util/topk: add package containing a probabilistic top-K tracker
This package uses a count-min sketch and a heap to track the top K items in a stream of data. Tracking a new item and adding a count to an existing item both require no memory allocations and is at worst O(log(k)) complexity. Change-Id: I0553381be3fef2470897e2bd806d43396f2dbb36 Signed-off-by: Andrew Dunham <andrew@du.nham.ca>
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Andrew Dunham
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// Copyright (c) Tailscale Inc & AUTHORS
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// SPDX-License-Identifier: BSD-3-Clause
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// Package topk defines a count-min sketch and a cheap probabilistic top-K data
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// structure that uses the count-min sketch to track the top K items in
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// constant memory and O(log(k)) time.
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package topk
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import (
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"container/heap"
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"hash/maphash"
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"math"
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"slices"
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"sync"
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)
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// TopK is a probabilistic counter of the top K items, using a count-min sketch
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// to keep track of item counts and a heap to track the top K of them.
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type TopK[T any] struct {
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heap minHeap[T]
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k int
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sf SerializeFunc[T]
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cms CountMinSketch
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}
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// HashFunc is responsible for providing a []byte serialization of a value,
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// appended to the provided byte slice. This is used for hashing the value when
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// adding to a CountMinSketch.
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type SerializeFunc[T any] func([]byte, T) []byte
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// New creates a new TopK that stores k values. Parameters for the underlying
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// count-min sketch are chosen for a 0.1% error rate and a 0.1% probability of
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// error.
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func New[T any](k int, sf SerializeFunc[T]) *TopK[T] {
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hashes, buckets := PickParams(0.001, 0.001)
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return NewWithParams(k, sf, hashes, buckets)
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}
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// NewWithParams creates a new TopK that stores k values, and additionally
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// allows customizing the parameters for the underlying count-min sketch.
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func NewWithParams[T any](k int, sf SerializeFunc[T], numHashes, numCols int) *TopK[T] {
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ret := &TopK[T]{
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heap: make(minHeap[T], 0, k),
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k: k,
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sf: sf,
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}
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ret.cms.init(numHashes, numCols)
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return ret
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}
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// Add calls AddN(val, 1).
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func (tk *TopK[T]) Add(val T) uint64 {
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return tk.AddN(val, 1)
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}
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var hashPool = &sync.Pool{
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New: func() any {
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buf := make([]byte, 0, 128)
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return &buf
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},
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}
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// AddN adds the given item to the set with the provided count, returning the
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// new estimated count.
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func (tk *TopK[T]) AddN(val T, count uint64) uint64 {
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buf := hashPool.Get().(*[]byte)
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defer hashPool.Put(buf)
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ser := tk.sf((*buf)[:0], val)
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vcount := tk.cms.AddN(ser, count)
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// If we don't have a full heap, just push it.
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if len(tk.heap) < tk.k {
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heap.Push(&tk.heap, mhValue[T]{
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count: vcount,
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val: val,
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})
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return vcount
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}
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// If this item's count surpasses the heap's minimum, update the heap.
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if vcount > tk.heap[0].count {
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tk.heap[0] = mhValue[T]{
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count: vcount,
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val: val,
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}
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heap.Fix(&tk.heap, 0)
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}
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return vcount
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}
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// Top returns the estimated top K items as stored by this TopK.
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func (tk *TopK[T]) Top() []T {
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ret := make([]T, 0, tk.k)
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for _, item := range tk.heap {
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ret = append(ret, item.val)
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}
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return ret
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}
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// AppendTop appends the estimated top K items as stored by this TopK to the
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// provided slice, allocating only if the slice does not have enough capacity
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// to store all items. The provided slice can be nil.
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func (tk *TopK[T]) AppendTop(sl []T) []T {
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sl = slices.Grow(sl, tk.k)
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for _, item := range tk.heap {
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sl = append(sl, item.val)
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}
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return sl
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}
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// CountMinSketch implements a count-min sketch, a probabilistic data structure
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// that tracks the frequency of events in a stream of data.
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//
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// See: https://en.wikipedia.org/wiki/Count%E2%80%93min_sketch
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type CountMinSketch struct {
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hashes []maphash.Seed
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nbuckets int
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matrix []uint64
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}
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// NewCountMinSketch creates a new CountMinSketch with the provided number of
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// hashes and buckets. Hashes and buckets are often called "depth" and "width",
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// or "d" and "w", respectively.
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func NewCountMinSketch(hashes, buckets int) *CountMinSketch {
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ret := &CountMinSketch{}
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ret.init(hashes, buckets)
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return ret
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}
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// PickParams provides good parameters for 'hashes' and 'buckets' when
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// constructing a CountMinSketch, given an estimated total number of counts
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// (i.e. the sum of all counts ever stored), the error factor ϵ as a float
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// (e.g. 1% is 0.001), and the probability factor δ.
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//
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// Parameters are chosen such that with a probability of 1−δ, the error is at
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// most ϵ∗totalCount. Or, in other words: if N is the true count of an event,
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// E is the estimate given by a sketch and T the total count of items in the
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// sketch, E ≤ N + T*ϵ with probability (1 - δ).
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func PickParams(err, probability float64) (hashes, buckets int) {
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d := math.Ceil(math.Log(1 / probability))
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w := math.Ceil(math.E / err)
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return int(d), int(w)
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}
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func (cms *CountMinSketch) init(hashes, buckets int) {
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for i := 0; i < hashes; i++ {
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cms.hashes = append(cms.hashes, maphash.MakeSeed())
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}
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// Need a matrix of hashes * buckets to store counts
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cms.nbuckets = buckets
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cms.matrix = make([]uint64, hashes*buckets)
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}
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// Add calls AddN(val, 1).
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func (cms *CountMinSketch) Add(val []byte) uint64 {
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return cms.AddN(val, 1)
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}
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// AddN increments the count for the given value by the provided count,
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// returning the new count.
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func (cms *CountMinSketch) AddN(val []byte, count uint64) uint64 {
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var (
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mh maphash.Hash
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ret uint64 = math.MaxUint64
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)
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for i, seed := range cms.hashes {
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mh.SetSeed(seed)
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// Generate a hash for this value using Lemire's alternative to modular reduction:
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// https://lemire.me/blog/2016/06/27/a-fast-alternative-to-the-modulo-reduction/
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mh.Write(val)
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hash := mh.Sum64()
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hash = multiplyHigh64(hash, uint64(cms.nbuckets))
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// The index in our matrix is (i * buckets) to move "down" i
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// rows in our matrix to the row for this hash, plus 'hash' to
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// move inside this row.
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idx := (i * cms.nbuckets) + int(hash)
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// Add to this row
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cms.matrix[idx] += count
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ret = min(ret, cms.matrix[idx])
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}
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return ret
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}
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// Get returns the count for the provided value.
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func (cms *CountMinSketch) Get(val []byte) uint64 {
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var (
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mh maphash.Hash
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ret uint64 = math.MaxUint64
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)
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for i, seed := range cms.hashes {
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mh.SetSeed(seed)
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// Generate a hash for this value using Lemire's alternative to modular reduction:
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// https://lemire.me/blog/2016/06/27/a-fast-alternative-to-the-modulo-reduction/
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mh.Write(val)
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hash := mh.Sum64()
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hash = multiplyHigh64(hash, uint64(cms.nbuckets))
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// The index in our matrix is (i * buckets) to move "down" i
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// rows in our matrix to the row for this hash, plus 'hash' to
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// move inside this row.
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idx := (i * cms.nbuckets) + int(hash)
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// Select the minimal value among all rows
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ret = min(ret, cms.matrix[idx])
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}
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return ret
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}
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// multiplyHigh64 implements (x * y) >> 64 "the long way" without access to a
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// 128-bit type. This function is adapted from something similar in Tensorflow:
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//
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// https://github.com/tensorflow/tensorflow/commit/a47a300185026fe7829990def9113bf3a5109fed
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//
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// TODO(andrew-d): this could be replaced with a single "MULX" instruction on
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// x86_64 platforms, which we can do if this ever turns out to be a performance
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// bottleneck.
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func multiplyHigh64(x, y uint64) uint64 {
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x_lo := x & 0xffffffff
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x_hi := x >> 32
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buckets_lo := y & 0xffffffff
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buckets_hi := y >> 32
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prod_hi := x_hi * buckets_hi
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prod_lo := x_lo * buckets_lo
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prod_mid1 := x_hi * buckets_lo
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prod_mid2 := x_lo * buckets_hi
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carry := ((prod_mid1 & 0xffffffff) + (prod_mid2 & 0xffffffff) + (prod_lo >> 32)) >> 32
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return prod_hi + (prod_mid1 >> 32) + (prod_mid2 >> 32) + carry
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}
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type mhValue[T any] struct {
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count uint64
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val T
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}
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// An minHeap is a min-heap of ints and associated values.
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type minHeap[T any] []mhValue[T]
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func (h minHeap[T]) Len() int { return len(h) }
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func (h minHeap[T]) Less(i, j int) bool { return h[i].count < h[j].count }
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func (h minHeap[T]) Swap(i, j int) { h[i], h[j] = h[j], h[i] }
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func (h *minHeap[T]) Push(x any) {
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// Push and Pop use pointer receivers because they modify the slice's length,
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// not just its contents.
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*h = append(*h, x.(mhValue[T]))
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}
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func (h *minHeap[T]) Pop() any {
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old := *h
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n := len(old)
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x := old[n-1]
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*h = old[0 : n-1]
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return x
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}
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@ -0,0 +1,135 @@
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// Copyright (c) Tailscale Inc & AUTHORS
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// SPDX-License-Identifier: BSD-3-Clause
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package topk
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import (
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"encoding/binary"
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"fmt"
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"slices"
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"testing"
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)
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func TestCountMinSketch(t *testing.T) {
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cms := NewCountMinSketch(4, 10)
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items := []string{"foo", "bar", "baz", "asdf", "quux"}
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for _, item := range items {
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cms.Add([]byte(item))
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}
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for _, item := range items {
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count := cms.Get([]byte(item))
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if count < 1 {
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t.Errorf("item %q should have count >= 1", item)
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} else if count > 1 {
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t.Logf("item %q has count > 1: %d", item, count)
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}
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}
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// Test that an item that's *not* in the set has a value lower than the
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// total number of items we inserted (in the case that all items
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// collided).
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noItemCount := cms.Get([]byte("doesn't exist"))
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if noItemCount > uint64(len(items)) {
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t.Errorf("expected nonexistent item to have value < %d; got %d", len(items), noItemCount)
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}
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}
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func TestTopK(t *testing.T) {
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// This is probabilistic, so we're going to try 10 times to get the
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// "right" value; the likelihood that we fail on all attempts is
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// vanishingly small since the number of hash buckets is drastically
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// larger than the number of items we're inserting.
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var (
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got []int
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want = []int{5, 6, 7, 8, 9}
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)
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for try := 0; try < 10; try++ {
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topk := NewWithParams[int](5, func(in []byte, val int) []byte {
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return binary.LittleEndian.AppendUint64(in, uint64(val))
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}, 4, 1000)
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// Add the first 10 integers with counts equal to 2x their value
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for i := 0; i < 10; i++ {
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topk.AddN(i, uint64(i*2))
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}
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got = topk.Top()
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t.Logf("top K items: %+v", got)
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slices.Sort(got)
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if slices.Equal(got, want) {
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// All good!
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return
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}
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// continue and retry or fail
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}
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t.Errorf("top K mismatch\ngot: %v\nwant: %v", got, want)
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}
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func TestPickParams(t *testing.T) {
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hashes, buckets := PickParams(
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0.001, // 0.1% error rate
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0.001, // 0.1% chance of having an error, or 99.9% chance of not having an error
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)
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t.Logf("hashes = %d, buckets = %d", hashes, buckets)
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}
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func BenchmarkCountMinSketch(b *testing.B) {
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cms := NewCountMinSketch(PickParams(0.001, 0.001))
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b.ResetTimer()
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b.ReportAllocs()
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var enc [8]byte
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for i := 0; i < b.N; i++ {
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binary.LittleEndian.PutUint64(enc[:], uint64(i))
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cms.Add(enc[:])
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}
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}
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func BenchmarkTopK(b *testing.B) {
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for _, n := range []int{
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10,
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128,
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256,
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1024,
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8192,
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} {
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b.Run(fmt.Sprintf("Top%d", n), func(b *testing.B) {
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out := make([]int, 0, n)
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topk := New[int](n, func(in []byte, val int) []byte {
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return binary.LittleEndian.AppendUint64(in, uint64(val))
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})
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b.ResetTimer()
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b.ReportAllocs()
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for i := 0; i < b.N; i++ {
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topk.Add(i)
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}
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out = topk.AppendTop(out[:0]) // should not allocate
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_ = out // appease linter
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})
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}
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}
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func TestMultiplyHigh64(t *testing.T) {
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testCases := []struct {
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x, y uint64
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want uint64
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}{
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{0, 0, 0},
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{0xffffffff, 0xffffffff, 0},
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{0x2, 0xf000000000000000, 1},
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{0x3, 0xf000000000000000, 2},
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{0x3, 0xf000000000000001, 2},
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{0x3, 0xffffffffffffffff, 2},
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{0xffffffffffffffff, 0xffffffffffffffff, 0xfffffffffffffffe},
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}
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for _, tc := range testCases {
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got := multiplyHigh64(tc.x, tc.y)
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if got != tc.want {
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t.Errorf("got multiplyHigh64(%x, %x) = %x, want %x", tc.x, tc.y, got, tc.want)
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}
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}
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}
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