# Parallel computations using RecursivelyEnumeratedSet and Map-Reduce¶

There is an efficient way to distribute computations on a set $$S$$ of objects defined by RecursivelyEnumeratedSet() (see sage.sets.recursively_enumerated_set for more details) over which one would like to perform the following kind of operations:

• Compute the cardinality of a (very large) set defined recursively (through a call to RecursivelyEnumeratedSet_forest)

• More generally, compute any kind of generating series over this set

• Test a conjecture, e.g. find an element of $$S$$ satisfying a specific property, or check that none does or that they all do

• Count/list the elements of $$S$$ that have a specific property

• Apply any map/reduce kind of operation over the elements of $$S$$

AUTHORS:

• Florent Hivert – code, documentation (2012–2016)

• Jean Baptiste Priez – prototype, debugging help on MacOSX (2011-June, 2016)

• Nathann Cohen – some documentation (2012)

## How is this different from usual MapReduce?¶

This implementation is specific to RecursivelyEnumeratedSet_forest, and uses its properties to do its job. Not only mapping and reducing but also generating the elements of $$S$$ is done on different processors.

## How can I use all that stuff?¶

First, you need to set the environment variable $$SAGE_NUM_THREADS$$ to the desired number of parallel threads to be used:

sage: import os # not tested sage: os.environ[“SAGE_NUM_THREADS”] = ‘8’ # not tested

Second, you need the information necessary to describe a RecursivelyEnumeratedSet_forest representing your set $$S$$ (see sage.sets.recursively_enumerated_set). Then, you need to provide a “map” function as well as a “reduce” function. Here are some examples:

• Counting the number of elements. In this situation, the map function can be set to lambda x: 1, and the reduce function just adds the values together, i.e. lambda x, y: x + y.

We count binary words of length $$\leq 16$$:

sage: seeds = [[]]
sage: succ = lambda l: [l + , l + ] if len(l) < 16 else []
sage: S = RecursivelyEnumeratedSet(seeds, succ,
....:     structure='forest', enumeration='depth')
sage: map_function = lambda x: 1
sage: reduce_function = lambda x, y: x + y
sage: reduce_init = 0
sage: S.map_reduce(map_function, reduce_function, reduce_init)
131071


This matches the number of binary words of length $$\leq 16$$:

sage: factor(131071 + 1)
2^17


Note that the map and reduce functions here have the default values of the sage.sets.recursively_enumerated_set.RecursivelyEnumeratedSet_forest.map_reduce() method so that the number of elements can be obtained more simply with:

sage: S.map_reduce()
131071


Instead of using RecursivelyEnumeratedSet(), one can directly use RESetMapReduce, which gives finer control over the parallel execution (see Advanced use below):

sage: from sage.parallel.map_reduce import RESetMapReduce
sage: S = RESetMapReduce(
....:     roots=[[]],
....:     children=lambda l: [l + , l + ] if len(l) < 16 else [],
....:     map_function=lambda x: 1,
....:     reduce_function=lambda x, y: x + y,
....:     reduce_init=0)
sage: S.run()
131071

• Generating series. For this, take a Map function that associates a monomial to each element of $$S$$, while the Reduce function is still equal to lambda x, y: x + y.

We compute the generating series for counting binary words of each length $$\leq 16$$:

sage: S = RecursivelyEnumeratedSet(
....:     [[]], lambda l: [l + , l + ] if len(l) < 16 else [],
....:     structure='forest', enumeration='depth')
sage: x = polygen(ZZ)
sage: sp = S.map_reduce(
....:     map_function=lambda z: x**len(z),
....:     reduce_function=lambda x, y: x + y,
....:     reduce_init=0)
sage: sp
65536*x^16 + 32768*x^15 + 16384*x^14 + 8192*x^13 + 4096*x^12
+ 2048*x^11 + 1024*x^10 + 512*x^9 + 256*x^8 + 128*x^7 + 64*x^6
+ 32*x^5 + 16*x^4 + 8*x^3 + 4*x^2 + 2*x + 1


This is of course $$\sum_{i=0}^{16} (2x)^i$$:

sage: sp == sum((2*x)^i for i in range(17))
True


Here is another example where we count permutations of size $$\leq 8$$ (here we use the default values):

sage: S = RecursivelyEnumeratedSet(
....:     [[]],
....:     lambda l: ([l[:i] + [len(l)] + l[i:]
....:                for i in range(len(l) + 1)] if len(l) < 8 else []),
....:     structure='forest',
....:     enumeration='depth')
sage: x = polygen(ZZ)
sage: sp = S.map_reduce(lambda z: x**len(z)); sp
40320*x^8 + 5040*x^7 + 720*x^6 + 120*x^5 + 24*x^4 + 6*x^3 + 2*x^2 + x + 1


This is of course $$\sum_{i=0}^{8} i! x^i$$:

sage: sp == sum(factorial(i)*x^i for i in range(9))
True

• Post Processing. We now demonstrate the use of post_process. We generate the permutation as previously, but we only perform the map/reduce computation on those of even len. Of course we get the even part of the previous generating series:

sage: S = RecursivelyEnumeratedSet(
....:     [[]],
....:     lambda l: ([l[:i] + [len(l) + 1] + l[i:]
....:                for i in range(len(l) + 1)] if len(l) < 8 else []),
....:     post_process=lambda l: l if len(l) % 2 == 0 else None,
....:     structure='forest',
....:     enumeration='depth')
sage: sp = S.map_reduce(lambda z: x**len(z)); sp
40320*x^8 + 720*x^6 + 24*x^4 + 2*x^2 + 1


This is also useful for example to call a constructor on the generated elements:

sage: S = RecursivelyEnumeratedSet(
....:     [[]],
....:     lambda l: ([l[:i] + [len(l) + 1] + l[i:]
....:     for i in range(len(l) + 1)] if len(l) < 5 else []),
....:     post_process=lambda l: Permutation(l) if len(l) == 5 else None,
....:     structure='forest',
....:     enumeration='depth')
sage: x = polygen(ZZ)
sage: sp = S.map_reduce(lambda z: x**z.number_of_inversions()); sp
x^10 + 4*x^9 + 9*x^8 + 15*x^7 + 20*x^6 + 22*x^5 + 20*x^4 + 15*x^3 + 9*x^2 + 4*x + 1


We get here a polynomial which is the $$q$$-factorial (in the variable $$x$$) of $$5$$, that is, $$\prod_{i=1}^{5} \frac{1-x^i}{1-x}$$:

sage: x = polygen(ZZ)
sage: prod((1-x^i)//(1-x) for i in range(1, 6))
x^10 + 4*x^9 + 9*x^8 + 15*x^7 + 20*x^6 + 22*x^5 + 20*x^4 + 15*x^3 + 9*x^2 + 4*x + 1


Compare:

sage: from sage.combinat.q_analogues import q_factorial
sage: q_factorial(5)
q^10 + 4*q^9 + 9*q^8 + 15*q^7 + 20*q^6 + 22*q^5 + 20*q^4 + 15*q^3 + 9*q^2 + 4*q + 1

• Listing the objects. One can also compute the list of objects in a RecursivelyEnumeratedSet_forest> using RESetMapReduce. As an example, we compute the set of numbers between 1 and 63, generated by their binary expansion:

sage: S = RecursivelyEnumeratedSet(
....:     ,
....:     lambda l: [(l<<1)|0, (l<<1)|1] if l < 1<<5 else [],
....:     structure='forest',
....:     enumeration='depth')


Here is the list computed without RESetMapReduce:

sage: serial = list(S)
sage: serial
[1, 2, 4, 8, 16, 32, 33, 17, 34, 35, 9, 18, 36, 37, 19, 38, 39, 5, 10,
20, 40, 41, 21, 42, 43, 11, 22, 44, 45, 23, 46, 47, 3, 6, 12, 24, 48,
49, 25, 50, 51, 13, 26, 52, 53, 27, 54, 55, 7, 14, 28, 56, 57, 29, 58,
59, 15, 30, 60, 61, 31, 62, 63]


Here is how to perform the parallel computation. The order of the lists depends on the synchronisation of the various computation processes and therefore should be considered as random:

sage: parall = S.map_reduce(lambda x: [x], lambda x, y: x + y, [])
sage: parall  # random
[1, 3, 7, 15, 31, 63, 62, 30, 61, 60, 14, 29, 59, 58, 28, 57, 56, 6, 13,
27, 55, 54, 26, 53, 52, 12, 25, 51, 50, 24, 49, 48, 2, 5, 11, 23, 47,
46, 22, 45, 44, 10, 21, 43, 42, 20, 41, 40, 4, 9, 19, 39, 38, 18, 37,
36, 8, 17, 35, 34, 16, 33, 32]
sage: sorted(serial) == sorted(parall)
True


Fine control over the execution of a map/reduce computation is achieved via parameters passed to the RESetMapReduce.run() method. The following three parameters can be used:

• max_proc – (integer, default: None) if given, the maximum number of worker processors to use. The actual number is also bounded by the value of the environment variable SAGE_NUM_THREADS (the number of cores by default).

• timeout – a timeout on the computation (default: None)

• reduce_locally – whether the workers should reduce locally their work or sends results to the master as soon as possible. See RESetMapReduceWorker for details.

Here is an example or how to deal with timeout:

sage: from sage.parallel.map_reduce import (RESetMPExample, AbortError)
sage: EX = RESetMPExample(maxl=100)
sage: try:
....:     res = EX.run(timeout=float(0.01))
....: except AbortError:
....:     print("Computation timeout")
....: else:
....:     print("Computation normally finished")
....:     res
Computation timeout


The following should not timeout even on a very slow machine:

sage: EX = RESetMPExample(maxl=8)
sage: try:
....:     res = EX.run(timeout=60)
....: except AbortError:
....:     print("Computation Timeout")
....: else:
....:     print("Computation normally finished")
....:     res
Computation normally finished
40320*x^8 + 5040*x^7 + 720*x^6 + 120*x^5 + 24*x^4 + 6*x^3 + 2*x^2 + x + 1


As for reduce_locally, one should not see any difference, except for speed during normal usage. Most of the time one should leave it set to True, unless one sets up a mechanism to consume the partial results as soon as they arrive. See RESetParallelIterator and in particular the __iter__ method for a example of consumer use.

## Profiling¶

It is possible to profile a map/reduce computation. First we create a RESetMapReduce object:

sage: from sage.parallel.map_reduce import RESetMapReduce
sage: S = RESetMapReduce(
....:     roots=[[]],
....:     children=lambda l: [l + , l + ] if len(l) < 16 else [],
....:     map_function=lambda x: 1,
....:     reduce_function=lambda x, y: x + y,
....:     reduce_init=0)


The profiling is activated by the profile parameter. The value provided should be a prefix (including a possible directory) for the profile dump:

sage: prof = tmp_dir('RESetMR_profile') + 'profcomp'
sage: res = S.run(profile=prof)  # random
[RESetMapReduceWorker-1:58] (20:00:41.444) Profiling in
/home/user/.sage/temp/.../32414/RESetMR_profilewRCRAx/profcomp1
...
[RESetMapReduceWorker-1:57] (20:00:41.444) Profiling in
/home/user/.sage/temp/.../32414/RESetMR_profilewRCRAx/profcomp0
...
sage: res
131071


In this example, the profiles have been dumped in files such as profcomp0. One can then load and print them as follows. See cProfile.Profile for more details:

sage: import cProfile, pstats
sage: st = pstats.Stats(prof+'0')
sage: st.strip_dirs().sort_stats('cumulative').print_stats()  # random
...
Ordered by: cumulative time

ncalls  tottime  percall  cumtime  percall filename:lineno(function)
1    0.023    0.023    0.432    0.432 map_reduce.py:1211(run_myself)
11968    0.151    0.000    0.223    0.000 map_reduce.py:1292(walk_branch_locally)
...
<pstats.Stats instance at 0x7fedea40c6c8>


The Python Profilers for more detail on profiling in python.

## Logging¶

The computation progress is logged through a logging.Logger in sage.parallel.map_reduce.logger together with logging.StreamHandler and a logging.Formatter. They are currently configured to print warning messages to the console.

Logging facility for Python for more detail on logging and log system configuration.

Note

Calls to logger which involve printing the node are commented out in the code, because the printing (to a string) of the node can be very time consuming depending on the node and it happens before the decision whether the logger should record the string or drop it.

## How does it work ?¶

The scheduling algorithm we use here is any adaptation of Wikipedia article Work_stealing:

In a work stealing scheduler, each processor in a computer system has a queue of work items (computational tasks, threads) to perform. […]. Each work items are initially put on the queue of the processor executing the work item. When a processor runs out of work, it looks at the queues of other processors and “steals” their work items. In effect, work stealing distributes the scheduling work over idle processors, and as long as all processors have work to do, no scheduling overhead occurs.

For communication we use Python’s basic multiprocessing module. We first describe the different actors and communication tools used by the system. The work is done under the coordination of a master object (an instance of RESetMapReduce) by a bunch of worker objects (instances of RESetMapReduceWorker).

Each running map reduce instance works on a RecursivelyEnumeratedSet_forest> called here $$C$$ and is coordinated by a RESetMapReduce object called the master. The master is in charge of launching the work, gathering the results and cleaning up at the end of the computation. It doesn’t perform any computation associated to the generation of the element $$C$$ nor the computation of the mapped function. It however occasionally perform a reduce, but most reducing is by default done by the workers. Also thanks to the work-stealing algorithm, the master is only involved in detecting the termination of the computation but all the load balancing is done at the level of the workers.

Workers are instances of RESetMapReduceWorker. They are responsible for doing the actual computations: element generation, mapping and reducing. They are also responsible for the load balancing thanks to work-stealing.

Here is a description of the attributes of the master relevant to the map-reduce protocol:

• _results – a SimpleQueue where the master gathers the results sent by the workers.

• _active_tasks – a Semaphore recording the number of active tasks. The work is complete when it reaches 0.

• _done – a Lock which ensures that shutdown is done only once.

• _aborted – a Value() storing a shared ctypes.c_bool which is True if the computation was aborted before all workers ran out of work.

• _workers – a list of RESetMapReduceWorker objects. Each worker is identified by its position in this list.

Each worker is a process (RESetMapReduceWorker inherits from Process) which contains:

• worker._iproc – the identifier of the worker that is its position in the master’s list of workers

• worker._todo – a collections.deque storing of nodes of the worker. It is used as a stack by the worker. Thiefs steal from the bottom of this queue.

• worker._request – a SimpleQueue storing steal request submitted to worker.

• worker._read_task, worker._write_task – a Pipe used to transfert node during steal.

• worker._thief – a Thread which is in charge of stealing from worker._todo.

Here is a schematic of the architecture: ## How thefts are performed¶

During normal time, that is, when all workers are active, a worker W is iterating though a loop inside RESetMapReduceWorker.walk_branch_locally(). Work nodes are taken from and new nodes W._todo are appended to W._todo. When a worker W runs out of work, that is, when worker._todo is empty, it tries to steal some work (i.e., a node) from another worker. This is performed in the RESetMapReduceWorker.steal() method.

From the point of view of W, here is what happens:

• W signals to the master that it is idle: master._signal_task_done;

• W chooses a victim V at random;

• W sends a request to V: it puts its identifier into V._request;

• W tries to read a node from W._read_task. Then three things may happen:

• a proper node is read. Then the theft was a success and W starts working locally on the received node.

• None is received. This means that V was idle. Then W tries another victim.

• AbortError is received. This means either that the computation was aborted or that it simply succeeded and that no more work is required by W. Therefore an AbortError exception is raised leading W to shutdown.

We now describe the protocol on the victim’s side. Each worker process contains a Thread which we call T for thief which acts like some kind of Troyan horse during theft. It is normally blocked waiting for a steal request.

From the point of view of V and T, here is what happens:

• during normal time, T is blocked waiting on V._request;

• upon steal request, T wakes up receiving the identification of W;

• T signals to the master that a new task is starting by master._signal_task_start;

• Two things may happen depending if the queue V._todo is empty or not. Remark that due to the GIL, there is no parallel execution between the victim V and its thief thread T.

• If V._todo is empty, then None is answered on W._write_task. The task is immediately signaled to end the master through master._signal_task_done.

• Otherwise, a node is removed from the bottom of V._todo. The node is sent to W on W._write_task. The task will be ended by W, that is, when finished working on the subtree rooted at the node, W will call master._signal_task_done.

## The end of the computation¶

To detect when a computation is finished, a synchronized integer is kept which counts the number of active tasks. This is essentially a semaphore but semaphores are broken on Darwin OSes so we ship two implementations depending on the OS (see ActiveTaskCounter and ActiveTaskCounterDarwin and the note below).

When a worker finishes working on a task, it calls master._signal_task_done. This decreases the task counter master._active_tasks. When it reaches 0, it means that there are no more nodes: the work is completed. The worker executes master._shutdown which sends AbortError to all worker._request and worker._write_task queues. Each worker or thief thread receiving such a message raises the corresponding exception, therefore stopping its work. A lock called master._done ensures that shutdown is only done once.

Finally, it is also possible to interrupt the computation before its ends, by calling master.abort(). This is achieved by setting master._active_tasks to 0 and calling master._shutdown.

Warning

The macOS Semaphore bug

Darwin OSes do not correctly implement POSIX’s semaphore semantic. Indeed, on these systems, acquire may fail and return False not only when the semaphore is equal to zero but also because someone else is trying to acquire at the same time. This makes using Semaphores impossible on macOS so that on these systems we use a synchronized integer instead.

## Are there examples of classes?¶

Yes! Here they are:

## Tests¶

Generating series for the sum of strictly decreasing lists of integers smaller than 15:

sage: y = polygen(ZZ, 'y')
sage: R = RESetMapReduce(
....:     roots=[([], 0, 0)] + [([i], i, i) for i in range(1, 15)],
....:     children=lambda list_sum_last:
....:         [(list_sum_last + [i], list_sum_last + i, i)
....:          for i in range(1, list_sum_last)],
....:     map_function=lambda li_sum_dummy: y**li_sum_dummy)
sage: sg = R.run()
sage: sg == prod((1 + y**i) for i in range(1, 15))
True


## Classes and methods¶

exception sage.parallel.map_reduce.AbortError

Bases: Exception

Exception for aborting parallel computations.

This is used both as exception or as abort message.

sage.parallel.map_reduce.ActiveTaskCounter
class sage.parallel.map_reduce.ActiveTaskCounterDarwin(task_number)

Bases: object

Handling the number of active tasks.

A class for handling the number of active tasks in a distributed computation process. This is essentially a semaphore, but Darwin OSes do not correctly implement POSIX’s semaphore semantic. So we use a shared integer with a lock.

abort()

Set the task counter to zero.

EXAMPLES:

sage: from sage.parallel.map_reduce import ActiveTaskCounterDarwin as ATC
sage: c = ATC(4); c
sage: c.abort()
sage: c

task_done()

Decrement the task counter by one.

OUTPUT:

Calling task_done() decrements the counter and returns its new value.

EXAMPLES:

sage: from sage.parallel.map_reduce import ActiveTaskCounterDarwin as ATC
sage: c = ATC(4); c
3
sage: c

sage: c = ATC(0)
-1

task_start()

Increment the task counter by one.

OUTPUT:

Calling task_start() on a zero or negative counter returns 0, otherwise increment the counter and returns its value after the incrementation.

EXAMPLES:

sage: from sage.parallel.map_reduce import ActiveTaskCounterDarwin as ATC
sage: c = ATC(4); c
5
sage: c


Calling task_start() on a zero counter does nothing:

sage: c = ATC(0)
0
sage: c

class sage.parallel.map_reduce.ActiveTaskCounterPosix(task_number)

Bases: object

Handling the number of active tasks.

A class for handling the number of active tasks in a distributed computation process. This is the standard implementation on POSIX compliant OSes. We essentially wrap a semaphore.

Note

A legitimate question is whether there is a need in keeping the two implementations. I ran the following experiment on my machine:

S = RecursivelyEnumeratedSet(
[[]],
lambda l: ([l[:i] + [len(l)] + l[i:]
for i in range(len(l) + 1)]
if len(l) < NNN else []),
structure='forest',
enumeration='depth')
%time sp = S.map_reduce(lambda z: x**len(z)); sp


For NNN = 10, averaging a dozen of runs, I got:

• Posix compliant implementation: 17.04 s

• Darwin implementation: 18.26 s

So there is a non negligible overhead. It will probably be worth it if we try to cythonize the code. So I’m keeping both implementations.

abort()

Set the task counter to zero.

EXAMPLES:

sage: from sage.parallel.map_reduce import ActiveTaskCounter as ATC
sage: c = ATC(4); c
sage: c.abort()
sage: c

task_done()

Decrement the task counter by one.

OUTPUT:

Calling task_done() decrements the counter and returns its new value.

EXAMPLES:

sage: from sage.parallel.map_reduce import ActiveTaskCounter as ATC
sage: c = ATC(4); c
3
sage: c

sage: c = ATC(0)
-1

task_start()

Increment the task counter by one.

OUTPUT:

Calling task_start() on a zero or negative counter returns 0, otherwise increment the counter and returns its value after the incrementation.

EXAMPLES:

sage: from sage.parallel.map_reduce import ActiveTaskCounterDarwin as ATC
sage: c = ATC(4); c
5
sage: c


Calling task_start() on a zero counter does nothing:

sage: c = ATC(0)
0
sage: c

class sage.parallel.map_reduce.RESetMPExample(maxl=9)

An example of map reduce class.

INPUT:

• maxl – the maximum size of permutations generated (default to $$9$$).

This computes the generating series of permutations counted by their size up to size maxl.

EXAMPLES:

sage: from sage.parallel.map_reduce import RESetMPExample
sage: EX = RESetMPExample()
sage: EX.run()
362880*x^9 + 40320*x^8 + 5040*x^7 + 720*x^6 + 120*x^5
+ 24*x^4 + 6*x^3 + 2*x^2 + x + 1


This is an example of RESetMapReduce

children(l)

Return the children of the permutation $$l$$.

INPUT:

• l – a list containing a permutation

OUTPUT:

The lists with len(l) inserted at all possible positions into l.

EXAMPLES:

sage: from sage.parallel.map_reduce import RESetMPExample
sage: RESetMPExample().children([1,0])
[[2, 1, 0], [1, 2, 0], [1, 0, 2]]

map_function(l)

The monomial associated to the permutation $$l$$.

INPUT:

• l – a list containing a permutation

OUTPUT:

The monomial x^len(l).

EXAMPLES:

sage: from sage.parallel.map_reduce import RESetMPExample
sage: RESetMPExample().map_function([1,0])
x^2

roots()

Return the empty permutation.

EXAMPLES:

sage: from sage.parallel.map_reduce import RESetMPExample
sage: RESetMPExample().roots()
[[]]

class sage.parallel.map_reduce.RESetMapReduce(roots=None, children=None, post_process=None, map_function=None, reduce_function=None, reduce_init=None, forest=None)

Bases: object

Map-Reduce on recursively enumerated sets.

INPUT:

Description of the set:

• either forest=f – where f is a RecursivelyEnumeratedSet_forest>

• or a triple roots, children, post_process as follows

• roots=r – The root of the enumeration

• children=c – a function iterating through children nodes, given a parent node

• post_process=p – a post-processing function

The option post_process allows for customizing the nodes that are actually produced. Furthermore, if post_process(x) returns None, then x won’t be output at all.

Description of the map/reduce operation:

• map_function=f – (default to None)

• reduce_function=red – (default to None)

• reduce_init=init – (default to None)

the Map/Reduce module for details and examples.

abort()

Abort the current parallel computation.

EXAMPLES:

sage: from sage.parallel.map_reduce import RESetParallelIterator
sage: S = RESetParallelIterator([[]],
....:     lambda l: [l + , l + ] if len(l) < 17 else [])
sage: it = iter(S)
sage: next(it)  # random
[]
sage: S.abort()
sage: hasattr(S, 'work_queue')
False


Cleanup:

sage: S.finish()

finish()

Destroy the workers and all the communication objects.

Communication statistics are gathered before destroying the workers.

get_results(timeout=None)

Get the results from the queue.

OUTPUT:

The reduction of the results of all the workers, that is, the result of the map/reduce computation.

EXAMPLES:

sage: from sage.parallel.map_reduce import RESetMapReduce
sage: S = RESetMapReduce()
sage: S.setup_workers(2)
sage: for v in [1, 2, None, 3, None]: S._results.put(v)
sage: S.get_results()
6


Cleanup:

sage: del S._results, S._active_tasks, S._done, S._workers

map_function(o)

Return the function mapped by self.

INPUT:

• o – a node

OUTPUT:

By default 1.

Note

This should be overloaded in applications.

EXAMPLES:

sage: from sage.parallel.map_reduce import RESetMapReduce
sage: S = RESetMapReduce()
sage: S.map_function(7)
1
sage: S = RESetMapReduce(map_function = lambda x: 3*x + 5)
sage: S.map_function(7)
26

post_process(a)

Return the image of a under the post-processing function for self.

INPUT:

• a – a node

With the default post-processing function, which is the identity function, this returns a itself.

Note

This should be overloaded in applications.

EXAMPLES:

sage: from sage.parallel.map_reduce import RESetMapReduce
sage: S = RESetMapReduce()
sage: S.post_process(4)
4
sage: S = RESetMapReduce(post_process=lambda x: x*x)
sage: S.post_process(4)
16

print_communication_statistics(blocksize=16)

Print the communication statistics in a nice way.

EXAMPLES:

sage: from sage.parallel.map_reduce import RESetMPExample
sage: S = RESetMPExample(maxl=6)
sage: S.run()
720*x^6 + 120*x^5 + 24*x^4 + 6*x^3 + 2*x^2 + x + 1

sage: S.print_communication_statistics()  # random
#proc:        0    1    2    3    4    5    6    7
reqs sent:    5    2    3   11   21   19    1    0
reqs rcvs:   10   10    9    5    1   11    9    2
- thefs:      1    0    0    0    0    0    0    0
+ thefs:      0    0    1    0    0    0    0    0

random_worker()

Return a random worker.

OUTPUT:

A worker for self chosen at random.

EXAMPLES:

sage: from sage.parallel.map_reduce import RESetMPExample, RESetMapReduceWorker
sage: EX = RESetMPExample(maxl=6)
sage: EX.setup_workers(2)
sage: EX.random_worker()
<RESetMapReduceWorker...RESetMapReduceWorker-... initial...>
sage: EX.random_worker() in EX._workers
True


Cleanup:

sage: del EX._results, EX._active_tasks, EX._done, EX._workers

reduce_function(a, b)

Return the reducer function for self.

INPUT:

• a, b – two values to be reduced

OUTPUT:

By default the sum of a and b.

Note

This should be overloaded in applications.

EXAMPLES:

sage: from sage.parallel.map_reduce import RESetMapReduce
sage: S = RESetMapReduce()
sage: S.reduce_function(4, 3)
7
sage: S = RESetMapReduce(reduce_function=lambda x,y: x*y)
sage: S.reduce_function(4, 3)
12

reduce_init()

Return the initial element for a reduction.

Note

This should be overloaded in applications.

roots()

Return the roots of self.

OUTPUT:

An iterable of nodes.

Note

This should be overloaded in applications.

EXAMPLES:

sage: from sage.parallel.map_reduce import RESetMapReduce
sage: S = RESetMapReduce(42)
sage: S.roots()
42

run(max_proc=None, reduce_locally=True, timeout=None, profile=None)

Run the computations.

INPUT:

• max_proc – (integer, default: None) if given, the maximum number of worker processors to use. The actual number is also bounded by the value of the environment variable SAGE_NUM_THREADS (the number of cores by default).

• reduce_locally – See RESetMapReduceWorker (default: True)

• timeout – a timeout on the computation (default: None)

• profile – directory/filename prefix for profiling, or None for no profiling (default: None)

OUTPUT:

The result of the map/reduce computation or an exception AbortError if the computation was interrupted or timeout.

EXAMPLES:

sage: from sage.parallel.map_reduce import RESetMPExample
sage: EX = RESetMPExample(maxl = 8)
sage: EX.run()
40320*x^8 + 5040*x^7 + 720*x^6 + 120*x^5 + 24*x^4 + 6*x^3 + 2*x^2 + x + 1


Here is an example or how to deal with timeout:

sage: from sage.parallel.map_reduce import AbortError
sage: EX = RESetMPExample(maxl = 100)
sage: try:
....:     res = EX.run(timeout=float(0.01))
....: except AbortError:
....:     print("Computation timeout")
....: else:
....:     print("Computation normally finished")
....:     res
Computation timeout


The following should not timeout even on a very slow machine:

sage: from sage.parallel.map_reduce import AbortError
sage: EX = RESetMPExample(maxl = 8)
sage: try:
....:     res = EX.run(timeout=60)
....: except AbortError:
....:     print("Computation Timeout")
....: else:
....:     print("Computation normally finished")
....:     res
Computation normally finished
40320*x^8 + 5040*x^7 + 720*x^6 + 120*x^5 + 24*x^4 + 6*x^3 + 2*x^2 + x + 1

run_serial()

Run the computation serially (mostly for tests).

EXAMPLES:

sage: from sage.parallel.map_reduce import RESetMPExample
sage: EX = RESetMPExample(maxl = 4)
sage: EX.run_serial()
24*x^4 + 6*x^3 + 2*x^2 + x + 1

setup_workers(max_proc=None, reduce_locally=True)

Setup the communication channels.

INPUT:

• max_proc – (integer) an upper bound on the number of worker processes.

• reduce_locally – whether the workers should reduce locally their work or sends results to the master as soon as possible. See RESetMapReduceWorker for details.

start_workers()

Launch the workers.

The workers should have been created using setup_workers().

class sage.parallel.map_reduce.RESetMapReduceWorker(mapred, iproc, reduce_locally)

Bases: multiprocessing.context.ForkProcess

Worker for generate-map-reduce.

This shouldn’t be called directly, but instead created by RESetMapReduce.setup_workers().

INPUT:

• mapred – the instance of RESetMapReduce for which this process is working.

• iproc – the id of this worker.

• reduce_locally – when reducing the results. Three possible values are supported:

• True – means the reducing work is done all locally, the result is only sent back at the end of the work. This ensure the lowest level of communication.

• False – results are sent back after each finished branches, when the process is asking for more work.

run()

The main function executed by the worker.

Calls run_myself() after possibly setting up parallel profiling.

EXAMPLES:

sage: from sage.parallel.map_reduce import RESetMPExample, RESetMapReduceWorker
sage: EX = RESetMPExample(maxl=6)
sage: EX.setup_workers(1)

sage: w = EX._workers
sage: w._todo.append(EX.roots())

sage: w.run()
sage: sleep(1)
sage: w._todo.append(None)

sage: EX.get_results()
720*x^6 + 120*x^5 + 24*x^4 + 6*x^3 + 2*x^2 + x + 1


Cleanups:

sage: del EX._results, EX._active_tasks, EX._done, EX._workers

run_myself()

The main function executed by the worker.

EXAMPLES:

sage: from sage.parallel.map_reduce import RESetMPExample, RESetMapReduceWorker
sage: EX = RESetMPExample(maxl=6)
sage: EX.setup_workers(1)

sage: w = EX._workers
sage: w._todo.append(EX.roots())
sage: w.run_myself()

sage: sleep(1)
sage: w._todo.append(None)

sage: EX.get_results()
720*x^6 + 120*x^5 + 24*x^4 + 6*x^3 + 2*x^2 + x + 1


Cleanups:

sage: del EX._results, EX._active_tasks, EX._done, EX._workers

send_partial_result()

Send results to the MapReduce process.

Send the result stored in self._res to the master an reinitialize it to master.reduce_init.

EXAMPLES:

sage: from sage.parallel.map_reduce import RESetMPExample, RESetMapReduceWorker
sage: EX = RESetMPExample(maxl=4)
sage: EX.setup_workers(1)
sage: w = EX._workers
sage: w._res = 4
sage: w.send_partial_result()
sage: w._res
0
sage: EX._results.get()
4

steal()

Steal some node from another worker.

OUTPUT:

A node stolen from another worker chosen at random.

EXAMPLES:

sage: from sage.parallel.map_reduce import RESetMPExample, RESetMapReduceWorker
sage: EX = RESetMPExample(maxl=6)
sage: EX.setup_workers(2)

sage: w0, w1 = EX._workers
sage: w0._todo.append(42)
sage: thief0 = Thread(target = w0._thief, name="Thief")
sage: thief0.start()  # known bug (Trac #27537)

sage: w1.steal()  # known bug (Trac #27537)
42
sage: w0._todo  # known bug (Trac #27537)
deque([])

walk_branch_locally(node)

Work locally.

Performs the map/reduce computation on the subtrees rooted at node.

INPUT:

• node – the root of the subtree explored.

OUTPUT:

Nothing, the result are stored in self._res.

This is where the actual work is performed.

EXAMPLES:

sage: from sage.parallel.map_reduce import RESetMPExample, RESetMapReduceWorker
sage: EX = RESetMPExample(maxl=4)
sage: w = RESetMapReduceWorker(EX, 0, True)
sage: def sync(): pass
sage: w.synchronize = sync
sage: w._res = 0

sage: w.walk_branch_locally([])
sage: w._res
x^4 + x^3 + x^2 + x + 1

sage: w.walk_branch_locally(w._todo.pop())
sage: w._res
2*x^4 + x^3 + x^2 + x + 1

sage: while True: w.walk_branch_locally(w._todo.pop())
Traceback (most recent call last):
...
IndexError: pop from an empty deque
sage: w._res
24*x^4 + 6*x^3 + 2*x^2 + x + 1

class sage.parallel.map_reduce.RESetParallelIterator(roots=None, children=None, post_process=None, map_function=None, reduce_function=None, reduce_init=None, forest=None)

A parallel iterator for recursively enumerated sets.

This demonstrates how to use RESetMapReduce to get an iterator on a recursively enumerated set for which the computations are done in parallel.

EXAMPLES:

sage: from sage.parallel.map_reduce import RESetParallelIterator
sage: S = RESetParallelIterator([[]],
....:     lambda l: [l + , l + ] if len(l) < 15 else [])
sage: sum(1 for _ in S)
65535

map_function(z)

Return a singleton tuple.

INPUT:

• z – a node

OUTPUT:

The singleton (z, ).

EXAMPLES:

sage: from sage.parallel.map_reduce import RESetParallelIterator
sage: S = RESetParallelIterator( [[]],
....:     lambda l: [l + , l + ] if len(l) < 15 else [])
sage: S.map_function([1, 0])
([1, 0],)

reduce_init

alias of builtins.tuple

sage.parallel.map_reduce.proc_number(max_proc=None)

Return the number of processes to use.

INPUT:

• max_proc – an upper bound on the number of processes or None.

EXAMPLES:

sage: from sage.parallel.map_reduce import proc_number
sage: proc_number()  # random
8
sage: proc_number(max_proc=1)
1
sage: proc_number(max_proc=2) in (1, 2)
True