Chapter 9 Parallel programming

In this chapter, we shall look at the parallel programming facilities in OCaml. The OCaml standard library exposes low-level primitives for parallel programming. We recommend the users to utilise higher-level parallel programming libraries such as domainslib. This tutorial will first cover the high-level parallel programming using domainslib followed by low-level primitives exposed by the compiler.

OCaml distinguishes concurrency and parallelism and provides distinct mechanisms for expressing them. Concurrency is overlapped execution of tasks (section 12.24.2) whereas parallelism is simultaneous execution of tasks. In particular, parallel tasks overlap in time but concurrent tasks may or may not overlap in time. Tasks may execute concurrently by yielding control to each other. While concurrency is a program structuring mechanism, parallelism is a mechanism to make your programs run faster. If you are interested in the concurrent programming mechanisms in OCaml, please refer to the section 12.24 on effect handlers and the chapter 34 on the threads library.

1 Domains

Domains are the units of parallelism in OCaml. The module Domain provides the primitives to create and manage domains. New domains can be spawned using the spawn function.

Domain.spawn (fun _ -> print_endline "I ran in parallel")
I ran in parallel - : unit Domain.t = <abstr>

The spawn function executes the given computation in parallel with the calling domain.

Domains are heavy-weight entities. Each domain maps 1:1 to an operating system thread. Each domain also has its own runtime state, which includes domain-local structures for allocating memory. Hence, they are relatively expensive to create and tear down.

It is recommended that the programs do not spawn more domains than cores available.

In this tutorial, we shall be implementing, running and measuring the performance of parallel programs. The results observed are dependent on the number of cores available on the target machine. This tutorial is being written on a 2.3 GHz Quad-Core Intel Core i7 MacBook Pro with 4 cores and 8 hardware threads. It is reasonable to expect roughly 4x performance on 4 domains for parallel programs with little coordination between the domains, and when the machine is not under load. Beyond 4 domains, the speedup is likely to be less than linear. We shall also use the command-line benchmarking tool hyperfine for benchmarking our programs.

1.1 Joining domains

We shall use the program to compute the nth Fibonacci number using recursion as a running example. The sequential program for computing the nth Fibonacci number is given below.

(* *) let n = try int_of_string Sys.argv.(1) with _ -> 1 let rec fib n = if n < 2 then 1 else fib (n - 1) + fib (n - 2) let main () = let r = fib n in Printf.printf "fib(%d) = %d\n%!" n r let _ = main ()

The program can be compiled and benchmarked as follows.

$ ocamlopt -o fib.exe
$ ./fib.exe 42
fib(42) = 433494437
$ hyperfine './fib.exe 42' # Benchmarking
Benchmark 1: ./fib.exe 42
  Time (mean ± sd):     1.193 s ±  0.006 s    [User: 1.186 s, System: 0.003 s]
  Range (min … max):    1.181 s …  1.202 s    10 runs

We see that it takes around 1.2 seconds to compute the 42nd Fibonacci number.

Spawned domains can be joined using the join function to get their results. The join function waits for target domain to terminate. The following program computes the nth Fibonacci number twice in parallel.

(* *) let n = int_of_string Sys.argv.(1) let rec fib n = if n < 2 then 1 else fib (n - 1) + fib (n - 2) let main () = let d1 = Domain.spawn (fun _ -> fib n) in let d2 = Domain.spawn (fun _ -> fib n) in let r1 = Domain.join d1 in Printf.printf "fib(%d) = %d\n%!" n r1; let r2 = Domain.join d2 in Printf.printf "fib(%d) = %d\n%!" n r2 let _ = main ()

The program spawns two domains which compute the nth Fibonacci number. The spawn function returns a Domain.t value which can be joined to get the result of the parallel computation. The join function blocks until the computation runs to completion.

$ ocamlopt -o fib_twice.exe
$ ./fib_twice.exe 42
fib(42) = 433494437
fib(42) = 433494437
$ hyperfine './fib_twice.exe 42'
Benchmark 1: ./fib_twice.exe 42
  Time (mean ± sd):     1.249 s ±  0.025 s    [User: 2.451 s, System: 0.012 s]
  Range (min … max):    1.221 s …  1.290 s    10 runs

As one can see that computing the nth Fibonacci number twice almost took the same time as computing it once thanks to parallelism.

2 Domainslib: A library for nested-parallel programming

Let us attempt to parallelise the Fibonacci function. The two recursive calls may be executed in parallel. However, naively parallelising the recursive calls by spawning domains for each one will not work as it spawns too many domains.

(* *) let n = try int_of_string Sys.argv.(1) with _ -> 1 let rec fib n = if n < 2 then 1 else begin let d1 = Domain.spawn (fun _ -> fib (n - 1)) in let d2 = Domain.spawn (fun _ -> fib (n - 2)) in Domain.join d1 + Domain.join d2 end let main () = let r = fib n in Printf.printf "fib(%d) = %d\n%!" n r let _ = main ()
fib(1) = 1 val n : int = 1 val fib : int -> int = <fun> val main : unit -> unit = <fun>
$ ocamlopt -o fib_par1.exe
$ ./fib_par1.exe 42
Fatal error: exception Failure("failed to allocate domain")

OCaml has a limit of 128 domains that can be active at the same time. An attempt to spawn more domains will raise an exception. How then can we parallelise the Fibonacci function?

2.1 Parallelising Fibonacci using domainslib

The OCaml standard library provides only low-level primitives for concurrent and parallel programming, leaving high-level programming libraries to be developed and distributed outside the core compiler distribution. Domainslib is such a library for nested-parallel programming, which is epitomised by the parallelism available in the recursive Fibonacci computation. Let us use domainslib to parallelise the recursive Fibonacci program. It is recommended that you install domainslib using the opam package manager. This tutorial uses domainslib version 0.5.0.

Domainslib provides an async/await mechanism for spawning parallel tasks and awaiting their results. On top of this mechanism, domainslib provides parallel iterators. At its core, domainslib has an efficient implementation of work-stealing queue in order to efficiently share tasks with other domains. A parallel implementation of the Fibonacci program is given below.

(* *)
let num_domains = int_of_string Sys.argv.(1)
let n = int_of_string Sys.argv.(2)

let rec fib n = if n < 2 then 1 else fib (n - 1) + fib (n - 2)

module T = Domainslib.Task

let rec fib_par pool n =
  if n > 20 then begin
    let a = T.async pool (fun _ -> fib_par pool (n-1)) in
    let b = T.async pool (fun _ -> fib_par pool (n-2)) in
    T.await pool a + T.await pool b
  end else fib n

let main () =
  let pool = T.setup_pool ~num_domains:(num_domains - 1) () in
  let res = pool (fun _ -> fib_par pool n) in
  T.teardown_pool pool;
  Printf.printf "fib(%d) = %d\n" n res

let _ = main ()

The program takes the number of domains and the input to the Fibonacci function as the first and the second command-line arguments respectively.

Let us start with the main function. First, we set up a pool of domains on which the nested parallel tasks will run. The domain invoking the run function will also participate in executing the tasks submitted to the pool. We invoke the parallel Fibonacci function fib_par in the run function. Finally, we tear down the pool and print the result.

For sufficiently large inputs (n > 20), the fib_par function spawns the left and the right recursive calls asynchronously in the pool using the async function. The async function returns a promise for the result. The result of an asynchronous computation is obtained by awaiting the promise using the await function. The await function call blocks until the promise is resolved.

For small inputs, the fib_par function simply calls the sequential Fibonacci function fib. It is important to switch to sequential mode for small problem sizes. If not, the cost of parallelisation will outweigh the work available.

For simplicity, we use ocamlfind to compile this program. It is recommended that the users use dune to build their programs that utilise libraries installed through opam.

$ ocamlfind ocamlopt -package domainslib -linkpkg -o fib_par2.exe
$ ./fib_par2.exe 1 42
fib(42) = 433494437
$ hyperfine './fib.exe 42' './fib_par2.exe 2 42' \
            './fib_par2.exe 4 42' './fib_par2.exe 8 42'
Benchmark 1: ./fib.exe 42
  Time (mean ± sd):     1.217 s ±  0.018 s    [User: 1.203 s, System: 0.004 s]
  Range (min … max):    1.202 s …  1.261 s    10 runs

Benchmark 2: ./fib_par2.exe 2 42
  Time (mean ± sd):    628.2 ms ±   2.9 ms    [User: 1243.1 ms, System: 4.9 ms]
  Range (min … max):   625.7 ms … 634.5 ms    10 runs

Benchmark 3: ./fib_par2.exe 4 42
  Time (mean ± sd):    337.6 ms ±  23.4 ms    [User: 1321.8 ms, System: 8.4 ms]
  Range (min … max):   318.5 ms … 377.6 ms    10 runs

Benchmark 4: ./fib_par2.exe 8 42
  Time (mean ± sd):    250.0 ms ±   9.4 ms    [User: 1877.1 ms, System: 12.6 ms]
  Range (min … max):   242.5 ms … 277.3 ms    11 runs

  './fib_par2.exe 8 42' ran
    1.35 ± 0.11 times faster than './fib_par2.exe 4 42'
    2.51 ± 0.10 times faster than './fib_par2.exe 2 42'
    4.87 ± 0.20 times faster than './fib.exe 42'

The results show that, with 8 domains, the parallel Fibonacci program runs 4.87 times faster than the sequential version.

2.2 Parallel iteration constructs

Many numerical algorithms use for-loops. The parallel-for primitive provides a straight-forward way to parallelise such code. Let us take the spectral-norm benchmark from the computer language benchmarks game and parallelise it. The sequential version of the program is given below.

(* *) let n = try int_of_string Sys.argv.(1) with _ -> 32 let eval_A i j = 1. /. float((i+j)*(i+j+1)/2+i+1) let eval_A_times_u u v = let n = Array.length v - 1 in for i = 0 to n do let vi = ref 0. in for j = 0 to n do vi := !vi +. eval_A i j *. u.(j) done; v.(i) <- !vi done let eval_At_times_u u v = let n = Array.length v - 1 in for i = 0 to n do let vi = ref 0. in for j = 0 to n do vi := !vi +. eval_A j i *. u.(j) done; v.(i) <- !vi done let eval_AtA_times_u u v = let w = Array.make (Array.length u) 0.0 in eval_A_times_u u w; eval_At_times_u w v let () = let u = Array.make n 1.0 and v = Array.make n 0.0 in for _i = 0 to 9 do eval_AtA_times_u u v; eval_AtA_times_u v u done; let vv = ref 0.0 and vBv = ref 0.0 in for i=0 to n-1 do vv := !vv +. v.(i) *. v.(i); vBv := !vBv +. u.(i) *. v.(i) done; Printf.printf "%0.9f\n" (sqrt(!vBv /. !vv))

Observe that the program has nested loops in eval_A_times_u and eval_At_times_u. Each iteration of the outer loop body reads from u but writes to disjoint memory locations in v. Hence, the iterations of the outer loop are not dependent on each other and can be executed in parallel.

The parallel version of spectral norm is shown below.

(* *)
let num_domains = try int_of_string Sys.argv.(1) with _ -> 1
let n = try int_of_string Sys.argv.(2) with _ -> 32

let eval_A i j = 1. /. float((i+j)*(i+j+1)/2+i+1)

module T = Domainslib.Task

let eval_A_times_u pool u v =
  let n = Array.length v - 1 in
  T.parallel_for pool ~start:0 ~finish:n ~body:(fun i ->
    let vi = ref 0. in
    for j = 0 to n do vi := !vi +. eval_A i j *. u.(j) done;
    v.(i) <- !vi

let eval_At_times_u pool u v =
  let n = Array.length v - 1 in
  T.parallel_for pool ~start:0 ~finish:n ~body:(fun i ->
    let vi = ref 0. in
    for j = 0 to n do vi := !vi +. eval_A j i *. u.(j) done;
    v.(i) <- !vi

let eval_AtA_times_u pool u v =
  let w = Array.make (Array.length u) 0.0 in
  eval_A_times_u pool u w; eval_At_times_u pool w v

let () =
  let pool = T.setup_pool ~num_domains:(num_domains - 1) () in
  let u = Array.make n 1.0  and  v = Array.make n 0.0 in pool (fun _ ->
  for _i = 0 to 9 do
    eval_AtA_times_u pool u v; eval_AtA_times_u pool v u

  let vv = ref 0.0  and  vBv = ref 0.0 in
  for i=0 to n-1 do
    vv := !vv +. v.(i) *. v.(i);
    vBv := !vBv +. u.(i) *. v.(i)
  T.teardown_pool pool;
  Printf.printf "%0.9f\n" (sqrt(!vBv /. !vv))

Observe that the parallel_for function is isomorphic to the for-loop in the sequential version. No other change is required except for the boiler plate code to set up and tear down the pools.

$ ocamlopt -o spectralnorm.exe
$ ocamlfind ocamlopt -package domainslib -linkpkg -o spectralnorm_par.exe \
$ hyperfine './spectralnorm.exe 4096' './spectralnorm_par.exe 2 4096' \
            './spectralnorm_par.exe 4 4096' './spectralnorm_par.exe 8 4096'
Benchmark 1: ./spectralnorm.exe 4096
  Time (mean ± sd):     1.989 s ±  0.013 s    [User: 1.972 s, System: 0.007 s]
  Range (min … max):    1.975 s …  2.018 s    10 runs

Benchmark 2: ./spectralnorm_par.exe 2 4096
  Time (mean ± sd):     1.083 s ±  0.015 s    [User: 2.140 s, System: 0.009 s]
  Range (min … max):    1.064 s …  1.102 s    10 runs

Benchmark 3: ./spectralnorm_par.exe 4 4096
  Time (mean ± sd):    698.7 ms ±  10.3 ms    [User: 2730.8 ms, System: 18.3 ms]
  Range (min … max):   680.9 ms … 721.7 ms    10 runs

Benchmark 4: ./spectralnorm_par.exe 8 4096
  Time (mean ± sd):    921.8 ms ±  52.1 ms    [User: 6711.6 ms, System: 51.0 ms]
  Range (min … max):   838.6 ms … 989.2 ms    10 runs

  './spectralnorm_par.exe 4 4096' ran
    1.32 ± 0.08 times faster than './spectralnorm_par.exe 8 4096'
    1.55 ± 0.03 times faster than './spectralnorm_par.exe 2 4096'
    2.85 ± 0.05 times faster than './spectralnorm.exe 4096'

On the author’s machine, the program scales reasonably well up to 4 domains but performs worse with 8 domains. Recall that the machine only has 4 physical cores. Debugging and fixing this performance issue is beyond the scope of this tutorial.

3 Parallel garbage collection

An important aspect of the scalability of parallel OCaml programs is the scalability of the garbage collector (GC). The OCaml GC is designed to have both low latency and good parallel scalability. OCaml has a generational garbage collector with a small minor heap and a large major heap. New objects (upto a certain size) are allocated in the minor heap. Each domain has its own domain-local minor heap arena into which new objects are allocated without synchronising with the other domains. When a domain exhausts its minor heap arena, it calls for a stop-the-world collection of the minor heaps. In the stop-the-world section, all the domains collect their minor heap arenas in parallel evacuating the survivors to the major heap.

For the major heap, each domain maintains domain-local, size-segmented pools of memory into which large objects and survivors from the minor collection are allocated. Having domain-local pools avoids synchronisation for most major heap allocations. The major heap is collected by a concurrent mark-and-sweep algorithm that involves a few short stop-the-world pauses for each major cycle.

Overall, the users should expect the garbage collector to scale well with increasing number of domains, with the latency remaining low. For more information on the design and evaluation of the garbage collector, please have a look at the ICFP 2020 paper on Retrofitting Parallelism onto OCaml.

4 Memory model: The easy bits

Modern processors and compilers aggressively optimise programs. These optimisations accelerate without otherwise affecting sequential programs, but cause surprising behaviours to be visible in parallel programs. To benefit from these optimisations, OCaml adopts a relaxed memory model that precisely specifies which of these relaxed behaviours programs may observe. While these models are difficult to program against directly, the OCaml memory model provides recipes that retain the simplicity of sequential reasoning.

Firstly, immutable values may be freely shared between multiple domains and may be accessed in parallel. For mutable data structures such as reference cells, arrays and mutable record fields, programmers should avoid data races. Reference cells, arrays and mutable record fields are said to be non-atomic data structures. A data race is said to occur when two domains concurrently access a non-atomic memory location without synchronisation and at least one of the accesses is a write. OCaml provides a number of ways to introduce synchronisation including atomic variables (section 9.7) and mutexes (section 9.5).

Importantly, for data race free (DRF) programs, OCaml provides sequentially consistent (SC) semantics – the observed behaviour of such programs can be explained by the interleaving of operations from different domains. This property is known as DRF-SC guarantee. Moreover, in OCaml, DRF-SC guarantee is modular – if a part of a program is data race free, then the OCaml memory model ensures that those parts have sequential consistency despite other parts of the program having data races. Even for programs with data races, OCaml provides strong guarantees. While the user may observe non sequentially consistent behaviours, there are no crashes.

For more details on the relaxed behaviours in the presence of data races, please have a look at the chapter on the hard bits of the memory model (chapter 10).

5 Blocking synchronisation

Domains may perform blocking synchronisation with the help of Mutex, Condition and Semaphore modules. These modules are the same as those used to synchronise threads created by the threads library (chapter 34). For clarity, in the rest of this chapter, we shall call the threads created by the threads library as systhreads. The following program implements a concurrent stack using mutex and condition variables.

module Blocking_stack : sig type 'a t val make : unit -> 'a t val push : 'a t -> 'a -> unit val pop : 'a t -> 'a end = struct type 'a t = { mutable contents: 'a list; mutex : Mutex.t; nonempty : Condition.t } let make () = { contents = []; mutex = Mutex.create (); nonempty = Condition.create () } let push r v = Mutex.lock r.mutex; r.contents <- v::r.contents; Condition.signal r.nonempty; Mutex.unlock r.mutex let pop r = Mutex.lock r.mutex; let rec loop () = match r.contents with | [] -> Condition.wait r.nonempty r.mutex; loop () | x::xs -> r.contents <- xs; x in let res = loop () in Mutex.unlock r.mutex; res end

The concurrent stack is implemented using a record with three fields: a mutable field contents which stores the elements in the stack, a mutex to control access to the contents field, and a condition variable nonempty, which is used to signal blocked domains waiting for the stack to become non-empty.

The push operation locks the mutex, updates the contents field with a new list whose head is the element being pushed and the tail is the old list. The condition variable nonempty is signalled while the lock is held in order to wake up any domains waiting on this condition. If there are waiting domains, one of the domains is woken up. If there are none, then the signal operation has no effect.

The pop operation locks the mutex and checks whether the stack is empty. If so, the calling domain waits on the condition variable nonempty using the wait primitive. The wait call atomically suspends the execution of the current domain and unlocks the mutex. When this domain is woken up again (when the wait call returns), it holds the lock on mutex. The domain tries to read the contents of the stack again. If the pop operation sees that the stack is non-empty, it updates the contents to the tail of the old list, and returns the head.

The use of mutex to control access to the shared resource contents introduces sufficient synchronisation between multiple domains using the stack. Hence, there are no data races when multiple domains use the stack in parallel.

5.1 Interaction with systhreads

How do systhreads interact with domains? The systhreads created on a particular domain remain pinned to that domain. Only one systhread at a time is allowed to run OCaml code on a particular domain. However, systhreads belonging to a particular domain may run C library or system code in parallel. Systhreads belonging to different domains may execute in parallel.

When using systhreads, the thread created for executing the computation given to Domain.spawn is also treated as a systhread. For example, the following program creates in total two domains (including the initial domain) with two systhreads each (including the initial systhread for each of the domains).

(* *)
let m = Mutex.create ()
let r = ref None (* protected by m *)

let task () =
  let my_thr_id = Thread.(id (self ())) in
  let my_dom_id :> int = Domain.self () in
  Mutex.lock m;
  begin match !r with
  | None ->
      Printf.printf "Thread %d running on domain %d saw initial write\n%!"
        my_thr_id my_dom_id
  | Some their_thr_id ->
      Printf.printf "Thread %d running on domain %d saw the write by thread %d\n%!"
        my_thr_id my_dom_id their_thr_id;
  r := Some my_thr_id;
  Mutex.unlock m

let task' () =
  let t = Thread.create task () in
  task ();
  Thread.join t

let main () =
  let d = Domain.spawn task' in
  task' ();
  Domain.join d

let _ = main ()
$ ocamlopt -I +threads unix.cmxa threads.cmxa -o dom_thr.exe
$ ./dom_thr.exe
Thread 1 running on domain 1 saw initial write
Thread 0 running on domain 0 saw the write by thread 1
Thread 2 running on domain 1 saw the write by thread 0
Thread 3 running on domain 0 saw the write by thread 2

This program uses a shared reference cell protected by a mutex to communicate between the different systhreads running on two different domains. The systhread identifiers uniquely identify systhreads in the program. The initial domain gets the domain id and the thread id as 0. The newly spawned domain gets domain id as 1.

6 Interaction with C bindings

During parallel execution with multiple domains, C code running on a domain may run in parallel with any C code running in other domains even if neither of them has released the “domain lock”. Prior to OCaml 5.0, C bindings may have assumed that if the OCaml runtime lock is not released, then it would be safe to manipulate global C state (e.g. initialise a function-local static value). This is no longer true in the presence of parallel execution with multiple domains.

7 Atomics

Mutex, condition variables and semaphores are used to implement blocking synchronisation between domains. For non-blocking synchronisation, OCaml provides Atomic variables. As the name suggests, non-blocking synchronisation does not provide mechanisms for suspending and waking up domains. On the other hand, primitives used in non-blocking synchronisation are often compiled to atomic read-modify-write primitives that the hardware provides. As an example, the following program increments a non-atomic counter and an atomic counter in parallel.

(* *) let twice_in_parallel f = let d1 = Domain.spawn f in let d2 = Domain.spawn f in Domain.join d1; Domain.join d2 let plain_ref n = let r = ref 0 in let f () = for _i=1 to n do incr r done in twice_in_parallel f; Printf.printf "Non-atomic ref count: %d\n" !r let atomic_ref n = let r = Atomic.make 0 in let f () = for _i=1 to n do Atomic.incr r done in twice_in_parallel f; Printf.printf "Atomic ref count: %d\n" (Atomic.get r) let main () = let n = try int_of_string Sys.argv.(1) with _ -> 1 in plain_ref n; atomic_ref n let _ = main ()
$ ocamlopt -o incr.exe
$ ./incr.exe 1_000_000
Non-atomic ref count: 1187193
Atomic ref count: 2000000

Observe that the result from using the non-atomic counter is lower than what one would naively expect. This is because the non-atomic incr function is equivalent to:

let incr r = let curr = !r in r := curr + 1

Observe that the load and the store are two separate operations, and the increment operation as a whole is not performed atomically. When two domains execute this code in parallel, both of them may read the same value of the counter curr and update it to curr + 1. Hence, instead of two increments, the effect will be that of a single increment. On the other hand, the atomic counter performs the load and the store atomically with the help of hardware support for atomicity. The atomic counter returns the expected result.

The atomic variables can be used for low-level synchronisation between the domains. The following example uses an atomic variable to exchange a message between two domains.

let r = Atomic.make None let sender () = Atomic.set r (Some "Hello") let rec receiver () = match Atomic.get r with | None -> Domain.cpu_relax (); receiver () | Some m -> print_endline m let main () = let s = Domain.spawn sender in let d = Domain.spawn receiver in Domain.join s; Domain.join d let _ = main ()
Hello val r : string option Atomic.t = <abstr> val sender : unit -> unit = <fun> val receiver : unit -> unit = <fun> val main : unit -> unit = <fun>

While the sender and the receiver compete to access r, this is not a data race since r is an atomic reference.

7.1 Lock-free stack

The Atomic module is used to implement non-blocking, lock-free data structures. The following program implements a lock-free stack.

module Lockfree_stack : sig type 'a t val make : unit -> 'a t val push : 'a t -> 'a -> unit val pop : 'a t -> 'a option end = struct type 'a t = 'a list Atomic.t let make () = Atomic.make [] let rec push r v = let s = Atomic.get r in if Atomic.compare_and_set r s (v::s) then () else (Domain.cpu_relax (); push r v) let rec pop r = let s = Atomic.get r in match s with | [] -> None | x::xs -> if Atomic.compare_and_set r s xs then Some x else (Domain.cpu_relax (); pop r) end

The atomic stack is represented by an atomic reference that holds a list. The push and pop operations use the compare_and_set primitive to attempt to atomically update the atomic reference. The expression compare_and_set r seen v sets the value of r to v if and only if its current value is physically equal to seen. Importantly, the comparison and the update occur atomically. The expression evaluates to true if the comparison succeeded (and the update happened) and false otherwise.

If the compare_and_set fails, then some other domain is also attempting to update the atomic reference at the same time. In this case, the push and pop operations call Domain.cpu_relax to back off for a short duration allowing competing domains to make progress before retrying the failed operation. This lock-free stack implementation is also known as Treiber stack.