I'm looking for information on how to implement binary heaps efficiently. I feel like there should be a nice article somewhere about implementing heaps efficiently, but I haven't found one. In fact I've been unable to find any resources on the matter of efficient implementation beyond the basics like storing the heap in an array. I'm looking for techniques for making a fast binary heap beyond the ones I describe below.
I've already written a C++ implementation that is faster than Microsoft Visual C++'s and GCC's std::priority_queue or using std::make_heap, std::push_heap and std::pop_heap. The following are the techniques I've already got covered in my implementation. I only came up with the last 2 myself, though I doubt that those are new ideas:
(Edit: added section on memory optimization)
Start indexes at 1
Look at the Wikipedia implementation notes for binary heaps. If the heap's root is placed at index 0, then the formulas for parent, left-child and right-child of the node at index n are respectively (n-1)/2, 2n+1 and 2n+2. If you use a 1-based array then the formulas become the simpler n/2, 2n and 2n + 1. So parent and left-child are more efficient when using a 1-based array. If p points to a 0-based array and q = p - 1 then we can access p[0] as q[1] so there is no overhead in using a 1-based array.
Make pop/removal move element to bottom of heap before replacement with leaf
Pop on a heap is frequently described by replacing the top element by the left-most bottom leaf and then moving it down until the heap property is restored. This requires 2 comparisons per level that we go by, and we are likely to go far down the heap since we moved a leaf to the top of the heap. So we should expect a little less than 2 log n comparisons.
Instead, we can leave a hole in the heap where the top element was. Then we move that hole down the heap by iteratively moving the larger child up. This requires only 1 comparison per level that we go past. In this way the hole will become a leaf. At this point we can move the right-most bottom leaf into the position of the hole and move that value up until the heap property is restored. Since the value we moved was a leaf we don't expect it to move very far up the tree. So we should expect a little more than log n comparisons, which is better than before.
Support replace-top
Suppose you want to remove the max element and also insert a new element. Then you can do either of the removal/pop implementations described above, but instead of moving the right-most bottom leaf, you use the new value you wish to insert/push. (When most operations are of this kind I've found that a tournament tree is better than a heap, but otherwise the heap is slightly better.)
Make sizeof(T) a power of 2
The parent, left-child and right-child formulas work on indexes and they cannot be made to work directly on pointer values. So we are going to be working with indexes and that implies looking up values p[i] in an array p from an index i. If p is a T* and i is an integer, then
&(p[i]) == static_cast<char*>(p) + sizeof(T) * i
and the compiler has to perform this computation to get p[i]. sizeof(T) is a compile-time constant, and the multiplication can be done more efficiently if sizeof(T) is a power of two. My implementation got faster by adding 8 padding bytes to increase sizeof(T) from 24 to 32. Reduced efficiency of the cache probably means that this is not a win for sufficiently large data sets.
Pre-multiply indexes
This was a 23% performance increase on my data set. The only thing we ever do with an index other than finding parent, left-child and right-child is to look the index up in an array. So if we keep track of j = sizeof(T) * i instead of an index i, then we could do a lookup p[i] without the multiplication that is otherwise implicit in evaluating p[i] because
&(p[i]) == static_cast<char*>(p) + sizeof(T) * i == static_cast<char*>(p) + j
Then the left-child and right-child formulas for j-values become respectively 2*j and 2*j + sizeof(T). The parent formula is a little more tricky, and I haven't found a way to do it other than converting the j-value to an i-value and back like so:
parentOnJ(j) = parent(j/sizeof(T))*sizeof(T) == (j/(2*sizeof(T))*sizeof(T)
If sizeof(T) is a power of 2 then this will compile to 2 shifts. That is 1 operation more than the usual parent using indexes i. However we then save 1 operation on lookup. So the net effect is that finding the parent takes the same amount of time this way, while lookup of left-child and right-child becomes faster.
Memory optimization
The answers of TokenMacGuy and templatetypedef point out memory based optimizations that reduce cache misses. For very large data sets or infrequently used priority queues, parts of the queue can be swapped out to disk by the OS. In that case it is worth it to add a lot of overhead to make optimal use of the cache because swapping in from disk is very slow. My data easily fits in memory and is in continuous use, so no part of the queue will likely be swapped to disk. I suspect that this is the case for most uses of priority queues.
There are other priority queues that are designed to make better use of the CPU cache. For example a 4-heap should have fewer cache misses and the amount of extra overhead is not that much. LaMarca and Ladner report in 1996 that they get 75% performance improvement from going to aligned 4-heaps. However, Hendriks reports in 2010 that:
The improvements to the implicit heap suggested by LaMarca and Ladner [17] to improve data locality and reduce cache misses were also tested. We implemented a four-way heap, that indeed shows a slightly better consistency than the two-way heap for very skewed input data, but only for very large queue sizes. Very large queue sizes are better handled by the hierarchical heap.
Question
Are there more techniques than these?
An interesting paper/article on this topic considers the behavior of caching/paging on the overall layout of the heap; The idea being that it's vastly more costly to pay for a cache miss or page in than nearly any other part of a datastructure's implementation. The paper discusses a heap layout that addresses this.
You're Doing It Wrong
by Poul-Henning Kamp
As an elaboration on #TokenMacGuy's post, you might want to look into cache-oblivious data structures. The idea is to build data structures that, for arbitrary caching systems, minimize the number of cache misses. They're tricky, but they actually might be useful from your perspective since they perform well even when dealing with multi-layer cache systems (for example, registers / L1 / L2 / VM).
There's actually a paper detailing an optimal cache-oblivious priority queue that might be of interest. This data structure would have all sorts of advantages in terms of speed, since it would try to minimize the number of cache misses at every level.
On the first point: even having a "spare spot" for your array based implementation isn't a waste. Many operations need a temporary element anyway. Rather than initializing a new element each time, having a dedicated element at index [0] is handy.
Related
I am writing a binary heap over an array arr.
Every node except the leaf nodes have two children.
The root can be at arr[0] or arr[1].
The accepted answer at Why in a heap implemented by array the index 0 is left unused? says arr[1] is faster.
But one comment below that answer says most implementation put root at arr[0].
What are the benefits of putting the root at arr[0]?
I am the person who answered the question you linked.
Creating a binary heap that has the root at arr[1] in a language that has 0-based arrays is idiotic. Not because it wastes a trivial amount of space, but because it creates unnecessarily confusing code for no benefit.
Why is the code confusing? Because it breaks a fundamental rule that we as programmers have been working under for years: arrays start at 0. If you want an array that holds 100 items, you declare it that way:
int a[100];
Except for a binary heap. Because some idiot who converted the original binary heap code from Algol (whose arrays are 1-based) to C (0-based arrays) back in 1973 didn't have the brains to change the child and parent calculations, we've ended up with this one special case where to hold 100 items you have to allocate 101:
int a[101];
And when somebody called that person on the inconsistency, he came up with a specious performance argument.
Yes, there is an extra increment instruction in the code for computing the left child index, and an extra decrement instruction when computing a child's parent index. In the wider context of what a binary heap does, those two instructions will make no practical difference to the running time of any program that uses the heap. None. If the difference is even measurable, it will definitely be noisy. There are many other things happening on your computer that will have much larger effects on your program's performance.
If you're writing a program that requires a high performance priority queue, what the heck are you doing with a binary heap in the first place? If you're really going to store huge numbers of things in your priority queue, you probably should be using something like a Pairing heap, which will outperform binary heap, although at a higher memory cost.
The C++ STL priority_queue, the Java PriorityQueue, and python's heapq all use 0-based binary heaps. The people who wrote those packages understand performance considerations. If there was a significant performance gain to going with a 1-based binary heap, they would have done so. That they went with 0-based heaps should tell you that any performance gain from a 1-based heap is illusory.
See my blog post But that's the way we've always done it! for a more complete discussion.
It is common knowledge in programming that memory locality improves performance a lot due to cache hits. I recently found out about boost::flat_map which is a vector based implementation of a map. It doesn't seem to be nearly as popular as your typical map/unordered_map so I haven't been able to find any performance comparisons. How does it compare and what are the best use cases for it?
Thanks!
I have run a benchmark on different data structures very recently at my company so I feel I need to drop a word. It is very complicated to benchmark something correctly.
Benchmarking
On the web, we rarely find (if ever) a well-engineered benchmark. Until today I only found benchmarks that were done the journalist way (pretty quickly and sweeping dozens of variables under the carpet).
1) You need to consider cache warming
Most people running benchmarks are afraid of timer discrepancy, therefore they run their stuff thousands of times and take the whole time, they just are careful to take the same thousand of times for every operation, and then consider that comparable.
The truth is, in the real world it makes little sense, because your cache will not be warm, and your operation will likely be called just once. Therefore you need to benchmark using RDTSC, and time stuff calling them once only.
Intel has made a paper describing how to use RDTSC (using a cpuid instruction to flush the pipeline, and calling it at least 3 times at the beginning of the program to stabilize it).
2) RDTSC accuracy measure
I also recommend doing this:
u64 g_correctionFactor; // number of clocks to offset after each measurement to remove the overhead of the measurer itself.
u64 g_accuracy;
static u64 const errormeasure = ~((u64)0);
#ifdef _MSC_VER
#pragma intrinsic(__rdtsc)
inline u64 GetRDTSC()
{
int a[4];
__cpuid(a, 0x80000000); // flush OOO instruction pipeline
return __rdtsc();
}
inline void WarmupRDTSC()
{
int a[4];
__cpuid(a, 0x80000000); // warmup cpuid.
__cpuid(a, 0x80000000);
__cpuid(a, 0x80000000);
// measure the measurer overhead with the measurer (crazy he..)
u64 minDiff = LLONG_MAX;
u64 maxDiff = 0; // this is going to help calculate our PRECISION ERROR MARGIN
for (int i = 0; i < 80; ++i)
{
u64 tick1 = GetRDTSC();
u64 tick2 = GetRDTSC();
minDiff = std::min(minDiff, tick2 - tick1); // make many takes, take the smallest that ever come.
maxDiff = std::max(maxDiff, tick2 - tick1);
}
g_correctionFactor = minDiff;
printf("Correction factor %llu clocks\n", g_correctionFactor);
g_accuracy = maxDiff - minDiff;
printf("Measurement Accuracy (in clocks) : %llu\n", g_accuracy);
}
#endif
This is a discrepancy measurer, and it will take the minimum of all measured values, to avoid getting a -10**18 (64 bits first negatives values) from time to time.
Notice the use of intrinsics and not inline assembly. First inline assembly is rarely supported by compilers nowadays, but much worse of all, the compiler creates a full ordering barrier around inline assembly because it cannot static analyze the inside, so this is a problem to benchmark real-world stuff, especially when calling stuff just once. So an intrinsic is suited here because it doesn't break the compiler free-re-ordering of instructions.
3) parameters
The last problem is people usually test for too few variations of the scenario.
A container performance is affected by:
Allocator
size of the contained type
cost of implementation of the copy operation, assignment operation, move operation, construction operation, of the contained type.
number of elements in the container (size of the problem)
type has trivial 3.-operations
type is POD
Point 1 is important because containers do allocate from time to time, and it matters a lot if they allocate using the CRT "new" or some user-defined operation, like pool allocation or freelist or other...
(for people interested about pt 1, join the mystery thread on gamedev about system allocator performance impact)
Point 2 is because some containers (say A) will lose time copying stuff around, and the bigger the type the bigger the overhead. The problem is that when comparing to another container B, A may win over B for small types, and lose for larger types.
Point 3 is the same as point 2, except it multiplies the cost by some weighting factor.
Point 4 is a question of big O mixed with cache issues. Some bad-complexity containers can largely outperform low-complexity containers for a small number of types (like map vs. vector, because their cache locality is good, but map fragments the memory). And then at some crossing point, they will lose, because the contained overall size starts to "leak" to main memory and cause cache misses, that plus the fact that the asymptotic complexity can start to be felt.
Point 5 is about compilers being able to elide stuff that are empty or trivial at compile time. This can optimize greatly some operations because the containers are templated, therefore each type will have its own performance profile.
Point 6 same as point 5, PODs can benefit from the fact that copy construction is just a memcpy, and some containers can have a specific implementation for these cases, using partial template specializations, or SFINAE to select algorithms according to traits of T.
About the flat map
Apparently, the flat map is a sorted vector wrapper, like Loki AssocVector, but with some supplementary modernizations coming with C++11, exploiting move semantics to accelerate insert and delete of single elements.
This is still an ordered container. Most people usually don't need the ordering part, therefore the existence of unordered...
Have you considered that maybe you need a flat_unorderedmap? which would be something like google::sparse_map or something like that—an open address hash map.
The problem of open address hash maps is that at the time of rehash they have to copy everything around to the new extended flat land, whereas a standard unordered map just has to recreate the hash index, while the allocated data stays where it is. The disadvantage of course is that the memory is fragmented like hell.
The criterion of a rehash in an open address hash map is when the capacity exceeds the size of the bucket vector multiplied by the load factor.
A typical load factor is 0.8; therefore, you need to care about that, if you can pre-size your hash map before filling it, always pre-size to: intended_filling * (1/0.8) + epsilon this will give you a guarantee of never having to spuriously rehash and recopy everything during filling.
The advantage of closed address maps (std::unordered..) is that you don't have to care about those parameters.
But the boost::flat_map is an ordered vector; therefore, it will always have a log(N) asymptotic complexity, which is less good than the open address hash map (amortized constant time). You should consider that as well.
Benchmark results
This is a test involving different maps (with int key and __int64/somestruct as value) and std::vector.
tested types information:
typeid=__int64 . sizeof=8 . ispod=yes
typeid=struct MediumTypePod . sizeof=184 . ispod=yes
Insertion
EDIT:
My previous results included a bug: they actually tested ordered insertion, which exhibited a very fast behavior for the flat maps.
I left those results later down this page because they are interesting.
This is the correct test:
I have checked the implementation, there is no such thing as a deferred sort implemented in the flat maps here. Each insertion sorts on the fly, therefore this benchmark exhibits the asymptotic tendencies:
map: O(N * log(N))
hashmaps: O(N)
vector and flatmaps: O(N * N)
Warning: hereafter the 2 tests for std::map and both flat_maps are buggy and actually test ordered insertion (vs random insertion for other containers. yes it's confusing sorry):
We can see that ordered insertion, results in back pushing, and is extremely fast. However, from the non-charted results of my benchmark, I can also say that this is not near the absolute optimality for a back-insertion. At 10k elements, perfect back-insertion optimality is obtained on a pre-reserved vector. Which gives us 3Million cycles; we observe 4.8M here for the ordered insertion into the flat_map (therefore 160% of the optimal).
Analysis: remember this is 'random insert' for the vector, so the massive 1 billion cycles come from having to shift half (in average) the data upward (one element by one element) at each insertion.
Random search of 3 elements (clocks renormalized to 1)
in size = 100
in size = 10000
Iteration
over size 100 (only MediumPod type)
over size 10000 (only MediumPod type)
Final grain of salt
In the end, I wanted to come back on "Benchmarking §3 Pt1" (the system allocator). In a recent experiment, I am doing around the performance of an open address hash map I developed, I measured a performance gap of more than 3000% between Windows 7 and Windows 8 on some std::unordered_map use cases (discussed here).
This makes me want to warn the reader about the above results (they were made on Win7): your mileage may vary.
From the docs it seems this is analogous to Loki::AssocVector which I'm a fairly heavy user of. Since it's based on a vector it has the characteristics of a vector, that is to say:
Iterators gets invalidated whenever size grows beyond capacity.
When it grows beyond capacity it needs to reallocated and move objects over, ie insertion isn't guaranteed constant time except for the special case of inserting at end when capacity > size
Lookup is faster than std::map due to cache locality, a binary search which has the same performance characteristics as std::map otherwise
Uses less memory because it isn't a linked binary tree
It never shrinks unless you forcibly tell it to ( since that triggers reallocation )
The best use is when you know the number of elements in advance ( so you can reserve upfront ), or when insertion / removal is rare but lookup is frequent. Iterator invalidation makes it a bit cumbersome in some use cases so they're not interchangeable in terms of program correctness.
I was reading up on the two different ways of implementing a stack: linked list and dynamic arrays. The main advantage of a linked list over a dynamic array was that the linked list did not have to be resized while a dynamic array had to be resized if too many elements were inserted hence wasting alot of time and memory.
That got me wondering if this is true for C++ (as there is a vector class which automatically resizes whenever new elements are inserted)?
It's difficult to compare the two, because the patterns of their memory usage are quite different.
Vector resizing
A vector resizes itself dynamically as needed. It does that by allocating a new chunk of memory, moving (or copying) data from the old chunk to the new chunk, the releasing the old one. In a typical case, the new chunk is 1.5x the size of the old (contrary to popular belief, 2x seems to be quite unusual in practice). That means for a short time while reallocating, it needs memory equal to roughly 2.5x as much as the data you're actually storing. The rest of the time, the "chunk" that's in use is a minimum of 2/3rds full, and a maximum of completely full. If all sizes are equally likely, we can expect it to average about 5/6ths full. Looking at it from the other direction, we can expect about 1/6th, or about 17% of the space to be "wasted" at any given time.
When we do resize by a constant factor like that (rather than, for example, always adding a specific size of chunk, such as growing in 4Kb increments) we get what's called amortized constant time addition. In other words, as the array grows, resizing happens exponentially less often. The average number of times items in the array have been copied tends to a constant (usually around 3, but depends on the growth factor you use).
linked list allocations
Using a linked list, the situation is rather different. We never see resizing, so we don't see extra time or memory usage for some insertions. At the same time, we do see extra time and memory used essentially all the time. In particular, each node in the linked list needs to contain a pointer to the next node. Depending on the size of the data in the node compared to the size of a pointer, this can lead to significant overhead. For example, let's assume you need a stack of ints. In a typical case where an int is the same size as a pointer, that's going to mean 50% overhead -- all the time. It's increasingly common for a pointer to be larger than an int; twice the size is fairly common (64-bit pointer, 32-bit int). In such a case, you have ~67% overhead -- i.e., obviously enough, each node devoting twice as much space to the pointer as the data being stored.
Unfortunately, that's often just the tip of the iceberg. In a typical linked list, each node is dynamically allocated individually. At least if you're storing small data items (such as int) the memory allocated for a node may be (usually will be) even larger than the amount you actually request. So -- you ask for 12 bytes of memory to hold an int and a pointer -- but the chunk of memory you get is likely to be rounded up to 16 or 32 bytes instead. Now you're looking at overhead of at least 75% and quite possibly ~88%.
As far as speed goes, the situation is rather similar: allocating and freeing memory dynamically is often quite slow. The heap manager typically has blocks of free memory, and has to spend time searching through them to find the block that's most suited to the size you're asking for. Then it (typically) has to split that block into two pieces, one to satisfy your allocation, and another of the remaining memory it can use to satisfy other allocations. Likewise, when you free memory, it typically goes back to that same list of free blocks and checks whether there's an adjoining block of memory already free, so it can join the two back together.
Allocating and managing lots of blocks of memory is expensive.
cache usage
Finally, with recent processors we run into another important factor: cache usage. In the case of a vector, we have all the data right next to each other. Then, after the end of the part of the vector that's in use, we have some empty memory. This leads to excellent cache usage -- the data we're using gets cached; the data we're not using has little or no effect on the cache at all.
With a linked list, the pointers (and probable overhead in each node) are distributed throughout our list. I.e., each piece of data we care about has, right next to it, the overhead of the pointer, and the empty space allocated to the node that we're not using. In short, the effective size of the cache is reduced by about the same factor as the overall overhead of each node in the list -- i.e., we might easily see only 1/8th of the cache storing the date we care about, and 7/8ths devoted to storing pointers and/or pure garbage.
Summary
A linked list can work well when you have a relatively small number of nodes, each of which is individually quite large. If (as is more typical for a stack) you're dealing with a relatively large number of items, each of which is individually quite small, you're much less likely to see a savings in time or memory usage. Quite the contrary, for such cases, a linked list is much more likely to basically waste a great deal of both time and memory.
Yes, what you say is true for C++. For this reason, the default container inside std::stack, which is the standard stack class in C++, is neither a vector nor a linked list, but a double ended queue (a deque). This has nearly all the advantages of a vector, but it resizes much better.
Basically, an std::deque is a linked list of arrays of sorts internally. This way, when it needs to resize, it just adds another array.
First, the performance trade-offs between linked-lists and dynamic arrays are a lot more subtle than that.
The vector class in C++ is, by requirement, implemented as a "dynamic array", meaning that it must have an amortized-constant cost for inserting elements into it. How this is done is usually by increasing the "capacity" of the array in a geometric manner, that is, you double the capacity whenever you run out (or come close to running out). In the end, this means that a reallocation operation (allocating a new chunk of memory and copying the current content into it) is only going to happen on a few occasions. In practice, this means that the overhead for the reallocations only shows up on performance graphs as little spikes at logarithmic intervals. This is what it means to have "amortized-constant" cost, because once you neglect those little spikes, the cost of the insert operations is essentially constant (and trivial, in this case).
In a linked-list implementation, you don't have the overhead of reallocations, however, you do have the overhead of allocating each new element on freestore (dynamic memory). So, the overhead is a bit more regular (not spiked, which can be needed sometimes), but could be more significant than using a dynamic array, especially if the elements are rather inexpensive to copy (small in size, and simple object). In my opinion, linked-lists are only recommended for objects that are really expensive to copy (or move). But at the end of the day, this is something you need to test in any given situation.
Finally, it is important to point out that locality of reference is often the determining factor for any application that makes extensive use and traversal of the elements. When using a dynamic array, the elements are packed together in memory one after the other and doing an in-order traversal is very efficient as the CPU can preemptively cache the memory ahead of the reading / writing operations. In a vanilla linked-list implementation, the jumps from one element to the next generally involves a rather erratic jumps between wildly different memory locations, which effectively disables this "pre-fetching" behavior. So, unless the individual elements of the list are very big and operations on them are typically very long to execute, this lack of pre-fetching when using a linked-list will be the dominant performance problem.
As you can guess, I rarely use a linked-list (std::list), as the number of advantageous applications are few and far between. Very often, for large and expensive-to-copy objects, it is often preferable to simply use a vector of pointers (you get basically the same performance advantages (and disadvantages) as a linked list, but with less memory usage (for linking pointers) and you get random-access capabilities if you need it).
The main case that I can think of, where a linked-list wins over a dynamic array (or a segmented dynamic array like std::deque) is when you need to frequently insert elements in the middle (not at either ends). However, such situations usually arise when you are keeping a sorted (or ordered, in some way) set of elements, in which case, you would use a tree structure to store the elements (e.g., a binary search tree (BST)), not a linked-list. And often, such trees store their nodes (elements) using a semi-contiguous memory layout (e.g., a breadth-first layout) within a dynamic array or segmented dynamic array (e.g., a cache-oblivious dynamic array).
Yes, it's true for C++ or any other language. Dynamic array is a concept. The fact that C++ has vector doesn't change the theory. The vector in C++ actually does the resizing internally so this task isn't the developers' responsibility. The actual cost doesn't magically disappear when using vector, it's simply offloaded to the standard library implementation.
std::vector is implemented using a dynamic array, whereas std::list is implemented as a linked list. There are trade-offs for using both data structures. Pick the one that best suits your needs.
As you indicated, a dynamic array can take a larger amount of time adding an item if it gets full, as it has to expand itself. However, it is faster to access since all of its members are grouped together in memory. This tight grouping also usually makes it more cache-friendly.
Linked lists don't need to resize ever, but traversing them takes longer as the CPU must jump around in memory.
That got me wondering if this is true for c++ as there is a vector class which automatically resizes whenever new elements are inserted.
Yes, it still holds, because a vector resize is a potentially expensive operation. Internally, if the pre-allocated size for the vector is reached and you attempt to add new elements, a new allocation takes place and the old data is moved to the new memory location.
From the C++ documentation:
vector::push_back - Add element at the end
Adds a new element at the end of the vector, after its current last element. The content of val is copied (or moved) to the new element.
This effectively increases the container size by one, which causes an automatic reallocation of the allocated storage space if -and only if- the new vector size surpasses the current vector capacity.
http://channel9.msdn.com/Events/GoingNative/GoingNative-2012/Keynote-Bjarne-Stroustrup-Cpp11-Style
Skip to 44:40. You should prefer std::vector whenever possible over a std::list, as explained in the video, by Bjarne himself. Since std::vector stores all of it's elements next to each other, in memory, and because of that it will have the advantage of being cached in memory. And this is true for adding and removing elements from std::vector and also searching. He states that std::list is 50-100x slower than a std::vector.
If you really want a stack, you should really use std::stack instead of making your own.
I would like to know what the best practice for efficiently storing (and subsequently accessing) sets of multi-dimensional data arrays with variable length. The focus is on performance, but I also need to be able to handle changing the length of an individual data set during runtime without too much overhead.
Note: I know this is a somewhat lengthy question, but I have looked around quite a lot and could not find a solution or example which describes the problem at hand with sufficient accuracy.
Background
The context is a computational fluid dynamics (CFD) code that is based on the discontinuous Galerkin spectral element method (DGSEM) (cf. Kopriva (2009), Implementing Spectral Methods for Partial Differential Equations). For the sake of simplicity, let us assume a 2D data layout (it is in fact in three dimensions, but the extension from 2D to 3D should be straightforward).
I have a grid that consists of K square elements k (k = 0,...,K-1) that can be of different (physical) sizes. Within each grid element (or "cell") k, I have N_k^2 data points. N_k is the number of data points in each dimension, and can vary between different grid cells.
At each data point n_k,i (where i = 0,...,N_k^2-1) I have to store an array of solution values, which has the same length nVars in the whole domain (i.e. everywhere), and which does not change during runtime.
Dimensions and changes
The number of grid cells K is of O(10^5) to O(10^6) and can change during runtime.
The number of data points N_k in each grid cell is between 2 and 8 and can change during runtime (and may be different for different cells).
The number of variables nVars stored at each grid point is around 5 to 10 and cannot change during runtime (it is also the same for every grid cell).
Requirements
Performance is the key issue here. I need to be able to regularly iterate in an ordered fashion over all grid points of all cells in an efficient manner (i.e. without too many cache misses). Generally, K and N_k do not change very often during the simulation, so for example a large contiguous block of memory for all cells and data points could be an option.
However, I do need to be able to refine or coarsen the grid (i.e. delete cells and create new ones, the new ones may be appended to the end) during runtime. I also need to be able to change the approximation order N_k, so the number of data points I store for each cell can change during runtime as well.
Conclusion
Any input is appreciated. If you have experience yourself, or just know a few good resources that I could look at, please let me know. However, while the solution will be crucial to the performance of the final program, it is just one of many problems, so the solution needs to be of an applied nature and not purely academic.
Should this be the wrong venue to ask this question, please let me know what a more suitable place would be.
Often, these sorts of dynamic mesh structures can be very tricky to deal with efficiently, but in block-structured adaptive mesh refinement codes (common in astrophysics, where complex geometries aren't important) or your spectral element code where you have large block sizes, it is often much less of an issue. You have so much work to do per block/element (with at least 10^5 cells x 2 points/cell in your case) that the cost of switching between blocks is comparitively minor.
Keep in mind, too, that you can't generally do too much of the hard work on each element or block until a substantial amount of that block's data is already in cache. You're already going to have to had flushed most of block N's data out of cache before getting much work done on block N+1's anyway. (There might be some operations in your code which are exceptions to this, but those are probably not the ones where you're spending much time anyway, cache or no cache, because there's not a lot of data reuse - eg, elementwise operations on cell values). So keeping each the blocks/elements beside each other isn't necessarily a huge deal; on the other hand, you definitely want the blocks/elements to be themselves contiguous.
Also notice that you can move blocks around to keep them contiguous as things get resized, but not only are all those memory copies also going to wipe your cache, but the memory copies themselves get very expensive. If your problem is filling a significant fraction of memory (and aren't we always?), say 1GB, and you have to move 20% of that around after a refinement to make things contiguous again, that's .2 GB (read + write) / ~20 GB/s ~ 20 ms compared to reloading (say) 16k cache lines at ~100ns each ~ 1.5 ms. And your cache is trashed after the shuffle anyway. This might still be worth doing if you knew that you were going to do the refinement/derefinement very seldom.
But as a practical matter, most adaptive mesh codes in astrophysical fluid dynamics (where I know the codes well enough to say) simply maintain a list of blocks and their metadata and don't worry about their contiguity. YMMV of course. My suggestion would be - before spending too much time crafting the perfect data structure - to first just test the operation on two elements, twice; the first, with the elements in order and computing on them 1-2, and the second, doing the operation in the "wrong" order, 2-1, and timing the two computations several times.
For each cell, store the offset in which to find the cell data in a contiguous array. This offset mapping is very efficient and widely used. You can reorder the cells for cache reuse in traversals. When the order or number of cells changes, create a new array and interpolate, then throw away the old arrays. This storage is much better for external analysis because operations like inner products in Krylov methods and stages in Runge-Kutta methods can be managed without reference to the mesh. It also requires minimal memory per vector (e.g. in Krylov bases and with time integration).
It is said that iterating through a vector (as in reading all it's element) is faster than iterating through a list, because of optimized cache.
Is there any ressource on the web that would quantify how much it impacts the performances ?
Also, would it be better to use a custom linked list, whom elements would be prealocated so that they are consecutive in memory?
The idea behind that is that I want to store elements in a certain order that won't change. I still need to be able to insert some at run time in the midle quickly, but most of them will still be consecutive, because the order won't change.
Does the fact that the elements are consecutive have an impact in the cache, or because I'll still call list_element->next instead of ++list_element it does not improve anything ?
The main difference between vector and lists is that in vector elements are constructed subsequently inside a preallocated buffer, while in a list elements are constructed one by one.
As a consequence, elements in a vector are granted to occupy a contiguous memory space, while list elements (unless some specific situations, like a custom allocator working that way) aren't granted to be so, and can be "sparse" around the memory.
Now, since the processor operates on a cache (that can be up to 1000 times faster than the main RAM) that remaps entire pages of the main memory, if elements are consecutive it is higly probable that they fits a same memory page and hence are moved all together in the cache when iteration begins. While proceeding, everything happens in the cache without further moving of data or further access to the slower RAM.
With list-s, since elements are sparse everywhere, "going to the next" means refer to an address that may not be in the same memory page of its previous, and hence, the cache needs to be updated upon every iteration step, accessing the slower RAM on each iteration.
The performance difference greatly depends on the processor and on the type of memory used for both the main RAM and the cache, and on the way the std::allocator (and ultimately operator new and malloc) are implemented, so a general number is impossible to be given.
(Note: great difference means bad RAM respect to to the cache, but may also means bad implementation on list-s)
The efficiency gains from cache coherency due to compact representation of data structures can be rather dramatic. In the case of vectors compared to lists, compact representation can be better not just for read but even for insertion (shifting in vectors) of elements up to the order of 500K elements for some particular architecture as demonstrated in Figure 3 of this article by Bjarne Stroustrup:
http://www2.research.att.com/~bs/Computer-Jan12.pdf
(Publisher site: http://www.computer.org/portal/web/csdl/doi/10.1109/MC.2011.353)
I think that if this is a critical factor for your program, you should profile it on your architecture.
Not sure if I can explain it right but here's my view(i'm thinking along the lines of translated machine instruction below:),
Vector iterator(contiguous memory):
When you increment a vector iterator, the iterator value is simply added the size of the object(known at compile time) to point to the next object. In most CPUs this is anything from one to three instructions at most.
List iterator(linked list http://www.sgi.com/tech/stl/List.html):
When you increment a list iterator(the pointed object), the location of the forward link is located by adding some number to the base of the object pointed and then loaded up as the new value of the iterator. There is more than one memory access for this and is slower than the vector iteration operation.