTime(TimeValue.seconds(10))
                .warmupIterations(10)
                .measurementIterations(10)
                .forks(1)
                .shouldFailOnError(true)
                .shouldDoGC(true)
                .jvmArgs(jvmArgs)
                .resultFormat(ResultFormatType.CSV) 
                .build();

        new Runner(opt).run();
    }

    @Benchmark
    public void Arrays_Sort(Data data, Blackhole bh) {
        int[] array = Arrays.copyOf(data.data, data.size);
        Arrays.sort(array);
        bh.consume(array);
    }

    @Benchmark
    public void Radix_Sort(Data data, Blackhole bh) {
        int[] array = Arrays.copyOf(data.data, data.size);
        data.radixSort.sort(array);
        bh.consume(array);
    }

    @State(Scope.Thread)
    public static class Data {

        @Param({"100", "1000", "10000", "100000", "1000000"})
        int size;

        int[] data;
        RadixSort radixSort = new RadixSort();

        @Setup(Level.Trial)
        public void init() {
            data = createArray(size);
        }
    }

    public static int[] createArray(int size) {
        int[] array = new int[size];
        Random random = new Random(0);
        for (int i = 0; i < size; ++i) {
            array[i] = Math.abs(random.nextInt());
        }
        return array;
    }

Benchmark Mode Threads Samples Score Score Error (99.9%) Unit Param: size
Arrays_Sort thrpt 1 10 1304.687189 147.380334 ops/ms 100
Arrays_Sort thrpt 1 10 78.518664 9.339994 ops/ms 1000
Arrays_Sort thrpt 1 10 1.700208 0.091836 ops/ms 10000
Arrays_Sort thrpt 1 10 0.133835 0.007146 ops/ms 100000
Arrays_Sort thrpt 1 10 0.010560 0.000409 ops/ms 1000000
Radix_Sort thrpt 1 10 404.807772 24.930898 ops/ms 100
Radix_Sort thrpt 1 10 51.787409 4.881181 ops/ms 1000
Radix_Sort thrpt 1 10 6.065590 0.576709 ops/ms 10000
Radix_Sort thrpt 1 10 0.620338 0.068776 ops/ms 100000
Radix_Sort thrpt 1 10 0.043098 0.004481 ops/ms 1000000
Arrays_Sort sample 1 3088586 0.000902 0.000018 ms/op 100
Arrays_Sort·p0.00 sample 1 1 0.000394 ms/op 100
Arrays_Sort·p0.50 sample 1 1 0.000790 ms/op 100
Arrays_Sort·p0.90 sample 1 1 0.000791 ms/op 100
Arrays_Sort·p0.95 sample 1 1 0.001186 ms/op 100
Arrays_Sort·p0.99 sample 1 1 0.001974 ms/op 100
Arrays_Sort·p0.999 sample 1 1 0.020128 ms/op 100
Arrays_Sort·p0.9999 sample 1 1 0.084096 ms/op 100
Arrays_Sort·p1.00 sample 1 1 4.096000 ms/op 100
Arrays_Sort sample 1 2127794 0.011876 0.000037 ms/op 1000
Arrays_Sort·p0.00 sample 1 1 0.007896 ms/op 1000
Arrays_Sort·p0.50 sample 1 1 0.009872 ms/op 1000
Arrays_Sort·p0.90 sample 1 1 0.015408 ms/op 1000
Arrays_Sort·p0.95 sample 1 1 0.024096 ms/op 1000
Arrays_Sort·p0.99 sample 1 1 0.033920 ms/op 1000
Arrays_Sort·p0.999 sample 1 1 0.061568 ms/op 1000
Arrays_Sort·p0.9999 sample 1 1 0.894976 ms/op 1000
Arrays_Sort·p1.00 sample 1 1 4.448256 ms/op 1000
Arrays_Sort sample 1 168991 0.591169 0.001671 ms/op 10000
Arrays_Sort·p0.00 sample 1 1 0.483840 ms/op 10000
Arrays_Sort·p0.50 sample 1 1 0.563200 ms/op 10000
Arrays_Sort·p0.90 sample 1 1 0.707584 ms/op 10000
Arrays_Sort·p0.95 sample 1 1 0.766976 ms/op 10000
Arrays_Sort·p0.99 sample 1 1 0.942080 ms/op 10000
Arrays_Sort·p0.999 sample 1 1 2.058273 ms/op 10000
Arrays_Sort·p0.9999 sample 1 1 7.526102 ms/op 10000
Arrays_Sort·p1.00 sample 1 1 46.333952 ms/op 10000
Arrays_Sort sample 1 13027 7.670135 0.021512 ms/op 100000
Arrays_Sort·p0.00 sample 1 1 6.356992 ms/op 100000
Arrays_Sort·p0.50 sample 1 1 7.634944 ms/op 100000
Arrays_Sort·p0.90 sample 1 1 8.454144 ms/op 100000
Arrays_Sort·p0.95 sample 1 1 8.742502 ms/op 100000
Arrays_Sort·p0.99 sample 1 1 9.666560 ms/op 100000
Arrays_Sort·p0.999 sample 1 1 12.916883 ms/op 100000
Arrays_Sort·p0.9999 sample 1 1 28.037900 ms/op 100000
Arrays_Sort·p1.00 sample 1 1 28.573696 ms/op 100000
Arrays_Sort sample 1 1042 96.278673 0.603645 ms/op 1000000
Arrays_Sort·p0.00 sample 1 1 86.114304 ms/op 1000000
Arrays_Sort·p0.50 sample 1 1 94.896128 ms/op 1000000
Arrays_Sort·p0.90 sample 1 1 104.293990 ms/op 1000000
Arrays_Sort·p0.95 sample 1 1 106.430464 ms/op 1000000
Arrays_Sort·p0.99 sample 1 1 111.223767 ms/op 1000000
Arrays_Sort·p0.999 sample 1 1 134.172770 ms/op 1000000
Arrays_Sort·p0.9999 sample 1 1 134.742016 ms/op 1000000
Arrays_Sort·p1.00 sample 1 1 134.742016 ms/op 1000000
Radix_Sort sample 1 2240042 0.002941 0.000033 ms/op 100
Radix_Sort·p0.00 sample 1 1 0.001578 ms/op 100
Radix_Sort·p0.50 sample 1 1 0.002368 ms/op 100
Radix_Sort·p0.90 sample 1 1 0.003556 ms/op 100
Radix_Sort·p0.95 sample 1 1 0.004344 ms/op 100
Radix_Sort·p0.99 sample 1 1 0.011056 ms/op 100
Radix_Sort·p0.999 sample 1 1 0.027232 ms/op 100
Radix_Sort·p0.9999 sample 1 1 0.731127 ms/op 100
Radix_Sort·p1.00 sample 1 1 5.660672 ms/op 100
Radix_Sort sample 1 2695825 0.018553 0.000038 ms/op 1000
Radix_Sort·p0.00 sample 1 1 0.013424 ms/op 1000
Radix_Sort·p0.50 sample 1 1 0.016576 ms/op 1000
Radix_Sort·p0.90 sample 1 1 0.025280 ms/op 1000
Radix_Sort·p0.95 sample 1 1 0.031200 ms/op 1000
Radix_Sort·p0.99 sample 1 1 0.050944 ms/op 1000
Radix_Sort·p0.999 sample 1 1 0.082944 ms/op 1000
Radix_Sort·p0.9999 sample 1 1 0.830295 ms/op 1000
Radix_Sort·p1.00 sample 1 1 6.660096 ms/op 1000
Radix_Sort sample 1 685589 0.145695 0.000234 ms/op 10000
Radix_Sort·p0.00 sample 1 1 0.112512 ms/op 10000
Radix_Sort·p0.50 sample 1 1 0.128000 ms/op 10000
Radix_Sort·p0.90 sample 1 1 0.196608 ms/op 10000
Radix_Sort·p0.95 sample 1 1 0.225792 ms/op 10000
Radix_Sort·p0.99 sample 1 1 0.309248 ms/op 10000
Radix_Sort·p0.999 sample 1 1 0.805888 ms/op 10000
Radix_Sort·p0.9999 sample 1 1 1.818141 ms/op 10000
Radix_Sort·p1.00 sample 1 1 14.401536 ms/op 10000
Radix_Sort sample 1 60843 1.641961 0.005783 ms/op 100000
Radix_Sort·p0.00 sample 1 1 1.251328 ms/op 100000
Radix_Sort·p0.50 sample 1 1 1.542144 ms/op 100000
Radix_Sort·p0.90 sample 1 1 2.002944 ms/op 100000
Radix_Sort·p0.95 sample 1 1 2.375680 ms/op 100000
Radix_Sort·p0.99 sample 1 1 3.447030 ms/op 100000
Radix_Sort·p0.999 sample 1 1 5.719294 ms/op 100000
Radix_Sort·p0.9999 sample 1 1 8.724165 ms/op 100000
Radix_Sort·p1.00 sample 1 1 13.074432 ms/op 100000
Radix_Sort sample 1 4846 20.640787 0.260926 ms/op 1000000
Radix_Sort·p0.00 sample 1 1 14.893056 ms/op 1000000
Radix_Sort·p0.50 sample 1 1 18.743296 ms/op 1000000
Radix_Sort·p0.90 sample 1 1 26.673152 ms/op 1000000
Radix_Sort·p0.95 sample 1 1 30.724915 ms/op 1000000
Radix_Sort·p0.99 sample 1 1 40.470446 ms/op 1000000
Radix_Sort·p0.999 sample 1 1 63.016600 ms/op 1000000
Radix_Sort·p0.9999 sample 1 1 136.052736 ms/op 1000000
Radix_Sort·p1.00 sample 1 1 136.052736 ms/op 1000000

The table tells an interesting story. Arrays.sort is vastly superior for small arrays (the arrays most people have), but for large arrays the custom implementation comes into its own. Interestingly, this is consistent with the computer science. If you need to sort large arrays of (unsigned) integers and care about performance, think about implementing radix sort.

Microsecond Latency Rules Engine with RoaringBitmap

Implementing a rules engine can shorten development time and remove a lot of tedious if statements from your business logic. Unfortunately they are almost always slow and often bloated. Simple rules engines can be implemented by assigning integer salience to each line in a truth table, with rule resolution treated as an iterative intersection of ordered sets of integers. Implemented in terms of sorted sets, it would be remiss not to consider RoaringBitmap for the engine’s core. The code is at github.

Classification Table and Syntax

This rules engine builds on the simple idea of a truth table usually used to teach predicate logic and computer hardware. Starting with a table and some attributes, interpreting one attribute as a classification, we get a list of rules. It is trivial to load such a table from a database. Since classifications can overlap, we prioritise by putting the rules we care about most – or the most salient rules – at the top of the table. When multiple rules match a fact, we take the last in the set ordered by salience. So we don’t always have to specify all of the attributes to get a classification, we can rank attributes by their importance left to right, where it’s required that all attributes to the left of a specified attribute are also specified when matching a fact against a rule set.

It’s possible to define rules containing wildcards. Wildcard rules will match any query (warning: if these are marked as high salience they will hide more specific rules with lower salience). It’s also possible to specify a prefix with a wildcard, which will match any query that matches at least the prefix.

Below is an example table consisting of rules for classification of regional English accents by phonetic feature.

English Accent Rules

thought cloth lot palm plant bath trap accent
/ɔ/ /ɒ/ /ɑ/ /ɑː/ /ɑː/ /ɑː/

/æ/ Received Pronunciation (UK)
/ɔ/ /ɔ/ /ɑ/ /ɑ/ /æ/ /æ/

/æ/ Georgian (US)
/ɑ/ /ɑ/ /ɑ/ /ɑ/ /æ/ /æ/

/æ/ Canadian
* * /ɑ/ /ɑ/ /æ/ /æ/

/æ/ North American
* * * * * *

/æ/ Non Native
* * * * * *

* French

 

In the example above, the vowel sounds used in words differentiating speakers of several English accents are configured as a classification table. The accent column is the classification of any speaker exhibiting the properties specified in the six leftmost columns. UK Received Pronunciation is the most specific rule and has high salience, whereas various North American accents differ from RP in their use of short A vowels. A catch all for North American accents would wild card the sounds in thought and caught (contrast Boston pronunciations with Texas). So long as trap has been pronounced with a short A (which all English speakers do), and no other rule would recognise the sounds used in the first six words, the rule engine would conclude the speaker is using English as a second language. If not even the word trap is recognisable, then the speaker is probably unintelligible, or could be French.

Implementation

A rule with a given salience can be represented by creating a bitmap index on salience by the attribute values of the rules. For instance, to store the rule {foo, bar} -> 42, with salience 10, create a bitmap index on the first attribute of the rule, and set the 10th bit of the “foo” bitmap; likewise for the “bar” bitmap of the second index. Finding rules which match both attributes is a bitwise intersection, and since we rank by salience, the rule that wins is the first in the set. An obvious choice for fast ordered sets is RoaringBitmap.

RoaringBitmap consists of containers, which are fast, cache-friendly sorted sets of integers, and can contain up to 2^16 shorts. In RoaringBitmap, containers are indexed by keys consisting of the most significant 16 bits of the integer. For a rules engine, if you have more than 2^16 rules you have a much bigger problem anyway, so a container could index all the rules you could ever need, so RoaringBitmap itself would be overkill. While RoaringBitmap indexes containers by shorts (it does so for the sake of compression), we can implement wildcard and prefix matching by associating containers with Strings rather than shorts. As the core data structure of the rules engine, a RoaringBitmap container is placed at each node of an Apache commons PatriciaTrie. It’s really that simple – see the source at github.

When the rules engine is queried, a set consisting of all the rules that match is intersected with the container found at the node in the trie matching the value specified for each attribute. When more than one rule matches, the rule with the highest salience is accessed via the Container.first() method, one of the features I have contributed to RoaringBitmap. See example usage at github.