I'm trying to match (linear assigment) two sets of elements according to some weight(i,j) function. I used munkres until now but the amount of memory used by the result alone (15000 x 15000 x sizeof(float)) is too big. My next bet would be the auction algorithm, but I'm not sure if it fits my criteria.
There may be elements which appear on one side only. An optimal and simple to implement solution is preferable. I just need a hint in the right direction, thank you very much.
Once a weight is calculated, it does not need to be stored with much precision. You can immediately cut your storage requirement from 858 MB to 429 MB by using half-precision floating point values, or some other 16-bit format. For example, depending on the range of weights you might want to take the logarithm of the weight and store that as a 16-bit integer. Or you could store only the exponent part of the original 32-bit floats, which is a mere 8 bits, cutting storage down to 215 MB.
Once the weights are transformed (or quantized), you can apply the algorithm as normal.
I need to partition a large set of 3D points (using C++). The points are stored on the HDD as binary float array, and the files are usually larger than 10GB.
I need to divide the set into smaller subsets that have a size less than 1GB.
The points in the subset should still have the same neighborhood because I need to perform certain algorithms on the data (e.g., object detection).
I thought I could use a KD-Tree. But how can I construct the KD-Tree efficiently if I can't load all the points into RAM? Maybe I could map the file as virtual memory. Then I could save a pointer to each 3D point that belongs to a segment and store it in a node of the KD-Tree. Would that work? Any other ideas?
Thank you for your help. I hope you can unterstand the problem :D
You basically need an out-of-core algorithm for computing (approximate) medians. Given a large file, find its median and then partition it into two smaller files. A k-d tree is the result of applying this process recursively along varying dimensions (and when the smaller files start to fit in memory, you don't have to bother with the out-of-core algorithm any more).
To approximate the median of a large file, use reservoir sampling to grab a large but in-memory sample, then run an in-core median finding algorithm. Alternatively, for an exact median, compute the (e.g.) approximate 45th and 55th percentiles, then make another pass to extract the data points between them and compute the median exactly (unless the sample was unusually non-random, in which case retry). Details are in the Motwani--Raghavan book on randomized algorithms.
I want to generate a huge weighted undirected graph, represented by a huge adjacency matrix AJM. So for the loop over i and j,
AJM[i][j] = AJM[j][i]
AJM[i][i] = 0
The weights are generated as random double numbers in the interval, say [0.01, 10.00]. If I have 10k vertices, the matrix would be 10k by 10k with double type entries, which is a huge chunk in the memory if I store it.
Now I want to set a threshold E for the wanted number of edges, and ignore all the edges with weight larger than some threshold T (T is determined by E, E is user-defined), just store the smallest E edges with weight under T in a vector for later use. Could you give me some suggestion how to achieve this in an efficient manner? It is best to avoid any kind of storage of the whole adjacency matrix, just use streaming structure. So I'm wondering how I should generate the matrix and do the thresholding?
I guess writing and reading file is needed, right?
One approach would be, after some kind of manipulation with file, I set the threshold E and do the following:
I read the element from the matrix one by one so I don't read in the whole matrix (could you show some lines of C++ code for achieving this?), and insert its weight into a min-heap, store its corresponding edge index in a vector. I stop when the size of the heap reaches E so that the vector of edge indices is what I want.
Do you think its the right way to do it? Any other suggestions? Pls point out any error I may have here. Thank you so much!
If there is no need to keep the original threshold-ed graph then it sounds like there is an easy way to save yourself a lot of work. You are given the number of vertices (V=10,000), and the number of edges (E) is user configurable. Just randomly select pairs of vertices until you have the required number of edges. Am I missing an obvious reason why this would not be equivalent?
I need to find for each point of the data set all its nearest neighbors. The data set contains approx. 10 million 2D points. The data are close to the grid, but do not form a precise grid...
This option excludes (in my opinion) the use of KD Trees, where the basic assumption is no points have same x coordinate and y coordinate.
I need a fast algorithm O(n) or better (but not too difficult for implementation :-)) ) to solve this problem ... Due to the fact that boost is not standardized, I do not want to use it ...
Thanks for your answers or code samples...
I would do the following:
Create a larger grid on top of the points.
Go through the points linearly, and for each one of them, figure out which large "cell" it belongs to (and add the points to a list associated with that cell).
(This can be done in constant time for each point, just do an integer division of the coordinates of the points.)
Now go through the points linearly again. To find the 10 nearest neighbors you only need to look at the points in the adjacent, larger, cells.
Since your points are fairly evenly scattered, you can do this in time proportional to the number of points in each (large) cell.
Here is an (ugly) pic describing the situation:
The cells must be large enough for (the center) and the adjacent cells to contain the closest 10 points, but small enough to speed up the computation. You could see it as a "hash-function" where you'll find the closest points in the same bucket.
(Note that strictly speaking it's not O(n) but by tweaking the size of the larger cells, you should get close enough. :-)
I have used a library called ANN (Approximate Nearest Neighbour) with great success. It does use a Kd-tree approach, although there was more than one algorithm to try. I used it for point location on a triangulated surface. You might have some luck with it. It is minimal and was easy to include in my library just by dropping in its source.
Good luck with this interesting task!
I'm looking for a data structure that would allow me to store an M-by-N 2D matrix of values contiguously in memory, such that the distance in memory between any two points approximates the Euclidean distance between those points in the matrix. That is, in a typical row-major representation as a one-dimensional array of M * N elements, the memory distance differs between adjacent cells in the same row (1) and adjacent cells in neighbouring rows (N).
I'd like a data structure that reduces or removes this difference. Really, the name of such a structure is sufficient—I can implement it myself. If answers happen to refer to libraries for this sort of thing, that's also acceptable, but they should be usable with C++.
I have an application that needs to perform fast image convolutions without hardware acceleration, and though I'm aware of the usual optimisation techniques for this sort of thing, I feel a specialised data structure or data ordering could improve performance.
Given the requirement that you want to store the values contiguously in memory, I'd strongly suggest you research space-filling curves, especially Hilbert curves.
To give a bit of context, such curves are sometimes used in database indexes to improve the locality of multidimensional range queries (e.g., "find all items with x/y coordinates in this rectangle"), thereby aiming to reduce the number of distinct pages accessed. A bit similar to the R-trees that have been suggested here already.
Either way, it looks that you're bound to an M*N array of values in memory, so the whole question is about how to arrange the values in that array, I figure. (Unless I misunderstood the question.)
So in fact, such orderings would probably still only change the characteristics of distance distribution.. average distance for any two randomly chosen points from the matrix should not change, so I have to agree with Oli there. Potential benefit depends largely on your specific use case, I suppose.
I would guess "no"! And if the answer happens to be "yes", then it's almost certainly so irregular that it'll be way slower for a convolution-type operation.
EDIT
To qualify my guess, take an example. Let's say we store a[0][0] first. We want a[k][0] and a[0][k] to be similar distances, and proportional to k, so we might choose to interleave the storage of first row and first column (i.e. a[0][0], a[1][0], a[0][1], a[2][0], a[0][2], etc.) But how do we now do the same for e.g. a[1][0]? All the locations near it in memory are now taken up by stuff that's near a[0][0].
Whilst there are other possibilities than my example, I'd wager that you always end up with this kind of problem.
EDIT
If your data is sparse, then there may be scope to do something clever (re Cubbi's suggestion of R-trees). However, it'll still require irregular access and pointer chasing, so will be significantly slower than straightforward convolution for any given number of points.
You might look at space-filling curves, in particular the Z-order curve, which (mostly) preserves spatial locality. It might be computationally expensive to look up indices, however.
If you are using this to try and improve cache performance, you might try a technique called "bricking", which is a little bit like one or two levels of the space filling curve. Essentially, you subdivide your matrix into nxn tiles, (where nxn fits neatly in your L1 cache). You can also store another level of tiles to fit into a higher level cache. The advantage this has over a space-filling curve is that indices can be fairly quick to compute. One reference is included in the paper here: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.30.8959
This sounds like something that could be helped by an R-tree. or one of its variants. There is nothing like that in the C++ Standard Library, but looks like there is an R-tree in the boost candidate library Boost.Geometry (not a part of boost yet). I'd take a look at that before writing my own.
It is not possible to "linearize" a 2D structure into an 1D structure and keep the relation of proximity unchanged in both directions. This is one of the fundamental topological properties of the world.
Having that that, it is true that the standard row-wise or column-wise storage order normally used for 2D array representation is not the best one when you need to preserve the proximity (as much as possible). You can get better result by using various discrete approximations of fractal curves (space-filling curves).
Z-order curve is a popular one for this application: http://en.wikipedia.org/wiki/Z-order_(curve)
Keep in mind though that regardless of which approach you use, there will always be elements that violate your distance requirement.
You could think of your 2D matrix as a big spiral, starting at the center and progressing to the outside. Unwind the spiral, and store the data in that order, and distance between addresses at least vaguely approximates Euclidean distance between the points they represent. While it won't be very exact, I'm pretty sure you can't do a whole lot better either. At the same time, I think even at very best, it's going to be of minimal help to your convolution code.
The answer is no. Think about it - memory is 1D. Your matrix is 2D. You want to squash that extra dimension in - with no loss? It's not going to happen.
What's more important is that once you get a certain distance away, it takes the same time to load into cache. If you have a cache miss, it doesn't matter if it's 100 away or 100000. Fundamentally, you cannot get more contiguous/better performance than a simple array, unless you want to get an LRU for your array.
I think you're forgetting that distance in computer memory is not accessed by a computer cpu operating on foot :) so the distance is pretty much irrelevant.
It's random access memory, so really you have to figure out what operations you need to do, and optimize the accesses for that.
You need to reconvert the addresses from memory space to the original array space to accomplish this. Also, you've stressed distance only, which may still cause you some problems (no direction)
If I have an array of R x C, and two cells at locations [r,c] and [c,r], the distance from some arbitrary point, say [0,0] is identical. And there's no way you're going to make one memory address hold two things, unless you've got one of those fancy new qubit machines.
However, you can take into account that in a row major array of R x C that each row is C * sizeof(yourdata) bytes long. Conversely, you can say that the original coordinates of any memory address within the bounds of the array are
r = (address / C)
c = (address % C)
so
r1 = (address1 / C)
r2 = (address2 / C)
c1 = (address1 % C)
c2 = (address2 % C)
dx = r1 - r2
dy = c1 - c2
dist = sqrt(dx^2 + dy^2)
(this is assuming you're using zero based arrays)
(crush all this together to make it run more optimally)
For a lot more ideas here, go look for any 2D image manipulation code that uses a calculated value called 'stride', which is basically an indicator that they're jumping back and forth between memory addresses and array addresses
This is not exactly related to closeness but might help. It certainly helps for minimation of disk accesses.
one way to get better "closness" is to tile the image. If your convolution kernel is less than the size of a tile you typical touch at most 4 tiles at worst. You can recursively tile in bigger sections so that localization improves. A Stokes-like (At least I thinks its Stokes) argument (or some calculus of variations ) can show that for rectangles the best (meaning for examination of arbitrary sub rectangles) shape is a smaller rectangle of the same aspect ratio.
Quick intuition - think about a square - if you tile the larger square with smaller squares the fact that a square encloses maximal area for a given perimeter means that square tiles have minimal boarder length. when you transform the large square I think you can show you should the transform the tile the same way. (might also be able to do a simple multivariate differentiation)
The classic example is zooming in on spy satellite data images and convolving it for enhancement. The extra computation to tile is really worth it if you keep the data around and you go back to it.
Its also really worth it for the different compression schemes such as cosine transforms. (That's why when you download an image it frequently comes up as it does in smaller and smaller squares until the final resolution is reached.
There are a lot of books on this area and they are helpful.