fullstack/hyracks/hyracks-storage-am-bloomfilter/src/main/java/edu/uci/ics/hyracks/storage/am/bloomfilter/impls/BloomCalculations.java

/*
 * Copyright 2009-2013 by The Regents of the University of California
 * Licensed under the Apache License, Version 2.0 (the "License");
 * you may not use this file except in compliance with the License.
 * you may obtain a copy of the License from
 * 
 *     http://www.apache.org/licenses/LICENSE-2.0
 * 
 * Unless required by applicable law or agreed to in writing, software
 * distributed under the License is distributed on an "AS IS" BASIS,
 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 * See the License for the specific language governing permissions and
 * limitations under the License.
 */

package edu.uci.ics.hyracks.storage.am.bloomfilter.impls;

/**
 * This class has been taken from cassandra source code with minor modifications.
 */

/**
 * The following calculations are taken from:
 * http://www.cs.wisc.edu/~cao/papers/summary-cache/node8.html
 * "Bloom Filters - the math"
 * This class's static methods are meant to facilitate the use of the Bloom
 * Filter class by helping to choose correct values of 'bits per element' and
 * 'number of hash functions, k'.
 */
public class BloomCalculations {

    private static final int minBuckets = 2;
    private static final int minK = 1;

    /**
     * In the following table, the row 'i' shows false positive rates if i buckets
     * per element are used. Column 'j' shows false positive rates if j hash
     * functions are used. The first row is 'i=0', the first column is 'j=0'.
     * Each cell (i,j) the false positive rate determined by using i buckets per
     * element and j hash functions.
     */
    static final double[][] probs = new double[][] {
            { 1.0 }, // dummy row representing 0 buckets per element
            { 1.0, 1.0 }, // dummy row representing 1 buckets per element
            { 1.0, 0.393, 0.400 },
            { 1.0, 0.283, 0.237, 0.253 },
            { 1.0, 0.221, 0.155, 0.147, 0.160 },
            { 1.0, 0.181, 0.109, 0.092, 0.092, 0.101 }, // 5
            { 1.0, 0.154, 0.0804, 0.0609, 0.0561, 0.0578, 0.0638 },
            { 1.0, 0.133, 0.0618, 0.0423, 0.0359, 0.0347, 0.0364 },
            { 1.0, 0.118, 0.0489, 0.0306, 0.024, 0.0217, 0.0216, 0.0229 },
            { 1.0, 0.105, 0.0397, 0.0228, 0.0166, 0.0141, 0.0133, 0.0135, 0.0145 },
            { 1.0, 0.0952, 0.0329, 0.0174, 0.0118, 0.00943, 0.00844, 0.00819, 0.00846 }, // 10
            { 1.0, 0.0869, 0.0276, 0.0136, 0.00864, 0.0065, 0.00552, 0.00513, 0.00509 },
            { 1.0, 0.08, 0.0236, 0.0108, 0.00646, 0.00459, 0.00371, 0.00329, 0.00314 },
            { 1.0, 0.074, 0.0203, 0.00875, 0.00492, 0.00332, 0.00255, 0.00217, 0.00199, 0.00194 },
            { 1.0, 0.0689, 0.0177, 0.00718, 0.00381, 0.00244, 0.00179, 0.00146, 0.00129, 0.00121, 0.0012 },
            { 1.0, 0.0645, 0.0156, 0.00596, 0.003, 0.00183, 0.00128, 0.001, 0.000852, 0.000775, 0.000744 }, // 15
            { 1.0, 0.0606, 0.0138, 0.005, 0.00239, 0.00139, 0.000935, 0.000702, 0.000574, 0.000505, 0.00047, 0.000459 },
            { 1.0, 0.0571, 0.0123, 0.00423, 0.00193, 0.00107, 0.000692, 0.000499, 0.000394, 0.000335, 0.000302,
                    0.000287, 0.000284 },
            { 1.0, 0.054, 0.0111, 0.00362, 0.00158, 0.000839, 0.000519, 0.00036, 0.000275, 0.000226, 0.000198,
                    0.000183, 0.000176 },
            { 1.0, 0.0513, 0.00998, 0.00312, 0.0013, 0.000663, 0.000394, 0.000264, 0.000194, 0.000155, 0.000132,
                    0.000118, 0.000111, 0.000109 },
            { 1.0, 0.0488, 0.00906, 0.0027, 0.00108, 0.00053, 0.000303, 0.000196, 0.00014, 0.000108, 8.89e-05,
                    7.77e-05, 7.12e-05, 6.79e-05, 6.71e-05 } // 20
    }; // the first column is a dummy column representing K=0.

    /**
     * The optimal number of hashes for a given number of bits per element.
     * These values are automatically calculated from the data above.
     */
    private static final int[] optKPerBuckets = new int[probs.length];

    static {
        for (int i = 0; i < probs.length; i++) {
            double min = Double.MAX_VALUE;
            double[] prob = probs[i];
            for (int j = 0; j < prob.length; j++) {
                if (prob[j] < min) {
                    min = prob[j];
                    optKPerBuckets[i] = Math.max(minK, j);
                }
            }
        }
    }

    /**
     * Given the number of buckets that can be used per element, return a
     * specification that minimizes the false positive rate.
     * 
     * @param bucketsPerElement
     *            The number of buckets per element for the filter.
     * @return A spec that minimizes the false positive rate.
     */
    public static BloomFilterSpecification computeBloomSpec(int bucketsPerElement) {
        assert bucketsPerElement >= 1;
        assert bucketsPerElement <= probs.length - 1;
        return new BloomFilterSpecification(optKPerBuckets[bucketsPerElement], bucketsPerElement);
    }

    /**
     * Given a maximum tolerable false positive probability, compute a Bloom
     * specification which will give less than the specified false positive rate,
     * but minimize the number of buckets per element and the number of hash
     * functions used. Because bandwidth (and therefore total bitvector size)
     * is considered more expensive than computing power, preference is given
     * to minimizing buckets per element rather than number of hash functions.
     * 
     * @param maxBucketsPerElement
     *            The maximum number of buckets available for the filter.
     * @param maxFalsePosProb
     *            The maximum tolerable false positive rate.
     * @return A Bloom Specification which would result in a false positive rate
     *         less than specified by the function call
     * @throws UnsupportedOperationException
     *             if a filter satisfying the parameters cannot be met
     */
    public static BloomFilterSpecification computeBloomSpec(int maxBucketsPerElement, double maxFalsePosProb) {
        assert maxBucketsPerElement >= 1;
        assert maxBucketsPerElement <= probs.length - 1;
        int maxK = probs[maxBucketsPerElement].length - 1;

        // Handle the trivial cases
        if (maxFalsePosProb >= probs[minBuckets][minK]) {
            return new BloomFilterSpecification(2, optKPerBuckets[2]);
        }
        if (maxFalsePosProb < probs[maxBucketsPerElement][maxK]) {
            throw new UnsupportedOperationException(String.format("Unable to satisfy %s with %s buckets per element",
                    maxFalsePosProb, maxBucketsPerElement));
        }

        // First find the minimal required number of buckets:
        int bucketsPerElement = 2;
        int K = optKPerBuckets[2];
        while (probs[bucketsPerElement][K] > maxFalsePosProb) {
            bucketsPerElement++;
            K = optKPerBuckets[bucketsPerElement];
        }
        // Now that the number of buckets is sufficient, see if we can relax K
        // without losing too much precision.
        while (probs[bucketsPerElement][K - 1] <= maxFalsePosProb) {
            K--;
        }

        return new BloomFilterSpecification(K, bucketsPerElement);
    }

    /**
     * Calculates the maximum number of buckets per element that this implementation
     * can support. Crucially, it will lower the bucket count if necessary to meet
     * BitSet's size restrictions.
     */
    public static int maxBucketsPerElement(long numElements) {
        numElements = Math.max(1, numElements);
        double v = Long.MAX_VALUE / (double) numElements;
        if (v < 1.0) {
            throw new UnsupportedOperationException("Cannot compute probabilities for " + numElements + " elements.");
        }
        return Math.min(BloomCalculations.probs.length - 1, (int) v);
    }
}