Cooperative Kernels: GPU Multitasking for Blocking Algorithms (Extended Version)

07/06/2017 ∙ by Tyler Sorensen, et al. ∙ Imperial College London 0

There is growing interest in accelerating irregular data-parallel algorithms on GPUs. These algorithms are typically blocking, so they require fair scheduling. But GPU programming models (e.g. OpenCL) do not mandate fair scheduling, and GPU schedulers are unfair in practice. Current approaches avoid this issue by exploiting scheduling quirks of today's GPUs in a manner that does not allow the GPU to be shared with other workloads (such as graphics rendering tasks). We propose cooperative kernels, an extension to the traditional GPU programming model geared towards writing blocking algorithms. Workgroups of a cooperative kernel are fairly scheduled, and multitasking is supported via a small set of language extensions through which the kernel and scheduler cooperate. We describe a prototype implementation of a cooperative kernel framework implemented in OpenCL 2.0 and evaluate our approach by porting a set of blocking GPU applications to cooperative kernels and examining their performance under multitasking. Our prototype exploits no vendor-specific hardware, driver or compiler support, thus our results provide a lower-bound on the efficiency with which cooperative kernels can be implemented in practice.



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1. Introduction

The Needs of Irregular Data-parallel Algorithms.

Many interesting data-parallel algorithms are irregular: the amount of work to be processed is unknown ahead of time and may change dynamically in a workload-dependent manner. There is growing interest in accelerating such algorithms on GPUs (Gupta et al., 2012; Kaleem et al., 2016; Davidson et al., 2014; Harish and Narayanan, 2007; Merrill et al., 2015; Vineet et al., 2009; Nobari et al., 2012; Solomon et al., 2010; Prabhu et al., 2011; Méndez-Lojo et al., 2012; Pai and Pingali, 2016; Sorensen et al., 2016; Cederman and Tsigas, 2008; Tzeng et al., 2010; Burtscher et al., 2012; Che et al., 2013). Irregular algorithms usually require blocking synchronization between workgroups, e.g. many graph algorithms use a level-by-level strategy, with a global barrier between levels; work stealing algorithms require each workgroup to maintain a queue, typically mutex-protected, to enable stealing by other workgroups.

To avoid starvation, a blocking concurrent algorithm requires fair scheduling of workgroups. For example, if one workgroup holds a mutex, an unfair scheduler may cause another workgroup to spin-wait forever for the mutex to be released. Similarly, an unfair scheduler can cause a workgroup to spin-wait indefinitely at a global barrier so that other workgroups do not reach the barrier.

A Degree of Fairness: Occupancy-bound Execution.

The current GPU programming models—OpenCL (Khronos Group, 2015), CUDA (Nvidia, 2016) and HSA (HSA Foundation, 2016), specify almost no guarantees regarding scheduling of workgroups, and current GPU schedulers are unfair in practice. Roughly speaking, each workgroup executing a GPU kernel is mapped to a hardware compute unit.111In practice, depending on the kernel, multiple workgroups might map to the same compute unit; we ignore this in our current discussion. The simplest way for a GPU driver to handle more workgroups being launched than there are compute units is via an occupancy-bound execution model (Gupta et al., 2012; Sorensen et al., 2016) where, once a workgroup has commenced execution on a compute unit (it has become occupant), the workgroup has exclusive access to the compute unit until it finishes execution. Experiments suggest that this model is widely employed by today’s GPUs (Gupta et al., 2012; Sorensen et al., 2016; Pai and Pingali, 2016; Burtscher et al., 2012).

Figure 1. Cooperative kernels can flexibly resize to let other tasks, e.g. graphics, run concurrently

The occupancy-bound execution model does not guarantee fair scheduling between workgroups: if all compute units are occupied then a not-yet-occupant workgroup will not be scheduled until some occupant workgroup completes execution. Yet the execution model does provide fair scheduling between occupant workgroups, which are bound to separate compute units that operate in parallel. Current GPU implementations of blocking algorithms assume the occupancy-bound execution model, which they exploit by launching no more workgroups than there are available compute units (Gupta et al., 2012).

Resistance to Occupancy-bound Execution.

Despite its practical prevalence, none of the current GPU programming models actually mandate occupancy-bound execution. Further, there are reasons why this model is undesirable. First, the execution model does not enable multitasking, since a workgroup effectively owns a compute unit until the workgroup has completed execution. The GPU cannot be used meanwhile for other tasks (e.g. rendering). Second, energy throttling is an important concern for battery-powered devices (Vallina-Rodriguez and Crowcroft, 2013). In the future, it will be desirable for a mobile GPU driver to power down some compute units, suspending execution of associated occupant workgroups, if the battery level is low.

Our assessment, informed by discussions with a number of industrial practitioners who have been involved in the OpenCL and/or HSA standardisation efforts (including (Richards, ; Howes, )), is that GPU vendors (1) will not commit to the occupancy-bound execution model they currently implement, for the above reasons, yet (2) will not guarantee fair scheduling using preemption. This is due to the high runtime cost of preempting workgroups, which requires managing thread local state (e.g. registers, program location) for all workgroup threads (up to 1024 on Nvidia GPUs), as well as shared memory, the workgroup local cache (up to 64 KB on Nvidia GPUs). Vendors instead wish to retain the essence of the simple occupancy-bound model, supporting preemption only in key special cases.

For example, preemption is supported by Nvidia’s Pascal architecture (NVIDIA, 2016), but on a GTX Titan X (Pascal) we still observe starvation: a global barrier executes successfully with 56 workgroups, but deadlocks with 57 workgroups, indicating unfair scheduling.

Our Proposal: Cooperative Kernels.

To summarise: blocking algorithms demand fair scheduling, but for good reasons GPU vendors will not commit to the guarantees of the occupancy-bound execution model. We propose cooperative kernels, an extension to the GPU programming model that aims to resolve this impasse.

A kernel that requires fair scheduling is identified as cooperative, and written using two additional language primitives, and , placed by the programmer. Where the cooperative kernel could proceed with fewer workgroups, a workgroup can execute , offering to sacrifice itself to the scheduler. This indicates that the workgroup would ideally continue executing, but that the scheduler may preempt the workgroup; the cooperative kernel must be prepared to deal with either scenario. Where the cooperative kernel could use additional resources, a workgroup can execute to indicate that the kernel is prepared to proceed with the existing set of workgroups, but is able to benefit from one or more additional workgroups commencing execution directly after the program point.

The use of and creates a contract between the scheduler and the cooperative kernel. Functionally, the scheduler must guarantee that the workgroups executing a cooperative kernel are fairly scheduled, while the cooperative kernel must be robust to workgroups leaving and joining the computation in response to and . Non-functionally, a cooperative kernel must ensure that is executed frequently enough such that the scheduler can accommodate soft-real time constraints, e.g. allowing a smooth frame-rate for graphics. In return, the scheduler should allow the cooperative kernel to utilise hardware resources where possible, killing workgroups only when demanded by other tasks, and forking additional workgroups when possible.

Cooperative kernels allow for cooperative multitasking (see Sec. 6), used historically when preemption was not available or too costly. Our approach avoids the cost of arbitrary preemption as the state of a workgroup killed via does not have to be saved. Previous cooperative multitasking systems have provided yield semantics, where a processing unit would temporarily give up its hardware resource. We deviate from this design as, in the case of a global barrier, adopting yield would force the cooperative kernel to block completely when a single workgroup yields, stalling the kernel until the given workgroup resumes. Instead, our allows a kernel to make progress with a smaller number of workgroups, with workgroups potentially joining again later via .

Figure 1 illustrates sharing of GPU compute units between a cooperative kernel and a graphics task. Workgroups 2 and 3 of the cooperative kernel are killed at an to make room for a graphics task. The workgroups are subsequently restored to the cooperative kernel when workgroup 0 calls . The gather time is the time between resources being requested and the application surrendering them via . To satisfy soft-real time constraints, this time should be low; our experimental study (Sec. 5.4) shows that, in practice, the gather-time for our applications is acceptable for a range of graphics workloads.

The cooperative kernels model has several appealing properties:

  1. [leftmargin=*]

  2. By providing fair scheduling between workgroups, cooperative kernels meet the needs of blocking algorithms, including irregular data-parallel algorithms.

  3. The model has no impact on the development of regular (non-cooperative) compute and graphics kernels.

  4. The model is backwards-compatible: and may be ignored, and a cooperative kernel will behave exactly as a regular kernel does on current GPUs.

  5. Cooperative kernels can be implemented over the occupancy/bound execution model provided by current GPUs: our prototype implementation uses no special hardware/driver support.

  6. If hardware support for preemption is available, it can be leveraged to implement cooperative kernels efficiently, and cooperative kernels can avoid unnecessary preemptions by allowing the programmer to communicate “smart” preemption points.

Placing the primitives manually is straightforward for the representative set of GPU-accelerated irregular algorithms we have ported so far. Our experiments show that the model can enable efficient multitasking of cooperative and non-cooperative tasks.

In summary, our main contributions are: cooperative kernels, an extended GPU programming model that supports the scheduling requirements of blocking algorithms (Sec. 3); a prototype implementation of cooperative kernels on top of OpenCL 2.0 (Sec. 4); and experiments assessing the overhead and responsiveness of the cooperative kernels approach over a set of irregular algorithms (Sec. 5), including a best-effort comparison with the efficiency afforded by hardware-supported preemption available on Nvidia GPUs.

We begin by providing background on OpenCL via two motivating examples (Sec. 2). At the end we discuss related work (Sec. 6) and avenues for future work (Sec. 7).

2. Background and Examples

We outline the OpenCL programming model on which we base cooperative kernels (Sec. 2.1), and illustrate OpenCL and the scheduling requirements of irregular algorithms using two examples: a work stealing queue and frontier-based graph traversal (Sec. 2.2).

2.1. OpenCL Background

An OpenCL program is divided into host and device components. A host application runs on the CPU and launches one or more kernels that run on accelerator devices—GPUs in the context of this paper. A kernel is written in OpenCL C, based on C99. All threads executing a kernel start at the same entry function with identical arguments. A thread can call to obtain a unique id, to access distinct data or follow different control flow paths.

The threads of a kernel are divided into workgroups. Functions and return a thread’s local id within its workgroup and the workgroup id. The number of threads per workgroup and number of workgroups are obtained via and . Execution of the threads in a workgroup can be synchronised via a workgroup barrier. A global barrier (synchronising all threads of a kernel) is not provided as a primitive.

Memory Spaces and Memory Model.

A kernel has access to four memory spaces. Shared virtual memory (SVM) is accessible to all device threads and the host concurrently. Global memory is shared among all device threads. Each workgroup has a portion of local memory for fast intra-workgroup communication. Every thread has a portion of very fast private memory for function-local variables.

Fine-grained communication within a workgroup, as well as inter-workgroup communication and communication with the host while the kernel is running, is enabled by a set of atomic data types and operations. In particular, fine-grained host/device communication is via atomic operations on SVM.

Execution Model.

OpenCL (Khronos Group, 2015, p. 31) and CUDA (Nvidia, 2016) specifically make no guarantees about fair scheduling between workgroups executing the same kernel. HSA provides limited, one-way guarantees, stating (HSA Foundation, 2016, p. 46): “Work-group A can wait for values written by work-group B without deadlock provided … (if) A comes after B in work-group flattened ID order”. This is not sufficient to support blocking algorithms that use mutexes and inter-workgroup barriers, both of which require symmetric communication between threads.

2.2. Motivating Examples

1kernel work_stealing(global Task * queues) {
2  int queue_id = get_group_id();
3  while (more_work(queues)) {
4    Task * t = pop_or_steal(queues, queue_id);
5    if (t)
6      process_task(t, queues, queue_id);
7  }
Figure 2. An excerpt of a work stealing algorithm in OpenCL

Work Stealing.

Work stealing enables dynamic balancing of tasks across processing units. It is useful when the number of tasks to be processed is dynamic, due to one task creating an arbitrary number of new tasks. Work stealing has been explored in the context of GPUs (Cederman and Tsigas, 2008; Tzeng et al., 2010). Each workgroup has a queue from which it obtains tasks to process, and to which it stores new tasks. If its queue is empty, a workgroup tries to steal a task from another queue.

Figure 2 illustrates a work stealing kernel. Each thread receives a pointer to the task queues, in global memory, initialized by the host to contain initial tasks. A thread uses its workgroup id (line 2) as a queue id to access the relevant task queue. The function (line 2) pops a task from the workgroup’s queue or tries to steal a task from other queues. Although not depicted here, concurrent accesses to queues inside and are guarded by a mutex per queue, implemented using atomic compare and swap operations on global memory.

If a task is obtained, then the workgroup processes it (line 2), which may lead to new tasks being created and pushed to the workgroup’s queue. The kernel presents two opportunities for spin-waiting: spinning to obtain a mutex, and spinning in the main kernel loop to obtain a task. Without fair scheduling, threads waiting for the mutex might spin indefinitely, causing the application to hang.

1kernel graph_app(global graph * g,
2       global nodes * n0, global nodes * n1) {
3  int level = 0;
4  global nodes * in_nodes = n0;
5  global nodes * out_nodes = n1;
6  int tid = get_global_id();
7  int stride = get_global_size();
8  while(in_nodes.size > 0) {
9    for (int i = tid; i < in_nodes.size; i += stride)
10      process_node(g, in_nodes[i], out_nodes, level);
11    swap(&in_nodes, &out_nodes);
12    global_barrier();
13    reset(out_nodes);
14    level++;
15    global_barrier();
16  }
Figure 3. An OpenCL graph traversal algorithm

Graph Traversal.

Figure 3 illustrates a frontier-based graph traversal algorithm; such algorithms have been shown to execute efficiently on GPUs (Burtscher et al., 2012; Pai and Pingali, 2016). The kernel is given three arguments in global memory: a graph structure, and two arrays of graph nodes. Initially, contains the starting nodes to process. Private variable records the current frontier level, and and point to distinct arrays recording the nodes to be processed during the current and next frontier, respectively.

The application iterates as long as the current frontier contains nodes to process (line 3). At each frontier, the nodes to be processed are evenly distributed between threads through stride based processing. In this case, the stride is the total number of threads, obtained via . A thread calls to process a node given the current level, with nodes to be processed during the next frontier being pushed to . After processing the frontier, the threads swap their node array pointers (line 3).

At this point, the GPU threads must wait for all other threads to finish processing the frontier. To achieve this, we use a global barrier construct (line 3). After all threads reach this point, the output node array is reset (line 3) and the level is incremented. The threads use another global barrier to wait until the output node is reset (line 3), after which they continue to the next frontier.

The global barrier used in this application is not provided as a GPU primitive, though previous works have shown that such a global barrier can be implemented (Xiao and Feng, 2010; Sorensen et al., 2016), based on CPU barrier designs (Herlihy and Shavit, 2008, ch. 17). These barriers employ spinning to ensure each thread waits at the barrier until all threads have arrived, thus fair scheduling between workgroups is required for the barrier to operate correctly. Without fair scheduling, the barrier threads may wait indefinitely at the barrier, causing the application to hang.

The mutexes and barriers used by these two examples appear to run reliably on current GPUs for kernels that are executed with no more workgroups than there are compute units. This is due to the fairness of the occupancy-bound execution model that current GPUs have been shown, experimentally, to provide. But, as discussed in Sec. 1, this model is not endorsed by language standards or vendor implementations, and may not be respected in the future.

In Sec. 3.2 we show how the work stealing and graph traversal examples of Figs. 2 and 3 can be updated to use our cooperative kernels programming model to resolve the scheduling issue.

3. Cooperative Kernels

We present our cooperative kernels programming model as an extension to OpenCL. We describe the semantics of the model (Sec. 3.1), presenting a more formal operational semantics in Appendix A and discussing possible alternative semantic choices in Appendix B, use our motivating examples to discuss programmability (Sec. 3.2) and outline important nonfunctional properties that the model requires to work successfully (Sec. 3.3).

3.1. Semantics of Cooperative Kernels

As with a regular OpenCL kernel, a cooperative kernel is launched by the host application, passing parameters to the kernel and specifying a desired number of threads and workgroups. Unlike in a regular kernel, the parameters to a cooperative kernel are immutable (though pointer parameters can refer to mutable data).

Cooperative kernels are written using the following extensions: , a qualifier on the variables of a thread; and , the key functions that enable cooperative scheduling; and and primitives for inter-workgroup synchronisation.

Transmitted Variables.

A variable declared in the root scope of the cooperative kernel can optionally be annotated with a new qualifier. Annotating a variable with means that when a workgroup uses to spawn new workgroups, the workgroup should transmit its current value for to the threads of the new workgroups. We detail the semantics for this when we describe below.

Active Workgroups.

If the host application launches a cooperative kernel requesting workgroups, this indicates that the kernel should be executed with a maximum of workgroups, and that as many workgroups as possible, up to this limit, are desired. However, the scheduler may initially schedule fewer than workgroups, and as explained below the number of workgroups that execute the cooperative kernel can change during the lifetime of the kernel.

The number of active workgroups—workgroups executing the kernel—is denoted . Active workgroups have consecutive ids in the range . Initially, at least one workgroup is active; if necessary the scheduler must postpone the kernel until some compute unit becomes available. For example, in Fig. 1: at the beginning of the execution ; while the graphics task is executing ; after the fork again.

When executed by a cooperative kernel, returns , the current number of active workgroups. This is in contrast to for regular kernels, which returns the fixed number of workgroups that execute the kernel (see Sec. 2.1).

Fair scheduling is guaranteed between active workgroups; i.e. if some thread in an active workgroup is enabled, then eventually this thread is guaranteed to execute an instruction.

Semantics for .

The primitive allows the cooperative kernel to return compute units to the scheduler by offering to sacrifice workgroups. The idea is as follows: allowing the scheduler to arbitrarily and abruptly terminate execution of workgroups might be drastic, yet the kernel may contain specific program points at which a workgroup could gracefully leave the computation.

Similar to the OpenCL workgroup primitive, , is a workgroup-level function—it must be encountered uniformly by all threads in a workgroup.

Suppose a workgroup with id executes . If the workgroup has the largest id among active workgroups then it can be killed by the scheduler, except that workgroup 0 can never be killed (to avoid early termination of the kernel). More formally, if or then is a no-op. If instead and , the scheduler can choose to ignore the offer, so that executes as a no-op, or accept the offer, so that execution of the workgroup ceases and the number of active workgroups is atomically decremented by one. Figure 1 illustrates this, showing that workgroup is killed before workgroup .

Semantics for .

Recall that a desired limit of workgroups was specified when the cooperative kernel was launched, but that the number of active workgroups, , may be smaller than , either because (due to competing workloads) the scheduler did not provide workgroups initially, or because the kernel has given up some workgroups via calls. Through the primitive (also a workgroup-level function), the kernel and scheduler can collaborate to allow new workgroups to join the computation at an appropriate point and with appropriate state.

Suppose a workgroup with id executes . Then the following occurs: an integer is chosen by the scheduler; new workgroups are spawned with consecutive ids in the range ; the active workgroup count is atomically incremented by .

The new workgroups commence execution at the program point immediately following the call. The variables that describe the state of a thread are all uninitialised for the threads in the new workgroups; reading from these variables without first initialising them is an undefined behaviour. There are two exceptions to this: (1) because the parameters to a cooperative kernel are immutable, the new threads have access to these parameters as part of their local state and can safely read from them; (2) for each variable annotated with , every new thread’s copy of is initialised to the value that thread 0 in workgroup held for at the point of the call. In effect, thread 0 of the forking workgroup transmits the relevant portion of its local state to the threads of the forked workgroups.

Figure 1 illustrates the behaviour of . After the graphics task finishes executing, workgroup calls , spawning the two new workgroups with ids and . Workgroups and join the computation where workgroup called .

Notice that is always a valid choice for the number of workgroups to be spawned by , and is guaranteed if is equal to the workgroup limit .

Global Barriers.

Because workgroups of a cooperative kernel are fairly scheduled, a global barrier primitive can be provided. We specify two variants: and .

Our primitive is a kernel-level function: if it appears in conditional code then it must be reached by all threads executing the cooperative kernel. On reaching a , a thread waits until all threads have arrived at the barrier. Once all threads have arrived, the threads may proceed past the barrier with the guarantee that all global memory accesses issued before the barrier have completed. The primitive can be implemented by adapting an inter-workgroup barrier design, e.g. (Xiao and Feng, 2010), to take account of a growing and shrinking number of workgroups, and the atomic operations provided by the OpenCL 2.0 memory model enable a memory-safe implementation (Sorensen et al., 2016).

The primitive is also a kernel-level function. It is identical to , except that it caters for cooperation with the scheduler: by issuing a the programmer indicates that the cooperative kernel is prepared to proceed after the barrier with more or fewer workgroups.

When all threads have reached , the number of active workgroups, , is atomically set to a new value, say, with . If then the active workgroups remain unchanged. If , workgroups are killed. If then new workgroups join the computation after the barrier, as if they were forked from workgroup 0. In particular, the -annotated local state of thread 0 in workgroup 0 is transmitted to the threads of the new workgroups.

The semantics of can be modelled via calling and , surrounded and separated by calls to a . The enclosing calls ensure that the change in number of active workgroups from to occurs entirely within the resizing barrier, so that changes atomically from a programmer’s perspective. The middle ensures that forking occurs before killing, so that workgroups are left intact.

Because can be implemented as above, we do not regard it conceptually as a primitive of our model. However, in Sec. 4.2 we show how a resizing barrier can be implemented more efficiently through direct interaction with the scheduler.

3.2. Programming with Cooperative Kernels

A Changing Workgroup Count.

Unlike in regular OpenCL, the value returned by is not fixed during the lifetime of a cooperative kernel: it corresponds to the active group count , which changes as workgroups execute , and . The value returned by is similarly subject to change. A cooperative kernel must thus be written in a manner that is robust to changes in the values returned by these functions.

In general, their volatility means that use of these functions should be avoided. However, the situation is more stable if a cooperative kernel does not call and directly, so that only can affect the number of active workgroups. Then, at any point during execution, the threads of a kernel are executing between some pair of resizing barrier calls, which we call a resizing barrier interval (considering the kernel entry and exit points conceptually to be special cases of resizing barriers). The active workgroup count is constant within each resizing barrier interval, so that and return stable values during such intervals. As we illustrate below for graph traversal, this can be exploited by algorithms that perform strided data processing.

Adapting Work Stealing.

In this example there is no state to transmit since a computation is entirely parameterised by a task, which is retrieved from a queue located in global memory. With respect to Fig.  2, we add and calls at the start of the main loop (below line 2) to let a workgroup offer itself to be killed or forked, respectively, before it processes a task. Note that a workgroup may be killed even if its associated task queue is not empty, since remaining tasks will be stolen by other workgroups. In addition, since may be the entry point of a workgroup, the queue id must now be computed after it, so we move line 2 to be placed just before line 2. In particular, the queue id cannot be transmitted since we want a newly spawned workgroup to read its own queue and not the one of the forking workgroup.

Adapting Graph Traversal.

Figure 4 shows a cooperative version of the graph traversal kernel of Fig. 3 from Sec. 2.2. On lines 4 and  4, we change the original global barriers into a resizing barriers. Several variables are marked to be transmitted in the case of workgroups joining at the resizing barriers (lines 4, 4 and 4): must be restored so that new workgroups know which frontier they are processing; and must be restored so that new workgroups know which of the node arrays to use for input and output. Lastly, the static work distribution of the original kernel is no longer valid in a cooperative kernel. This is because the stride (which is based on ) may change after each resizing barrier call. To fix this, we re-distribute the work after each resizing barrier call by recomputing the thread id and stride (lines 4 and 4). This example exploits the fact that the cooperative kernel does not issue nor directly: the value of obtained from at line 4 is stable until the next resizing barrier at line 4.

1kernel graph_app(global graph *g,
2       global nodes *n0, global nodes *n1) {
3  transmit int level = 0;
4  transmit global nodes *in_nodes = n0;
5  transmit global nodes *out_nodes = n1;
6  while(in_nodes.size > 0) {
7    int tid = get_global_id();
8    int stride = get_global_size();
9    for (int i = tid; i < in_nodes.size; i += stride)
10      process_node(g, in_nodes[i], out_nodes, level);
11    swap(&in_nodes, &out_nodes);
12    resizing_global_barrier();
13    reset(out_nodes);
14    level++;
15    resizing_global_barrier();
16  }
Figure 4. Cooperative version of the graph traversal kernel of Fig. 3, using a resizing barrier and annotations

Patterns for Irregular Algorithms.

In Sec. 5.1 we describe the set of irregular GPU algorithms used in our experiments, which largely captures the irregular blocking algorithms that are available as open source GPU kernels. These all employ either work stealing or operate on graph data structures, and placing our new constructs follows a common, easy-to-follow pattern in each case. The work stealing algorithms have a transactional flavour and require little or no state to be carried between transactions. The point at which a workgroup is ready to process a new task is a natural place for and , and few or no annotations are required. Figure 4 is representative of most level-by-level graph algorithms. It is typically the case that on completing a level of the graph algorithm, the next level could be processed by more or fewer workgroups, which facilitates. Some level-specific state must be transmitted to new workgroups.

3.3. Non-Functional Requirements

The semantics presented in Sec. 3.1 describe the behaviours that a developer of a cooperative kernel should be prepared for. However, the aim of cooperative kernels is to find a balance that allows efficient execution of algorithms that require fair scheduling, and responsive multitasking, so that the GPU can be shared between cooperative kernels and other shorter tasks with soft real-time constraints. To achieve this balance, an implementation of the cooperative kernels model, and the programmer of a cooperative kernel, must strive to meet the following non-functional requirements.

The purpose of is to let the scheduler destroy a workgroup in order to schedule higher-priority tasks. The scheduler relies on the cooperative kernel to execute sufficiently frequently that soft real-time constraints of other workloads can be met. Using our work stealing example: a workgroup offers itself to the scheduler after processing each task. If tasks are sufficiently fast to process then the scheduler will have ample opportunities to de-schedule workgroups. But if tasks are very time-consuming to process then it might be necessary to rewrite the algorithm so that tasks are shorter and more numerous, to achieve a higher rate of calls to . Getting this non-functional requirement right is GPU- and application-dependent. In Sec. 5.2 we conduct experiments to understand the response rate that would be required to co-schedule graphics rendering with a cooperative kernel, maintaining a smooth frame rate.

Recall that, on launch, the cooperative kernel requests workgroups. The scheduler should thus aim to provide workgroups if other constraints allow it, by accepting an only if a compute unit is required for another task, and responding positively to calls if compute units are available.

4. Prototype Implementation

Our vision is that cooperative kernel support will be integrated in the runtimes of future GPU implementations of OpenCL, with driver support for our new primitives. To experiment with our ideas on current GPUs, we have developed a prototype that mocks up the required runtime support via a megakernel, and exploits the occupancy-bound execution model that these GPUs provide to ensure fair scheduling between workgroups. We emphasise that an aim of cooperative kernels is to avoid depending on the occupancy-bound model. Our prototype exploits this model simply to allow us to experiment with current GPUs whose proprietary drivers we cannot change. We describe the megakernel approach (Sec. 4.1) and detail various aspects of the scheduler component of our implementation (Sec. 4.2).

4.1. The Megakernel Mock Up

Instead of multitasking multiple separate kernels, we merge a set of kernels into a megakernel—a single, monolithic kernel. The megakernel is launched with as many workgroups as can be occupant concurrently. One workgroup takes the role of the scheduler,222We note that the scheduler requirements given in Sec. 3 are agnostic to whether the scheduling logic takes place on the CPU or GPU. To avoid expensive communication between GPU and host, we choose to implement the scheduler on the GPU. and the scheduling logic is embedded as part of the megakernel. The remaining workgroups act as a pool of workers. A worker repeatedly queries the scheduler to be assigned a task. A task corresponds to executing a cooperative or non-cooperative kernel. In the non-cooperative case, the workgroup executes the relevant kernel function uninterrupted, then awaits further work. In the cooperative case, the workgroup either starts from the kernel entry point or immediately jumps to a designated point within the kernel, depending on whether the workgroup is an initial workgroup of the kernel, or a forked workgroup. In the latter case, the new workgroup also receives a struct containing the values of all relevant -annotated variables.

Simplifying Assumptions.

For ease of implementation, our prototype supports multitasking a single cooperative kernel with a single non-cooperative kernel (though the non-cooperative kernel can be invoked many times). We require that , and are called from the entry function of a cooperative kernel. This allows us to use and to direct threads into and out of the kernel. With these restrictions we can experiment with interesting irregular algorithms (see Sec. 5). A non-mock implementation of cooperative kernels would not use the megakernel approach, so we did not deem the engineering effort associated with lifting these restrictions in our prototype to be worthwhile.

4.2. Scheduler Design

To enable multitasking through cooperative kernels, the runtime (in our case, the megakernel) must track the state of workgroups, i.e. whether a workgroup is waiting or computing a kernel; maintain consistent context states for each kernel, e.g. tracking the number of active workgroups; and provide a safe way for these states to be modified in response to /. We discuss these issues, and describe the implementation of an efficient resizing barrier. We describe how the scheduler would handle arbitrary combinations of kernels, though as noted above our current implementation is restricted to the case of two kernels.

Scheduler Contexts.

To dynamically manage workgroups executing cooperative kernels, our framework must track the state of each workgroup and provide a channel of communication from the scheduler workgroup to workgroups executing and . To achieve this, we use a scheduler context structure, mapping a primitive workgroup id the workgroup’s status, which is either available or the id of the kernel that the workgroup is currently executing. The scheduler can then send cooperative kernels a resource message, commanding workgroups to exit at , or spawn additional workgroups at . Thus, the scheduler context needs a communication channel for each cooperative kernel. We implement the communication channels using atomic variables in global memory.

Launching Kernels and Managing Workgroups.

To launch a kernel, the host sends a data packet to the GPU scheduler consisting of a kernel to execute, kernel inputs, and a flag indicating whether the kernel is cooperative. In our prototype, this host-device communication channel is built using fine-grained SVM atomics.

On receiving a data packet describing a kernel launch , the GPU scheduler must decide how to schedule . Suppose requests workgroups. The scheduler queries the scheduler context. If there are at least available workgroups, can be scheduled immediately. Suppose instead that there are only available workgroups, but a cooperative kernel is executing. The scheduler can use ’s channel in the scheduler context to command to provide workgroups via . Once workgroups are available, the scheduler then sends workgroups from the available workgroups to execute kernel . If the new kernel is itself a cooperative kernel, the scheduler would be free to provide with fewer than active workgroups initially.

If a cooperative kernel is executing with fewer workgroups than it initially requested, the scheduler may decide make extra workgroups available to , to be obtained next time calls . To do this, the scheduler asynchronously signals through ’s channel to indicate the number of workgroups that should join at the next command. When a workgroup of subsequently executes , thread 0 of updates the kernel and scheduler contexts so that the given number of new workgroups are directed to the program point after the call. This involves selecting workgroups whose status is available, as well as copying the values of -annotated variables to the new workgroups.

An Efficient Resizing Barrier.

In Sec. 3.1, we defined the semantics of a resizing barrier in terms of calls to other primitives. It is possible, however, to implement the resizing barrier with only one call to a global barrier with and inside.

We consider barriers that use the master/slave model (Xiao and Feng, 2010): one workgroup (master) collects signals from the other workgroups (slaves) indicating that they have arrived at the barrier and are waiting for a reply indicating that they may leave the barrier. Once the master has received a signal from all slaves, it replies with a signal saying that they may leave.

Incorporating and into such a barrier is straightforward. Upon entering the barrier, the slaves first execute , possibly exiting. The master then waits for slaves (the number of active workgroups), which may decrease due to calls by the slaves, but will not increase. Once the master observes that slaves have arrived, it knows that all other workgroups are waiting to be released. The master executes , and the statement immediately following this is a conditional that forces newly spawned workgroups to join the slaves in waiting to be released. Finally, the master releases all the slaves: the original slaves and the new slaves that joined at .

This barrier implementation is sub-optimal because workgroups only execute once per barrier call and, depending on order of arrival, it is possible that only one workgroup is killed per barrier call, preventing the scheduler from gathering workgroups quickly.

We can reduce the gather time by providing a new function for cooperative kernels, which returns the number of workgroups that the scheduler needs to obtain from the cooperative kernel. A resizing barrier can now be implemented as follows: (1) the master waits for all slaves to arrive; (2) the master calls and commands the new workgroups to be slaves; (3) the master calls , obtaining a value ; (4) the master releases the slaves, broadcasting the value to them; (5) workgroups with ids larger than spin, calling repeatedly until the scheduler claims them—we know from that the scheduler will eventually do so. We show in Sec. 5.4 that the barrier using greatly reduces the gather time in practice.

5. Applications and Experiments

We discuss our experience porting irregular algorithms to cooperative kernels and describe the GPUs on which we evaluate these applications (Sec. 5.1). For these GPUs, we report on experiments to determine non-cooperative workloads that model the requirements of various graphics rendering tasks (Sec. 5.2). We then examine the overhead associated with moving to cooperative kernels when multitasking is not required (Sec. 5.3), as well as the responsiveness and throughput observed when a cooperative kernel is multi-tasked with non-cooperative workloads (Sec. 5.4). Finally, we compare against a performance model of kernel-level preemption, which we understand to be what current Nvidia GPUs provide (Sec. 5.5).

5.1. Applications and GPUs

App. barriers kill fork transmit LoC inputs
color 2 / 2 0 0 4 55 2
mis 3 / 3 0 0 0 71 2
p-sssp 3 / 3 0 0 0 42 1
bfs 2 / 2 0 0 4 185 2
l-sssp 2 / 2 0 0 4 196 2
octree 0 / 0 1 1 0 213 1
game 0 / 0 1 1 0 308 1

  Pannotia        Lonestar GPU        work stealing

Table 1. Blocking GPU applications investigated

Table 1 gives an overview of the 7 irregular algorithms that we ported to cooperative kernels. Among them, 5 are graph algorithms, based on the Pannotia (Che et al., 2013) and Lonestar (Burtscher et al., 2012) GPU application suites, using global barriers. We indicate how many of the original number of barriers are changed to resizing barriers (all of them), and how many variables need to be transmitted. The remaining two algorithms are work stealing applications: each required the addition of and at the start of the main loop, and no variables needed to be transmitted (similar to example discussed in Sec. 3.2). Most graph applications come with 2 different data sets as input, leading to 11 application/input pairs in total.

Our prototype implementation (Sec. 4) requires two optional features of OpenCL 2.0: SVM fine-grained buffers and SVM atomics. Out of the GPUs available to us, from ARM, AMD, Nvidia, and Intel, only Intel GPUs provided robust support of these features.

We thus ran our experiments on three Intel GPUs: HD 520, HD 5500 and Iris 6100. The results were similar across the GPUs, so for conciseness, we report only on the Iris 6100 GPU (driver with a host CPU i3-5157U. The Iris has a reported 47 compute units. Results for the other Intel GPUs are presented in Appendix C.

5.2. Sizing Non-cooperative Kernels

Enabling rendering of smooth graphics in parallel with irregular algorithms is an important use case for our approach. Because our prototype implementation is based on a megakernel that takes over the entire GPU (see Sec. 4), we cannot assess this directly.

We devised the following method to determine OpenCL workloads that simulate the computational intensity of various graphics rendering workloads. We designed a synthetic kernel that occupies all workgroups of a GPU for a parameterised time period , invoked in an infinite loop by a host application. We then searched for a maximum value for that allowed the synthetic kernel to execute without having an observable impact on graphics rendering. Using the computed value, we ran the host application for seconds, measuring the time dedicated to GPU execution during this period and the number of kernel launches that were issued. We used in all experiments. The values and estimate the average time spent using the GPU to render the display between kernel calls (call this ) and the period at which the OS requires the GPU for display rendering (call this ), respectively.

We used this approach to measure the GPU availability required for three types of rendering: light, whereby desktop icons are smoothly emphasised under the mouse pointer; medium, whereby window dragging over the desktop is smoothly animated; and heavy, which requires smooth animation of a WebGL shader in a browser. For heavy we used WebGL demos from the Chrome experiments (Google, ).

Our results are the following: and for light; , for medium; and , for heavy. For medium and heavy, the period coincides with the human persistence of vision. The execution duration of both light and medium configurations indicates that GPU computation is cheaper for basic display rendering compared with more complex rendering.

5.3. The Overhead of Cooperative Kernels

Experimental Setup.

Invoking the cooperative scheduling primitives incurs some overhead even if no killing, forking or resizing actually occurs, because the cooperative kernel still needs to interact with the scheduler to determine this. We assess this overhead by measuring the slowdown in execution time between the original and cooperative versions of a kernel, forcing the scheduler to never modify the number of active workgroups in the cooperative case.

Recall that our mega kernel-based implementation merges the code of a cooperative and a non-cooperative kernel. This can reduce the occupancy for the merged kernel, e.g. due to higher register pressure, This is an artifact of our prototype implementation, and would not be a problem if our approach was implemented inside the GPU driver. We thus launch both the original and cooperative versions of a kernel with the reduced occupancy bound in order to meaningfully compare execution times.

Figure 5. Example gather time and non-cooperative timing results
overall barrier wk.steal.
mean max mean max mean max

octree, color G3_circuit

Table 2. Cooperative kernel slowdown w/o multitasking


Tab. 2

shows the geometric mean and maximum slowdown across all applications and inputs, with averages and maxima computed over 10 runs per benchmark. For the maximum slowdowns, we indicate which application and input was responsible. The slowdown is below 1.25 even in the worst case, and closer to 1 on average. We consider these results encouraging, especially since the performance of our prototype could clearly be improved upon in a native implementation.

5.4. Multitasking via Cooperative Scheduling

We now assess the responsiveness of multitasking between a long-running cooperative kernel and a series of short, non-cooperative kernel launches, and the performance impact of multitasking on the cooperative kernel.

Experimental Setup.

For a given cooperative kernel and its input, we launch the kernel and then repeatedly schedule a non-cooperative kernel that aims to simulate the intensity of one of the three classes of graphics rendering workload discussed in Sec. 5.2. In practice, we use matrix multiplication as the non-cooperative workload, with matrix dimensions tailored to reach the appropriate execution duration. We conduct separate runs where we vary the number of workgroups requested by the non-cooperative kernel, considering the cases where one, a quarter, a half, and all-but-one, of the total number of workgroups are requested. For the graph algorithms we try both regular and query barrier implementations.

Our experiments span 11 pairs of cooperative kernels and inputs, 3 classes of non-cooperative kernel workloads, 4 quantities of workgroups claimed for the non-cooperative kernel and 2 variations of resizing barriers for graph algorithms, leading to 240 configurations. We run each configuration 10 times, in order to report averaged performance numbers. For each run, we record the execution time of the cooperative kernel. For each scheduling of the non-cooperative kernel during the run, we also record the gather time needed by the scheduler to collect workgroups to launch the non-cooperative kernel, and the non-cooperative kernel execution time.


Figure 5 reports, on three configurations, the average gather and execution times for the non-cooperative kernel with respect to the quantity of workgroups allocated to it. A logarithmic scale is used for time since gather times tend to be much smaller than execution times. The horizontal grey lines indicates the desired period for non-cooperative kernels. These graphs show a representative sample of our results; the full set of graphs for all configurations is provided in Appendix C.

The left-most graph illustrates a work stealing example. When the non-cooperative kernel is given only one workgroup, its execution is so long that it cannot complete within the period required for a screen refresh. The gather time is very good though, since the scheduler needs to collect only one workgroup. The more workgroups are allocated to the non-cooperative kernels, the faster it can compute: here the non-cooperative kernel becomes fast enough with a quarter (resp. half) of available workgroups for light (resp. heavy) graphics workload. Inversely, the gather time increases since the scheduler must collect more and more workgroups.

The middle and right graphs show results for graph algorithms. These algorithms use barriers, and we experimented with the regular and query barrier implementations described in Sec. 4.2. The execution times for the non-cooperative task are averaged across all runs, including with both types of barrier. We show separately the average gather time associated with each type of barrier. The graphs show a similar trend to the left-most graph: as the number of non-cooperative workgroups grows, the execution time decreases and the gather time increases. The gather time is higher on the rightmost figure as the G3 circuit input graph is rather wide than deep, so the graph algorithm reaches resizing barriers less often than for the USA road input of the middle figure for instance. The scheduler thus has fewer opportunities to collect workgroups and gather time increases. Nonetheless, scheduling responsiveness can benefit from the query barrier: when used, this barrier lets the scheduler collect all needed workgroups as soon as they hit a resizing barrier. As we can see, the gather time of the query barrier is almost stable with respect to the number of workgroups that needs to be collected.

Figure 6. Performance impact of multitasking cooperative and non-cooperative workloads, and the period with which non-cooperative kernels execute


Figure 6

reports the overhead brought by the scheduling of non-cooperative kernels over the cooperative kernel execution time. This is the slowdown associated with running the cooperative kernel in the presence of multitasking, vs. running the cooperative kernel in isolation (median over all applications and inputs). We also show the period at which non-cooperative kernels can be scheduled (median over all applications and inputs). Our data included some outliers that occur with benchmarks in which the resizing barrier are not called very frequently and the graphics task requires half or more workgroups. For example, a medium graphics workload for bfs on the rmat input has over an 8

overhead when asking for all but one of the workgroups. As Figure 6 shows, most of our benchmarks are much better behaved than this. In future work is required to examine the problematic benchmarks in more detail, possibly inserting more resizing calls.

We show results for the three workloads listed in Sec.  5.2. The horizontal lines in the period graph correspond to the goals of the workloads: the higher (resp. lower) line corresponds to a period of (resp. ) for the light (resp. medium and heavy) workload.

Co-scheduling non-cooperative kernels that request a single workgroup leads to almost no overhead, but the period is far too high to meet the needs of any of our three workloads; e.g. a heavy workload averages a period of . As more workgroups are dedicated to non-cooperative kernels, they execute quickly enough to be scheduled at the expected period. For the light and medium workloads, a quarter of the workgroups executing the non-cooperative kernel are able to meet their goal period (70 and resp.). However, this is not sufficient to meet the goal for the heavy workload (giving a median period of ). If half of the workgroups are allocated to the non-cooperative kernel, the heavy workload achieves its goal period (median of ). Yet, as expected, allocating more non-cooperative workgroups increases the overhead of the cooperative kernel. Still, heavy workloads meet their period by allocating half of the workgroups, incurring a slow down of less than 1.5 (median). Light and medium workloads meet their period with only a small overhead; 1.04 and 1.08 median slowdown respectively.

5.5. Comparison with Kernel-Level Preemption

g. workload kernel-level cooperative resources
light 1.04 1.04
medium 1.08 1.08
heavy 1.33 1.47

Table 3. Overhead of kernel level preemption vs cooperative kernels for three graphics workloads

Nvidia’s recent Pascal architecture provides hardware support for instruction-level preemption (NVIDIA, 2016; Smith and Anandtech, 2016), however, preemption of entire kernels, but not of individual workgroups is supported. Intel GPUs do not provide this feature, and our OpenCL prototype of cooperative kernels cannot run on Nvidia GPUs, making a direct comparison impossible. We present here a theoretical analysis of the overheads associated with sharing the GPU between graphics and compute tasks via kernel-level preemption.

Suppose a graphics workload is required to be scheduled with period and duration , and that a compute kernel requires time to execute without interruption. If we assume the cost of preemption is negligible (e.g. Nvidia have reported preemption times of 0.1 for Pascal (Smith and Anandtech, 2016), because of special hardware support), then the overhead associated with switching between compute and graphics every time steps is .

We compare this task-level preemption overhead model with our experimental results per graphics workload in Tab. 3. We report the overhead of the configuration that allowed us to meet the deadline of the graphics task. Based on the above assumptions, our approach provides similar overhead for low and medium graphics workloads, however, has a higher overhead for the high workload.

Our low performance for heavy workloads is because the graphics task requires half of the workgroups, crippling the cooperative kernel enough that calls are not issued as frequently. Future work may examine how to insert more resizing calls in these applications to address this. These results suggest that a hybrid preemption scheme may work well. That is, the cooperative approach works well for light and medium tasks; on the other hand, heavy graphics tasks benefit from the coarser grained, kernel-level preemption strategy. However, the preemption strategy requires specialised hardware support in order to be efficient.

6. Related Work

Irregular Algorithms and Persistent kernels.

There has been a lot of work on accelerating blocking irregular algorithms using GPUs, and on the persistent threads programming style for long-running kernels (Gupta et al., 2012; Kaleem et al., 2016; Davidson et al., 2014; Harish and Narayanan, 2007; Merrill et al., 2015; Vineet et al., 2009; Nobari et al., 2012; Solomon et al., 2010; Prabhu et al., 2011; Méndez-Lojo et al., 2012; Pai and Pingali, 2016; Sorensen et al., 2016; Cederman and Tsigas, 2008; Tzeng et al., 2010; Burtscher et al., 2012; Che et al., 2013). These approaches rely on the occupancy-bound execution model, flooding available compute units with work, so that the GPU is unavailable for other tasks, and assuming fair scheduling between occupant workgroups, which is unlikely to be guaranteed on future GPU platforms. As our experiments demonstrate, our cooperative kernels model allows blocking algorithms to be upgraded to run in a manner that facilitates responsive multitasking.

GPU Multitasking and Scheduling.

Hardware support for preemption has been proposed for Nvidia GPUs, as well as SM-draining whereby workgroups occupying a symmetric multiprocessor (SM; a compute unit using our terminology) are allowed to complete until the SM becomes free for other tasks (Tanasic et al., 2014). SM draining is limited the presence of blocking constructs, since it may not be possible to drain a blocked workgroup. A follow-up work adds the notion of SM flushing, where a workgroup can be re-scheduled from scratch if it has not yet committed side-effects (Park et al., 2015). Both approaches have been evaluated using simulators, over sets of regular GPU kernels. Very recent Nvidia GPUs (i.e. the Pascal architecture) support preemption, though, as discussed in Sec. 1 and Sec. 5.5, it is not clear whether they guarantee fairness or allow tasks to share GPU resources at the workgroup level (NVIDIA, 2016).

CUDA and OpenCL provide the facility for a kernel to spawn further kernels (Nvidia, 2016). This dynamic parallelism can be used to implement a GPU-based scheduler, by having an initial scheduler kernel repeatedly spawn further kernels as required, according to some scheduling policy (Muyan-Özçelik and Owens, 2016). However, kernels that uses dynamic parallelism are still prone to unfair scheduling of workgroups, and thus does not help in deploying blocking algorithms on GPUs.

Cooperative Multitasking.

Cooperative multitasking was offered in older operating systems (e.g. pre 1995 Windows) and is still used by some operating systems, such as RISC OS (RISC OS, ). Additionally, cooperative multitasking can be efficiently implemented in today’s high-level languages for domains in which preemptive multitasking is either too costly or not supported on legacy systems (Tarpenning, 1991).

7. Conclusions and Future Work

We have proposed cooperative kernels, a small set of GPU programming extensions that allow long-running, blocking kernels to be fairly scheduled and to share GPU resources with other workloads. Experimental results using our megakernel-based prototype show that the model is a good fit for current GPU-accelerated irregular algorithms. The performance that could be gained through a native implementation with driver support would be even better. Avenues for future work include seeking additional classes of irregular algorithms to which the model might (be extended to) apply (to), investigating implementing native support in open source drivers, and integrating cooperative kernels into template- and compiler-based programming models for graph algorithms on GPUs (Wang et al., 2016; Pai and Pingali, 2016).


We are grateful to Lee Howes, Bernhard Kainz, Paul Kelly, Christopher Lidbury, Steven McDonagh, Sreepathi Pai, and Andrew Richards for insightful comments throughout the work. We thank the FSE reviewers for their thorough evaluations and feedback. This work is supported in part by EPSRC Fellowship EP/N026314, and a gift from Intel Corporation.


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Appendix A Operational Semantics for Cooperative Kernels

In Sec. 3.1 we presented the semantics of cooperative kernels relatively informally, using English, to provide the intuition behind our programming model. We now back this up with a more formal presentation as an operational semantics for an abstract GPU programming model.


Let be a set of local states that abstractly captures the private memory associated with a thread executing a GPU kernel. Let denote the set of all possible statements that a thread can execute. We do not detail the structure of these statements, except that we assume sequential composition of statements is provided by the ; separator, and that the , , and primitives from our cooperative kernels programming model are valid statements.

A thread state is then a pair , where and . The component captures the valuation of all the thread’s private memory, and the component captures the remaining statements to be executed by the thread. Let denote the set of all thread states.

Assuming that threads per workgroup were requested on kernel launch, a workgroup state is a -tuple , where each is the thread state for the th thread in the workgroup ).

Assuming that was specified as the maximum number of workgroups that should execute the cooperative kernel, a kernel state is then a pair

where: represents the shared state of the kernel; is the number of active workgroups; is the workgroup state for active workgroup (); and occurrences of indicate absent workgroups. Let denote the set of all possible shared states. We regard workgroup-local storage as being part of the shared state of a kernel.

Thread-level transitions.

We leave the semantics for thread-level transitions abstract, assuming a binary relation on . If , this indicates that if a thread is in local state , the thread can transition to local state , changing the shared state from to in the process.

All we require is that if has the form , where is one of , , or . This is because we shall specifically define the meaning of the new primitives introduced by our programming model.

Memory synchronisation.

GPUs are known to have relaxed memory models (Alglave et al., 2015). To abstractly account for this, we assume that the shared state component is not simply a mapping from locations to values, but instead captures all the intricacies of GPU memory that can lead to this relaxed behaviour. We also assume a function which, given a kernel state , returns a set of kernel states. The idea is that each is a possible kernel state that can be reached from by stalling until all stores to memory and loads from memory to thread-local storage have completed. All we require is that does not modify the number of active workgroups nor the component of a thread state that determines which statements remain to be executed.

w_i(j) = (l, ss)
(σ, (l, ss)) →_τ (σ’, (l’, ss’))
w_i’ = w_i[j ↦(l’, ss’)] (σ, (…, w_i, …)) →(σ’, (…, w_i’, …)) (Thread-Step)

∀j  .  w_i(j) = (l_j, offer_kill();ss)
w_i’ = ((l_0, ss), …, (l_d-1, ss)) (σ, (…, w_i, …)) →(σ, (…, w_i’, …)) (Kill-No-Op)

∀j  .  w_M-1(j) = (l_j, offer_kill();ss)
M > 0 (σ, (…, w_M-2, w_M-1, ⊥, …, ⊥)) →(σ, (…, w_M-2, ⊥, ⊥, …, ⊥)) (Kill)

∀j  .  w_i(j) = (l_j, request_fork();ss)
w_i’ = ((l_0, ss), …, (l_d-1, ss))
k ∈[0, N - M]
∀a ∈[0, k - 1]  .  w_M+a = ((l_0, ss), …, (l_0, ss)) (σ, (…, w_i, …, w_M-1, ⊥, …, ⊥)) →(σ, (…, w_i’, …, w_M-1, w_M, …, w_M+k-1, ⊥, …, ⊥)) (Fork)

∀i  . ∀j . w_i(j) = (l_i,j, global_barrier();ss)
∀i  . ∀j . w_i’(j) = (l_i,j, ss)

κsync((σ, (w_0’, …, w_M-1’, ⊥, …, ⊥)) (σ, (w_0, …, w_M-1, ⊥, …, ⊥)) →κ (Barrier)

∀i  . ∀j . w_i(j) = (l_i,j, resizing_global_barrier();ss)
∀j . w_0’(j) = (l_1,j, global_barrier(); request_fork(); global_barrier(); global_barrier();ss)
∀i ≠1  . ∀j . w_i’(j) = (l_i,j, global_barrier(); global_barrier(); offer_kill(); global_barrier();ss) (σ, (w_0, …, w_M-1, ⊥, …, ⊥)) →(σ, (w_0’, …, w_M-1’, ⊥, …, ⊥)) (Resizing-Barrier)

Figure 7. Abstract operational semantics for our cooperative kernels language extensions

Operational semantics rules.

Figure 7 presents the rules of our operational semantics, defining a relation on kernel states.

Rule Thread-Step defines the semantics for thread making a single execution step, delegating to the abstract relation to determine how the thread’s local state and the shared state component change. For simplicity, this rule ignores the semantics of intra-workgroup barriers, which are not our focus here.

Rule Kill-No-Op reflects the fact that when a workgroup reaches , this can always be treated as a no-op. Whether a scheduler implementation accepts calls or not depends on competing workloads and how the scheduler has been designed to meet the non-functional requirements discussed in Sec. 3.3, but in general the programmer should always be prepared for the possibility that a workgroup survives after calling .

The case where a workgroup’s offer to be killed is accepted by the scheduler is captured by rule Kill. Because we have adopted a semantics where workgroup 0 is never killed and where only the workgroup with the highest id can be killed, the rule only fires if and workgroup has reached . The rule has the effect of replacing the workgroup state for with .

Recall that is a workgroup-level function: the same syntactic call must be reached by all threads in a workgroup. This is captured in our rules by requiring in both Kill-No-Op and Kill that every thread is ready to execute followed by an identical statement . Neither rule applies until all threads in a workgroup reach , and the workgroup gets stuck if multiple threads in a workgroup reach with different subsequent statements.

The Fork rule similarly requires all threads in a workgroup to reach with identical following statements. A nondeterministic number of new workgroups, , is selected to be forked, where . Importantly, is always a legitimate choice, in which case has no effect on the number of workgroups that are executing. In the case where , new workgroup states are created, where each workgroup inherits the local state of thread 0 in , the workgroup executing the fork call. After , all threads in all workgroups, including the new threads, proceed to execute the sequence of statements that followed .

A simplification here is that we do not model transmission of particular annotated variables from thread 0 of the forking workgroup, instead specifying that the entire local state of the workgroup is transmitted. Extending the semantics to transmit only annotated variables would be straightforward but verbose, requiring the local memory component of a thread state to be split into two: the part of the local state to be transmitted (modelling -annotated variables), and the rest of the local state.

The Barrier rule requires that every thread executing the kernel reaches an identical following statement. This reflects that fact that is a kernel-level function. Each thread then skips over the call, and the function is applied to yield a set of kernel states that can arise due to memory synchronization taking place. An arbitrary member of this set is selected as the next kernel state.

Despite its apparent complexity, the Resizing-Barrier rule simply implements the rewriting of in terms of , and discussed in Sec. 3.1.

Appendix B Alternative Semantic Choices

The semantics of cooperative kernels has been guided by the practical applications we have studied (described in Sec. 5.1). We now discuss several cases where we might have taken different and also reasonable semantic decisions.

Killing order.

We opted for a semantics whereby only the active workgroup with the highest id can be killed. This has an appealing property: it means that the ids of active workgroups are contiguous, which is important for processing of contiguous data. The cooperative graph traversal algorithm of Fig. 4 illustrates this: the algorithm is prepared for to change after each resizing barrier call, but depends on the fact that returns a contiguous range of thread ids.

A disadvantage of this decision is that it may provide sub-optimal responsiveness from the point of view of the scheduler. Suppose the scheduler requires an additional compute unit, but the active thread with the largest id is processing some computationally intensive work and will take a while to reach . Our semantics means that the scheduler cannot take advantage of the fact that another active workgroup may invoke sooner.

Cooperative kernels that do not require contiguous thread ids might me more suited to a semantics in which workgroups can be killed in any order, but where workgroup ids (and thus thread global ids) are not guaranteed to be contiguous.

Keeping one workgroup alive.

Our semantics dictate that the workgroup with id 0 will not be killed if it invokes . This avoids the possibility of the cooperative kernel terminating early due to the programmer inadvertently allowing all workgroups to be killed, and the decision to keep workgroup 0 alive fits well with our choice to kill workgroups in descending order of id.

However, there might be a use case for a cooperative kernel reaching a point where it would be acceptable for the kernel to exit, although desirable for some remaining computation to be performed if competing workloads allow it. In this case, a semantics where all workgroups can be killed via would be appropriate, and the programmer would need to guard each with an id check in cases where killing all workgroups would be unacceptable. For example:

  if( != 0) ();

would ensure that at least workgroup 0 is kept alive.

Transmission of partial state from a single thread.

Recall from the semantics of that newly forked workgroups inherit the variable valuation associated with thread 0 of the forking workgroup, but only for -annotated variables. Alternative choices here would be to have forked workgroups inherit values for all variables from the forking workgroup, and to have thread in the forking workgroup provide the valuation for thread in each spawned workgroup, rather than having thread 0 transmit the valuation to all new threads.

We opted for transmitting only selected variables based on the observation that many of a thread’s private variables are dead at the point of issuing or , thus it would be wasteful to transmit them. A live variable analysis could instead be employed to over-approximate the variables that might be accessed by newly arriving workgroups, so that these are automatically transmitted.

In all cases, we found that a variable that needed to be transmitted had the property of being uniform across the workgroup. That is, despite each thread having its own copy of the variable, each thread is in agreement on the variable’s value. As an example, the , and variables used in Fig. 4 are all stored in thread-private memory, but all threads in a workgroup agree on the values of these variables at each call. As a result, transmitting the thread 0’s valuation of the annotated variables is equivalent to (and more efficient than) transmitting values on a thread-by-thread basis. We have not yet encountered a real-world example where our current semantics would not suffice.

Appendix C Graphs for Multitasking Experiments

We present the full set of graphs exemplified by the examples in Fig. 5; for completeness we reproduce those graphs here too.