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// Package quantile computes approximate quantiles over an unbounded data
// stream within low memory and CPU bounds.
//
// A small amount of accuracy is traded to achieve the above properties.
//
// Multiple streams can be merged before calling Query to generate a single set
// of results. This is meaningful when the streams represent the same type of
// data. See Merge and Samples.
//
// For more detailed information about the algorithm used, see:
//
// Effective Computation of Biased Quantiles over Data Streams
//
// http://www.cs.rutgers.edu/~muthu/bquant.pdf
package quantile
import (
"math"
"sort"
)
// Sample holds an observed value and meta information for compression. JSON
// tags have been added for convenience.
type Sample struct {
Value float64 `json:",string"`
Width float64 `json:",string"`
Delta float64 `json:",string"`
}
// Samples represents a slice of samples. It implements sort.Interface.
type Samples []Sample
func (a Samples) Len() int { return len(a) }
func (a Samples) Less(i, j int) bool { return a[i].Value < a[j].Value }
func (a Samples) Swap(i, j int) { a[i], a[j] = a[j], a[i] }
type invariant func(s *stream, r float64) float64
// NewLowBiased returns an initialized Stream for low-biased quantiles
// (e.g. 0.01, 0.1, 0.5) where the needed quantiles are not known a priori, but
// error guarantees can still be given even for the lower ranks of the data
// distribution.
//
// The provided epsilon is a relative error, i.e. the true quantile of a value
// returned by a query is guaranteed to be within (1±Epsilon)*Quantile.
//
// See http://www.cs.rutgers.edu/~muthu/bquant.pdf for time, space, and error
// properties.
func NewLowBiased(epsilon float64) *Stream {
ƒ := func(s *stream, r float64) float64 {
return 2 * epsilon * r
}
return newStream(ƒ)
}
// NewHighBiased returns an initialized Stream for high-biased quantiles
// (e.g. 0.01, 0.1, 0.5) where the needed quantiles are not known a priori, but
// error guarantees can still be given even for the higher ranks of the data
// distribution.
//
// The provided epsilon is a relative error, i.e. the true quantile of a value
// returned by a query is guaranteed to be within 1-(1±Epsilon)*(1-Quantile).
//
// See http://www.cs.rutgers.edu/~muthu/bquant.pdf for time, space, and error
// properties.
func NewHighBiased(epsilon float64) *Stream {
ƒ := func(s *stream, r float64) float64 {
return 2 * epsilon * (s.n - r)
}
return newStream(ƒ)
}
// NewTargeted returns an initialized Stream concerned with a particular set of
// quantile values that are supplied a priori. Knowing these a priori reduces
// space and computation time. The targets map maps the desired quantiles to
// their absolute errors, i.e. the true quantile of a value returned by a query
// is guaranteed to be within (Quantile±Epsilon).
//
// See http://www.cs.rutgers.edu/~muthu/bquant.pdf for time, space, and error properties.
func NewTargeted(targetMap map[float64]float64) *Stream {
// Convert map to slice to avoid slow iterations on a map.
// ƒ is called on the hot path, so converting the map to a slice
// beforehand results in significant CPU savings.
targets := targetMapToSlice(targetMap)
ƒ := func(s *stream, r float64) float64 {
var m = math.MaxFloat64
var f float64
for _, t := range targets {
if t.quantile*s.n <= r {
f = (2 * t.epsilon * r) / t.quantile
} else {
f = (2 * t.epsilon * (s.n - r)) / (1 - t.quantile)
}
if f < m {
m = f
}
}
return m
}
return newStream(ƒ)
}
type target struct {
quantile float64
epsilon float64
}
func targetMapToSlice(targetMap map[float64]float64) []target {
targets := make([]target, 0, len(targetMap))
for quantile, epsilon := range targetMap {
t := target{
quantile: quantile,
epsilon: epsilon,
}
targets = append(targets, t)
}
return targets
}
// Stream computes quantiles for a stream of float64s. It is not thread-safe by
// design. Take care when using across multiple goroutines.
type Stream struct {
*stream
b Samples
sorted bool
}
func newStream(ƒ invariant) *Stream {
x := &stream{ƒ: ƒ}
return &Stream{x, make(Samples, 0, 500), true}
}
// Insert inserts v into the stream.
func (s *Stream) Insert(v float64) {
s.insert(Sample{Value: v, Width: 1})
}
func (s *Stream) insert(sample Sample) {
s.b = append(s.b, sample)
s.sorted = false
if len(s.b) == cap(s.b) {
s.flush()
}
}
// Query returns the computed qth percentiles value. If s was created with
// NewTargeted, and q is not in the set of quantiles provided a priori, Query
// will return an unspecified result.
func (s *Stream) Query(q float64) float64 {
if !s.flushed() {
// Fast path when there hasn't been enough data for a flush;
// this also yields better accuracy for small sets of data.
l := len(s.b)
if l == 0 {
return 0
}
i := int(math.Ceil(float64(l) * q))
if i > 0 {
i -= 1
}
s.maybeSort()
return s.b[i].Value
}
s.flush()
return s.stream.query(q)
}
// Merge merges samples into the underlying streams samples. This is handy when
// merging multiple streams from separate threads, database shards, etc.
//
// ATTENTION: This method is broken and does not yield correct results. The
// underlying algorithm is not capable of merging streams correctly.
func (s *Stream) Merge(samples Samples) {
sort.Sort(samples)
s.stream.merge(samples)
}
// Reset reinitializes and clears the list reusing the samples buffer memory.
func (s *Stream) Reset() {
s.stream.reset()
s.b = s.b[:0]
}
// Samples returns stream samples held by s.
func (s *Stream) Samples() Samples {
if !s.flushed() {
return s.b
}
s.flush()
return s.stream.samples()
}
// Count returns the total number of samples observed in the stream
// since initialization.
func (s *Stream) Count() int {
return len(s.b) + s.stream.count()
}
func (s *Stream) flush() {
s.maybeSort()
s.stream.merge(s.b)
s.b = s.b[:0]
}
func (s *Stream) maybeSort() {
if !s.sorted {
s.sorted = true
sort.Sort(s.b)
}
}
func (s *Stream) flushed() bool {
return len(s.stream.l) > 0
}
type stream struct {
n float64
l []Sample
ƒ invariant
}
func (s *stream) reset() {
s.l = s.l[:0]
s.n = 0
}
func (s *stream) insert(v float64) {
s.merge(Samples{{v, 1, 0}})
}
func (s *stream) merge(samples Samples) {
// TODO(beorn7): This tries to merge not only individual samples, but
// whole summaries. The paper doesn't mention merging summaries at
// all. Unittests show that the merging is inaccurate. Find out how to
// do merges properly.
var r float64
i := 0
for _, sample := range samples {
for ; i < len(s.l); i++ {
c := s.l[i]
if c.Value > sample.Value {
// Insert at position i.
s.l = append(s.l, Sample{})
copy(s.l[i+1:], s.l[i:])
s.l[i] = Sample{
sample.Value,
sample.Width,
math.Max(sample.Delta, math.Floor(s.ƒ(s, r))-1),
// TODO(beorn7): How to calculate delta correctly?
}
i++
goto inserted
}
r += c.Width
}
s.l = append(s.l, Sample{sample.Value, sample.Width, 0})
i++
inserted:
s.n += sample.Width
r += sample.Width
}
s.compress()
}
func (s *stream) count() int {
return int(s.n)
}
func (s *stream) query(q float64) float64 {
t := math.Ceil(q * s.n)
t += math.Ceil(s.ƒ(s, t) / 2)
p := s.l[0]
var r float64
for _, c := range s.l[1:] {
r += p.Width
if r+c.Width+c.Delta > t {
return p.Value
}
p = c
}
return p.Value
}
func (s *stream) compress() {
if len(s.l) < 2 {
return
}
x := s.l[len(s.l)-1]
xi := len(s.l) - 1
r := s.n - 1 - x.Width
for i := len(s.l) - 2; i >= 0; i-- {
c := s.l[i]
if c.Width+x.Width+x.Delta <= s.ƒ(s, r) {
x.Width += c.Width
s.l[xi] = x
// Remove element at i.
copy(s.l[i:], s.l[i+1:])
s.l = s.l[:len(s.l)-1]
xi -= 1
} else {
x = c
xi = i
}
r -= c.Width
}
}
func (s *stream) samples() Samples {
samples := make(Samples, len(s.l))
copy(samples, s.l)
return samples
}
