How to use Morton Order in range search?
From the wiki, In the paragraph "Use with one-dimensional data structures for range searching",
it says
"the range being queried (x = 2, ..., 3, y = 2, ..., 6) is indicated
by the dotted rectangle. Its highest Z-value (MAX) is 45. In this
example, the value F = 19 is encountered when searching a data
structure in increasing Z-value direction. ......BIGMIN (36 in the
example).....only search in the interval between BIGMIN and MAX...."
My questions are:
1) why the F is 19? Why the F should not be 16?
2) How to get the BIGMIN?
3) Are there any web blogs demonstrate how to do the range search?
EDIT: The AWS Database Blog now has a detailed introduction to this subject.
This blog post does a reasonable job of illustrating the process.
When searching the rectangular space x=[2,3], y=[2,6]:
The minimum Z Value (12) is found by interleaving the bits of the lowest x and y values: 2 and 2, respectively.
The maximum Z value (45) is found by interleaving the bits of the highest x and y values: 3 and 6, respectively.
Having found the min and max Z values (12 and 45), we now have a linear range that we can iterate across that is guaranteed to contain all of the entries inside of our rectangular space. The data within the linear range is going to be a superset of the data we actually care about: the data in the rectangular space. If we simply iterate across the entire range, we are going to find all of the data we care about and then some. You can test each value you visit to see if it's relevant or not.
An obvious optimization is to try to minimize the amount of superfluous data that you must traverse. This is largely a function of the number of 'seams' that you cross in the data -- places where the 'Z' curve has to make large jumps to continue its path (e.g. from Z value 31 to 32 below).
This can be mitigated by employing the BIGMIN and LITMAX functions to identify these seams and navigate back to the rectangle. To minimize the amount of irrelevant data we evaluate, we can:
Keep a count of the number of consecutive pieces of junk data we've visited.
Decide on a maximum allowable value (maxConsecutiveJunkData) for this count. The blog post linked at the top uses 3 for this value.
If we encounter maxConsecutiveJunkData pieces of irrelevant data in a row, we initiate BIGMIN and LITMAX. Importantly, at the point at which we've decided to use them, we're now somewhere within our linear search space (Z values 12 to 45) but outside the rectangular search space. In the Wikipedia article, they appear to have chosen a maxConsecutiveJunkData value of 4; they started at Z=12 and walked until they were 4 values outside of the rectangle (beyond 15) before deciding that it was now time to use BIGMIN. Because maxConsecutiveJunkData is left to your tastes, BIGMIN can be used on any value in the linear range (Z values 12 to 45). Somewhat confusingly, the article only shows the area from 19 on as crosshatched because that is the subrange of the search that will be optimized out when we use BIGMIN with a maxConsecutiveJunkData of 4.
When we realize that we've wandered outside of the rectangle too far, we can conclude that the rectangle in non-contiguous. BIGMIN and LITMAX are used to identify the nature of the split. BIGMIN is designed to, given any value in the linear search space (e.g. 19), find the next smallest value that will be back inside the half of the split rectangle with larger Z values (i.e. jumping us from 19 to 36). LITMAX is similar, helping us to find the largest value that will be inside the half of the split rectangle with smaller Z values. The implementations of BIGMIN and LITMAX are explained in depth in the zdivide function explanation in the linked blog post.
It appears that the quoted example in the Wikipedia article has not been edited to clarify the context and assumptions. The approach used in that example is applicable to linear data structures that only allow sequential (forward and backward) seeking; that is, it is assumed that one cannot randomly seek to a storage cell in constant time using its morton index alone.
With that constraint, one's strategy begins with a full range that is the mininum morton index (16) and the maximum morton index (45). To make optimizations, one tries to find and eliminate large swaths of subranges that are outside the query rectangle. The hatched area in the diagram refers to what would have been accessed (sequentially) if such optimization (eliminating subranges) had not been applied.
After discussing the main optimization strategy for linear sequential data structures, it goes on to talk about other data structures with better seeking capability.
Related
I'm looking for a bit of guidance on using CONVN to calculate moving averages in one dimension on a 3d matrix. I'm getting a little caught up on the flipping of the kernel under the hood and am hoping someone might be able to clarify the behaviour for me.
A similar post that still has me a bit confused is here:
CONVN example about flipping
The Problem:
I have daily river and weather flow data for a watershed at different source locations.
So the matrix is as so,
dim 1 (the rows) represent each site
dim 2 (the columns) represent the date
dim 3 (the pages) represent the different type of measurement (river height, flow, rainfall, etc.)
The goal is to try and use CONVN to take a 21 day moving average at each site, for each observation point for each variable.
As I understand it, I should just be able to use a a kernel such as:
ker = ones(1,21) ./ 21.;
mat = randn(150,365*10,4);
avgmat = convn(mat,ker,'valid');
I tried playing around and created another kernel which should also work (I think) and set ker2 as:
ker2 = [zeros(1,21); ker; zeros(1,21)];
avgmat2 = convn(mat,ker2,'valid');
The question:
The results don't quite match and I'm wondering if I have the dimensions incorrect here for the kernel. Any guidance is greatly appreciated.
Judging from the context of your question, you have a 3D matrix and you want to find the moving average of each row independently over all 3D slices. The code above should work (the first case). However, the valid flag returns a matrix whose size is valid in terms of the boundaries of the convolution. Take a look at the first point of the post that you linked to for more details.
Specifically, the first 21 entries for each row will be missing due to the valid flag. It's only when you get to the 22nd entry of each row does the convolution kernel become completely contained inside a row of the matrix and it's from that point where you get valid results (no pun intended). If you'd like to see these entries at the boundaries, then you'll need to use the 'same' flag if you want to maintain the same size matrix as the input or the 'full' flag (which is default) which gives you the size of the output starting from the most extreme outer edges, but bear in mind that the moving average will be done with a bunch of zeroes and so the first 21 entries wouldn't be what you expect anyway.
However, if I'm interpreting what you are asking, then the valid flag is what you want, but bear in mind that you will have 21 entries missing to accommodate for the edge cases. All in all, your code should work, but be careful on how you interpret the results.
BTW, you have a symmetric kernel, and so flipping should have no effect on the convolution output. What you have specified is a standard moving averaging kernel, and so convolution should work in finding the moving average as you expect.
Good luck!
I'm about to start using geometry or geography data types for the first time now that we have a development baseline of 2008R2 (!)
I'm struggling to find how to store the representation for a circle. We currently have the lat and long of the centre of the circle along with the radius, something like :-
[Lat] [float] NOT NULL,
[Long] [float] NOT NULL,
[Radius] [decimal](9, 4) NOT NULL,
Does anyone know the equivalent way to store this using the STGeomFromText method, ie which Well-Known Text (WKT) representation to use? I've looked at circular string (LINESTRING) and curve, but can't find any examples....
Thanks.
One thing you can do if you are using SQL Server 2008 is to buffer a point and store the resulting Polygon (as well-known binary, internally). For example,
declare #g geometry
set #g=geometry::STGeomFromText('POINT(0 0)', 4326).STBuffer(1)
select #g.ToString()
select #g.STNumPoints()
select #g.STArea()
This outputs, the WKT,
POLYGON ((0 -1, 0.051459848880767822 -0.99869883060455322, 0.10224419832229614 -0.99483710527420044, 0.15229016542434692 -0.98847776651382446, 0.20153486728668213 -0.97968357801437378, 0.24991559982299805 -0.96851736307144165,... , 0 -1))
the number of points, 129, from which it can be seen that buffering a circle uses 128 points plus a repeated start point and and the area, 3.1412, which is accurate to 3 decimal places, and differs from the real value by 0.01%, which would be acceptable for many use cases.
If you want less accuracy (ie, less points), you can use the Reduce function to decrease the number of points, eg,
declare #g geometry
set #g=geometry::STGeomFromText('POINT(0 0)', 4326).STBuffer(1).Reduce(0.01)
which now produces a circle approximation with 33 points and an area of 3.122 (now 0.6% less than the real value of PI).
Less points will reduce storage and make queries such as STIntersects and STIntersection faster, but, obviously, at the cost of accuracy.
EDIT 1: As Jon Bellamy has pointed out, if you choose to use the Reduce function, the parameter needs to be scaled proportionally to the circle/buffer radius, as it is a sensitivity factor for removing points, based on the Ramer-Douglas-Peucker algorithm
EDIT 2: There is also a function, BufferWithTolerance, which can be used to approximate a circle with a polygon. The second parameter, tolerance effects how close this approximation will be: the lower the value, the more points and better approximation. The 3rd parameter is a bit, indicating whether the tolerance is relative or absolute in relation to the buffer radius. This function could be used instead of the STBuffer, Reduce combination to create a circle with more points.
The following query produces,
declare #g geometry
set #g=geometry::STGeomFromText('POINT(0 0)', 4326).BufferWithTolerance(1,0.0001,1)
select #g.STNumPoints()
select #g.STArea()
a "circle" of 321 points with an area of 3.1424, ie, within 0.02% of the true value of PI (but now larger) and actually less accurate than the simple buffer above. Increasing the tolerance further does not lead to any significant improvement in accuracy, which suggests there is an upper limit to this approach.
As MvG has said, there is no CircularString or CompoundCurve until SQL Server 2012, which would allow you to store circles more compactly and accurately, by building a CompoundCurve made up of two semi-circles, ie, using two CircularStrings.
As far as I can tell from the docs, CircularString was only added for SQL Server 2012. The only other instantiable curve appears to be LineString which, as the name suggests, encodes a sequence of line segments. So your best bet would be approximating the circle as a (possibly regular) polygon with a sufficient number of corners. If that is not acceptable, you might have to keep your current data structures in place, either exclusively or in addition to spatial data types to verify that a match there indeed matches the circle.
This answer was written purely from the docs, with no experience to support it.
In the Kademlia paper by Petar Maymounkov and David Mazières, it is said that the XOR distance is a valid non-Euclidian metric with limited explanations as to why each of the properties of a valid metric are necessary or interesting, namely:
d(x,x) = 0
d(x,y) > 0, if x != y
forall x,y : d(x,y) = d(y,x) -- symmetry
d(x,z) <= d(x,y) + d(y,z) -- triangle inequality
Why is it important for a metric to have these properties in general? Why is each of these properties necessary in the context of routing queries in the Kademlia Distributed Hash Table implementation?
In addition, the paper mentions that unidirectionality (for a given x, and a distance l, there exist only a single y for which d(x,y) = l) guarantees that all queries will converge along the same path. Why is that so?
I can only speak for Kademlia, maybe someone else can provide a more general answer. In the meantime...
d(x,x) = 0
d(x,y) > 0, if x != y
These two points together effectively mean that the closest point to x is x itself; every other point is further away. (This may seem intuitive, but other aspects of the XOR metric aren't.)
In the context of Kademlia, this is important since a lookup for node with ID x will yield that node as the closest. It would be awkward if that were not the case, since a search converging towards x might not find node x.
forall x,y : d(x,y) = d(y,x)
The structure of the Kademlia routing table is such that nodes maintain detailed knowledge of the address space closest to them, and exponentially decreasing knowledge of more distant address space. In short, a node tries to keep all the k closest contacts it hears about.
The symmetry is useful since it means that each of these closest contacts will be maintaining detailed knowledge of a similar part of the address space, rather than a remote part.
If we didn't have this property, it might be helpful to think of the search as more like the hands of a clock moving in one direction round a clockface. The node at 1 o'clock (Node1) is close to Node2 at 2 o'clock (30°), but Node2 is far from Node1 (330°). So imagine we're looking for the two closest to 3 o'clock (i.e. Node1 and Node2). If the search reaches Node2, it won't know about Node1 since it's far away. The whole lookup and topology would have to change.
d(x,z) <= d(x,y) + d(y,z)
If this weren't the case, it would be impossible for a node to know which contacts from its routing table to return during a lookup. It would know the k closest to the target, but there would be no guarantee that one of the other more distant contacts wouldn't yield a shorter overall path.
Because of this property and unidirectionality, different searches starting from vastly separated points will tend to converge down the same path.
The unidirectionality means that no two nodes can have the same distance from a given point. If that weren't the case, then the target point could be encircled by a bunch of nodes all the same distance from it. Then various different searches would be free pick any of those to pass through. However, unidirectionality guarantees that exactly one of this bunch will be the closest, and any search which chooses between this group will always select the same one.
I've been bashing my head on this for quite some time: how can the XOR - as in the number of differing bits, a proper Hamming distance - be the basis of a total order?
Well it can't, such a metric on its own is not enough for a comparable relationship, all it can do is dump nodes in circles around a point.
Then I read the paper more closely and noticed that it says "the XOR as an integer value" and it dawned on me: the crux is not the "XOR metric", but the length of the common prefix of the ID (of which XOR is a derivation mechanism.)
Take two nodes with the same Hamming distance from "self" and the length of their prefix common to "self": the one with shortest common prefix is the furthest node.
The paper uses "XOR distance metric" but it really should read "ID prefix length total ordering"
I think this may explain it a wee bit, let me know http://metaquestions.me/2014/08/01/shortest-distance-between-two-points-is-not-always-a-straight-line/
Basically each hop if it were only one bit at a time in a fully populated network (extreme) then would have twice the knowledge of the previous hop. As you converge the knowledge is greater until you get to the closest nodes whose knowledge is ultimate in the network.
We are looking for the computationally simplest function that will enable an indexed look-up of a function to be determined by a high frequency input stream of widely distributed integers and ranges of integers.
It is OK if the hash/map function selection itself varies based on the specific integer and range requirements, and the performance associated with the part of the code that selects this algorithm is not critical. The number of integers/ranges of interest in most cases will be small (zero to a few thousand). The performance critical portion is in processing the incoming stream and selecting the appropriate function.
As a simple example, please consider the following pseudo-code:
switch (highFrequencyIntegerStream)
case(2) : func1();
case(3) : func2();
case(8) : func3();
case(33-122) : func4();
...
case(10,000) : func40();
In a typical example, there would be only a few thousand of the "cases" shown above, which could include a full range of 32-bit integer values and ranges. (In the pseudo code above 33-122 represents all integers from 33 to 122.) There will be a large number of objects containing these "switch statements."
(Note that the actual implementation will not include switch statements. It will instead be a jump table (which is an array of function pointers) or maybe a combination of the Command and Observer patterns, etc. The implementation details are tangential to the request, but provided to help with visualization.)
Many of the objects will contain "switch statements" with only a few entries. The values of interest are subject to real time change, but performance associated with managing these changes is not critical. Hash/map algorithms can be re-generated slowly with each update based on the specific integers and ranges of interest (for a given object at a given time).
We have searched around the internet, looking at Bloom filters, various hash functions listed on Wikipedia's "hash function" page and elsewhere, quite a few Stack Overflow questions, abstract algebra (mostly Galois theory which is attractive for its computationally simple operands), various ciphers, etc., but have not found a solution that appears to be targeted to this problem. (We could not even find a hash or map function that considered these types of ranges as inputs, much less a highly efficient one. Perhaps we are not looking in the right places or using the correct vernacular.)
The current plan is to create a custom algorithm that preprocesses the list of interesting integers and ranges (for a given object at a given time) looking for shifts and masks that can be applied to input stream to help delineate the ranges. Note that most of the incoming integers will be uninteresting, and it is of critical importance to make a very quick decision for as large a percentage of that portion of the stream as possible (which is why Bloom filters looked interesting at first (before we starting thinking that their implementation required more computational complexity than other solutions)).
Because the first decision is so important, we are also considering having multiple tables, the first of which would be inverse masks (masks to select uninteresting numbers) for the easy to find large ranges of data not included in a given "switch statement", to be followed by subsequent tables that would expand the smaller ranges. We are thinking this will, for most cases of input streams, yield something quite a bit faster than a binary search on the bounds of the ranges.
Note that the input stream can be considered to be randomly distributed.
There is a pretty extensive theory of minimal perfect hash functions that I think will meet your requirement. The idea of a minimal perfect hash is that a set of distinct inputs is mapped to a dense set of integers in 1-1 fashion. In your case a set of N 32-bit integers and ranges would each be mapped to a unique integer in a range of size a small multiple of N. Gnu has a perfect hash function generator called gperf that is meant for strings but might possibly work on your data. I'd definitely give it a try. Just add a length byte so that integers are 5 byte strings and ranges are 9 bytes. There are some formal references on the Wikipedia page. A literature search in ACM and IEEE literature will certainly turn up more.
I just ran across this library I had not seen before.
Addition
I see now that you are trying to map all integers in the ranges to the same function value. As I said in the comment, this is not very compatible with hashing because hash functions deliberately try to "erase" the magnitude information in a bit's position so that values with similar magnitude are unlikely to map to the same hash value.
Consequently, I think that you will not do better than an optimal binary search tree, or equivalently a code generator that produces an optimal "tree" of "if else" statements.
If we wanted to construct a function of the type you are asking for, we could try using real numbers where individual domain values map to consecutive integers in the co-domain and ranges map to unit intervals in the co-domain. So a simple floor operation will give you the jump table indices you're looking for.
In the example you provided you'd have the following mapping:
2 -> 0.0
3 -> 1.0
8 -> 2.0
33 -> 3.0
122 -> 3.99999
...
10000 -> 42.0 (for example)
The trick is to find a monotonically increasing polynomial that interpolates these points. This is certainly possible, but with thousands of points I'm certain you'ed end up with something much slower to evaluate than the optimal search would be.
Perhaps our thoughts on hashing integers can help a little bit. You will also find there a hashing library (hashlib.zip) based on Bob Jenkins' work which deals with integer numbers in a smart way.
I would propose to deal with larger ranges after the single cases have been rejected by the hashing mechanism.
I read that it's possible to make quicksort run at O(nlogn)
the algorithm says on each step choose the median as a pivot
but, suppose we have this array:
10 8 39 2 9 20
which value will be the median?
In math if I remember correct the median is (39+2)/2 = 41/2 = 20.5
I don't have a 20.5 in my array though
thanks in advance
You can choose either of them; if you consider the input as a limit, it does not matter as it scales up.
We're talking about the exact wording of the description of an algorithm here, and I don't have the text you're referring to. But I think in context by "median" they probably meant, not the mathematical median of the values in the list, but rather the middle point in the list, i.e. the median INDEX, which in this cade would be 3 or 4. As coffNjava says, you can take either one.
The median is actually found by sorting the array first, so in your example, the median is found by arranging the numbers as 2 8 9 10 20 39 and the median would be the mean of the two middle elements, (9+10)/2 = 9.5, which doesn't help you at all. Using the median is sort of an ideal situation, but would work if the array were at least already partially sorted, I think.
With an even numbered array, you can't find an exact pivot point, so I believe you can use either of the middle numbers. It'll throw off the efficiency a bit, but not substantially unless you always ended up sorting even arrays.
Finding the median of an unsorted set of numbers can be done in O(N) time, but it's not really necessary to find the true median for the purposes of quicksort's pivot. You just need to find a pivot that's reasonable.
As the Wikipedia entry for quicksort says:
In very early versions of quicksort, the leftmost element of the partition would often be chosen as the pivot element. Unfortunately, this causes worst-case behavior on already sorted arrays, which is a rather common use-case. The problem was easily solved by choosing either a random index for the pivot, choosing the middle index of the partition or (especially for longer partitions) choosing the median of the first, middle and last element of the partition for the pivot (as recommended by R. Sedgewick).
Finding the median of three values is much easier than finding it for the whole collection of values, and for collections that have an even number of elements, it doesn't really matter which of the two 'middle' elements you choose as the potential pivot.