Reading that article I came up with an idea that doesn’t require a multimap. It handles colliding map keys by moving the colliding key forward by its prime value again and again until it reaches a key that isn’t in the map. Below primes is a map with keys of the next iterator value and values that are primes.
let primes =
let rec nextPrime n p primes =
if primes |> Map.containsKey n then
nextPrime (n + p) p primes
else
primes.Add(n, p)
let rec prime n primes =
seq {
if primes |> Map.containsKey n then
let p = primes.Item n
yield! prime (n + 1) (nextPrime (n + p) p (primes.Remove n))
else
yield n
yield! prime (n + 1) (primes.Add(n * n, n))
}
prime 2 Map.empty
Here’s the priority queue based algorithm from that paper without the square optimization. I placed the generic priority queue functions at the top. I used a tuple to represent the lazy list iterators.
let primes() =
// the priority queue functions
let insert = Heap.Insert
let findMin = Heap.Min
let insertDeleteMin = Heap.DeleteInsert
// skips primes 2, 3, 5, 7
let wheelData = [|2L;4L;2L;4L;6L;2L;6L;4L;2L;4L;6L;6L;2L;6L;4L;2L;6L;4L;6L;8L;4L;2L;4L;2L;4L;8L;6L;4L;6L;2L;4L;6L;2L;6L;6L;4L;2L;4L;6L;2L;6L;4L;2L;4L;2L;10L;2L;10L|]
// increments iterator
let wheel (composite, n, prime) =
composite + wheelData.[n % 48] * prime, n + 1, prime
let insertPrime prime n table =
insert (prime * prime, n, prime) table
let rec adjust x (table : Heap) =
let composite, n, prime = findMin table
if composite <= x then
table
|> insertDeleteMin (wheel (composite, n, prime))
|> adjust x
else
table
let rec sieve iterator table =
seq {
let x, n, _ = iterator
let composite, _, _ = findMin table
if composite <= x then
yield! sieve (wheel iterator) (adjust x table)
else
if x = 13L then
yield! [2L; 3L; 5L; 7L; 11L]
yield x
yield! sieve (wheel iterator) (insertPrime x n table)
}
sieve (13L, 1, 1L) (insertPrime 11L 0 (Heap(0L, 0, 0L)))
Here’s the priority queue based algorithm with the square optimization. In order to facilitate lazy adding primes to the lookup table, the wheel offsets had to be returned along with prime values. This version of the algorithm has O(sqrt(n)) memory usage where the none optimized one is O(n).
let rec primes2() : seq<int64 * int> =
// the priority queue functions
let insert = Heap.Insert
let findMin = Heap.Min
let insertDeleteMin = Heap.DeleteInsert
// increments iterator
let wheel (composite, n, prime) =
composite + wheelData.[n % 48] * prime, n + 1, prime
let insertPrime enumerator composite table =
// lazy initialize the enumerator
let enumerator =
if enumerator = null then
let enumerator = primes2().GetEnumerator()
enumerator.MoveNext() |> ignore
// skip primes that are a part of the wheel
while fst enumerator.Current < 11L do
enumerator.MoveNext() |> ignore
enumerator
else
enumerator
let prime = fst enumerator.Current
// Wait to insert primes until their square is less than the tables current min
if prime * prime < composite then
enumerator.MoveNext() |> ignore
let prime, n = enumerator.Current
enumerator, insert (prime * prime, n, prime) table
else
enumerator, table
let rec adjust x table =
let composite, n, prime = findMin table
if composite <= x then
table
|> insertDeleteMin (wheel (composite, n, prime))
|> adjust x
else
table
let rec sieve iterator (enumerator, table) =
seq {
let x, n, _ = iterator
let composite, _, _ = findMin table
if composite <= x then
yield! sieve (wheel iterator) (enumerator, adjust x table)
else
if x = 13L then
yield! [2L, 0; 3L, 0; 5L, 0; 7L, 0; 11L, 0]
yield x, n
yield! sieve (wheel iterator) (insertPrime enumerator composite table)
}
sieve (13L, 1, 1L) (null, insert (11L * 11L, 0, 11L) (Heap(0L, 0, 0L)))
Here’s my test program.
type GenericHeap<'T when 'T : comparison>(defaultValue : 'T) =
let mutable capacity = 1
let mutable values = Array.create capacity defaultValue
let mutable size = 0
let swap i n =
let temp = values.[i]
values.[i] <- values.[n]
values.[n] <- temp
let rec rollUp i =
if i > 0 then
let parent = (i - 1) / 2
if values.[i] < values.[parent] then
swap i parent
rollUp parent
let rec rollDown i =
let left, right = 2 * i + 1, 2 * i + 2
if right < size then
if values.[left] < values.[i] then
if values.[left] < values.[right] then
swap left i
rollDown left
else
swap right i
rollDown right
elif values.[right] < values.[i] then
swap right i
rollDown right
elif left < size then
if values.[left] < values.[i] then
swap left i
member this.insert (value : 'T) =
if size = capacity then
capacity <- capacity * 2
let newValues = Array.zeroCreate capacity
for i in 0 .. size - 1 do
newValues.[i] <- values.[i]
values <- newValues
values.[size] <- value
size <- size + 1
rollUp (size - 1)
member this.delete () =
values.[0] <- values.[size]
size <- size - 1
rollDown 0
member this.deleteInsert (value : 'T) =
values.[0] <- value
rollDown 0
member this.min () =
values.[0]
static member Insert (value : 'T) (heap : GenericHeap<'T>) =
heap.insert value
heap
static member DeleteInsert (value : 'T) (heap : GenericHeap<'T>) =
heap.deleteInsert value
heap
static member Min (heap : GenericHeap<'T>) =
heap.min()
type Heap = GenericHeap<int64 * int * int64>
let wheelData = [|2L;4L;2L;4L;6L;2L;6L;4L;2L;4L;6L;6L;2L;6L;4L;2L;6L;4L;6L;8L;4L;2L;4L;2L;4L;8L;6L;4L;6L;2L;4L;6L;2L;6L;6L;4L;2L;4L;6L;2L;6L;4L;2L;4L;2L;10L;2L;10L|]
let primes() =
// the priority queue functions
let insert = Heap.Insert
let findMin = Heap.Min
let insertDeleteMin = Heap.DeleteInsert
// increments iterator
let wheel (composite, n, prime) =
composite + wheelData.[n % 48] * prime, n + 1, prime
let insertPrime prime n table =
insert (prime * prime, n, prime) table
let rec adjust x (table : Heap) =
let composite, n, prime = findMin table
if composite <= x then
table
|> insertDeleteMin (wheel (composite, n, prime))
|> adjust x
else
table
let rec sieve iterator table =
seq {
let x, n, _ = iterator
let composite, _, _ = findMin table
if composite <= x then
yield! sieve (wheel iterator) (adjust x table)
else
if x = 13L then
yield! [2L; 3L; 5L; 7L; 11L]
yield x
yield! sieve (wheel iterator) (insertPrime x n table)
}
sieve (13L, 1, 1L) (insertPrime 11L 0 (Heap(0L, 0, 0L)))
let rec primes2() : seq<int64 * int> =
// the priority queue functions
let insert = Heap.Insert
let findMin = Heap.Min
let insertDeleteMin = Heap.DeleteInsert
// increments iterator
let wheel (composite, n, prime) =
composite + wheelData.[n % 48] * prime, n + 1, prime
let insertPrime enumerator composite table =
// lazy initialize the enumerator
let enumerator =
if enumerator = null then
let enumerator = primes2().GetEnumerator()
enumerator.MoveNext() |> ignore
// skip primes that are a part of the wheel
while fst enumerator.Current < 11L do
enumerator.MoveNext() |> ignore
enumerator
else
enumerator
let prime = fst enumerator.Current
// Wait to insert primes until their square is less than the tables current min
if prime * prime < composite then
enumerator.MoveNext() |> ignore
let prime, n = enumerator.Current
enumerator, insert (prime * prime, n, prime) table
else
enumerator, table
let rec adjust x table =
let composite, n, prime = findMin table
if composite <= x then
table
|> insertDeleteMin (wheel (composite, n, prime))
|> adjust x
else
table
let rec sieve iterator (enumerator, table) =
seq {
let x, n, _ = iterator
let composite, _, _ = findMin table
if composite <= x then
yield! sieve (wheel iterator) (enumerator, adjust x table)
else
if x = 13L then
yield! [2L, 0; 3L, 0; 5L, 0; 7L, 0; 11L, 0]
yield x, n
yield! sieve (wheel iterator) (insertPrime enumerator composite table)
}
sieve (13L, 1, 1L) (null, insert (11L * 11L, 0, 11L) (Heap(0L, 0, 0L)))
let mutable i = 0
let compare a b =
i <- i + 1
if a = b then
true
else
printfn "%A %A %A" a b i
false
Seq.forall2 compare (Seq.take 50000 (primes())) (Seq.take 50000 (primes2() |> Seq.map fst))
|> printfn "%A"
primes2()
|> Seq.map fst
|> Seq.take 10
|> Seq.toArray
|> printfn "%A"
primes2()
|> Seq.map fst
|> Seq.skip 999999
|> Seq.take 10
|> Seq.toArray
|> printfn "%A"
System.Console.ReadLine() |> ignore