executor.hpp

#pragma once
 
#include "../observer/tfprof.hpp"
#include "taskflow.hpp"
#include "async_task.hpp"

 
namespace tf {
 
// ----------------------------------------------------------------------------
// Executor Definition
// ----------------------------------------------------------------------------

class Executor {
 
  friend class FlowBuilder;
  friend class Subflow;
  friend class Runtime;
  friend class NonpreemptiveRuntime;
  friend class Algorithm;
  friend class TaskGroup;
 
  public:

  explicit Executor(
    size_t N = std::thread::hardware_concurrency(),
    std::shared_ptr<WorkerInterface> wif = nullptr
  );

  ~Executor();

  tf::Future<void> run(Taskflow& taskflow);

  tf::Future<void> run(Taskflow&& taskflow);

  template<typename C>
  tf::Future<void> run(Taskflow& taskflow, C&& callable);

  template<typename C>
  tf::Future<void> run(Taskflow&& taskflow, C&& callable);

  tf::Future<void> run_n(Taskflow& taskflow, size_t N);

  tf::Future<void> run_n(Taskflow&& taskflow, size_t N);

  template<typename C>
  tf::Future<void> run_n(Taskflow& taskflow, size_t N, C&& callable);

  template<typename C>
  tf::Future<void> run_n(Taskflow&& taskflow, size_t N, C&& callable);

  template<typename P>
  tf::Future<void> run_until(Taskflow& taskflow, P&& pred);

  template<typename P>
  tf::Future<void> run_until(Taskflow&& taskflow, P&& pred);

  template<typename P, typename C>
  tf::Future<void> run_until(Taskflow& taskflow, P&& pred, C&& callable);

  template<typename P, typename C>
  tf::Future<void> run_until(Taskflow&& taskflow, P&& pred, C&& callable);

  template <typename T>
  void corun(T& target);

  template <typename P>
  void corun_until(P&& predicate);

  void wait_for_all();

  size_t num_workers() const noexcept;
  
  size_t num_waiters() const noexcept;
  
  size_t num_queues() const noexcept;

  size_t num_topologies() const;

  Worker* this_worker();
  
  int this_worker_id() const;
 
  // --------------------------------------------------------------------------
  // Observer methods
  // --------------------------------------------------------------------------

  template <typename Observer, typename... ArgsT>
  std::shared_ptr<Observer> make_observer(ArgsT&&... args);

  template <typename Observer>
  void remove_observer(std::shared_ptr<Observer> observer);

  size_t num_observers() const noexcept;
 
  // --------------------------------------------------------------------------
  // Async Task Methods
  // --------------------------------------------------------------------------
  
  template <typename P, typename F>
  auto async(P&& params, F&& func);

  template <typename F>
  auto async(F&& func);

  template <typename P, typename F>
  void silent_async(P&& params, F&& func);
  
  template <typename F>
  void silent_async(F&& func);
 
  // --------------------------------------------------------------------------
  // Silent Dependent Async Methods
  // --------------------------------------------------------------------------
  
  template <typename F, typename... Tasks>
requires (std::same_as<std::decay_t<Tasks>, AsyncTask> && ...)
  tf::AsyncTask silent_dependent_async(F&& func, Tasks&&... tasks);
  
  template <TaskParamsLike P, typename F, typename... Tasks>
      requires (std::same_as<std::decay_t<Tasks>, AsyncTask> && ...)
  tf::AsyncTask silent_dependent_async(P&& params, F&& func, Tasks&&... tasks);
  
  template <typename F, typename I>
requires (!std::same_as<std::decay_t<I>, AsyncTask>)
  tf::AsyncTask silent_dependent_async(F&& func, I first, I last);
  
  template <TaskParamsLike P, typename F, typename I>
      requires (!std::same_as<std::decay_t<I>, AsyncTask>)
  tf::AsyncTask silent_dependent_async(P&& params, F&& func, I first, I last);
  
  // --------------------------------------------------------------------------
  // Dependent Async Methods
  // --------------------------------------------------------------------------
  
  template <typename F, typename... Tasks>
requires (std::same_as<std::decay_t<Tasks>, AsyncTask> && ...)
  auto dependent_async(F&& func, Tasks&&... tasks);
  
  template <TaskParamsLike P, typename F, typename... Tasks>
      requires (std::same_as<std::decay_t<Tasks>, AsyncTask> && ...)
  auto dependent_async(P&& params, F&& func, Tasks&&... tasks);
  
  template <typename F, typename I>
requires (!std::same_as<std::decay_t<I>, AsyncTask>)
  auto dependent_async(F&& func, I first, I last);
  
  template <TaskParamsLike P, typename F, typename I>
      requires (!std::same_as<std::decay_t<I>, AsyncTask>)
  auto dependent_async(P&& params, F&& func, I first, I last);
 
  // ----------------------------------------------------------------------------------------------
  // Task Group
  // ----------------------------------------------------------------------------------------------
  
  TaskGroup task_group();
 
  private:
  
  struct Buffer {
    std::mutex mutex;
    UnboundedWSQ<Node*> queue;
  };  
  
  std::vector<Worker> _workers;
  std::vector<Buffer> _buffers;
  
  // notifier's state variable and num_topologies should sit on different cachelines
  // or the false sharing can cause serious performance drop
  alignas(TF_CACHELINE_SIZE) DefaultNotifier _notifier;
  alignas(TF_CACHELINE_SIZE) std::atomic<size_t> _num_topologies {0};
  
  std::unordered_map<std::thread::id, Worker*> _t2w;
  std::unordered_set<std::shared_ptr<ObserverInterface>> _observers;
 
  void _shutdown();
  void _observer_prologue(Worker&, Node*);
  void _observer_epilogue(Worker&, Node*);
  void _spawn(size_t, std::shared_ptr<WorkerInterface>);
  void _exploit_task(Worker&, Node*&);
  bool _explore_task(Worker&, Node*&);
  void _schedule(Worker&, Node*);
  void _schedule(Node*);
  void _schedule_graph(Worker&, Graph&, Topology*, NodeBase*);
  void _spill(Node*);
  void _set_up_topology(Worker*, Topology*);
  void _tear_down_topology(Worker&, Topology*, Node*&);
  void _tear_down_async(Worker&, Node*, Node*&);
  void _tear_down_dependent_async(Worker&, Node*, Node*&);
  void _tear_down_nonasync(Worker&, Node*, Node*&);
  void _tear_down_invoke(Worker&, Node*, Node*&);
  void _increment_topology();
  void _decrement_topology();
  void _invoke(Worker&, Node*);
  void _invoke_static_task(Worker&, Node*);
  void _invoke_nonpreemptive_runtime_task(Worker&, Node*);
  void _invoke_condition_task(Worker&, Node*, SmallVector<int>&);
  void _invoke_multi_condition_task(Worker&, Node*, SmallVector<int>&);
  void _process_dependent_async(Node*, tf::AsyncTask&, size_t&);
  void _process_exception(Worker&, Node*);
  void _update_cache(Worker&, Node*&, Node*);
  void _corun_graph(Worker&, Graph&, Topology*, NodeBase*);
 
  bool _wait_for_task(Worker&, Node*&);
  bool _invoke_subflow_task(Worker&, Node*);
  bool _invoke_module_task(Worker&, Node*);
  bool _invoke_adopted_module_task(Worker&, Node*);
  bool _invoke_module_task_impl(Worker&, Node*, Graph&);
  bool _invoke_async_task(Worker&, Node*);
  bool _invoke_dependent_async_task(Worker&, Node*);
  bool _invoke_runtime_task(Worker&, Node*);
  bool _invoke_runtime_task_impl(Worker&, Node*, std::function<void(Runtime&)>&);
  bool _invoke_runtime_task_impl(Worker&, Node*, std::function<void(Runtime&, bool)>&);
 
  size_t _set_up_graph(Graph&, Topology*, NodeBase*);
  
  template <typename P>
  void _corun_until(Worker&, P&&);
 
  template <typename I>
  void _bulk_schedule(Worker&, I, size_t);
 
  template <typename I>
  void _bulk_schedule(I, size_t);
 
  template <typename I>
  void _bulk_spill(I, size_t);
  
  template <typename I>
  void _bulk_spill_round_robin(I, size_t);
 
  template <size_t N>
  void _bulk_update_cache(Worker&, Node*&, Node*, std::array<Node*, N>&, size_t&);
 
  template <typename P, typename F>
  auto _async(P&&, F&&, Topology*, NodeBase*);
 
  template <typename P, typename F>
  void _silent_async(P&&, F&&, Topology*, NodeBase*);
 
  template <TaskParamsLike P, typename F, typename I>
  requires (!std::same_as<std::decay_t<I>, AsyncTask>)
  auto _dependent_async(P&&, F&&, I, I, Topology*, NodeBase*);
  
  template <TaskParamsLike P, typename F, typename I>
  requires (!std::same_as<std::decay_t<I>, AsyncTask>)
  auto _silent_dependent_async(P&&, F&&, I, I, Topology*, NodeBase*);
  
  template <typename... ArgsT>
  void _schedule_async_task(ArgsT&&...);
  
  template <typename I, typename... ArgsT>
  AsyncTask _schedule_dependent_async_task(I, I, size_t, ArgsT&&...);
};
 
#ifndef DOXYGEN_GENERATING_OUTPUT
 
// Constructor
inline Executor::Executor(size_t N, std::shared_ptr<WorkerInterface> wif) :
  _workers  (N),
  _buffers  (std::bit_width(N)), // Empirically, we find that log2(N) performs best.
  _notifier (N) {
 
  if(N == 0) {
    TF_THROW("executor must define at least one worker");
  }
  
  // If spawning N threads fails, shut down any created threads before 
  // rethrowing the exception.
#ifndef TF_DISABLE_EXCEPTION_HANDLING
  try {
#endif
    _spawn(N, std::move(wif));
#ifndef TF_DISABLE_EXCEPTION_HANDLING
  }
  catch(...) {
    _shutdown();
    std::rethrow_exception(std::current_exception());
  }
#endif
 
  // initialize the default observer if requested
  if(has_env(TF_ENABLE_PROFILER)) {
    TFProfManager::get()._manage(make_observer<TFProfObserver>());
  }
}
 
// Destructor
inline Executor::~Executor() {
  _shutdown();
}
 
// Function: _shutdown
inline void Executor::_shutdown() {
 
  // wait for all topologies to complete
  wait_for_all();
 
  // shut down the scheduler
  for(size_t i=0; i<_workers.size(); ++i) {
    _workers[i]._done.test_and_set(std::memory_order_relaxed);
  }
  
  _notifier.notify_all();
  
  // Only join the thread if it is joinable, as std::thread construction 
  // may fail and throw an exception.
  for(auto& w : _workers) {
    if(w._thread.joinable()) {
      w._thread.join();
    }
  }
}
 
// Function: num_workers
inline size_t Executor::num_workers() const noexcept {
  return _workers.size();
}
 
// Function: num_waiters
inline size_t Executor::num_waiters() const noexcept {
  return _notifier.num_waiters();
}
 
// Function: num_queues
inline size_t Executor::num_queues() const noexcept {
  return _workers.size() + _buffers.size();
}
 
// Function: num_topologies
inline size_t Executor::num_topologies() const {
  return _num_topologies.load(std::memory_order_relaxed);
}
 
// Function: this_worker
inline Worker* Executor::this_worker() {
  auto itr = _t2w.find(std::this_thread::get_id());
  return itr == _t2w.end() ? nullptr : itr->second;
}
 
// Function: this_worker_id
inline int Executor::this_worker_id() const {
  auto i = _t2w.find(std::this_thread::get_id());
  return i == _t2w.end() ? -1 : static_cast<int>(i->second->_id);
}
 
// Procedure: _spawn
inline void Executor::_spawn(size_t N, std::shared_ptr<WorkerInterface> wif) {
 
  for(size_t id=0; id<N; ++id) {
    _workers[id]._thread = std::thread([&, id, wif] () {
 
      auto& worker = _workers[id];
  
      worker._id = id;
      worker._sticky_victim = id;
      worker._rdgen.seed(static_cast<uint32_t>(std::hash<std::thread::id>()(std::this_thread::get_id())));
 
      // before entering the work-stealing loop, call the scheduler prologue
      if(wif) {
        wif->scheduler_prologue(worker);
      }
 
      Node* t = nullptr;
      std::exception_ptr ptr = nullptr;
 
      // must use 1 as condition instead of !done because
      // the previous worker may stop while the following workers
      // are still preparing for entering the scheduling loop
#ifndef TF_DISABLE_EXCEPTION_HANDLING
      try {
#endif
        // work-stealing loop
        while(1) {
 
          // drains out the local queue first
          _exploit_task(worker, t);
 
          // steals and waits for tasks
          if(_wait_for_task(worker, t) == false) {
            break;
          }
        }
 
#ifndef TF_DISABLE_EXCEPTION_HANDLING
      } 
      catch(...) {
        ptr = std::current_exception();
      }
#endif
      
      // call the user-specified epilogue function
      if(wif) {
        wif->scheduler_epilogue(worker, ptr);
      }
 
    });
    
    // We avoid using thread-local storage to track the mapping between a thread
    // and its corresponding worker in an executor. On Windows, thread-local
    // storage can be unreliable in certain situations (see issue #727).
    //
    // Instead, we maintain a per-executor mapping from threads to workers.
    // This approach has an additional advantage: according to the C++ Standard,
    // std::thread::id uniquely identifies a thread object. Therefore, once the map
    // returns a valid worker, we can be certain that the worker belongs to this
    // executor. This eliminates the need for additional executor validation 
    // required by using thread-local storage.
    //
    // Example:
    //
    //   Worker* w = this_worker();
    //   // Using thread-local storage, we would need additional executor validation:
    //   if (w == nullptr || w->_executor != this) { /* caller is not a worker of this executor */ }
    //
    //   // Using per-executor mapping, it suffices to check:
    //   if (w == nullptr) { /* caller is not a worker of this executor */ }
    //
    _t2w.emplace(_workers[id]._thread.get_id(), &_workers[id]);
  }
}
 
// Function: _explore_task
inline bool Executor::_explore_task(Worker& w, Node*& t) {
 
  // Fast path: if no topologies are live, all queues are guaranteed empty
  // by the executor's invariant (num_topologies reaches zero only after all
  // nodes have been scheduled and their queues flushed). Skip the entire
  // steal loop and return immediately so the caller enters _wait_for_task
  // to sleep. relaxed ordering is sufficient — this is a hint, and any
  // missed update is caught safely by the 2PC guard in _wait_for_task.
  if(_num_topologies.load(std::memory_order_relaxed) == 0) {
    return true;
  }
 
  const size_t MAX_VICTIM    = num_queues();  // guaranteed >= 2 by constructor
  const size_t MAX_STEALS    = ((MAX_VICTIM + 1) << 1);
  const size_t STICKY_THRESH = 4;  // max retries on a contended victim
 
  // local aliases for steal protocol sentinels — these are properties of the
  // steal protocol, not of any specific queue type
  constexpr Node* empty_steal     = wsq_empty_value<Node*>();
  const     Node* contended_steal = wsq_contended_value<Node*>();
 
  size_t num_steals    = 0;
  size_t num_contended = 0;
  size_t vtm           = w._sticky_victim;
 
  while(true) {
 
    Node* result = (vtm < _workers.size())
      ? _workers[vtm]._wsq.steal_with_feedback()
      : _buffers[vtm - _workers.size()].queue.steal_with_feedback();
 
    if(result != empty_steal && result != contended_steal) {
      // STOLEN: successfully acquired a task — reinforce sticky victim
      t = result;
      w._sticky_victim = vtm;
      break;
    }
 
    if(result == contended_steal) {
      // CONTENDED: victim has work but we lost the CAS race — retry the
      // same victim up to STICKY_THRESH times before moving on
      if(++num_contended < STICKY_THRESH) {
        continue;  // stay on vtm, skip victim switch and num_steals increment
      }
    }
    // EMPTY or CONTENDED-exhausted: pick a new victim excluding self
    // since our own queue is empty by invariant. map [0, MAX_VICTIM-1)
    // over [0, MAX_VICTIM) \ {w._id} — always safe since MAX_VICTIM >= 2.
    num_contended = 0;
    vtm = w._rdgen() % (MAX_VICTIM - 1);
    if(vtm >= w._id) vtm++;
 
    if(++num_steals > MAX_STEALS) {
      std::this_thread::yield();
      if(num_steals > 150 + MAX_STEALS) {
        break;
      }
    }
 
    if(w._done.test(std::memory_order_relaxed)) {
      return false;
    }
  }
 
  return true;
} 
 
/*
// Function: _explore_task
inline bool Executor::_explore_task(Worker& w, Node*& t) {
 
  // Fast path: if no topologies are live, all queues are guaranteed empty
  // by the executor's invariant (num_topologies reaches zero only after all
  // nodes have been scheduled and their queues flushed). Skip the entire
  // steal loop and return immediately so the caller enters _wait_for_task
  // to sleep. relaxed ordering is sufficient — this is a hint, and any
  // missed update is caught safely by the 2PC guard in _wait_for_task.
  if(_num_topologies.load(std::memory_order_relaxed) == 0) {
    return true;
  }
 
  //assert(!t);
  const size_t MAX_VICTIM = num_queues();
  const size_t MAX_STEALS = ((MAX_VICTIM + 1) << 1);
 
  size_t num_steals = 0;
  size_t vtm = w._sticky_victim;
 
  // Make the worker steal immediately from the assigned victim.
  while(true) {
    
    // If the worker's victim thread is within the worker pool, steal from the worker's queue.
    // Otherwise, steal from the buffer, adjusting the victim index based on the worker pool size.
    t = (vtm < _workers.size())
      ? _workers[vtm]._wsq.steal()
      : _buffers[vtm - _workers.size()].queue.steal();
 
    if(t) {
      w._sticky_victim = vtm;
      break;
    }
 
    // Increment the steal count, and if it exceeds MAX_STEALS, yield the thread.
    // If the number of empty steals reaches MAX_STEALS, exit the loop.
    if (++num_steals > MAX_STEALS) {
      std::this_thread::yield();
      if(num_steals > 150 + MAX_STEALS) {
        break;
      }
    }
 
    if(w._done.test(std::memory_order_relaxed)) {
      return false;
    } 
 
    // Randomely generate a next victim.
    vtm = w._rdgen() % MAX_VICTIM;
  } 
  return true;
}
*/
 
// Procedure: _exploit_task
inline void Executor::_exploit_task(Worker& w, Node*& t) {
  while(t) {
    _invoke(w, t);
    t = w._wsq.pop();
  }
}
 
// Function: _wait_for_task
inline bool Executor::_wait_for_task(Worker& w, Node*& t) {
 
  explore_task:
 
  if(_explore_task(w, t) == false) {
    return false;
  }
  
  // Go exploit the task if we successfully steal one.
  if(t) {
    return true;
  }
 
  // Entering the 2PC guard as all queues are likely empty after many stealing attempts.
  _notifier.prepare_wait(w._id);
 
  // Fast path: if no topologies are live, all queues are guaranteed empty.
  // Skip the O(N) buffer and worker queue scans and go directly to sleep.
  // This is safe because prepare_wait has already been called — any notify
  // that arrives after this check but before commit_wait will be caught by
  // the 2PC guarantee of the notifier.
  if(_num_topologies.load(std::memory_order_relaxed) == 0) {
    // still check done flag before committing to sleep
    if(w._done.test(std::memory_order_relaxed)) {
      _notifier.cancel_wait(w._id);
      return false;
    }
    _notifier.commit_wait(w._id);
    goto explore_task;
  }
  
  // Condition #1: buffers should be empty
  for(size_t b=0; b<_buffers.size(); ++b) {
    if(!_buffers[b].queue.empty()) {
      _notifier.cancel_wait(w._id);
      w._sticky_victim = b + _workers.size();
      goto explore_task;
    }
  }
  
  // Condition #2: worker queues should be empty
  // Note: We need to use index-based looping to avoid data race with _spawn
  // which initializes other worker data structure at the same time.
  // Also, due to the property of a work-stealing queue, we don't need to check 
  // this worker's work-stealing queue.
  for(size_t k=0; k<_workers.size()-1; ++k) {
    if(size_t vtm = k + (k >= w._id); !_workers[vtm]._wsq.empty()) {
      _notifier.cancel_wait(w._id);
      w._sticky_victim = vtm;
      goto explore_task;
    }
  }
  
  // Condition #3: worker should be alive
  if(w._done.test(std::memory_order_relaxed)) {
    _notifier.cancel_wait(w._id);
    return false;
  }
  
  // Now I really need to relinquish myself to others.
  _notifier.commit_wait(w._id);
  goto explore_task;
}
 
// Function: make_observer
template<typename Observer, typename... ArgsT>
std::shared_ptr<Observer> Executor::make_observer(ArgsT&&... args) {
 
  static_assert(
    std::is_base_of_v<ObserverInterface, Observer>,
    "Observer must be derived from ObserverInterface"
  );
 
  // use a local variable to mimic the constructor
  auto ptr = std::make_shared<Observer>(std::forward<ArgsT>(args)...);
 
  ptr->set_up(_workers.size());
 
  _observers.emplace(std::static_pointer_cast<ObserverInterface>(ptr));
 
  return ptr;
}
 
// Procedure: remove_observer
template <typename Observer>
void Executor::remove_observer(std::shared_ptr<Observer> ptr) {
 
  static_assert(
    std::is_base_of_v<ObserverInterface, Observer>,
    "Observer must be derived from ObserverInterface"
  );
 
  _observers.erase(std::static_pointer_cast<ObserverInterface>(ptr));
}
 
// Function: num_observers
inline size_t Executor::num_observers() const noexcept {
  return _observers.size();
}
 
// Procedure: _spill
inline void Executor::_spill(Node* item) {
  // Since pointers are aligned to 8 bytes, we perform a simple hash to avoid 
  // contention caused by hashing to the same slot.
  auto b = (reinterpret_cast<uintptr_t>(item) >> 16) % _buffers.size();
  std::scoped_lock lock(_buffers[b].mutex);
  _buffers[b].queue.push(item);
}
 
// Procedure: _bulk_spill (single batch to one buffer)
// Uses Knuth multiplicative hash on the first pointer to select a buffer,
// providing better bit diffusion than the shift-based approach, especially
// when the allocator returns pointers with regular low-bit patterns.
template <typename I>
void Executor::_bulk_spill(I first, size_t N) {
  //assert(N != 0);
  auto b = ((reinterpret_cast<uintptr_t>(*first) * 2654435761ULL) >> 32) % _buffers.size();
  std::scoped_lock lock(_buffers[b].mutex);
  _buffers[b].queue.bulk_push(first, N);
}
 
// Procedure: _bulk_spill
// Distributes a batch of N spilled nodes across all buffers in round-robin
// order starting from a hash of the first node's pointer. Each buffer's lock
// is held only for its chunk, reducing contention compared to sending the
// entire batch to a single buffer.
template <typename I>
void Executor::_bulk_spill_round_robin(I first, size_t N) {
 
  // assert(N != 0);
  const size_t B     = _buffers.size();
  const size_t start = ((reinterpret_cast<uintptr_t>(*first) * 2654435761ULL) >> 32) % B;
  const size_t per_buf = (N + B - 1) / B;
  size_t remaining = N;
  for(size_t i = 0; i < B && remaining > 0; ++i) {
    size_t b     = (start + i) % B;
    size_t chunk = std::min(per_buf, remaining);
    {
      std::scoped_lock lock(_buffers[b].mutex);
      _buffers[b].queue.bulk_push(first, chunk);
    }
    // terminates early via remaining > 0, so we don't acquire unnecessary locks on empty chunks.
    remaining -= chunk;
  }
}
 
// Procedure: _schedule
inline void Executor::_schedule(Worker& worker, Node* node) {
  // starting at v3.5 we do not use any complicated notification mechanism 
  // as the experimental result has shown no significant advantage.
  if(worker._wsq.try_push(node) == false) {
    _spill(node);
  }
  _notifier.notify_one();
}
 
// Procedure: _schedule
inline void Executor::_schedule(Node* node) {
  _spill(node);
  _notifier.notify_one();
}
 
// Procedure: _schedule
template <typename I>
void Executor::_bulk_schedule(Worker& worker, I first, size_t num_nodes) {
 
  if(num_nodes == 0) {
    return;
  }
 
  // NOTE: We cannot use first/last in the for-loop (e.g., for(; first != last; ++first)).
  // This is because when a node v is inserted into the queue, v can run and finish 
  // immediately. If v is the last node in the graph, it will tear down the parent task vector
  // which cause the last ++first to fail. This problem is specific to MSVC which has a stricter
  // iterator implementation in std::vector than GCC/Clang.
  if(auto n = worker._wsq.try_bulk_push(first, num_nodes); n != num_nodes) {
    _bulk_spill(first, num_nodes - n);
  }
  _notifier.notify_n(num_nodes);
    
  // notify first before spilling to hopefully wake up workers earlier 
  // however, the experiment does not show any benefit for doing this.
  //auto n = worker._wsq.try_bulk_push(first, num_nodes);
  //_notifier.notify_n(n);
  //_bulk_schedule(first + n, num_nodes - n);
}
 
// Procedure: _schedule
template <typename I>
inline void Executor::_bulk_schedule(I first, size_t num_nodes) {
  
  if(num_nodes == 0) {
    return;
  }
  
  // NOTE: We cannot use first/last in the for-loop (e.g., for(; first != last; ++first)).
  // This is because when a node v is inserted into the queue, v can run and finish 
  // immediately. If v is the last node in the graph, it will tear down the parent task vector
  // which cause the last ++first to fail. This problem is specific to MSVC which has a stricter
  // iterator implementation in std::vector than GCC/Clang.
  _bulk_spill(first, num_nodes);
  _notifier.notify_n(num_nodes);
}
 
// Function: _update_cache
TF_FORCE_INLINE void Executor::_update_cache(Worker& worker, Node*& cache, Node* node) {
  if(cache) {
    _schedule(worker, cache);
  }
  cache = node;
}
 
// Function: _bulk_update_cache
template <size_t N>
TF_FORCE_INLINE void Executor::_bulk_update_cache(
  Worker& worker, Node*& cache, Node* node, std::array<Node*, N>& array, size_t& n
) {
  // experimental results show no benefit of using bulk_update_cache
  if(cache) {
    array[n++] = cache;
    if(n == N) {
      _bulk_schedule(worker, array, n);
      n = 0;
    }
  }
  cache = node;
}
  
// Procedure: _invoke
inline void Executor::_invoke(Worker& worker, Node* node) {
 
  #define TF_INVOKE_CONTINUATION()  \
  if (cache) {                      \
    node = cache;                   \
    goto begin_invoke;              \
  }
 
  begin_invoke:
 
  Node* cache {nullptr};
  
  // if this is the second invoke due to preemption, directly jump to invoke task
  if(node->_nstate & NSTATE::PREEMPTED) {
    goto invoke_task;
  }
 
  // If the work has been cancelled, there is no need to continue.
  // Here, we do tear_down_invoke since async tasks may also get cancelled where
  // we need to recycle the node.
  if(node->_is_parent_cancelled()) {
    _tear_down_invoke(worker, node, cache);
    TF_INVOKE_CONTINUATION();
    return;
  }
 
  // if acquiring semaphore(s) exists, acquire them first
  if(node->_semaphores && !node->_semaphores->to_acquire.empty()) {
    SmallVector<Node*> waiters;
    if(!node->_acquire_all(waiters)) {
      _bulk_schedule(worker, waiters.begin(), waiters.size());
      return;
    }
  }
  
  invoke_task:
  
  SmallVector<int> conds;
 
  // switch is faster than nested if-else due to jump table
  switch(node->_handle.index()) {
    // static task
    case Node::STATIC:{
      _invoke_static_task(worker, node);
    }
    break;
    
    // runtime task
    case Node::RUNTIME:{
      if(_invoke_runtime_task(worker, node)) {
        return;
      }
    }
    break;
    
    // non-preemptive runtime task
    case Node::NONPREEMPTIVE_RUNTIME:{
      _invoke_nonpreemptive_runtime_task(worker, node);
    }
    break;
 
    // subflow task
    case Node::SUBFLOW: {
      if(_invoke_subflow_task(worker, node)) {
        return;
      }
    }
    break;
 
    // condition task
    case Node::CONDITION: {
      _invoke_condition_task(worker, node, conds);
    }
    break;
 
    // multi-condition task
    case Node::MULTI_CONDITION: {
      _invoke_multi_condition_task(worker, node, conds);
    }
    break;
 
    // module task
    case Node::MODULE: {
      if(_invoke_module_task(worker, node)) {
        return;
      }
    }
    break;
    
    // adopted module task
    case Node::ADOPTED_MODULE: {
      if(_invoke_adopted_module_task(worker, node)) {
        return;
      }
    }
    break;
 
    // async task
    case Node::ASYNC: {
      if(_invoke_async_task(worker, node)) {
        return;
      }
      _tear_down_async(worker, node, cache);
      TF_INVOKE_CONTINUATION();
      return;
    }
    break;
 
    // dependent async task
    case Node::DEPENDENT_ASYNC: {
      if(_invoke_dependent_async_task(worker, node)) {
        return;
      }
      _tear_down_dependent_async(worker, node, cache);
      TF_INVOKE_CONTINUATION();
      return;
    }
    break;
 
    // monostate (placeholder)
    default:
    break;
  }
 
  // if releasing semaphores exist, release them
  if(node->_semaphores && !node->_semaphores->to_release.empty()) {
    SmallVector<Node*> waiters;
    node->_release_all(waiters);
    _bulk_schedule(worker, waiters.begin(), waiters.size());
  }
 
  // Reset the join counter with strong dependencies to support cycles.
  // + We must do this before scheduling the successors to avoid race
  //   condition on _predecessors.
  // + We must use fetch_add instead of direct assigning
  //   because the user-level call on "invoke" may explicitly schedule 
  //   this task again (e.g., pipeline) which can access the join_counter.
  node->_join_counter.fetch_add(
    node->_nstate & NSTATE::STRONG_DEPENDENCIES_MASK, std::memory_order_relaxed
  );
 
  // Invoke the task based on the corresponding type
  switch(node->_handle.index()) {
 
    // condition and multi-condition tasks
    case Node::CONDITION:
    case Node::MULTI_CONDITION: {
      for(auto cond : conds) {
        if(cond >= 0 && static_cast<size_t>(cond) < node->_num_successors) {
          auto s = node->_edges[cond]; 
          // zeroing the join counter for invariant
          s->_join_counter.store(0, std::memory_order_relaxed);
          node->_parent->_join_counter.fetch_add(1, std::memory_order_relaxed);
          _update_cache(worker, cache, s);
        }
      }
    }
    break;
 
    // non-condition task
    default: {
      for(size_t i=0; i<node->_num_successors; ++i) {
        if(auto s = node->_edges[i]; s->_join_counter.fetch_sub(1, std::memory_order_acq_rel) == 1) {
          node->_parent->_join_counter.fetch_add(1, std::memory_order_relaxed);
          _update_cache(worker, cache, s);
        }
      }
    }
    break;
  }
 
  // clean up the node after execution
  _tear_down_nonasync(worker, node, cache);
  TF_INVOKE_CONTINUATION();
}
 
// Procedure: _tear_down_nonasync
inline void Executor::_tear_down_nonasync(Worker& worker, Node* node, Node*& cache) {
 
  // we must check parent first before subtracting the join counter,
  // or it can introduce data race
  if(auto parent = node->_parent; parent == node->_topology) {
    if(parent->_join_counter.fetch_sub(1, std::memory_order_acq_rel) == 1) {
      _tear_down_topology(worker, node->_topology, cache);
    }
  }
  else {  
    // needs to fetch every data before join counter becomes zero at which
    // the node may be deleted
    auto state = parent->_nstate;
    if(parent->_join_counter.fetch_sub(1, std::memory_order_acq_rel) == 1) {
      // this task is spawned from a preempted parent, so we need to resume it
      if(state & NSTATE::PREEMPTED) {
        _update_cache(worker, cache, static_cast<Node*>(parent));
      }
    }
  }
}
 
// Procedure: _tear_down_invoke
inline void Executor::_tear_down_invoke(Worker& worker, Node* node, Node*& cache) {
  switch(node->_handle.index()) {
    case Node::ASYNC:
      _tear_down_async(worker, node, cache);
    break;
 
    case Node::DEPENDENT_ASYNC:
      _tear_down_dependent_async(worker, node, cache);
    break;
 
    default:
      _tear_down_nonasync(worker, node, cache);
    break;
  }
}
 
// Procedure: _observer_prologue
inline void Executor::_observer_prologue(Worker& worker, Node* node) {
  for(auto& observer : _observers) {
    observer->on_entry(WorkerView(worker), TaskView(*node));
  }
}
 
// Procedure: _observer_epilogue
inline void Executor::_observer_epilogue(Worker& worker, Node* node) {
  for(auto& observer : _observers) {
    observer->on_exit(WorkerView(worker), TaskView(*node));
  }
}
 
// Procedure: _process_exception
inline void Executor::_process_exception(Worker&, Node* node) {
 
  // Finds the anchor and mark the entire path with exception, 
  // so recursive tasks can be cancelled properly.
  // Since exception can come from asynchronous task (with runtime), the node itself can be anchored.
  NodeBase* ea = node;     // explicit anchor
  NodeBase* ia = nullptr;  // implicit anchor
  
  while(ea && (ea->_estate.load(std::memory_order_relaxed) & ESTATE::EXPLICITLY_ANCHORED) == 0) {
    ea->_estate.fetch_or(ESTATE::EXCEPTION, std::memory_order_relaxed);
    // we only want the inner-most implicit anchor
    if(ia == nullptr && (ea->_nstate & NSTATE::IMPLICITLY_ANCHORED)) {
      ia = ea;
    }
    ea = ea->_parent;
  }
  
  // flag used to ensure execution is caught in a thread-safe manner
  constexpr static auto flag = ESTATE::EXCEPTION | ESTATE::CAUGHT;
 
  // The exception occurs under a blocking call (e.g., corun, join).
  if(ea) {
    // multiple tasks may throw, and we only take the first thrown exception
    if((ea->_estate.fetch_or(flag, std::memory_order_relaxed) & ESTATE::CAUGHT) == 0) {
      ea->_exception_ptr = std::current_exception();
      return;
    }
  }
  // Implicit anchor has the lowest priority
  else if(ia){
    if((ia->_estate.fetch_or(flag, std::memory_order_relaxed) & ESTATE::CAUGHT) == 0) {
      ia->_exception_ptr = std::current_exception();
      return;
    }
  }
  
  // For now, we simply store the exception in this node; this can happen in an 
  // execution that does not have any external control to capture the exception,
  // such as silent async task without any parent.
  node->_exception_ptr = std::current_exception();
}
 
// Procedure: _invoke_static_task
inline void Executor::_invoke_static_task(Worker& worker, Node* node) {
  _observer_prologue(worker, node);
  TF_EXECUTOR_EXCEPTION_HANDLER(worker, node, {
    std::get_if<Node::Static>(&node->_handle)->work();
  });
  _observer_epilogue(worker, node);
}
 
// Procedure: _invoke_subflow_task
inline bool Executor::_invoke_subflow_task(Worker& worker, Node* node) {
    
  auto& h = *std::get_if<Node::Subflow>(&node->_handle);
  auto& g = h.subgraph;
 
  if((node->_nstate & NSTATE::PREEMPTED) == 0) {
    
    // set up the subflow
    Subflow sf(*this, worker, node, g);
 
    // invoke the subflow callable
    _observer_prologue(worker, node);
    TF_EXECUTOR_EXCEPTION_HANDLER(worker, node, {
      h.work(sf);
    });
    _observer_epilogue(worker, node);
    
    // spawn the subflow if it is joinable and its graph is non-empty
    // implicit join is faster than Subflow::join as it does not involve corun
    if(sf.joinable() && !g.empty()) {
 
      // signal the executor to preempt this node
      node->_nstate |= NSTATE::PREEMPTED;
 
      // set up and schedule the graph
      _schedule_graph(worker, g, node->_topology, node);
      return true;
    }
  }
  else {
    node->_nstate &= ~NSTATE::PREEMPTED;
  }
 
  // The subflow has finished or joined.
  // By default, we clear the subflow storage as applications can perform recursive
  // subflow tasking which accumulates a huge amount of memory overhead, hampering 
  // the performance.
  if((node->_nstate & NSTATE::RETAIN_SUBFLOW) == 0) {
    g.clear();
  }
 
  return false;
}
 
// Procedure: _invoke_condition_task
inline void Executor::_invoke_condition_task(
  Worker& worker, Node* node, SmallVector<int>& conds
) {
  _observer_prologue(worker, node);
  TF_EXECUTOR_EXCEPTION_HANDLER(worker, node, {
    auto& work = std::get_if<Node::Condition>(&node->_handle)->work;
    conds = { work() };
  });
  _observer_epilogue(worker, node);
}
 
// Procedure: _invoke_multi_condition_task
inline void Executor::_invoke_multi_condition_task(
  Worker& worker, Node* node, SmallVector<int>& conds
) {
  _observer_prologue(worker, node);
  TF_EXECUTOR_EXCEPTION_HANDLER(worker, node, {
    conds = std::get_if<Node::MultiCondition>(&node->_handle)->work();
  });
  _observer_epilogue(worker, node);
}
 
// Procedure: _invoke_module_task
inline bool Executor::_invoke_module_task(Worker& w, Node* node) {
  return _invoke_module_task_impl(w, node, std::get_if<Node::Module>(&node->_handle)->graph);  
}
 
// Procedure: _invoke_adopted_module_task
inline bool Executor::_invoke_adopted_module_task(Worker& w, Node* node) {
  return _invoke_module_task_impl(w, node, std::get_if<Node::AdoptedModule>(&node->_handle)->graph);  
}
 
// Procedure: _invoke_module_task_impl
inline bool Executor::_invoke_module_task_impl(Worker& w, Node* node, Graph& graph) {
 
  // No need to do anything for empty graph
  if(graph.empty()) {
    return false;
  }
 
  // first entry - not spawned yet
  if((node->_nstate & NSTATE::PREEMPTED) == 0) {
    // signal the executor to preempt this node
    node->_nstate |= NSTATE::PREEMPTED;
    _schedule_graph(w, graph, node->_topology, node);
    return true;
  }
 
  // second entry - already spawned
  node->_nstate &= ~NSTATE::PREEMPTED;
 
  return false;
}
 
 
// Procedure: _invoke_async_task
inline bool Executor::_invoke_async_task(Worker& worker, Node* node) {
  auto& work = std::get_if<Node::Async>(&node->_handle)->work;
  switch(work.index()) {
    // void()
    case 0:
      _observer_prologue(worker, node);
      TF_EXECUTOR_EXCEPTION_HANDLER(worker, node, {
        std::get_if<0>(&work)->operator()();
      });
      _observer_epilogue(worker, node);
    break;
    
    // void(Runtime&)
    case 1:
      if(_invoke_runtime_task_impl(worker, node, *std::get_if<1>(&work))) {
        return true;
      }
    break;
    
    // void(Runtime&, bool)
    case 2:
      if(_invoke_runtime_task_impl(worker, node, *std::get_if<2>(&work))) {
        return true;
      }
    break;
  }
 
  return false;
}
 
// Procedure: _invoke_dependent_async_task
inline bool Executor::_invoke_dependent_async_task(Worker& worker, Node* node) {
  auto& work = std::get_if<Node::DependentAsync>(&node->_handle)->work;
  switch(work.index()) {
    // void()
    case 0:
      _observer_prologue(worker, node);
      TF_EXECUTOR_EXCEPTION_HANDLER(worker, node, {
        std::get_if<0>(&work)->operator()();
      });
      _observer_epilogue(worker, node);
    break;
    
    // void(Runtime&) - silent async
    case 1:
      if(_invoke_runtime_task_impl(worker, node, *std::get_if<1>(&work))) {
        return true;
      }
    break;
 
    // void(Runtime&, bool) - async
    case 2:
      if(_invoke_runtime_task_impl(worker, node, *std::get_if<2>(&work))) {
        return true;
      }
    break;
  }
  return false;
}
 
// Function: run
inline tf::Future<void> Executor::run(Taskflow& f) {
  return run_n(f, 1, [](){});
}
 
// Function: run
inline tf::Future<void> Executor::run(Taskflow&& f) {
  return run_n(std::move(f), 1, [](){});
}
 
// Function: run
template <typename C>
tf::Future<void> Executor::run(Taskflow& f, C&& c) {
  return run_n(f, 1, std::forward<C>(c));
}
 
// Function: run
template <typename C>
tf::Future<void> Executor::run(Taskflow&& f, C&& c) {
  return run_n(std::move(f), 1, std::forward<C>(c));
}
 
// Function: run_n
inline tf::Future<void> Executor::run_n(Taskflow& f, size_t repeat) {
  return run_n(f, repeat, [](){});
}
 
// Function: run_n
inline tf::Future<void> Executor::run_n(Taskflow&& f, size_t repeat) {
  return run_n(std::move(f), repeat, [](){});
}
 
// Function: run_n
template <typename C>
tf::Future<void> Executor::run_n(Taskflow& f, size_t repeat, C&& c) {
  return run_until(
    f, [repeat]() mutable { return repeat-- == 0; }, std::forward<C>(c)
  );
}
 
// Function: run_n
template <typename C>
tf::Future<void> Executor::run_n(Taskflow&& f, size_t repeat, C&& c) {
  return run_until(
    std::move(f), [repeat]() mutable { return repeat-- == 0; }, std::forward<C>(c)
  );
}
 
// Function: run_until
template<typename P>
tf::Future<void> Executor::run_until(Taskflow& f, P&& pred) {
  return run_until(f, std::forward<P>(pred), [](){});
}
 
// Function: run_until
template<typename P>
tf::Future<void> Executor::run_until(Taskflow&& f, P&& pred) {
  return run_until(std::move(f), std::forward<P>(pred), [](){});
}
 
// Function: run_until
template <typename P, typename C>
tf::Future<void> Executor::run_until(Taskflow& f, P&& p, C&& c) {
 
  // No need to create a real topology but returns an dummy future for invariant.
  if(f.empty() || p()) {
    c();
    std::promise<void> promise;
    promise.set_value();
    return tf::Future<void>(promise.get_future());
  }
  
  _increment_topology();
 
  // create a topology for this run
  auto t = std::make_shared<Topology>(f, std::forward<P>(p), std::forward<C>(c));
  //auto t = std::make_shared<DerivedTopology<P, C>>(f, std::forward<P>(p), std::forward<C>(c));
 
  // need to create future before the topology got torn down quickly
  tf::Future<void> future(t->_promise.get_future(), t);
 
  // modifying topology needs to be protected under the lock
  if(f._fetch_enqueue(t) == 0) {
    _set_up_topology(this_worker(), t.get());
  }
 
  return future;
}
 
 
 
// Function: corun_until
template <typename P>
void Executor::corun_until(P&& predicate) {
  
  Worker* w = this_worker();
  if(w == nullptr) {
    TF_THROW("corun_until must be called by a worker of the executor");
  }
 
  _corun_until(*w, std::forward<P>(predicate));
}
 
// Function: _corun_until
template <typename P>
void Executor::_corun_until(Worker& w, P&& stop_predicate) {
 
  const size_t MAX_VICTIM    = num_queues();
  const size_t MAX_STEALS    = ((MAX_VICTIM + 1) << 1);
  const size_t STICKY_THRESH = 8;
 
  constexpr Node* empty_steal     = wsq_empty_value<Node*>();
  const     Node* contended_steal = wsq_contended_value<Node*>();
 
  bool stop = false;
 
  while(!stop && !(stop = stop_predicate())) {
 
    // try local queue first — only one task at a time to avoid deep
    // recursive corun calls causing stack overflow
    if(auto t = w._wsq.pop(); t) {
      _invoke(w, t);
      continue;
    }
 
    // local queue empty: steal from others until stop_predicate or stolen.
    // stop is set by the inner loop condition so when predicate becomes true
    // the outer loop exits immediately without calling stop_predicate again.
    size_t num_steals    = 0;
    size_t num_contended = 0;
    size_t vtm           = w._sticky_victim;
 
    while(!(stop = stop_predicate())) {
 
      Node* result = (vtm < _workers.size())
        ? _workers[vtm]._wsq.steal_with_feedback()
        : _buffers[vtm - _workers.size()].queue.steal_with_feedback();
 
      if(result != empty_steal && result != contended_steal) {
        // STOLEN: invoke task then return to outer loop to re-check
        // local queue and stop_predicate
        _invoke(w, result);
        w._sticky_victim = vtm;
        break;
      }
 
      if(result == contended_steal) {
        // CONTENDED: victim has work, retry same victim up to STICKY_THRESH
        if(++num_contended < STICKY_THRESH) {
          continue;
        }
      }
 
      // EMPTY or CONTENDED-exhausted: pick a new victim excluding self
      num_contended = 0;
      vtm = w._rdgen() % (MAX_VICTIM - 1);
      if(vtm >= w._id) vtm++;
 
      if(++num_steals > MAX_STEALS) {
        // unlike _explore_task we cannot sleep here — the calling worker
        // is blocked inside a task and must keep making progress to avoid
        // deadlock. yield to let other threads run and make progress.
        std::this_thread::yield();
      }
    }
  }
}
 
/*
// Function: _corun_until
template <typename P>
void Executor::_corun_until(Worker& w, P&& stop_predicate) {
 
  const size_t MAX_VICTIM = num_queues();
  const size_t MAX_STEALS = ((MAX_VICTIM + 1) << 1);
    
  exploit:
 
  while(!stop_predicate()) {
    
    // here we don't do while-loop to drain out the local queue as it can
    // potentially enter a very deep recursive corun, cuasing stack overflow
    if(auto t = w._wsq.pop(); t) {
      _invoke(w, t);
    }
    else {
      size_t num_steals = 0;
      size_t vtm = w._sticky_victim;
 
      explore:
 
      t = (vtm < _workers.size()) 
        ? _workers[vtm]._wsq.steal()
        : _buffers[vtm-_workers.size()].queue.steal();
 
      if(t) {
        _invoke(w, t);
        w._sticky_victim = vtm;
        goto exploit;
      }
      else if(!stop_predicate()) {
        if(++num_steals > MAX_STEALS) {
          std::this_thread::yield();
        }
        vtm = w._rdgen() % MAX_VICTIM;
        goto explore;
      }
      else {
        break;
      }
    }
  }
}*/
 
// Function: corun
template <typename T>
void Executor::corun(T& target) {
 
  Worker* w = this_worker();
  if(w == nullptr) {
    TF_THROW("corun must be called by a worker of the executor");
  }
 
  NodeBase anchor;
  _corun_graph(*w, retrieve_graph(target), nullptr, &anchor);
}
 
// Procedure: _corun_graph
inline void Executor::_corun_graph(Worker& w, Graph& g, Topology* tpg, NodeBase* p) {
 
  // empty graph
  if(g.empty()) {
    return;
  }
  
  // anchor this parent as the blocking point
  {
    ExplicitAnchorGuard anchor(p);
    _schedule_graph(w, g, tpg, p);
    _corun_until(w, [p] () -> bool { 
      return p->_join_counter.load(std::memory_order_acquire) == 0; }
    );
  }
 
  // rethrow the exception to the caller
  p->_rethrow_exception();
}
 
// Procedure: _increment_topology
inline void Executor::_increment_topology() {
  _num_topologies.fetch_add(1, std::memory_order_relaxed);
}
 
// Procedure: _decrement_topology
inline void Executor::_decrement_topology() {
  if(_num_topologies.fetch_sub(1, std::memory_order_acq_rel) == 1) {
    _num_topologies.notify_all();
  }
}
 
// Procedure: wait_for_all
inline void Executor::wait_for_all() {
  size_t n = _num_topologies.load(std::memory_order_acquire);
  while(n != 0) {
    _num_topologies.wait(n, std::memory_order_acquire);
    n = _num_topologies.load(std::memory_order_acquire);
  }
}
  
// Function: _schedule_graph
inline void Executor::_schedule_graph(
  Worker& worker, Graph& graph, Topology* tpg, NodeBase* parent
) {
  size_t num_srcs = _set_up_graph(graph, tpg, parent);
  parent->_join_counter.fetch_add(num_srcs, std::memory_order_relaxed);
  _bulk_schedule(worker, graph.begin(), num_srcs);
}
 
// Function: _set_up_topology
inline void Executor::_set_up_topology(Worker* w, Topology* tpg) {
  // ---- under taskflow lock ----
  auto& g = tpg->_taskflow._graph;
  size_t num_srcs = _set_up_graph(g, tpg, tpg);
  tpg->_join_counter.store(num_srcs, std::memory_order_relaxed);
  w ? _bulk_schedule(*w, g.begin(), num_srcs) : _bulk_schedule(g.begin(), num_srcs);
}
 
// Function: _set_up_graph
inline size_t Executor::_set_up_graph(Graph& graph, Topology* tpg, NodeBase* parent) {
 
  auto first = graph.begin();
  auto last  = graph.end();
  auto send  = first;
  for(; first != last; ++first) {
 
    auto node = *first;
    node->_topology = tpg;
    node->_parent = parent;
    node->_nstate = NSTATE::NONE;
    node->_estate.store(ESTATE::NONE, std::memory_order_relaxed);
    node->_set_up_join_counter();
    node->_exception_ptr = nullptr;
 
    // move source to the first partition
    // root, root, root, v1, v2, v3, v4, ...
    if(node->num_predecessors() == 0) {
      std::iter_swap(send++, first);
    }
  }
  return send - graph.begin();
}
 
// Function: _tear_down_topology
inline void Executor::_tear_down_topology(Worker& worker, Topology* tpg, Node*& cache) {
 
  auto &f = tpg->_taskflow;
 
  //assert(&tpg == &(f._topologies.front()));
 
  // case 1: we still need to run the topology again
  //if(!tpg->_exception_ptr && !tpg->cancelled() && !tpg->predicate()) {
  if(!tpg->cancelled() && !tpg->_predicate()) {
    //assert(tpg->_join_counter == 0);
    //std::lock_guard<std::mutex> lock(f._mutex);
    _schedule_graph(worker, tpg->_taskflow._graph, tpg, tpg);
  }
  // case 2: the final run of this topology
  else {
 
    // invoke the callback after each run
    tpg->_on_finish();
 
    // there is another topologies to run
    if(std::unique_lock<std::mutex> lock(f._mutex); f._topologies.size()>1) {
 
      auto fetched_tpg {std::move(f._topologies.front())};
      //assert(fetched_tpg.get() == tpg);
 
      f._topologies.pop();
      tpg = f._topologies.front().get();
 
      lock.unlock();
      
      // Soon after we carry out the promise, the associate taskflow may got destroyed
      // from the user side, and we should never tough it again.
      fetched_tpg->_carry_out_promise();
 
      // decrement the topology
      _decrement_topology();
 
      _schedule_graph(worker, tpg->_taskflow._graph, tpg, tpg);
    }
    else {
      //assert(f._topologies.size() == 1);
 
      auto fetched_tpg {std::move(f._topologies.front())};
      //assert(fetched_tpg.get() == tpg);
 
      f._topologies.pop();
 
      lock.unlock();
      
      // Soon after we carry out the promise, the associate taskflow may got destroyed
      // from the user side, and we should never tough it again.
      fetched_tpg->_carry_out_promise();
 
      _decrement_topology();
      
      // remove the parent that owns the moved taskflow so the storage can be freed
      if(auto parent = fetched_tpg->_parent; parent) {
        //auto state = parent->_nstate;
        if(parent->_join_counter.fetch_sub(1, std::memory_order_acq_rel) == 1) {
          // this async is spawned from a preempted parent, so we need to resume it
          //if(state & NSTATE::PREEMPTED) { 
            _update_cache(worker, cache, static_cast<Node*>(parent));
          //}
        }
      }
    }
  }
}
 
// ############################################################################
// Forward Declaration: Subflow
// ############################################################################
 
inline void Subflow::join() {
 
  if(!joinable()) {
    TF_THROW("subflow already joined");
  }
    
  _executor._corun_graph(_worker, _graph, _node->_topology, _node);
  
  // join here since corun graph may throw exception
  _node->_nstate |= NSTATE::JOINED_SUBFLOW;
}
 
#endif
 
 
 
 
}  // end of namespace tf -----------------------------------------------------