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344 lines
11 KiB
C++
344 lines
11 KiB
C++
#pragma once
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#include <cmath>
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#include <Common/RadixSort.h>
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#include <Common/PODArray.h>
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#include <IO/WriteBuffer.h>
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#include <IO/ReadBuffer.h>
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#include <IO/VarInt.h>
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namespace DB
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{
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namespace ErrorCodes
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{
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extern const int TOO_LARGE_ARRAY_SIZE;
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}
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/** The algorithm was implemented by Alexei Borzenkov https://github.com/snaury
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* He owns the authorship of the code and half the comments in this namespace,
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* except for merging, serialization, and sorting, as well as selecting types and other changes.
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* We thank Alexei Borzenkov for writing the original code.
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*/
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/** Implementation of t-digest algorithm (https://github.com/tdunning/t-digest).
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* This option is very similar to MergingDigest on java, however the decision about
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* the union is accepted based on the original condition from the article
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* (via a size constraint, using the approximation of the quantile of each
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* centroid, not the distance on the curve of the position of their boundaries). MergingDigest
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* on java gives significantly fewer centroids than this variant, that
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* negatively affects accuracy with the same compression factor, but gives
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* size guarantees. The author himself on the proposal for this variant said that
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* the size of the digest grows like O(log(n)), while the version on java
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* does not depend on the expected number of points. Also an variant on java
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* uses asin, which slows down the algorithm a bit.
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*/
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template <typename T>
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class QuantileTDigest
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{
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using Value = Float32;
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using Count = Float32;
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/** The centroid stores the weight of points around their mean value
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*/
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struct Centroid
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{
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Value mean;
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Count count;
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Centroid() = default;
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explicit Centroid(Value mean_, Count count_)
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: mean(mean_)
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, count(count_)
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{}
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Centroid & operator+=(const Centroid & other)
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{
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count += other.count;
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mean += other.count * (other.mean - mean) / count;
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return *this;
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}
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bool operator<(const Centroid & other) const
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{
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return mean < other.mean;
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}
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};
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/** :param epsilon: value \delta from the article - error in the range
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* quantile 0.5 (default is 0.01, i.e. 1%)
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* :param max_unmerged: when accumulating count of new points beyond this
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* value centroid compression is triggered
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* (default is 2048, the higher the value - the
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* more memory is required, but amortization of execution time increases)
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*/
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struct Params
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{
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Value epsilon = 0.01;
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size_t max_unmerged = 2048;
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};
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Params params;
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/// The memory will be allocated to several elements at once, so that the state occupies 64 bytes.
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static constexpr size_t bytes_in_arena = 128 - sizeof(PODArray<Centroid>) - sizeof(Count) - sizeof(UInt32);
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using Summary = PODArrayWithStackMemory<Centroid, bytes_in_arena>;
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Summary summary;
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Count count = 0;
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UInt32 unmerged = 0;
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/** Linear interpolation at the point x on the line (x1, y1)..(x2, y2)
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*/
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static Value interpolate(Value x, Value x1, Value y1, Value x2, Value y2)
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{
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double k = (x - x1) / (x2 - x1);
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return y1 + k * (y2 - y1);
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}
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struct RadixSortTraits
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{
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using Element = Centroid;
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using Key = Value;
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using CountType = UInt32;
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using KeyBits = UInt32;
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static constexpr size_t PART_SIZE_BITS = 8;
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using Transform = RadixSortFloatTransform<KeyBits>;
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using Allocator = RadixSortMallocAllocator;
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/// The function to get the key from an array element.
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static Key & extractKey(Element & elem) { return elem.mean; }
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};
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/** Adds a centroid `c` to the digest
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*/
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void addCentroid(const Centroid & c)
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{
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summary.push_back(c);
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count += c.count;
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++unmerged;
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if (unmerged >= params.max_unmerged)
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compress();
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}
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/** Performs compression of accumulated centroids
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* When merging, the invariant is retained to the maximum size of each
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* centroid that does not exceed `4 q (1 - q) \ delta N`.
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*/
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void compress()
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{
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if (unmerged > 0)
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{
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RadixSort<RadixSortTraits>::executeLSD(summary.data(), summary.size());
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if (summary.size() > 3)
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{
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/// A pair of consecutive bars of the histogram.
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auto l = summary.begin();
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auto r = std::next(l);
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Count sum = 0;
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while (r != summary.end())
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{
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// we use quantile which gives us the smallest error
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/// The ratio of the part of the histogram to l, including the half l to the entire histogram. That is, what level quantile in position l.
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Value ql = (sum + l->count * 0.5) / count;
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Value err = ql * (1 - ql);
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/// The ratio of the portion of the histogram to l, including l and half r to the entire histogram. That is, what level is the quantile in position r.
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Value qr = (sum + l->count + r->count * 0.5) / count;
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Value err2 = qr * (1 - qr);
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if (err > err2)
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err = err2;
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Value k = 4 * count * err * params.epsilon;
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/** The ratio of the weight of the glued column pair to all values is not greater,
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* than epsilon multiply by a certain quadratic coefficient, which in the median is 1 (4 * 1/2 * 1/2),
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* and at the edges decreases and is approximately equal to the distance to the edge * 4.
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*/
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if (l->count + r->count <= k)
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{
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// it is possible to merge left and right
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/// The left column "eats" the right.
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*l += *r;
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}
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else
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{
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// not enough capacity, check the next pair
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sum += l->count;
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++l;
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/// We skip all the values "eaten" earlier.
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if (l != r)
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*l = *r;
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}
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++r;
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}
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/// At the end of the loop, all values to the right of l were "eaten".
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summary.resize(l - summary.begin() + 1);
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}
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unmerged = 0;
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}
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}
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public:
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/** Adds to the digest a change in `x` with a weight of `cnt` (default 1)
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*/
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void add(T x, UInt64 cnt = 1)
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{
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addCentroid(Centroid(Value(x), Count(cnt)));
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}
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void merge(const QuantileTDigest & other)
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{
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for (const auto & c : other.summary)
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addCentroid(c);
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}
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void serialize(WriteBuffer & buf)
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{
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compress();
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writeVarUInt(summary.size(), buf);
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buf.write(reinterpret_cast<const char *>(summary.data()), summary.size() * sizeof(summary[0]));
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}
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void deserialize(ReadBuffer & buf)
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{
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size_t size = 0;
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readVarUInt(size, buf);
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if (size > params.max_unmerged)
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throw Exception("Too large t-digest summary size", ErrorCodes::TOO_LARGE_ARRAY_SIZE);
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summary.resize(size);
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buf.read(reinterpret_cast<char *>(summary.data()), size * sizeof(summary[0]));
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count = 0;
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for (const auto & c : summary)
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count += c.count;
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}
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/** Calculates the quantile q [0, 1] based on the digest.
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* For an empty digest returns NaN.
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*/
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template <typename ResultType>
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ResultType getImpl(Float64 level)
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{
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if (summary.empty())
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return std::is_floating_point_v<ResultType> ? NAN : 0;
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compress();
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if (summary.size() == 1)
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return summary.front().mean;
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Float64 x = level * count;
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Float64 prev_x = 0;
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Count sum = 0;
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Value prev_mean = summary.front().mean;
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for (const auto & c : summary)
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{
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Float64 current_x = sum + c.count * 0.5;
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if (current_x >= x)
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return interpolate(x, prev_x, prev_mean, current_x, c.mean);
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sum += c.count;
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prev_mean = c.mean;
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prev_x = current_x;
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}
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return summary.back().mean;
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}
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/** Get multiple quantiles (`size` parts).
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* levels - an array of levels of the desired quantiles. They are in a random order.
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* levels_permutation - array-permutation levels. The i-th position will be the index of the i-th ascending level in the `levels` array.
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* result - the array where the results are added, in order of `levels`,
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*/
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template <typename ResultType>
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void getManyImpl(const Float64 * levels, const size_t * levels_permutation, size_t size, ResultType * result)
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{
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if (summary.empty())
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{
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for (size_t result_num = 0; result_num < size; ++result_num)
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result[result_num] = std::is_floating_point_v<ResultType> ? NAN : 0;
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return;
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}
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compress();
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if (summary.size() == 1)
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{
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for (size_t result_num = 0; result_num < size; ++result_num)
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result[result_num] = summary.front().mean;
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return;
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}
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Float64 x = levels[levels_permutation[0]] * count;
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Float64 prev_x = 0;
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Count sum = 0;
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Value prev_mean = summary.front().mean;
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size_t result_num = 0;
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for (const auto & c : summary)
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{
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Float64 current_x = sum + c.count * 0.5;
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while (current_x >= x)
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{
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result[levels_permutation[result_num]] = interpolate(x, prev_x, prev_mean, current_x, c.mean);
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++result_num;
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if (result_num >= size)
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return;
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x = levels[levels_permutation[result_num]] * count;
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}
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sum += c.count;
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prev_mean = c.mean;
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prev_x = current_x;
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}
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auto rest_of_results = summary.back().mean;
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for (; result_num < size; ++result_num)
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result[levels_permutation[result_num]] = rest_of_results;
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}
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T get(Float64 level)
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{
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return getImpl<T>(level);
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}
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Float32 getFloat(Float64 level)
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{
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return getImpl<Float32>(level);
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}
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void getMany(const Float64 * levels, const size_t * indices, size_t size, T * result)
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{
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getManyImpl(levels, indices, size, result);
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}
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void getManyFloat(const Float64 * levels, const size_t * indices, size_t size, Float32 * result)
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{
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getManyImpl(levels, indices, size, result);
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}
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};
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}
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