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The Apache Cassandra Project develops a highly scalable second-generation distributed database, bringing together Dynamo's fully distributed design and Bigtable's ColumnFamily-based data model.
/*
* Licensed to the Apache Software Foundation (ASF) under one
* or more contributor license agreements. See the NOTICE file
* distributed with this work for additional information
* regarding copyright ownership. The ASF licenses this file
* to you 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 at
*
* 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 org.apache.cassandra.utils;
/**
* 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;
private static final int EXCESS = 20;
/**
* In the following keyspaceName, the row 'i' shows false positive rates if i buckets
* per element are used. Cell '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 BloomSpecification computeBloomSpec(int bucketsPerElement)
{
assert bucketsPerElement >= 1;
assert bucketsPerElement <= probs.length - 1;
return new BloomSpecification(optKPerBuckets[bucketsPerElement], bucketsPerElement);
}
/**
* A wrapper class that holds two key parameters for a Bloom Filter: the
* number of hash functions used, and the number of buckets per element used.
*/
public static class BloomSpecification
{
final int K; // number of hash functions.
final int bucketsPerElement;
public BloomSpecification(int k, int bucketsPerElement)
{
K = k;
this.bucketsPerElement = bucketsPerElement;
}
public String toString()
{
return String.format("BloomSpecification(K=%d, bucketsPerElement=%d)", K, 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 BloomSpecification 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 BloomSpecification(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 BloomSpecification(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 - EXCESS) / (double)numElements;
if (v < 1.0)
{
throw new UnsupportedOperationException("Cannot compute probabilities for " + numElements + " elements.");
}
return Math.min(BloomCalculations.probs.length - 1, (int)v);
}
/**
* Retrieves the minimum supported BloomFilterFpChance value
* @return Minimum supported value for BloomFilterFpChance
*/
public static double minSupportedBloomFilterFpChance()
{
int maxBuckets = probs.length - 1;
int maxK = probs[maxBuckets].length - 1;
return probs[maxBuckets][maxK];
}
}
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