Background:
I need to process some hundred thousand events (producing results) given a hard time limit. The clock is literally ticking, and when the timer fires, whatever is done at that point must be flushed out.
What isn't ready by that time is either discarded (depending on an importance metric) or processed during the next time quantum (with an "importance boost", i.e. adding a constant to the importance metric).
Now ideally, the CPU is much faster than needed, and the whole set is ready a long time before the end of the time slice. Unluckily, the world is rarely ever ideal, and "hundred thousands" becomes "tens of millions" before you know.
Events are added to the back of a queue (which is really a vector) as they come in, and are processed from the front during the respective next quantum (so the program always processes the last quantum's input).
However, not all events are equally important. In case the available time is not sufficient, it would be preferrable to drop unimportant events rather than important ones (this is not a strict requirement, since important events will be copied to the next time quantum's queue, but doing so further adds to the load so it isn't a perfect solution).
The obvious thing to use would be, of course, a priority queue / heap. Unluckily, heapifying 100k elements isn't precisely a free operation either (or parallel), and then I end up with objects being in some non-obvious and not necessarily cache-friendly memory locations, and pulling elements from a priority queue doesn't parallelize nicely.
What I would really like is somewhat like a vector that is sorted or at least "somewhat approximately sorted", which one can traverse sequentially afterwards. This would trivially allow me to create e.g. 12 threads (or any other number, one per CPU) that process e.g. 1/64 of the range (or another size) each, slowly advancing from the front to the end, and eventually dropping/postponing what's left over -- which will be events of little importantance that can be discarded.
Simply sorting the complete range using std::sort would be the easiest, most straightforward solution. However, the time it takes to sort items reduces the time available to actually process elements within the fixed time budget, and sorting time is for the most part single-CPU time (and parallel sort isn't that great either).
Also, doing a perfect sort (which isn't really needed) may bring forth worst case complexity whereas an approximate sort should ideally perform at its optimum and have a very predictable cost.
tl;dr
So, what I'm looking for is a way to sort an array/vector only approximately, but fast, and with a predictable (or guaranteed) runtime.
The sort key would be a small integer typically between 10 and 1000. Being postponed to the next time quantum might increase ("priority boost") that value by a small amount, e.g. 100 or 200.
In a different question where humans are supposed to do an approximate sort using "subjective compare"(?) shell sort was proposed. On various sorting demo applets, it seems like at least for the "random shuffle" input that's typical in these, shell sort can indeed do an "approximate sort" that doesn't look too bad with 3-4 passes over the data (and at least the read-tap is strictly sequential). Unluckily it seems to be somewhat of a black art to choose gap values that work well, and runtime estimates seem to involve a lot of looking into the crystal ball as well.
Comb sort with a relatively large shrink factor (such as 2 or 3?) seems tempting as well, since it visits memory strictly sequentially (on both taps) and is able to move far out elements by a great distance quickly. Again, judging from sorting demo applets, it seems like 3-4 passes already give a rather reasonable "approximate sort".
MSD radix sort comes to mind, though I am not sure how it would perform given typical 16/32bit integers in which most of the most significant bits are all zero! One would probably have to do an initial pass to find the most significant bit in the whole set, followed by 2-3 actual sort passes?
Is there a better algorithm or a well-known working approach with one of the algorithms I mentioned?
What comes to mind is to iterate over the vector and if some event is less important, don't process it but put it aside. As soon as the entire vector has been read, have a look at the events put aside. Of course you can use several buckets with different priorities. And only store references there, you don't want to move megabytes of data. (posted as an answer now as requested by Damon)
Use a separate vector for each priority. Then you don't need to sort them.
Sounds like a nice example where near-sort algorithms can be useful.
Back a decade Chazelle has developed a nice data-structure that somewhat works like a heap. The key difference is the time complexity though. It has constant time for all important operations, e.g. insert, remove, find lowest element etc.
The trick of this data-structure is, that it breaks the O(n*log n) complexity barrier by allowing for some error in the sort order.
To me that sounds pretty much what you need. The data-structure is called soft heap and explained on wikipedia:
https://en.wikipedia.org/wiki/Soft_heap
There are other algorithms that allow for some error in favor to speed as well. You'll find them if you google for Near Sort Algorithms
If you try that algorithm please give some feedback how it works in practice. I'm really eager to hear from you how the algorithm performs in practice.
Sounds like you want to use std::partition: move the part that interests you to the front, and the others to the back. Its complexity is in the order of O(n), but it is cache-friendly, so it's probably a lot faster than sorting.
If you have limited "bandwidth" in processing events (say a 128K per time quantum), you could use std::nth_element to select the 128K (minus some percentage lost due to making that computation) most promising events (assuming you have an operator< that compares priorities) in O(N) time. Then you process those in parallel, and when you are done, you reprioritize the remainder (again in O(N) time).
std::vector<Event> events;
auto const guaranteed_bandwidth = 1<<17; // 128K is always possible to process
if (events.size() <= guaranteed_bandwidth) {
// let all N workers loose on [begin(events), end(events)) range
} else {
auto nth = guaranteed_bandwidth * loss_from_nth_element;
std::nth_element(begin(events), begin(events) + nth);
// let all N workers loose on [begin(events), nth) range
// reprioritize [nth, end(events)) range and append to events for next time quantum
}
This guarantees that in the case that your bandwith threshold is reached, you process the most valuable elements first. You could even speed up the nth_element by a poor man's parallelization (e.g. let each of N workers compute M*128K/N best elements for small M in parallel, and then do a final merge and another nth_element on the M*128K elements).
The only weakness is that in case your system is really overloaded (billions of events, maybe due to some DOS attack) it could take more than the entire quantum to run nth_element (even when quasi-parallized) and you actually process nothing. But if the processing time per event is much larger (say a few 1,000 cycles) than a priority comparison (say a dozen cycles), this should not happen under regular loads.
NOTE: for performance reasons, it's of course better to sort pointers/indices into the main event vector, this is not shown for brevity.
If you have N worker threads, give each worker thread 1/Nth of the original unsorted array. The first thing the worker will do is your approximate fast sorting algorithm of preference on it's individual piece of the array. Then, they can each process their array peice in order - roughly performing higher priority items first, and also being very cache friendly. This way, you don't take a hit for trying to sort the entire array, or even trying to approximately sort the entire array; and what little sorting there is, is entirely parallelized. Sorting 10 pieces individually is much cheaper than sorting the whole thing.
This would work best if the priorities of items to process are randomly distributed. If there is some ordering to them you'll wind up with a thread being flooded by or starved of high priority items to process.
Related
Does anyone have any tips for efficiently parallelizing std::partition using TBB? Has this been done already?
Here is what I'm thinking:
if the array is small, std::partition it (serial) and return
else, treat the array as 2 interleaved arrays using custom iterators (interleave in cache-sized blocks)
start a parallel partition task for each pair of iterators (recurse to step 1)
swap elements between the two partition/middle pointers*
return the merged partition/middle pointer
*I am hoping in the average case this region will be small compared to the length of the array or compared to the swaps required if partitioning the array in contiguous chunks.
Any thoughts before I try it?
I'd treat it as a degenerate case of parallel sample sort. (Parallel code for sample sort can be found here.) Let N be the number of items. The degenerate sample sort will require Θ(N) temporary space, has Θ(N) work, and Θ(P+ lg N) span (critical path). The last two values are important for analysis, since speedup is limited to work/span.
I'm assuming the input is a random-access sequence. The steps are:
Allocate a temporary array big enough to hold a copy of the input sequence.
Divide the input into K blocks. K is a tuning parameter. For a system with P hardware threads, K=max(4*P,L) might be good, where L is a constant for avoiding ridiculously small blocks. The "4*P" allows some load balancing.
Move each block to its corresponding position in the temporary array and partition it using std::partition. Blocks can be processed in parallel. Remember the offset of the "middle" for each block. You might want to consider writing a custom routine that both moves (in the C++11 sense) and partitions a block.
Compute the offset to where each part of a block should go in the final result. The offsets for the first part of each block can be done using an exclusive prefix sum over the offsets of the middles from step 3. The offsets for the second part of each block can be computed similarly by using the offset of each middle relative to the end of its block. The running sums in the latter case become offsets from the end of the final output sequence. Unless you're dealing with more than 100 hardware threads, I recommend using a serial exclusive scan.
Move the two parts of each block from the temporary array back to the appropriate places in the original sequence. Copying each block can be done in parallel.
There is a way to embed the scan of step 4 into steps 3 and 5, so that the span can be reduced to Θ(lg N), but I doubt it's worth the additional complexity.
If using tbb::parallel_for loops to parallelize steps 3 and 5, consider using affinity_partitioner to help threads in step 5 pick up what they left in cache from step 3.
Note that partitioning requires only Θ(N) work for Θ(N) memory loads and stores. Memory bandwidth could easily become the limiting resource for speedup.
Why not to parallel something similar to std::partition_copy instead? The reasons are:
for std::partition, in-place swaps as in Adam's solution require logarithmic complexity due to recursive merge of the results.
you'll pay memory for parallelism anyway when using the threads and tasks.
if the objects are heavy, it is more reasonable to swap (shared) pointers anyway
if the results can be stored concurrently then threads can work independently.
It's pretty straight-forward to apply a parallel_for (for random-access iterators) or tbb::parallel_for_each (for non-random-access iterators) to start processing the input range. each task can store the 'true' and 'false' results independently. There are lots of ways to store the results, some from the top of my head:
using tbb::parallel_reduce (only for random-access iterators), store the results locally to the task body and move-append them in join() from another task
use tbb::concurrent_vector's method grow_by() to copy local results in a bunch or just push() each result separately on arrival.
cache thread-local results in tbb::combinable TLS container and combine them later
The exact semantics of std::partition_copy can be achieved by copy from the temporary storage from above or
(only for random-access output iterators) use atomic<size_t> cursors to synchronize where to store the results (assuming there is enough space)
Your approach should be correct, but why not follow the regular divide-and-conquer (or parallel_for) method? For two threads:
split the array in two. Turn your [start, end) into [start, middle), [middle, end).
run std::partition on both ranges in parallel.
merge the partitioned results. This can be done with a parallel_for.
This should make better use of the cache.
It seems to me like this should parallelize nicely, any thoughts before I try it?
Well... maybe a few:
There's no real reason to create more tasks than you have cores. Since your algorithm is recursive, you also need to keep track not to create additional threads, after you reach your limit, cause it'll just be a needless effort.
Keep in mind that splitting and merging the arrays costs you processing power, so set the split size in a way, which won't actually slow your calculations down. Splitting a 10-element array can be tempting, but wont get you where you want to be. Since the complexity of std::partition is linear, it's fairly easy to overestimate the speed of the task.
Since you asked and gave an algorithm, I hope you actually need parallelization here. If so - there's nothing much to add, the algorithm itself looks really fine :)
I need to sort object that arrive to a process X before treating them.
Object are sorted given to a time-stamp - a 64 bit number.
When object are timed out(a few milli) and sorted, the process X start to look at them.
Most of the time the object arrive sorted, between 3% and 5% the objects are out of order.
So i need a structure that allow me to:
- insert quickly elements
- remove quickly timed out elements
What should be the best to sort them, in term of performance?
I started to implement with boost::circular_buffer.
If boost::heap is better for this, which boost::heap should i use? Because there are a few(fibonacci, binomial, priority queue...)
I am using boost 1_49 but i can use a newer version also.
With circular buffer i insert most of the elements at the beginning of the buffer. But it can be O(n) in some case.
But to take timed out element it is O(1)
You will need to measure but I would guess that a std::priority_queue<T> has a fair chance to be most efficient. Using any of the other heaps won't do you much good because you don't really need the additional operations (changing the priority of an element) but being able to use these operations increases the overhead quite significantly.
Especially when the size of T is a bit bigger but even if it is just an int, you may want to use a d-heap with d == 8: although this results in more comparisons it reduces the number of times an object is moved.
I do not have formal CS training, so bear with me.
I need to do a simulation, which can abstracted away to the following (omitting the details):
We have a list of real numbers representing the times of events. In
each step, we
remove the first event, and
as a result of "processing" it, a few other events may get inserted into the list at a strictly later time
and repeat this many times.
Questions
What data structure / algorithm can I use to implement this as efficiently as possible? I need to increase the number of events/numbers in the list significantly. The priority is to make this as fast as possible for a long list.
Since I'm doing this in C++, what data structures are already available in the STL or boost that will make it simple to implement this?
More details:
The number of events in the list is variable, but it's guaranteed to be between n and 2*n where n is some simulation parameter. While the event times are increasing, the time-difference of the latest and earliest events is also guaranteed to be less than a constant T. Finally, I suspect that the density of events in time, while not constant, also has an upper and lower bound (i.e. all the events will never be strongly clustered around a single point in time)
Efforts so far:
As the title of the question says, I was thinking of using a sorted list of numbers. If I use a linked list for constant time insertion, then I have trouble finding the position where to insert new events in a fast (sublinear) way.
Right now I am using an approximation where I divide time into buckets, and keep track of how many event are there in each bucket. Then process the buckets one-by-one as time "passes", always adding a new bucket at the end when removing one from the front, thus keeping the number of buckets constant. This is fast, but only an approximation.
A min-heap might suit your needs. There's an explanation here and I think STL provides the priority_queue for you.
Insertion time is O(log N), removal is O(log N)
It sounds like you need/want a priority queue. If memory serves, the priority queue adapter in the standard library is written to retrieve the largest items instead of the smallest, so you'll have to specify that it use std::greater for comparison.
Other than that, it provides just about exactly what you've asked for: the ability to quickly access/remove the smallest/largest item, and the ability to insert new items quickly. While it doesn't maintain all the items in order, it does maintain enough order that it can still find/remove the one smallest (or largest) item quickly.
I would start with a basic priority queue, and see if that's fast enough.
If not, then you can look at writing something custom.
http://en.wikipedia.org/wiki/Priority_queue
A binary tree is always sorted and has faster access times than a linear list. Search, insert and delete times are O(log(n)).
But it depends whether the items have to be sorted all the time, or only after the process is finished. In the latter case a hash table is probably faster. At the end of the process you then would copy the items to an array or a list and sort it.
In an optimization problem I keep in a queue a lot of candidate solutions which I examine according to their priority.
Each time I handle one candidate, it is removed form the queue but it produces several new candidates making the number of cadidates to grow exponentially. To handle this I assign a relevancy to each candidate, whenever a candidate is added to the queue, if there is no more space avaliable, I replace (if appropiate) the least relevant candidate currently in the queue with the new one.
In order to do this efficiently I keep a large (fixed size) array with the candidates and two linked indirect binary heaps: one handles the candidates in decreasing priority order, and the other in ascending relevancy.
This is efficient enough for my purposes and the supplementary space needed is about 4 ints/candidate which is also reasonable. However it is complicated to code, and it doesn't seem optimal.
My question is if you know of a more adequate data structure or of a more natural way to perform this task without losing efficiency.
Here's an efficient solution that doesn't change the time or space complexity over a normal heap:
In a min-heap, every node is less than both its children. In a max-heap, every node is greater than its children. Let's alternate between a min and max property for each level making it: every odd row is less than its children and its grandchildren, and the inverse for even rows. Then finding the smallest node is the same as usual, and finding the largest node requires that we look at the children of the root and take the largest. Bubbling nodes (for insertion) becomes a bit tricker, but it's still the same O(logN) complexity.
Keeping track of capacity and popping the smallest (least relevant) node is the easy part.
EDIT: This appears to be a standard min-max heap! See here for a description. There's a C implementation: header, source and example. Here's an example graph:
(source: chonbuk.ac.kr)
"Optimal" is hard to judge (near impossible) without profiling.
Sometimes a 'dumb' algorithm can be the fastest because intel CPUs are incredibly fast at dumb array scans on contiguous blocks of memory especially if the loop and the data can fit on-chip. By contrast, jumping around following pointers in a larger block of memory that doesn't fit on-chip can be tens or hundreds or times slower.
You may also have the issues when you try to parallelize your code if the 'clever' data structure introduces locking thus preventing multiple threads from progressing simultaneously.
I'd recommend profiling both your current, the min-max approach and a simple array scan (no linked lists = less memory) to see which performs best. Odd as it may seem, I have seen 'clever' algorithms with linked lists beaten by simple array scans in practice often because the simpler approach uses less memory, has a tighter loop and benefits more from CPU optimizations. You also potentially avoid memory allocations and garbage collection issues with a fixed size array holding the candidates.
One option you might want to consider whatever the solution is to prune less frequently and remove more elements each time. For example, removing 100 elements on each prune operation means you only need to prune 100th of the time. That may allow a more asymmetric approach to adding and removing elements.
But overall, just bear in mind that the computer-science approach to optimization isn't always the practical approach to the highest performance on today and tomorrow's hardware.
If you use skip-lists instead of heaps you'll have O(1) time for dequeuing elements while still doing searches in O(logn).
On the other hand a skip list is harder to implement and uses more space than a binary heap.
I need a fast container with only two operations. Inserting keys on from a very sparse domain (all 32bit integers, and approx. 100 are set at a given time), and iterating over the inserted keys. It should deal with a lot of insertions which hit the same entries (like, 500k, but only 100 different ones).
Currently, I'm using a std::set (only insert and the iterating interface), which is decent, but still not fast enough. std::unordered_set was twice as slow, same for the Google Hash Maps. I wonder what data structure is optimized for this case?
Depending on the distribution of the input, you might be able to get some improvement without changing the structure.
If you tend to get a lot of runs of a single value, then you can probably speed up insertions by keeping a record of the last value you inserted, and don't bother doing the insertion if it matches. It costs an extra comparison per input, but saves a lookup for each element in a run beyond the first. So it could improve things no matter what data structure you're using, depending on the frequency of repeats and the relative cost of comparison vs insertion.
If you don't get runs, but you tend to find that values aren't evenly distributed, then a splay tree makes accessing the most commonly-used elements cheaper. It works by creating a deliberately-unbalanced tree with the frequent elements near the top, like a Huffman code.
I'm not sure I understand "a lot of insertions which hit the same entries". Do you mean that there are only 100 values which are ever members, but 500k mostly-duplicate operations which insert one of those 100 values?
If so, then I'd guess that the fastest container would be to generate a collision-free hash over those 100 values, then maintain an array (or vector) of flags (int or bit, according to what works out fastest on your architecture).
I leave generating the hash as an exercise for the reader, since it's something that I'm aware exists as a technique, but I've never looked into it myself. The point is to get a fast hash over as small a range as possible, such that for each n, m in your 100 values, hash(n) != hash(m).
So insertion looks like array[hash(value)] = 1;, deletion looks like array[hash(value)] = 0; (although you don't need that), and to enumerate you run over the array, and for each set value at index n, inverse_hash(n) is in your collection. For a small range you can easily maintain a lookup table to perform the inverse hash, or instead of scanning the whole array looking for set flags, you can run over the 100 potentially-in values checking each in turn.
Sorry if I've misunderstood the situation and this is useless to you. And to be honest, it's not very much faster than a regular hashtable, since realistically for 100 values you can easily size the table such that there will be few or no collisions, without using so much memory as to blow your caches.
For an in-use set expected to be this small, a non-bucketed hash table might be OK. If you can live with an occasional expansion operation, grow it in powers of 2 if it gets more than 70% full. Cuckoo hashing has been discussed on Stackoverflow before and might also be a good approach for a set this small. If you really need to optimise for speed, you can implement the hashing function and lookup in assembler - on linear data structures this will be very simple so the coding and maintenance effort for an assembler implementation shouldn't be unduly hard to maintain.
You might want to consider implementing a HashTree using a base 10 hash function at each level instead of a binary hash function. You could either make it non-bucketed, in which case your performance would be deterministic (log10) or adjust your bucket size based on your expected distribution so that you only have a couple of keys/bucket.
A randomized data structure might be perfect for your job. Take a look at the skip list – though I don't know any decend C++ implementation of it. I intended to submit one to Boost but never got around to do it.
Maybe a set with a b-tree (instead of binary tree) as internal data structure. I found this article on codeproject which implements this.
Note that while inserting into a hash table is fast, iterating over it isn't particularly fast, since you need to iterate over the entire array.
Which operation is slow for you? Do you do more insertions or more iteration?
How much memory do you have? 32-bits take "only" 4GB/8 bytes, which comes to 512MB, not much for a high-end server. That would make your insertions O(1). But that could make the iteration slow. Although skipping all words with only zeroes would optimize away most iterations. If your 100 numbers are in a relatively small range, you can optimize even further by keeping the minimum and maximum around.
I know this is just brute force, but sometimes brute force is good enough.
Since no one has explicitly mentioned it, have you thought about memory locality? A really great data structure with an algorithm for insertion that causes a page fault will do you no good. In fact a data structure with an insert that merely causes a cache miss would likely be really bad for perf.
Have you made sure a naive unordered set of elements packed in a fixed array with a simple swap to front when an insert collisides is too slow? Its a simple experiment that might show you have memory locality issues rather than algorithmic issues.