Fourth-Year Design Project, Part 4: The Overlapping Guided Improvement Algorithm
In the previous post, I finished discussing all the relevant background material for our group’s design project. I also described two approaches we were exploring: incremental and checkpointed solving.
However, these approaches are single-threaded and do not take advantage of multi-core processors. Therefore, our group is also interested in multi-threaded^{1} approaches. In this post, I will be discussing the overlapping guided improvement algorithm, which is the first of our two ideas. I also want to describe what problems we ran into and fixed, and what new ideas we have.
If you need to refresh your memory on terminology and the guided improvement algorithm, please take some time to review the two previous posts. Again, I am also happy to answer any questions in the comments.
The overlapping guided improvement algorithm
The idea behind the overlapping guided improvement algorithm (OGIA) is actually quite straightforward: we run multiple instances of GIA in parallel. The main program will start up multiple threads, and each thread will follow the old algorithm of finding a starting point, climbing up to the Pareto front, and then repeating.
In the best case, all the threads do useful work, finding unique solutions, and we speed up the algorithm immensely. In the worst case, only one thread does useful work and the other threads find only duplicate solutions. This wasted work is acceptable for our purposes, since we want to reduce the total time between the start and finish of the program. Since the wasted work is done in parallel, OGIA is no worse^{2} than GIA.
Implementation details
As described above, the idea behind OGIA is rather straightforward. However, there are many implementation details to consider.
For example, we need a way to identify duplicate solutions so that we never yield the same solution twice. We also want to be smart about our magnifier task^{3} —– if we can easily identify duplicate solutions, then there is no point having multiple threads run the magnifier task on the same Pareto point. To do this, we have a solution deduplicator^{4} that keeps track of all the Pareto points found, and also maintains the global exclusion constraints^{5}.
When a thread finds a Pareto point, it reports it to the solution deduplicator. If the Pareto point is a new solution, the thread adds a magnifier task for that Pareto point to the task queue^{6}. Next, whether we have a duplicate or not, the thread asks the solution deduplicator for the updated global exclusion constraints to find a new starting solution.
Thus, the solution deduplicator is responsible for keeping the set of unique Pareto points found, and informing threads if the solution they found was a duplicate or not. Since a magnifier task is queued only once per Pareto point, we can be sure that the magnifier tasks will not waste any work.
Below, I’ve listed some pseudocode for the solution deduplicator. Our implementation is thread-safe, but I have omitted those details from the pseudocode.
SolutionDeduplicator:
globalExclusionConstraints = empty
solutionHashTable = empty
# Called when a thread has found a new solution.
PushNewSolution(solution)
if solutionHashTable.contains(solution)
# We have a duplicate solution
return false
else
# Not a duplicate, so add the solution.
solutionHashTable.add(solution)
# Update global exclusion constraints.
globalExclusionConstraints = globalExclusionConstraints AND
not dominated by solution
return true
end
end
Here is how we have modified the double-nested loop from GIA, and packaged it as a solution finder task for the OGIA threads.
SolutionFinder(solution):
while solution exists
# Climb up to the Pareto front.
while solution exists
prevSolution = solution
# Find a better solution.
solution = Solve(problemConstraints AND dominates prevSolution)
end
# Nothing dominates prevSolution; it's a Pareto point.
# Report to the solution deduplicator.
isUniqueSolution = PushNewSolution(prevSolution)
# Queue magnifier task if solution was unique.
if isUniqueSolution
tasks.queue(Magnifier(solution))
end
# Throw the dart. Find a new solution.
solution = Solve(problemConstraints AND globalExclusionConstraints)
end
This was the idea for our first implementation. However, we discovered a serious problem: all the threads would start at the same starting solution, and then find the same better solutions. Apart from the magnifier tasks, only one thread was doing useful work. There was almost no improvement over the single-threaded GIA.
While the idea to fix this problem was straightforward —– have each thread find a different starting solution —– the actual implementation was less so. Originally, we had hoped that the non-determinism in our SAT solver would find different starting solutions, but this was not the case. We were also unsuccessful in forcing this behaviour with different random seeds. In the end, our only option was to find starting solutions in sequence, allowing us to use exclusion constraints from the previous solutions. Unfortunately, this means we have some unavoidable sequential work at the beginning of the algorithm.
Here is the initialization code from the main thread, which is responsible for finding the starting solutions and starting the other worker threads.
Main:
tasks = empty
intialPointConstraint = problemConstraints
for 1..numberOfThreadsToUse
# Get a unique starting point.
solution = Solve(initialPointConstraint)
if solution exists
initialPointConstraint = initialPointConstraint AND
not dominated by solution
# Queue a solution finder task.
tasks.queue(SolutionFinder(solution))
end
end
# Wait until all threads have finished.
tasks.wait
For the full implementation details, you can find our code on GitHub.
Preliminary results
Here is the table from last time, with OGIA results added, and GIA results removed, since our new work is built on top of IGIA. Recall that IGIA uses incremental solving, while CGIA uses checkpointed solving. Again, these are informal test results, run on the undergraduate computer science servers.
IGIA | CGIA | OGIA | |
---|---|---|---|
9 queens, 5 metrics | 2 hours, 0 min | 51 min | 50 min |
9 queens, 6 metrics | 4 days, 16 hours, 43 min | 1 hour, 39 min | 1 hour, 53 min |
9 queens, 7 metrics | Never attempted^{7} | 3 hours, 37 min | 4 hours, 4 min |
Search and rescue, 5 metrics | 3 hours, 0 min | 1 hour, 26 min | 2 hours, 38 min |
Search and rescue, 6 metrics | 2 hours, 8 min | 59 min | 1 hour, 11 min |
Search and rescue, 7 metrics | 5 hours, 1 min | 2 hours, 16 min | 2 hours, 46 min |
Once again, we see a dramatic improvement over IGIA. Interestingly, CGIA, a single-threaded approach, performs better than OGIA, a multi-threaded approach. However, in the same way we built OGIA on top of IGIA, we can build OGIA on top of CGIA.
Future work
At the time of writing, we have four open issues to explore.
As noted above, we can build OGIA on top of CGIA. This would seem to be a very easy way of getting significant improvements, but the situation is actually more complicated. Both CGIA and OGIA greatly increase the algorithm’s memory usage. We would probably have to reduce memory usage, before we could combine and use both of them effectively.
The next issue is how we queue up magnifier tasks. Due to our implementation, the magnifier tasks do not run until all the (unique) Pareto points have been found^{8}. This is not a serious problem, but a nice property of GIA is that it yields solutions as they are found. With the magnifier tasks deferred, we lose this property, and do not yield solutions until the first magnifier task runs. We should be able to fix this easily, by prioritizing magnifier tasks over the solution finder tasks.
Another minor issue is that the algorithm waits until every thread has finished, meaning we are as fast as the slowest thread. However, we can do better than this. If a solution finder task fails to find a new starting solution, then this means all Pareto points have been found. The other solution finder tasks are either trying to find a new starting solution (in which case, they will fail like the first task), or they are climbing up to a Pareto point (which will be a duplicate, since no new Pareto points exist). Therefore, we can cancel the other solution finder tasks.
The final issue is much more open-ended. Our group has noticed that, even with our dramatic improvements, there is still a lot of wasted work. Reducing this wasted work could improve OGIA even more.
One idea we want to explore is how we handle the global exclusion constraints. Currently, these constraints are based on the set of (known) Pareto points. We suspect that since it can take a long time before Pareto points are found, the global exclusion constraints are not updated frequently enough. This leads to other tasks using “stale” information and performing duplicate work. If the global exclusion constraints were based on non-optimal solutions, they would be updated more frequently, and other threads would have “fresher” information.
Conclusion
In this post, I described the overlapping guided improvement algorithm, highlighted some of its implementation details, and discussed some of the open issues our group will be addressing in the future.
In the next part, I will be discussing our other multi-threaded approach, the partitioned guided improvement algorithm. It lets us eliminate all the duplicate work we had with OGIA, but the price is a far more complicated algorithm.
I would like to thank Chris Kleynhans, Zameer Manji, and Arjun Sondhi for proofreading this post.
Notes
^ Loosely put, a thread is a portion of a program that can be executed concurrently with other portions of the program. On a multi-core processor, each core can execute a separate thread simultaneously. In contrast, a single-core processor must switch between the different threads. With a multi-threaded approach, we can run portions of the program in parallel.
^ There will be some overhead when we check for duplicates. However, this is a very cheap operation.
^ Recall that multiple solutions may exist at the same Pareto point. The magnifier task examines a Pareto point and yields all the solutions at that point.
^ The solution deduplicator is backed by a hash table, which means adding solutions and checking for duplicates is extremely cheap.
^ Recall that in GIA, when a Pareto point is found, the exclusion constraints are updated so that new starting solutions are not dominated by the existing Pareto points. We do the same thing here, except that the exclusion constraints are based on all Pareto points found, not just the ones found by the current thread.
^ When a thread is finished its task, it will check the queue for a new task. As a result, magnifier tasks are deferred until all Pareto points have been found.
^ IGIA had very little improvement over GIA for the 9-queens problems, so we never attempted this case.
^ Recall the argument for GIA: the starting solution must not be dominated by any existing Pareto point. If we cannot find such a solution, then all solutions are either Pareto points or dominated by existing Pareto points.