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Apache License
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The Java version of the t-digest was originally authored by Ted Dunning
A number of small but very helpful changes have been contributed by Adrien Grand (https://github.com/jpountz) to the Java version.
The C++ version herein is a derivative of the Java version. It was written by Derrick R. Burns (https:://github.com/derrickburns).
The main modifications are 1) higher performance multi- t-digest merging and 2) faster quantile() and cdf() computation.

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/*
* Licensed to Derrick R. Burns under one or more
* contributor license agreements. See the NOTICES 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.
*/
#pragma once
#include <algorithm>
#include <cfloat>
#include <cmath>
#include <queue>
#include <utility>
#include <vector>
#ifdef min
#undef min
#endif
#ifdef max
#undef max
#endif
namespace duckdb_tdigest {
using Value = double;
using Weight = double;
using Index = size_t;
const size_t kHighWater = 40000;
const double pi = 3.14159265358979323846;
class Centroid {
public:
Centroid() : Centroid(0.0, 0.0) {
}
Centroid(Value mean, Weight weight) : mean_(mean), weight_(weight) {
}
inline Value mean() const noexcept {
return mean_;
}
inline Weight weight() const noexcept {
return weight_;
}
inline void add(const Centroid &c) {
// CHECK_GT(c.weight_, 0);
if (weight_ != 0.0) {
weight_ += c.weight_;
mean_ += c.weight_ * (c.mean_ - mean_) / weight_;
} else {
weight_ = c.weight_;
mean_ = c.mean_;
}
}
private:
Value mean_ = 0;
Weight weight_ = 0;
};
struct CentroidList {
explicit CentroidList(const std::vector<Centroid> &s) : iter(s.cbegin()), end(s.cend()) {
}
std::vector<Centroid>::const_iterator iter;
std::vector<Centroid>::const_iterator end;
bool advance() {
return ++iter != end;
}
};
class CentroidListComparator {
public:
CentroidListComparator() {
}
bool operator()(const CentroidList &left, const CentroidList &right) const {
return left.iter->mean() > right.iter->mean();
}
};
using CentroidListQueue = std::priority_queue<CentroidList, std::vector<CentroidList>, CentroidListComparator>;
struct CentroidComparator {
bool operator()(const Centroid &a, const Centroid &b) const {
return a.mean() < b.mean();
}
};
class TDigest {
class TDigestComparator {
public:
TDigestComparator() {
}
bool operator()(const TDigest *left, const TDigest *right) const {
return left->totalSize() > right->totalSize();
}
};
using TDigestQueue = std::priority_queue<const TDigest *, std::vector<const TDigest *>, TDigestComparator>;
public:
TDigest() : TDigest(1000) {
}
explicit TDigest(Value compression) : TDigest(compression, 0) {
}
TDigest(Value compression, Index bufferSize) : TDigest(compression, bufferSize, 0) {
}
TDigest(Value compression, Index unmergedSize, Index mergedSize)
: compression_(compression), maxProcessed_(processedSize(mergedSize, compression)),
maxUnprocessed_(unprocessedSize(unmergedSize, compression)) {
processed_.reserve(maxProcessed_);
unprocessed_.reserve(maxUnprocessed_ + 1);
}
TDigest(std::vector<Centroid> &&processed, std::vector<Centroid> &&unprocessed, Value compression,
Index unmergedSize, Index mergedSize)
: TDigest(compression, unmergedSize, mergedSize) {
processed_ = std::move(processed);
unprocessed_ = std::move(unprocessed);
processedWeight_ = weight(processed_);
unprocessedWeight_ = weight(unprocessed_);
if (!processed_.empty()) {
min_ = std::min(min_, processed_[0].mean());
max_ = std::max(max_, (processed_.cend() - 1)->mean());
}
updateCumulative();
}
static Weight weight(std::vector<Centroid> &centroids) noexcept {
Weight w = 0.0;
for (auto centroid : centroids) {
w += centroid.weight();
}
return w;
}
TDigest &operator=(TDigest &&o) {
compression_ = o.compression_;
maxProcessed_ = o.maxProcessed_;
maxUnprocessed_ = o.maxUnprocessed_;
processedWeight_ = o.processedWeight_;
unprocessedWeight_ = o.unprocessedWeight_;
processed_ = std::move(o.processed_);
unprocessed_ = std::move(o.unprocessed_);
cumulative_ = std::move(o.cumulative_);
min_ = o.min_;
max_ = o.max_;
return *this;
}
TDigest(TDigest &&o)
: TDigest(std::move(o.processed_), std::move(o.unprocessed_), o.compression_, o.maxUnprocessed_,
o.maxProcessed_) {
}
static inline Index processedSize(Index size, Value compression) noexcept {
return (size == 0) ? static_cast<Index>(2 * std::ceil(compression)) : size;
}
static inline Index unprocessedSize(Index size, Value compression) noexcept {
return (size == 0) ? static_cast<Index>(8 * std::ceil(compression)) : size;
}
// merge in another t-digest
inline void merge(const TDigest *other) {
std::vector<const TDigest *> others {other};
add(others.cbegin(), others.cend());
}
const std::vector<Centroid> &processed() const {
return processed_;
}
const std::vector<Centroid> &unprocessed() const {
return unprocessed_;
}
Index maxUnprocessed() const {
return maxUnprocessed_;
}
Index maxProcessed() const {
return maxProcessed_;
}
inline void add(std::vector<const TDigest *> digests) {
add(digests.cbegin(), digests.cend());
}
// merge in a vector of tdigests in the most efficient manner possible
// in constant space
// works for any value of kHighWater
void add(std::vector<const TDigest *>::const_iterator iter, std::vector<const TDigest *>::const_iterator end) {
if (iter != end) {
auto size = std::distance(iter, end);
TDigestQueue pq(TDigestComparator {});
for (; iter != end; iter++) {
pq.push((*iter));
}
std::vector<const TDigest *> batch;
batch.reserve(size_t(size));
size_t totalSize = 0;
while (!pq.empty()) {
auto td = pq.top();
batch.push_back(td);
pq.pop();
totalSize += td->totalSize();
if (totalSize >= kHighWater || pq.empty()) {
mergeProcessed(batch);
mergeUnprocessed(batch);
processIfNecessary();
batch.clear();
totalSize = 0;
}
}
updateCumulative();
}
}
Weight processedWeight() const {
return processedWeight_;
}
Weight unprocessedWeight() const {
return unprocessedWeight_;
}
bool haveUnprocessed() const {
return unprocessed_.size() > 0;
}
size_t totalSize() const {
return processed_.size() + unprocessed_.size();
}
long totalWeight() const {
return static_cast<long>(processedWeight_ + unprocessedWeight_);
}
// return the cdf on the t-digest
Value cdf(Value x) {
if (haveUnprocessed() || isDirty()) {
process();
}
return cdfProcessed(x);
}
bool isDirty() {
return processed_.size() > maxProcessed_ || unprocessed_.size() > maxUnprocessed_;
}
// return the cdf on the processed values
Value cdfProcessed(Value x) const {
if (processed_.empty()) {
// no data to examin_e
return 0.0;
} else if (processed_.size() == 1) {
// exactly one centroid, should have max_==min_
auto width = max_ - min_;
if (x < min_) {
return 0.0;
} else if (x > max_) {
return 1.0;
} else if (x - min_ <= width) {
// min_ and max_ are too close together to do any viable interpolation
return 0.5;
} else {
// interpolate if somehow we have weight > 0 and max_ != min_
return (x - min_) / (max_ - min_);
}
} else {
auto n = processed_.size();
if (x <= min_) {
return 0;
}
if (x >= max_) {
return 1;
}
// check for the left tail
if (x <= mean(0)) {
// note that this is different than mean(0) > min_ ... this guarantees interpolation works
if (mean(0) - min_ > 0) {
return (x - min_) / (mean(0) - min_) * weight(0) / processedWeight_ / 2.0;
} else {
return 0;
}
}
// and the right tail
if (x >= mean(n - 1)) {
if (max_ - mean(n - 1) > 0) {
return 1.0 - (max_ - x) / (max_ - mean(n - 1)) * weight(n - 1) / processedWeight_ / 2.0;
} else {
return 1;
}
}
CentroidComparator cc;
auto iter = std::upper_bound(processed_.cbegin(), processed_.cend(), Centroid(x, 0), cc);
auto i = size_t(std::distance(processed_.cbegin(), iter));
auto z1 = x - (iter - 1)->mean();
auto z2 = (iter)->mean() - x;
return weightedAverage(cumulative_[i - 1], z2, cumulative_[i], z1) / processedWeight_;
}
}
// this returns a quantile on the t-digest
Value quantile(Value q) {
if (haveUnprocessed() || isDirty()) {
process();
}
return quantileProcessed(q);
}
// this returns a quantile on the currently processed values without changing the t-digest
// the value will not represent the unprocessed values
Value quantileProcessed(Value q) const {
if (q < 0 || q > 1) {
return NAN;
}
if (processed_.size() == 0) {
// no sorted means no data, no way to get a quantile
return NAN;
} else if (processed_.size() == 1) {
// with one data point, all quantiles lead to Rome
return mean(0);
}
// we know that there are at least two sorted now
auto n = processed_.size();
// if values were stored in a sorted array, index would be the offset we are Weighterested in
const auto index = q * processedWeight_;
// at the boundaries, we return min_ or max_
if (index <= weight(0) / 2.0) {
return min_ + 2.0 * index / weight(0) * (mean(0) - min_);
}
auto iter = std::lower_bound(cumulative_.cbegin(), cumulative_.cend(), index);
if (iter + 1 != cumulative_.cend()) {
auto i = size_t(std::distance(cumulative_.cbegin(), iter));
auto z1 = index - *(iter - 1);
auto z2 = *(iter)-index;
// LOG(INFO) << "z2 " << z2 << " index " << index << " z1 " << z1;
return weightedAverage(mean(i - 1), z2, mean(i), z1);
}
auto z1 = index - processedWeight_ - weight(n - 1) / 2.0;
auto z2 = weight(n - 1) / 2 - z1;
return weightedAverage(mean(n - 1), z1, max_, z2);
}
Value compression() const {
return compression_;
}
void add(Value x) {
add(x, 1);
}
inline void compress() {
process();
}
// add a single centroid to the unprocessed vector, processing previously unprocessed sorted if our limit has
// been reached.
inline bool add(Value x, Weight w) {
if (std::isnan(x)) {
return false;
}
unprocessed_.push_back(Centroid(x, w));
unprocessedWeight_ += w;
processIfNecessary();
return true;
}
inline void add(std::vector<Centroid>::const_iterator iter, std::vector<Centroid>::const_iterator end) {
while (iter != end) {
const size_t diff = size_t(std::distance(iter, end));
const size_t room = maxUnprocessed_ - unprocessed_.size();
auto mid = iter + int64_t(std::min(diff, room));
while (iter != mid) {
unprocessed_.push_back(*(iter++));
}
if (unprocessed_.size() >= maxUnprocessed_) {
process();
}
}
}
private:
Value compression_;
Value min_ = std::numeric_limits<Value>::max();
Value max_ = std::numeric_limits<Value>::min();
Index maxProcessed_;
Index maxUnprocessed_;
Value processedWeight_ = 0.0;
Value unprocessedWeight_ = 0.0;
std::vector<Centroid> processed_;
std::vector<Centroid> unprocessed_;
std::vector<Weight> cumulative_;
// return mean of i-th centroid
inline Value mean(size_t i) const noexcept {
return processed_[i].mean();
}
// return weight of i-th centroid
inline Weight weight(size_t i) const noexcept {
return processed_[i].weight();
}
// append all unprocessed centroids into current unprocessed vector
void mergeUnprocessed(const std::vector<const TDigest *> &tdigests) {
if (tdigests.size() == 0) {
return;
}
size_t total = unprocessed_.size();
for (auto &td : tdigests) {
total += td->unprocessed_.size();
}
unprocessed_.reserve(total);
for (auto &td : tdigests) {
unprocessed_.insert(unprocessed_.end(), td->unprocessed_.cbegin(), td->unprocessed_.cend());
unprocessedWeight_ += td->unprocessedWeight_;
}
}
// merge all processed centroids together into a single sorted vector
void mergeProcessed(const std::vector<const TDigest *> &tdigests) {
if (tdigests.size() == 0) {
return;
}
size_t total = 0;
CentroidListQueue pq(CentroidListComparator {});
for (auto &td : tdigests) {
auto &sorted = td->processed_;
auto size = sorted.size();
if (size > 0) {
pq.push(CentroidList(sorted));
total += size;
processedWeight_ += td->processedWeight_;
}
}
if (total == 0) {
return;
}
if (processed_.size() > 0) {
pq.push(CentroidList(processed_));
total += processed_.size();
}
std::vector<Centroid> sorted;
sorted.reserve(total);
while (!pq.empty()) {
auto best = pq.top();
pq.pop();
sorted.push_back(*(best.iter));
if (best.advance()) {
pq.push(best);
}
}
processed_ = std::move(sorted);
if (processed_.size() > 0) {
min_ = std::min(min_, processed_[0].mean());
max_ = std::max(max_, (processed_.cend() - 1)->mean());
}
}
inline void processIfNecessary() {
if (isDirty()) {
process();
}
}
void updateCumulative() {
const auto n = processed_.size();
cumulative_.clear();
cumulative_.reserve(n + 1);
auto previous = 0.0;
for (Index i = 0; i < n; i++) {
auto current = weight(i);
auto halfCurrent = current / 2.0;
cumulative_.push_back(previous + halfCurrent);
previous = previous + current;
}
cumulative_.push_back(previous);
}
// merges unprocessed_ centroids and processed_ centroids together and processes them
// when complete, unprocessed_ will be empty and processed_ will have at most maxProcessed_ centroids
inline void process() {
CentroidComparator cc;
std::sort(unprocessed_.begin(), unprocessed_.end(), cc);
auto count = unprocessed_.size();
unprocessed_.insert(unprocessed_.end(), processed_.cbegin(), processed_.cend());
std::inplace_merge(unprocessed_.begin(), unprocessed_.begin() + int64_t(count), unprocessed_.end(), cc);
processedWeight_ += unprocessedWeight_;
unprocessedWeight_ = 0;
processed_.clear();
processed_.push_back(unprocessed_[0]);
Weight wSoFar = unprocessed_[0].weight();
Weight wLimit = processedWeight_ * integratedQ(1.0);
auto end = unprocessed_.end();
for (auto iter = unprocessed_.cbegin() + 1; iter < end; iter++) {
auto &centroid = *iter;
Weight projectedW = wSoFar + centroid.weight();
if (projectedW <= wLimit) {
wSoFar = projectedW;
(processed_.end() - 1)->add(centroid);
} else {
auto k1 = integratedLocation(wSoFar / processedWeight_);
wLimit = processedWeight_ * integratedQ(k1 + 1.0);
wSoFar += centroid.weight();
processed_.emplace_back(centroid);
}
}
unprocessed_.clear();
min_ = std::min(min_, processed_[0].mean());
max_ = std::max(max_, (processed_.cend() - 1)->mean());
updateCumulative();
}
inline size_t checkWeights() {
return checkWeights(processed_, processedWeight_);
}
size_t checkWeights(const std::vector<Centroid> &sorted, Value total) {
size_t badWeight = 0;
auto k1 = 0.0;
auto q = 0.0;
for (auto iter = sorted.cbegin(); iter != sorted.cend(); iter++) {
auto w = iter->weight();
auto dq = w / total;
auto k2 = integratedLocation(q + dq);
if (k2 - k1 > 1 && w != 1) {
badWeight++;
}
if (k2 - k1 > 1.5 && w != 1) {
badWeight++;
}
q += dq;
k1 = k2;
}
return badWeight;
}
/**
* Converts a quantile into a centroid scale value. The centroid scale is nomin_ally
* the number k of the centroid that a quantile point q should belong to. Due to
* round-offs, however, we can't align things perfectly without splitting points
* and sorted. We don't want to do that, so we have to allow for offsets.
* In the end, the criterion is that any quantile range that spans a centroid
* scale range more than one should be split across more than one centroid if
* possible. This won't be possible if the quantile range refers to a single point
* or an already existing centroid.
* <p/>
* This mapping is steep near q=0 or q=1 so each centroid there will correspond to
* less q range. Near q=0.5, the mapping is flatter so that sorted there will
* represent a larger chunk of quantiles.
*
* @param q The quantile scale value to be mapped.
* @return The centroid scale value corresponding to q.
*/
inline Value integratedLocation(Value q) const {
return compression_ * (std::asin(2.0 * q - 1.0) + pi / 2) / pi;
}
inline Value integratedQ(Value k) const {
return (std::sin(std::min(k, compression_) * pi / compression_ - pi / 2) + 1) / 2;
}
/**
* Same as {@link #weightedAverageSorted(Value, Value, Value, Value)} but flips
* the order of the variables if <code>x2</code> is greater than
* <code>x1</code>.
*/
static Value weightedAverage(Value x1, Value w1, Value x2, Value w2) {
return (x1 <= x2) ? weightedAverageSorted(x1, w1, x2, w2) : weightedAverageSorted(x2, w2, x1, w1);
}
/**
* Compute the weighted average between <code>x1</code> with a weight of
* <code>w1</code> and <code>x2</code> with a weight of <code>w2</code>.
* This expects <code>x1</code> to be less than or equal to <code>x2</code>
* and is guaranteed to return a number between <code>x1</code> and
* <code>x2</code>.
*/
static Value weightedAverageSorted(Value x1, Value w1, Value x2, Value w2) {
const Value x = (x1 * w1 + x2 * w2) / (w1 + w2);
return std::max(x1, std::min(x, x2));
}
static Value interpolate(Value x, Value x0, Value x1) {
return (x - x0) / (x1 - x0);
}
/**
* Computes an interpolated value of a quantile that is between two sorted.
*
* Index is the quantile desired multiplied by the total number of samples - 1.
*
* @param index Denormalized quantile desired
* @param previousIndex The denormalized quantile corresponding to the center of the previous centroid.
* @param nextIndex The denormalized quantile corresponding to the center of the following centroid.
* @param previousMean The mean of the previous centroid.
* @param nextMean The mean of the following centroid.
* @return The interpolated mean.
*/
static Value quantile(Value index, Value previousIndex, Value nextIndex, Value previousMean, Value nextMean) {
const auto delta = nextIndex - previousIndex;
const auto previousWeight = (nextIndex - index) / delta;
const auto nextWeight = (index - previousIndex) / delta;
return previousMean * previousWeight + nextMean * nextWeight;
}
};
} // namespace tdigest