#### Count Min Sketch

S
```use std::collections::hash_map::RandomState;
use std::fmt::{Debug, Formatter};
use std::hash::{BuildHasher, Hash, Hasher};

/// A probabilistic data structure holding an approximate count for diverse items efficiently (using constant space)
///
/// Let's imagine we want to count items from an incoming (unbounded) data stream
/// One way to do this would be to hold a frequency hashmap, counting element hashes
/// This works extremely well, but unfortunately would require a lot of memory if we have a huge diversity of incoming items in the data stream
///
/// CountMinSketch aims at solving this problem, trading off the exact count for an approximate one, but getting from potentially unbounded space complexity to constant complexity
/// See the implementation below for more details
///
/// Here is the definition of the different allowed operations on a CountMinSketch:
///     * increment the count of an item
///     * retrieve the count of an item
pub trait CountMinSketch {
type Item;

fn increment(&mut self, item: Self::Item);
fn increment_by(&mut self, item: Self::Item, count: usize);
fn get_count(&self, item: Self::Item) -> usize;
}

/// The common implementation of a CountMinSketch
/// Holding a DEPTH x WIDTH matrix of counts
///
/// The idea behind the implementation is the following:
/// Let's start from our problem statement above. We have a frequency map of counts, and want to go reduce its space complexity
/// The immediate way to do this would be to use a Vector with a fixed size, let this size be `WIDTH`
/// We will be holding the count of each item `item` in the Vector, at index `i = hash(item) % WIDTH` where `hash` is a hash function: `item -> usize`
/// We now have constant space.
///
/// The problem though is that we'll potentially run into a lot of collisions.
/// Taking an extreme example, if `WIDTH = 1`, all items will have the same count, which is the sum of counts of every items
/// We could reduce the amount of collisions by using a bigger `WIDTH` but this wouldn't be way more efficient than the "big" frequency map
/// How do we improve the solution, but still keeping constant space?
///
/// The idea is to use not just one vector, but multiple (`DEPTH`) ones and attach different `hash` functions to each vector
/// This would lead to the following data structure:
///             <- WIDTH = 5 ->
///  D   hash1: [0, 0, 0, 0, 0]
///  E   hash2: [0, 0, 0, 0, 0]
///  P   hash3: [0, 0, 0, 0, 0]
///  T   hash4: [0, 0, 0, 0, 0]
///  H   hash5: [0, 0, 0, 0, 0]
///  =   hash6: [0, 0, 0, 0, 0]
///  7   hash7: [0, 0, 0, 0, 0]
/// Every hash function must return a different value for the same item.
/// Let's say we hash "TEST" and:
///     hash1("TEST") = 42 => idx = 2
///     hash2("TEST") = 26 => idx = 1
///     hash3("TEST") = 10 => idx = 0
///     hash4("TEST") = 33 => idx = 3
///     hash5("TEST") = 54 => idx = 4
///     hash6("TEST") = 11 => idx = 1
///     hash7("TEST") = 50 => idx = 0
/// This would lead our structure to become:
///             <- WIDTH = 5 ->
///  D   hash1: [0, 0, 1, 0, 0]
///  E   hash2: [0, 1, 0, 0, 0]
///  P   hash3: [1, 0, 0, 0, 0]
///  T   hash4: [0, 0, 0, 1, 0]
///  H   hash5: [0, 0, 0, 0, 1]
///  =   hash6: [0, 1, 0, 0, 0]
///  7   hash7: [1, 0, 0, 0, 0]
///
/// Now say we hash "OTHER" and:
///     hash1("OTHER") = 23 => idx = 3
///     hash2("OTHER") = 11 => idx = 1
///     hash3("OTHER") = 52 => idx = 2
///     hash4("OTHER") = 25 => idx = 0
///     hash5("OTHER") = 31 => idx = 1
///     hash6("OTHER") = 24 => idx = 4
///     hash7("OTHER") = 30 => idx = 0
/// Leading our data structure to become:
///             <- WIDTH = 5 ->
///  D   hash1: [0, 0, 1, 1, 0]
///  E   hash2: [0, 2, 0, 0, 0]
///  P   hash3: [1, 0, 1, 0, 0]
///  T   hash4: [1, 0, 0, 1, 0]
///  H   hash5: [0, 1, 0, 0, 1]
///  =   hash6: [0, 1, 0, 0, 1]
///  7   hash7: [2, 0, 0, 0, 0]
///
/// We actually can witness some collisions (invalid counts of `2` above in some rows).
/// This means that if we have to return the count for "TEST", we'd actually fetch counts from every row and return the minimum value
///
/// This could potentially be overestimated if we have a huge number of entries and a lot of collisions.
/// But an interesting property is that the count we return for "TEST" cannot be underestimated
pub struct HashCountMinSketch<Item: Hash, const WIDTH: usize, const DEPTH: usize> {
phantom: std::marker::PhantomData<Item>, // just a marker for Item to be used
counts: [[usize; WIDTH]; DEPTH],
hashers: [RandomState; DEPTH],
}

impl<Item: Hash, const WIDTH: usize, const DEPTH: usize> Debug
for HashCountMinSketch<Item, WIDTH, DEPTH>
{
fn fmt(&self, f: &mut Formatter<'_>) -> std::fmt::Result {
f.debug_struct("Item").field("vecs", &self.counts).finish()
}
}

impl<T: Hash, const WIDTH: usize, const DEPTH: usize> Default
for HashCountMinSketch<T, WIDTH, DEPTH>
{
fn default() -> Self {
let hashers = std::array::from_fn(|_| RandomState::new());

Self {
phantom: Default::default(),
counts: [[0; WIDTH]; DEPTH],
hashers,
}
}
}

impl<Item: Hash, const WIDTH: usize, const DEPTH: usize> CountMinSketch
for HashCountMinSketch<Item, WIDTH, DEPTH>
{
type Item = Item;

fn increment(&mut self, item: Self::Item) {
self.increment_by(item, 1)
}

fn increment_by(&mut self, item: Self::Item, count: usize) {
for (row, r) in self.hashers.iter_mut().enumerate() {
let mut h = r.build_hasher();
item.hash(&mut h);
let hashed = h.finish();
let col = (hashed % WIDTH as u64) as usize;
self.counts[row][col] += count;
}
}

fn get_count(&self, item: Self::Item) -> usize {
self.hashers
.iter()
.enumerate()
.map(|(row, r)| {
let mut h = r.build_hasher();
item.hash(&mut h);
let hashed = h.finish();
let col = (hashed % WIDTH as u64) as usize;
self.counts[row][col]
})
.min()
.unwrap()
}
}

#[cfg(test)]
mod tests {
use crate::data_structures::probabilistic::count_min_sketch::{
CountMinSketch, HashCountMinSketch,
};
use quickcheck::{Arbitrary, Gen};
use std::collections::HashSet;

#[test]
fn hash_functions_should_hash_differently() {
let mut sketch: HashCountMinSketch<&str, 50, 50> = HashCountMinSketch::default(); // use a big DEPTH
sketch.increment("something");
// We want to check that our hash functions actually produce different results, so we'll store the indices where we encounter a count=1 in a set
let mut indices_of_ones: HashSet<usize> = HashSet::default();
for counts in sketch.counts {
let ones = counts
.into_iter()
.enumerate()
.filter_map(|(idx, count)| (count == 1).then_some(idx))
.collect::<Vec<_>>();
assert_eq!(1, ones.len());
indices_of_ones.insert(ones[0]);
}
// Given the parameters (WIDTH = 50, DEPTH = 50) it's extremely unlikely that all hash functions hash to the same index
assert!(indices_of_ones.len() > 1); // but we want to avoid a bug where all hash functions would produce the same hash (or hash to the same index)
}

#[test]
fn inspect_counts() {
let mut sketch: HashCountMinSketch<&str, 5, 7> = HashCountMinSketch::default();
sketch.increment("test");
// Inspect internal state:
for counts in sketch.counts {
let zeroes = counts.iter().filter(|count| **count == 0).count();
assert_eq!(4, zeroes);
let ones = counts.iter().filter(|count| **count == 1).count();
assert_eq!(1, ones);
}
sketch.increment("test");
for counts in sketch.counts {
let zeroes = counts.iter().filter(|count| **count == 0).count();
assert_eq!(4, zeroes);
let twos = counts.iter().filter(|count| **count == 2).count();
assert_eq!(1, twos);
}

// This one is actually deterministic
assert_eq!(2, sketch.get_count("test"));
}

#[derive(Debug, Clone, Eq, PartialEq, Hash)]
struct TestItem {
item: String,
count: usize,
}

const MAX_STR_LEN: u8 = 30;
const MAX_COUNT: usize = 20;

impl Arbitrary for TestItem {
fn arbitrary(g: &mut Gen) -> Self {
let str_len = u8::arbitrary(g) % MAX_STR_LEN;
let mut str = String::with_capacity(str_len as usize);
for _ in 0..str_len {
str.push(char::arbitrary(g));
}
let count = usize::arbitrary(g) % MAX_COUNT;
TestItem { item: str, count }
}
}

#[quickcheck_macros::quickcheck]
fn must_not_understimate_count(test_items: Vec<TestItem>) {
let test_items = test_items.into_iter().collect::<HashSet<_>>(); // remove duplicated (would lead to weird counts)
let n = test_items.len();
let mut sketch: HashCountMinSketch<String, 50, 10> = HashCountMinSketch::default();
let mut exact_count = 0;
for TestItem { item, count } in &test_items {
sketch.increment_by(item.clone(), *count);
}
for TestItem { item, count } in test_items {
let stored_count = sketch.get_count(item);
assert!(stored_count >= count);
if count == stored_count {
exact_count += 1;
}
}
if n > 20 {
// if n is too short, the stat isn't really relevant
let exact_ratio = exact_count as f64 / n as f64;
assert!(exact_ratio > 0.7); // the proof is quite hard, but this should be OK
}
}
}
```