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BSP split plane determination

It's been a while since my last blog entry, but the problem I posed on my earlier blog entry still persists - how to efficiently choose a good split plane for an n-vector data structure. To summarize the structure, geographic points are stored as n-vectors (unit vectors) in a binary tree. Branch nodes of this tree define a plane that bisects/halves the unit sphere - one child of the branch contains points that are below the plane, the other child contains points that are above. Like all trees, leaf nodes are turned into branch nodes as they are filled beyond a threshold. That is the basic idea of it.   Split plane determination must occur when a leaf becomes a branch or when a branch becomes unbalanced. In either case, the method called to determine the split plane is agnostic as to why it was called - it simply receives a bunch of points that it must determine a good split for. My initial implementation was naive:   bestsplit = ... bestscore = infinite; for (i=0; i<numpoints; i++) { for (j=i+1; j<numpoints; j++) { split = (points[i]-points[j]).normalize(); score = 0; for (k=0; k<numpoints; k++) { dot = split.dot(points[k]); // accumulate some scoring heuristics, such as is each point above/below the split, how far above/below, etc. score += ...; } if ( bestscore > score ) { bestscore = score; bestsplit = split; } } } So basically - for all points, determine the point between it and every other point, normalize it, and consider this difference as the normal to the split plane, then test it to see how it "scores" using some heuristics. Probably you can see how this will perform miserably for any significant number of points. The big-O notation is something like n^3 which doesn't include the expensive normalization that is called n^2 times. Several other methods were tested, such as fixing the normal of the split plane to be perpendicular to the x, y, or z axis, but these also proved too expensive and/or also had test cases where the split determination was unsatisfactory.   Heuristics Enter calculus. If we can represent each of the heuristics as a mathematical function, we can determine when the function reaches what is called a "critical point". Specifically we are interested in the critical point that is the global maximum or minimum, depending on which heuristic it is for. So far we have three of these. 1. Similar Distance We don't want a split plane where, for example, all points above are nearby and all points below are far away. The points should be as evenly distributed as possible on either side. Given that the dot product of a plane normal and any other point is negative when the point is below the plane and positive when the point is above, and the absolute value of the dot product increases as distance to the plane increases, the sum of all dot products for a good split will be at or close to zero. If we let \(P\) be the array of points, \(N\) be the number of points in the array, and \(S\) the split plane, the following function will add all dot products: \(SumOfDots = \displaystyle\sum_{i=1}^{N} P_i \cdot S\) The summation here is not really part of a mathematical function, at least not one we can perform meaningful calculus on, since the calculation must be done in the code. We don't know ahead of time how many points there will be or what their values are, so the function should be agnostic in this regard. As written we cannot use it without inordinate complication, but consider that it is really doing this: \(SumOfDots = \displaystyle\sum_{i=1}^{N} P_{ix} * S_x + P_{iy} * S_y + P_{iz} * S_z\) This summation expanded will look something like: \(SumOfDots = P_{1x} * S_x + P_{1y} * S_y + P_{1z} * S_z + P_{2x} * S_x + P_{2y} * S_y + P_{2z} * S_z + P_{3x} * S_x + P_{3y} * S_y + P_{3z} * S_z + \cdots\) We can then rewrite this as: \(SumOfDots = S_x*(P_{1x} + P_{2x} + P_{3x} + \cdots) + S_y*(P_{1y} + P_{2y} + P_{3y} + \cdots) + S_z*(P_{1z} + P_{2z} + P_{3z} + \cdots)\) Or similarly: \(SumOfDots = S_x*(\displaystyle\sum_{i=1}^{N} P_{ix}) + S_y*(\displaystyle\sum_{i=1}^{N} P_{iy}) + S_z*(\displaystyle\sum_{i=1}^{N} P_{iz})\) As far as the mathematical function is concerned, the sums are constants and we can replace them with single characters to appear concise: \(A = \displaystyle\sum_{i=1}^{N} P_{ix}\) \(B = \displaystyle\sum_{i=1}^{N} P_{iy}\) \(C = \displaystyle\sum_{i=1}^{N} P_{iz}\) We can pre-calculate these in the code as such: double A = 0, B = 0, C = 0; for (i=0; i<N; i++) { A += points[i].x; B += points[i].y; C += points[i].z; } We will rewrite the function with these constants: \(SumOfDots = S_x*A + S_y*B + S_z*C\) This is great so far. We are interested in when this function reaches zero. To make it simpler, we can square it which makes the negative values positive, and then we become interested in when this function reaches a global minimum: \(SquaredSumOfDots = (S_x*A + S_y*B + S_z*C)^2\) So again, when this function reaches zero it means that the points on either side of the split plane - as denoted by \(S\) - are spread apart evenly. This does not mean that \(S\) is a good split plane overall, as the points could all lie on the plane, or some other undesirable condition occurs. For that we have other heuristics. As a final note, since the vector \(S\) represents a unit normal to a plane, any determination of it must be constrained to space of the unit sphere: \(S_x^2 + S_y^2 + S_z^2 = 1\)   2. Large Distance Practically speaking, the points will not be random and will originate from a grid of points or combination of grids. But random or not, if the points form a shape that is not equilateral - in other words if they form a rectangular shape instead of a square or an ellipse instead of a circle - the larger axis of the shape should be split so that child areas are not even less equilateral. To achieve this we can emphasize that the sum of the absolute value of all the dot products is large, meaning that the points are farther away from the split plane. To do this, we want to find the global maximum of a function that calculates the sum: \(SumOfAbsoluteDots=\displaystyle\sum_{i=1}^{N} |P_i \cdot S|\) Unfortunately there is no way, that I know of, to handle absolute values and still determine critical points, so we need to rewrite this function without the absolute value operator. Really all we are interested in is when this function reaches a maximum, so we can replace it with a square: \(SumOfSquaredDots=\displaystyle\sum_{i=1}^{N} (P_i \cdot S)^2\) Not unlike the previous heuristic function, we need to rewrite this so that \(S\) is not contained in the summation, and we can extract the constant values. If we skip some of the expanding and reducing the squared dot product we will arrive at this step: \(SumOfSquaredDots=(S_x^2*\displaystyle\sum_{i=1}^{N} P_{ix}^2) + (S_y^2*\displaystyle\sum_{i=1}^{N} P_{iy}^2) + (S_z^2*\displaystyle\sum_{i=1}^{N} P_{iz}^2) + (S_x * S_y * 2 * \displaystyle\sum_{i=1}^{N} P_{ix}*P_{iy})+ (S_x * S_z * 2 * \displaystyle\sum_{i=1}^{N} P_{ix}*P_{iz})+ (S_y * S_z * 2 * \displaystyle\sum_{i=1}^{N} P_{iy}*P_{iz})\) Again we will create some named constants to be concise: \(D = \displaystyle\sum_{i=1}^{N} P_{ix}^2\) \(E = \displaystyle\sum_{i=1}^{N} P_{iy}^2\) \(F = \displaystyle\sum_{i=1}^{N} P_{iz}^2\) \(G = 2 * \displaystyle\sum_{i=1}^{N} P_{ix}*P_{iy}\) \(H = 2 * \displaystyle\sum_{i=1}^{N} P_{ix}*P_{iz}\) \(I = 2 * \displaystyle\sum_{i=1}^{N} P_{iy}*P_{iz}\) As before, these can be pre-calculated: double D = 0, E = 0, F = 0, G = 0, H = 0, I = 0; for (i=0; i<N; i++) { D += points[i].x * points[i].x; E += points[i].y * points[i].y; F += points[i].z * points[i].z; G += points[i].x * points[i].y; H += points[i].x * points[i].z; I += points[i].y * points[i].z; } G *= 2.0; H *= 2.0; I *= 2.0; And then the function becomes: \(SumOfSquaredDots=(S_x^2*D) + (S_y^2*E) + (S_z^2*F) + (S_x * S_y * G)+ (S_x * S_z * H) + (S_y * S_z * I)\) Again when this function maximizes, the points are farthest away from the split plane, which is what we want. 3. Similar number of points A good split plane will also have the same number of points on either side. We can again use the dot product since it is negative for points below the plane and positive for points above. But we cannot simply sum the dot products themselves, since a large difference for a point on one side will cancel out several smaller differences on the other. To account for this we normalize the distance to be either +1 or -1: \(SumOfNormalizedDots=\displaystyle\sum_{i=1}^{N} \frac{P_i \cdot S}{\sqrt{(P_i \cdot S)^2}}\) Expanding this becomes: \(SumOfNormalizedDots=\displaystyle\sum_{i=1}^{N} \frac{P_{ix} * S_x + P_{iy} * S_y + P_{iz} * S_z}{\sqrt{(S_x^2*P_{ix}^2) + (S_y^2*P_{iy}^2) + (S_z^2*P_{iz}^2) + (S_x*S_y*2*P_{ix}*P_{iy})+ (S_x*S_z*2*P_{ix}*P_{iz})+ (S_y*S_z*2*P_{iy}*P_{iz})}}\) Unfortunately there is no way to reduce this function so that we can extract all \(S\) references out of the summation, and as with the previous heuristics, put all \(P\) references into constants that we pre-calculate and use to simplify the function. So at present, this heuristic is not able to be used. I am still working on it. The problem is that each iteration of the sum is dependent on the values of \(S\) in such a way that it cannot be extracted. Putting it all together What we ultimately want to do is combine the heuristic functions into one function, then use this one function find the critical point - either the global minimum or global maximum depending on how we combine them. The issue is that for \(SquaredSumOfDots\) we want the global minimum and for \(SumOfSquaredDots\) we want the global maximum. We can account for this by inverting the former so that we instead want the global maximum for it. The combined function then becomes: \(Combined = SumOfSquaredDots - SquaredSumOfDots\) Applying the terms from each function, we get: \(Combined = (S_x^2*D) + (S_y^2*E) + (S_z^2*F) + (S_x * S_y * G)+ (S_x * S_z * H) + (S_y * S_z * I) - (S_x*A + S_y*B + S_z*C)^2\) Expanding the square on the right side and combining some terms will give us: \(Combined = (S_x^2*(D-A^2)) + (S_y^2*(E - B^2)) + (S_z^2*(F - C^2)) + (S_x * S_y * (G - (2 * A * B)))+ (S_x * S_z * (H - (2 * A * C))) + (S_y * S_z * (I - (2 * B * C)))\) Again we can combine/pre-calculate some constants to simplify: \(J = D-A^2\) \(K = E-B^2\) \(L = F-C^2\) \(M = G - (2 * A * B)\) \(N = H - (2 * A * C)\) \(O = I - (2 * B * C)\) double J = D - (A * A); double K = E - (B * B); double L = F - (C * C); double M = G - (2 * A * B); double N = H - (2 * A * C); double O = I - (2 * B * C); And then apply these to the function: \(Combined = (S_x^2*J) + (S_y^2*K) + (S_z^2*L) + (S_x * S_y * M)+ (S_x * S_z * N) + (S_y * S_z * O)\) Because we are lazy, we then plug this into a tool that does the calculus for us. And this is where I am currently blocked on this issue, as I have found no program that can perform this computation. I've actually purchased Wolfram Mathematica and attempted it with the following command: Maximize[{((x^2)*j) + ((y^2)*k) + ((z^2)*l) + (x*y*m) + (x*z*n) + (y*z*o), ((x^2) + (y^2) + (z^2)) == 1}, {x, y, z}] After 5 or 6 days this had not finished and I had to restart the computer it was running on. I assumed that if it took that long, it would not complete in a reasonable amount of time. I will update this blog entry if I make any progress on this problem as well as the 3rd heuristic function.   Further Optimizations While I have not gotten this far yet, it may be ultimately necessary to emphasize (or de-emphasize) one heuristic over the others in order to further optimize the split determination. This could be done by simply multiplying each heuristic function by a scalar value to either increase or decrease it's emphases on the final result. The reason I haven't researched this yet is because if I cannot find the global maximum without these extra variables, I certainly cannot do it with them.




N-vector data structure to store and query for scatter points

Introduction The impetus for this research topic is concerned with a data structure that efficiently stores and queries scatter point data. These points may originally be neatly arranged on a grid, or randomly scattered across the surface of a sphere (e.g. the earth), but it makes no assumption in that regard. The data structure could have member functions such as add(point) or add(points, amount). The points themselves contain three scalar values: the longitude, the latitude, and the data value itself. This data value could be geographical data such as land height or weather data such as air temperature, but again, the data structure is agnostic about the data itself other than as a scalar value. Data Structure The purpose of the data structure is ultimately to efficiently and accurately query for point(s) that is(are) nearby another point, the reason being to compute the average value at that point. Suppose a requirement is to create an image representing the air temperature (or rainfall, cloud cover, whatever the scalar value of each point represents) - this data structure could be queried for each point (i.e. pixel) in the image. Once nearby points are queried, the algorithm to average the scalar values of each point is outside the scope of this, but a good candidate is the inverse distance weighed algorithm. Within the data structure, the data points will not be stored in spherical coordinates, but as n-vectors. The reason for this is that the sampling algorithm - finding N closest data points - is more efficient and accurate using vector algebra. While it is outside the scope of this, to summarize: it is easier to determine how close two points are on a sphere if these points are represented by unit vectors than by latitude and longitude. For example, two points sharing the same longitude are farther apart at the equator than above or below the equator. Even if we are just concerned with relative distances, and do not need to compute the square root of the distance formula, for this reason it is still not as accurate. Also the international date line, where longitude "resets", presents a problem that would have to be dealt with. Also either pole. But you get the idea - vectors are more stable, accurate, and efficient. I've always been a fan of Binary Space Partitions - each branch node of a BSP has two areas, one below and one above (or one left and one right, or one in and one out, however you like to think of it). To go along with storing points as vectors, branch nodes within the structure partitions their space also using a vector, which in this case forms the normal of a plane that bisects the unit sphere. All data points in a node (or children of the node) will either be above or below this plane (points that are on the plane can be allocated to either side as is appropriate). Points that are below the plane are placed in the first child of the node, and points that are above placed in the second child. We call this plane normal the split axis of the node. Querying the structure for the closest N points then becomes trivial. For any branch node, compute the dot product of the point in question and the split axis of the node. If it is negative (the point is below the split plane), recursively query with the first child node, and if positive (the point is above the split plane), with the second child node. For a leaf node, compute the dot product of the point in question with each data point contained in the node, keeping a sorted list of the closest N points. The one caveat is that in branch nodes, after recursing into one child node, it may be necessary to recurse into the other child if the farthest point found so far is farther than the other child node, since there may be closer points in the other child node. But this is trivial as we are comparing dot products. No expensive computations are necessary to compute the N closest points - just a bunch of dot products. As dot products of unit vectors range from -1 to 1 (-1 being farthest apart and 1 being equal), two points are closer if their dot product is higher. Once complete, and the list of N points found, the actual distances can be calculated if necessary, which in this case is done by calculating the angles using the already-calculated dot products. This angle can then be multiplied by the radius of the earth to get an exact distance, again only if necessary. But for averaging, these extra calculations are not necessary. As a final note, the data structure lends itself well to graphics hardware. In my particular case, rendering an image using such a data structure on the CPU may take several minutes, but on the GPU takes a fraction of a second.  Problem The problem - common to any space partitioning tree - is how to initially create the data structure. Again, the points are not assumed to be arranged in any specific way, and as far as we are concerned, are a "point soup". They can be specified one at a time - addPoint(point) - or all at once - addPoints(points) - or both. Specifically, how can the determination of the split axis of any branch be efficient and provide an even split (the same or similar number of points on either side). The problem is unique here because the points are not arranged on a two-dimensional surface or in three-dimensional space, but lie on a unit sphere. My current solution to the problem is not terribly elegant. For each two data points, compute the axis that splits them evenly. This is done by computing the point between the two (subtract one from the other) and normalizing it, then crossing this with the cross product of the two points. This cross product comprises the normal of the plane that evenly splits the two points. This normal is then tested against each other point in the node to get 1) the number of points on either side of the plane and 2) the distances of the points to the plane. A good split plane is one that 1) splits points evenly, 2) has the same/similar distances on other side, and 3) has large distances on other side. As this test is performed for each two data points, some big-O notation shows that determination of the split plane for nodes containing a large number of points will become prohibitive.  However, I have not been able to determine a better solution. But the advantages of such a data structure still outweigh this expense. In my particular case, the time spent initially creating the tree is worth the time saved during queries. I should mention that if the data points are known ahead of time, it is faster to specify them all at once, so the tree can re-build itself once, rather than one at a time which may cause the tree to re-build itself multiple times.




Efficient thread synchronization

*See the EDIT below*

The impetus for this research topic is the necessity for threads to exchange data in an efficient manner, namely between the network and main threads. Within the context of networking, we want data to be quickly available to, and dispensable from, the main thread, and that means calling the corresponding system calls (e.g. sendto/recvfrom) as often as possible. In addition to this, there is necessarily some translation done between native and network formats, as well lookups and other operations that could be off-loaded to the network thread if possible.

However, this introduces a problem; as data arrives and is picked up by the network thread, or as logic in the main thread requires data be sent out to the network, how do the separate threads share this data? If we create a mutex for it, we introduce an expensive system call that causes delays and is generally unacceptable in a tight loop. Whatever solution we implement should not involve systems calls or any significant pause in execution.

If our shared data is simple enough to be just one variable - just one *instance* of something, if you will - then nothing fancy is needed. As an example, if the network thread needs to notify the main thread that a player has sent a QUIT message, and that this player be removed from all operations and deleted, a "player * quitting" member can be added, which is updated as necessary. The main thread will watch this member, and take appropriate action when it sees that it is non-null. So, the course of action for both threads is:

Network thread:
1. QUIT message received by player
2. update member "quitting" to refer to this player

Main thread:
1. check member "quitting"
2. if non-null, delete all associated resourced and updating member "quitting" to null

But there is a problem with this: what if another QUIT message is received by the network thread before the main thread has processed the first one? The network thread may update member "quitting" before the main thread has seen the first one. To address this, we can add a safety check in step 2 of the Network thread - only update member "quitting" if it is null. The reasoning behind this is that after the Network thread has updated this member, it should not update it again until the Main thread has processed it, and the Network thread can tell this by whether or not it is null.

And this introduces yet another problem, as you can probably guess: what should the Network thread do with second QUIT message that it has received before the Main thread has finished processing the first one? It can simply queue this second message, and any others it receives, in a buffer. Each loop iteration, it will check to see if the member "quitting" is null, and if so, assign it to one of the items in the buffer (also removing this item from the buffer).

This solution works and is great in that it allows each thread to produce or consume data without waiting for the other. If data arrives in the producer, and the consumer is busy with previously shared data, it is simply queued, and the producer continues execution. The only cost incurred is one conditional in each thread, plus a buffer in the producer thread for queueing purposes.

The drawback of this approach is that only one item can be shared at a time. While this may be ok for QUIT messages - it would be uncommon for many players to quit in the few milliseconds it takes per each loop iteration - it would not be efficient for messages that are transferred more often. For this approach, two buffers can be used that swap their contents. Instead of explaining all the details, I will simply post the code I've written to support this://// Thread synchronization w/out semaphores//// the data producer can always insert data, and should call Commit() once during it's event loop// (the Commit() call will only commit produced data if the consumer has processed existing data)//// the data consumer will process data if there is any has been committed by the producer//template class Queue{private: std::vector m_clsIn; std::vector m_clsOut; bool m_blnData; bool m_blnProcessed;public: Queue() : m_blnData(false), m_blnProcessed(true) {} void Add(T clsData) { //ASSERT(thread == producer) m_clsIn.push_back(clsData); } void Commit() { //ASSERT(thread == producer) if ( m_blnProcessed && m_clsIn.size() > 0 ) { m_clsIn.swap(m_clsOut); m_blnProcessed = false; m_blnData = true; } } template void Process(Functor & refFunctor) { //ASSERT(thread == consumer) if ( m_blnData ) { for (size_t i=0; i); m_clsOut.clear(); m_blnData = false; m_blnProcessed = true; } }};
As incoming messages are received by the Network thread, it will call the Add() method. The Network thread will also call Commit() each loop iteration. The Main thread will call Process() each loop iteration to retrieve the message data. For outgoing messages, there will be a separate Queue object in which the above is reversed - the Main thread will call Add() and Commit(), and the Network thread will call Process().

Note: I've not tested this class yet. Specialized methods may be added to support certain data types or operations.

EDIT: Per [color=rgb(51,51,51)][font=helvetica] Josh's comment and the article he has linked, I've made some changes. Instead of swapping two vectors, one linked list is used with two pointers that refer to active and inactive portions of the list. Nodes that contain produced and un-consumed data constitute the active portion, while nodes containing consumed but un-produced data constitute the inactive portion. [/font][/color]

[color=rgb(51,51,51)][font=helvetica]Even though it is implemented as a list, Queue was kept as a name as it is processed in a first-in, first-out basis. I would have simply used the queue written by the author of the article Josh linked, but it has more functionality than I need; for example I do not need "try to put" or "try to get" semantics. [/font][/color]template class Queue{private: struct Node : public T { Node * m_ptrNext; Node(Node * ptrNext = NULL) : m_ptrNext(ptrNext) {} ~Node() {delete m_ptrNext;} } * m_ptrActive, * m_ptrInactive; public: Queue() { // // Start the queue with two elements // m_ptrActive = m_ptrInactive = new Node(); m_ptrInactive->m_ptrNext = new Node(m_ptrInactive); // // Make sure all threads have updated memory // std::atomic_signal_fence(std::memory_order_seq_cst); } ~Queue() { // // Make sure all threads have updated memory // std::atomic_signal_fence(std::memory_order_seq_cst); // // Set the 'next' pointer to NULL so the recursive nature of the Node d'tor will not // cause the destruction process to wrap around // Node * ptrNode = m_ptrInactive->m_ptrNext; m_ptrInactive->m_ptrNext = NULL; delete ptrNode; } template void Put(Functor functor) { //ASSERT(thread == producer) // // Check if the queue is full, in which case we must add a node // if ( m_ptrInactive->m_ptrNext == m_ptrActive ) { // // Between the above conditional and the following statement, the consumer can process // data and move the 'active' pointer, making the following node addition unnecessary. // However, there is no way to avoid it except with a mutex lock or other expensive // operation, and the extra node will get used anyway in the next call to Put() // // In the unlikely case that the 'active' pointer is advanced between the preceding // conditional and the Node c'tor below, we use 'inactive->next' as a parameter instead // of 'active' // m_ptrInactive->m_ptrNext = new Node(m_ptrInactive->m_ptrNext); } functor(static_cast(*m_ptrInactive)); std::atomic_signal_fence(std::memory_order_release); m_ptrInactive = m_ptrInactive->m_ptrNext; } template void Get(Functor refFunctor) { //ASSERT(thread == consumer) // // These is no [good] reason to fence here, as it is optimal to let the CPU handle memory as it // naturally does. The producer thread may not see our memory updates right away and subsequently // add extra nodes when it doesn't have to, but any extra nodes will be used anyway in subsequent // calls to Put(). // while ( m_ptrActive != m_ptrInactive ) { refFunctor(static_cast(*m_ptrActive)); m_ptrActive = m_ptrActive->m_ptrNext; } } // // For accurate results, call this from the producer thread // size_t SizeAll() { // Count from inactive -> inactive size_t count = 1; for (Node * ptrNode = m_ptrActive->m_ptrNext; ptrNode != m_ptrActive; ptrNode = ptrNode->m_ptrNext) ++count; return count; } // // Results may be inaccurate regardless of which thread calls this // size_t SizeActive() { // Count from active -> inactive size_t count = 0; for (Node * ptrNode = m_ptrActive; ptrNode != m_ptrInactive; ptrNode = ptrNode->m_ptrNext) ++count; return count; }};
The calls to [font='courier new']std::atomic_signal_fence(...)[/font] will have to be rewritten for compilers that do not support C++11. In my code I do not directly call these, but instead call a library that mimics the code in the article linked by Josh. I changed them here as I do not want to include my entire library.

An assumption made is that a [font='courier new']Queue[/font] object will not be destroyed without coordination between threads (e.g. one waits for the other to finish). In other words, the destructor will clobber concurrent calls to [font='courier new']Put()[/font]/[font='courier new']Get()[/font].

A caveat is that T objects are not destroyed until the [font='courier new']Queue[/font] object is destroyed. This is by design, since my particular implementation benefits from it. However, it can be trivially changed by:
In [font='courier new']Node[/font]: adding [font='courier new']char object[sizeof(T)][/font] and removing the T inheritance
in [font='courier new']Put()[/font]: calling [font='courier new']new (node->object) T()[/font] and changing the functor parameter to [font='courier new']static_cast(node->object)[/font]
in [font='courier new']Get()[/font]: changing the functor parameter to [font='courier new']static_cast(node->object)[/font] and calling [font='courier new']static_cast(node->object)->T::~T()[/font]

(syntax may not be correct, but you get the idea)



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