I am writing a generic hash map in C++ which uses chaining to deal with collisions.
Say if I have a hash map with 11 buckets, and I insert 8 items. The hash function will distribute it as follows:
bucket[0] = empty
bucket[1] = 2 elements
bucket[2] = empty
bucket[3] = 1 element
bucket[4] = 1 element
bucket[5] = 3 elements
bucket[6] = empty
bucket[7] = 1 element
bucket[8] = empty
bucket[9] = empty
bucket[10] = empty
Calculating the spread over the buckets is 5/8 = 0.625.
But how do I calculate the spread taking the depth of the buckets into account?
I want to know this because:
Say if I added 20 elements, and every bucket has 1 element and the last bucket has 11 elements.
then the spread would be 1 if i calculate it the easy way, but this is obviously not correct! (the table resizes to avoid this of course, but I want to be able to show the spread) I want to use this information to be able to tune hash functions.
Thanks in advance!
If you're only using this to tune the hash functions themselves, you could compute a genuine measure of statistical dispersion, such as the Gini coefficient. On the other hand, if you're trying to make this a feature of the hash-map itself, I would recommend against it - computing a complicated benchmark as part of the 'is resize necessary' logic has its own performance costs; something naïve is probably better.
When I've worked on improving hash functions, I've use the sum of the squares of the lengths divided by the number of items inserted (and attempted to minimize the result). In your first example, you've inserted 8 items and the sum of the squares of the lengths is 16, so your "figure of merit" is 2.
In the second, you've inserted 20 items, and the sum of the squares is 130, so your figure of merit would be 6.5. I'd say the first was likely to be a better hash function in general (though I generally prefer to compare results from identical inputs).
You probably care about the answer because you want to know how much work you're doing with chaining. Thus, you probably should instrument your hash map to output how much work it's doing (a few #ifdefs that increment a counter in the key methods will probably do the trick). You then can use the amount of work (# compares, #nodes followed, etc.) as a metric for your hash function, and as a bonus you get a nifty tool for performance tuning. Once you figure things out, you can remove the instrumentation.
Related
If the input data entries are around 10 raised to power of 9, do we keep the size of the hash table the same as input size or reduce the size? how to decide the table size?
if we are using numbers in the range of 10 raised to power of 6 as the key, how do we hash these numbers to smaller values? I know we use the modulo operator but module with what?
Kindly explain how these two things work. Its getting quite confusing. Thanks!!
I tried to make the table size around 75% of the input data size, that you can call as X. Then I did key%(X) to get the hash code. But I am not sure if this is correct.
If the input data entries are around 10 raised to power of 9, do we keep the size of the hash table the same as input size or reduce the size? how to decide the table size?
The ratio of the number of elements stored to the number of buckets in the hash table is known as the load factor. In a separate chaining implementation, I'd suggest doing what std::unordered_set et al do and keeping it roughly in the range 0.5 to 1.0. So, for 10^9 elements have 10^9 to 2x10^9 buckets. Luckily, with separate chaining nothing awful happens if you go a bit outside this range (lower load factors just waste some memory on extra unused buckets, and higher load factors lead to increased collisions, longer lists and search times, but at load factors under 5 or 10 with an ok hash function the slow down will be roughly linear on average (so 5 or 10x slower than at load factor 1).
One important decision you should make is whether to pick a number around this magnitude that is a power of two, or a prime number. Explaining the implications is tedious, and anyway - which will work best for you is best determined by trying both and measuring the performance (if you really have to care about smallish differences in performance; if not - a prime number is the safer bet).
if we are using numbers in the range of 10 raised to power of 6 as the key, how do we hash these numbers to smaller values? I know we use the modulo operator but module with what?
Are these keys unsigned integers? In general, you can't have only 10^6 potential keys and end up with 10^9 hash table entries, as hash tables don't normally store duplicates (std::unordered_multiset/multi_map can, but it'll be easier for you to model that kind of thing as being a hash table from distinct keys to a container or values). More generally, it's best to separate the act of hashing (which usually is expected to generate a size_t result), from the "folding" of the hash value over the number of buckets in the hash table. That folding can be done using % in the general case, or by bitwise-ANDing with a bitmask for power-of-two bucket counts (e.g. for 256 buckets, & 255 is the same as % 256, but may execute faster on the CPU when those 255/256 values aren't known at compile time).
I tried to make the table size around 75% of the input data size, that you can call as X.
So that's a load factor around 1.33, which is ok.
Then I did key%(X) to get the hash code. But I am not sure if this is
correct.
It ends up being the same thing, but I'd suggest thinking of that as having a hash function hash(key) = key, followed by mod-ing into the bucket count. Such a hash function is known as an identity hash function, and is the implementation used for integers by all major C++ compiler Standard Libraries, though no particular hash functions are specified in the C++ Standard. It tends to work ok, but if your integer keys are particularly prone to collisions (for example, if they were all distinct multiples of 16 and your bucket count was a power of two they'd tend to only map to every 16th bucket) then it'd be better to use a stronger hash function. There are other questions about that - e.g. What integer hash function are good that accepts an integer hash key?
Rehashing
If the number of elements may increase dramatically beyond your initial expectations at run-time, then you'll want to increase the number of buckets to keep the load factor reasonable (in the range discussed above). Implementing support for that can easily be done by first writing a hash table class that doesn't support rehashing - simply taking the number of buckets to use as a constructor argument. Then write an outer rehashing-capable hash table class with a data member of the above type, and when an insert would push the load factor too high (Standard Library containers have a max_load_factor member which defaults to 1.0), you can construct an additional inner hash table object telling the constructor a new larger bucket count to use, then iterate over the smaller hash table inserting (or - better - moving, see below) the elements to the new hash table, then swap the two hash tables so the data member ends up with the new larger content and the smaller one is destructed. By "moving" above With a little I mean simply relink linked list elements from the smaller hash table into the lists in the larger one, instead of deep copying the elements, which will be dramatically faster and use less memory momentarily while rehashing.
I'm working on a problem where I have an entire table from a database in memory at all times, with a low range and high range of 9-digit numbers. I'm given a 9-digit number that I need to use to lookup the rest of the columns in the table based on whether that number falls in the range. For example, if the range was 100,000,000 to 125,000,000 and I was given a number 117,123,456, then I would know that I'm in the 100-125 mil range, and whatever vector of data that points to is what I will be using.
Now the best I can think of for lookup time is log(n) run time. This is OK, at best, but still pretty slow. The table has at least 100,000 entries and I will need to look up values in this table tens-of-thousands, if not hundred-thousands of times, per execution of this application (10+ times/day).
So I was wondering if it was possible to use an unordered_set instead, writing my own Hash function that ALWAYS returns the same hash-value for every number in range. Using the same example above, 100,000,000 through 125,000,000 will always return, for example, a hash value of AB12CD. Then when I use the lookup value of 117,123,456, I will get that same AB12CD hash and have a lookup time of O(1).
Is this possible, and if so, any ideas how?
Thanks in advance.
Yes. Assuming that you can number your intervals in order, you could fit a polynomial to your cutoff values, and receive an index value from the polynomial. For instance, with cutoffs of 100,000,000, 125,000,000, 250,000,000, and 327,000,000, you could use points (100, 0), (125, 1), (250, 2), and (327, 3), restricting the first derivative to [0, 1]. Assuming that you have decently-behaved intervals, you'll be able to fit this with an (N+2)th-degree polynomial for N cutoffs.
Have a table of desired hash values; use floor[polynomial(i)] for the index into the table.
Can you write such a hash function? Yes. Will evaluating it be slower than a search? Well there's the catch...
I would personally solve this problem as follows. I'd have a sorted vector of all values. And then I'd have a jump table of indexes into that vector based on the value of n >> 8.
So now your logic is that you look in the jump table to figure out where you are jumping to and how many values you should consider. (Just look at where you land versus the next index to see the size of the range.) If the whole range goes to the same vector, you're done. If there are only a few entries, do a linear search to find where you belong. If they are a lot of entries, do a binary search. Experiment with your data to find when binary search beats a linear search.
A vague memory suggests that the tradeoff is around 100 or so because predicting a branch wrong is expensive. But that is a vague memory from many years ago, so run the experiment for yourself.
Problem: I need to sample from a discrete distribution constructed of certain weights e.g. {w1,w2,w3,..}, and thus probability distribution {p1,p2,p3,...}, where pi=wi/(w1+w2+...).
some of wi's change very frequently, but only a very low proportion of all wi's. But the distribution itself thus has to be renormalised every time it happens, and therefore I believe Alias method does not work efficiently because one would need to build the whole distribution from scratch every time.
The method I am currently thinking is a binary tree (heap method), where all wi's are saved in the lowest level, and then the sum of each two in higher level and so on. The sum of all of them will be in the highest level, which is also a normalisation constant. Thus in order to update the tree after change in wi, one needs to do log(n) changes, as well as the same amount to get the sample from the distribution.
Question:
Q1. Do you have a better idea on how to achieve it faster?
Q2. The most important part: I am looking for a library which has already done this.
explanation: I have done this myself several years ago, by building heap structure in a vector, but since then I have learned many things including discovering libraries ( :) ), and containers such as map... Now I need to rewrite that code with higher functionality, and I want to make it right this time:
so Q2.1 is there a nice way to make a c++ map ordered and searched not by index, but by a cumulative sum of it's elements (this is how we sample, right?..). (that is my current theory how I would like to do it, but it doesnt have to be this way...)
Q2.2 Maybe there is some even nicer way to do the same? I would believe this problem is so frequent that I am very surprised I could not find some sort of library which would do it for me...
Thank you very much, and I am very sorry if this has been asked in some other form, please direct me towards it, but I have spent a good while looking...
-z
Edit: There is a possibility that I might need to remove or add the elements as well, but I think I could avoid it, if that makes a huge difference, thus leaving only changing the value of the weights.
Edit2: weights are reals in general, I would have to think if I could make them integers...
I would actually use a hash set of strings (don't remember the C++ container for it, you might need to implement your own though). Put wi elements for each i, with the values "w1_1", "w1_2",... all through "w1_[w1]" (that is, w1 elements starting with "w1_").
When you need to sample, pick an element at random using a uniform distribution. If you picked w5_*, say you picked element 5. Because of the number of elements in the hash, this will give you the distribution you were looking for.
Now, when wi changes from A to B, just add B-A elements to the hash (if B>A), or remove the last A-B elements of wi (if A>B).
Adding new elements and removing old elements is trivial in this case.
Obviously the problem is 'pick an element at random'. If your hash is a closed hash, you pick an array cell at random, if it's empty - just pick one at random again. If you keep your hash 3 or 4 times larger than the total sum of weights, your complexity will be pretty good: O(1) for retrieving a random sample, O(|A-B|) for modifying the weights.
Another option, since only a small part of your weights change, is to split the weights into two - the fixed part and the changed part. Then you only need to worry about changes in the changed part, and the difference between the total weight of changed parts and the total weight of unchanged parts. Then for the fixed part your hash becomes a simple array of numbers: 1 appears w1 times, 2 appears w2 times, etc..., and picking a random fixed element is just picking a random number.
Updating your normalisation factor when you change a value is trivial. This might suggest an algorithm.
w_sum = w_sum_old - w_i_old + w_i_new;
If you leave p_i as a computed property p_i = w_i / w_sum you would avoid recalculating the entire p_i array at the cost of calculating p_i every time they are needed. You would, however, be able to update many statistical properties without recalculating the entire sum
expected_something = (something_1 * w_1 + something_2 * w_2 + ...) / w_sum;
With a bit of algebra you can update expected_something by subtracting the contribution with the old weight and add the contribution with the new weight, multiplying and dividing with the normalization factors as required.
If you during the sampling keep track of which outcomes that are part of the sample, it would be possible to propagate how the probabilities were updated to the generated sample. Would this make it possible for you to update rather than recalculate values related to the sample? I think a bitmap could provide an efficient way to store an index of which outcomes that were used to build the sample.
One way of storing the probabilities together with the sums is to start with all probabilities. In the next N/2 positions you store the sums of the pairs. After that N/4 sums of the pairs etc. Where the sums are located can, obviously, be calculate in O(1) time. This data-structure is sort of a heap, but upside down.
I have around 400.000 "items".
Each "item" consists of 16 double values.
At runtime I need to compare items with each other. Therefore I am muplicating their double values. This is quite time-consuming.
I have made some tests, and I found out that there are only 40.000 possible return values, no matter which items I compare with each other.
I would like to store these values in a look-up table so that I can easily retrieve them without doing any real calculation at runtime.
My question would be how to efficiently store the data in a look-up table.
The problem is that if I create a look-up table, it gets amazingly huge, for example like this:
item-id, item-id, compare return value
1 1 499483,49834
1 2 -0.0928
1 3 499483,49834
(...)
It would sum up to around 120 million combinations.
That just looks too big for a real-world application.
But I am not sure how to avoid that.
Can anybody please share some cool ideas?
Thank you very much!
Assuming I understand you correctly, You have two inputs with 400K possibilities, so 400K * 400K = 160B entries... assuming you have them indexed sequentially, and the you stored your 40K possibilities in a way that allowed 2-octets each, you're looking at a table size of roughly 300GB... pretty sure that's beyond current every-day computing. So, you might instead research if there is any correlation between the 400K "items", and if so, if you can assign some kind of function to that correlation that gives you a clue (read: hash function) as to which of the 40K results might/could/should result. Clearly your hash function and lookup needs to be shorter than just doing the multiplication in the first place. Or maybe you can reduce the comparison time with some kind of intelligent reduction, like knowing the result under certain scenarios. Or perhaps some of your math can be optimized using integer math or boolean comparisons. Just a few thoughts...
To speed things up, you should probably compute all of the possible answers, and store the inputs to each answer.
Then, I would recommend making some sort of look up table that uses the answer as the key(since the answers will all be unique), and then storing all of the possible inputs that get that result.
To help visualize:
Say you had the table 'Table'. Inside Table you have keys, and associated to those keys are values. What you do is you make the keys have the type of whatever format your answers are in(the keys will be all of your answers). Now, give your 400k inputs each a unique identifier. You then store the unique identifiers for a multiplication as one value associated to that particular key. When you compute that same answer again, you just add it as another set of inputs that can calculate that key.
Example:
Table<AnswerType, vector<Input>>
Define Input like:
struct Input {IDType one, IDType two}
Where one 'Input' might have ID's 12384, 128, meaning that the objects identified by 12384 and 128, when multiplied, will give the answer.
So, in your lookup, you'll have something that looks like:
AnswerType lookup(IDType first, IDType second)
{
foreach(AnswerType k in table)
{
if table[k].Contains(first, second)
return k;
}
}
// Defined elsewhere
bool Contains(IDType first, IDType second)
{
foreach(Input i in [the vector])
{
if( (i.one == first && i.two == second ) ||
(i.two == first && i.one == second )
return true;
}
}
I know this isn't real C++ code, its just meant as pseudo-code, and it's a rough cut as-is, but it might be a place to start.
While the foreach is probably going to be limited to a linear search, you can make the 'Contains' method run a binary search by sorting how the inputs are stored.
In all, you're looking at a run-once application that will run in O(n^2) time, and a lookup that will run in nlog(n). I'm not entirely sure how the memory will look after all of that, though. Of course, I don't know much about the math behind it, so you might be able to speed up the linear search if you can somehow sort the keys as well.
For my application I have to handle a bunch of objects (let's say ints) that gets subsequently divided and sorted into smaller buckets. To this end, I store the elements in a single continuous array
arr = {0,1,2,3,4,5,6,7,8,9,10,11,12,13,14...}
and the information about the buckets (sublists) is given by offsets to the first element in the respective bucket and the lengths of the sublist.
So, for instance, given
offsets = {0,3,8,..}
sublist_lengths = {3,5,2,...}
would result in the following splits:
0 1 2 || 3 4 5 6 7 || 8 9 || ...
What I am looking for is a somewhat general and efficient way to run algorithms, like reductions, on the buckets only using either custom kernels or the thrust library. Summing the buckets should give:
3 || 25 || 17 || ...
What I've come up with:
option 1: custom kernels require a quite a bit of tinkering, copies into shared memory, proper choice of block and grid sizes and an own implementation of the algorithms, like scan, reduce, etc. Also, every single operation would require an own custom kernel. In general it is clear to me how to do this, but after having used thrust for the last couple of days I have the impression that there might be a smarter way
option 2: generate an array of keys from the offsets ({0,0,0,1,1,1,1,1,2,2,3,...} in the above example) and use thrust::reduce_by_key. I don't like the extra list generation, though.
option 3: Use thrust::transform_iterator together with thrust::counting_iterator to generate the above given key list on the fly. Unfortunately, I can't come up with an implementation that doesn't require increments of indices to the offset list on the device and defeats parallelism.
What would be the most sane way to implement this?
Within Thrust, I can't think of a better solution than Option 2. The performance will not be terrible, but it's certainly not optimal.
Your data structure bears similarity to the Compressed Sparse Row (CSR) format for storing sparse matrices, so you could use techniques developed for computing sparse matrix-vector multiplies (SpMV) for such matrices if you want better performance. Note that the "offsets" array of the CSR format has length (N+1) for a matrix with N rows (i.e. buckets in your case) where the last offset value is the length of arr. The CSR SpMV code in Cusp is a bit convoluted, but it serves as a good starting point for your kernel. Simply remove any reference to Aj or x from the code and pass offsets and arr into the Ap and Av arguments respectively.
You didn't mention how big the buckets are. If the buckets are big enough, maybe you can get away with copying the offsets and sublist_lengths to the host, iterating over them and doing a separate Thrust call for each bucket. Fermi can have 16 kernels in flight at the same time, so on that architecture you might be able to handle smaller buckets and still get good utilization.