问题
I'm trying to figure out if it's possible to efficiently calculate the conditional entropy of a set of numbers using CUDA. You can calculate the conditional entropy by dividing an array into windows, then counting the number of matching subarrays/substrings for different lengths. For each subarray length, you calculate the entropy by adding together the matching subarray counts times the log of those counts. Then, whatever you get as the minimum entropy is the conditional entropy.
To give a more clear example of what I mean, here is full calculation:
The initial array is [1,2,3,5,1,2,5]. Assuming the window size is 3, this must be divided into five windows: [1,2,3], [2,3,5], [3,5,1], [5,1,2], and [1,2,5].
Next, looking at each window, we want to find the matching subarrays for each length.
The subarrays of length 1 are [1],[2],[3],[5],[1]. There are two 1s, and one of each other number. So the entropy is log(2)2 + 4(log(1)*1) = 0.6.
The subarrays of length 2 are [1,2], [2,3], [3,5], [5,1], and [1,2]. There are two [1,2]s, and four unique subarrays. The entropy is the same as length 1, log(2)2 + 4(log(1)*1) = 0.6.
The subarrays of length 3 are the full windows: [1,2,3], [2,3,5], [3,5,1], [5,1,2], and [1,2,5]. All five windows are unique, so the entropy is 5*(log(1)*1) = 0.
The minimum entropy is 0, meaning it is the conditional entropy for this array.
This can also be presented as a tree, where the counts at each node represent how many matches exist. The entropy for each subarray length is equivalent to the entropy for each level of the tree.
If possible, I'd like to perform this calculation on many arrays at once, and also perform the calculation itself in parallel. Does anyone have suggestions on how to accomplish this? Could thrust be useful? Please let me know if there is any additional information I should provide.
回答1:
I tried solving your problem using thrust. It works, but it results in a lot of thrust calls. Since your input size is rather small, you should process multiple arrays in parallel. However, doing this results in a lot of book-keeping effort, you will see this in the following code.
Your input range is limited to [1,5], which is equivalent to [0,4]. The general idea is that (theoretically) any tuple out of this range (e.g. {1,2,3} can be represented as a number in base 4 (e.g. 1+2*4+3*16 = 57).
In practice we are limited by the size of the integer type. For a 32bit unsigned integer this will lead to a maximum tuple size of 16. This is also the maximum window size the following code can handle (changing to a 64bit unsigned integer will lead to a maximum tuple size of 32).
Let's assume the input data is structured like this:
We have 2 arrays we want to process in parallel, each array is of size 5 and window size is 3.
{{0,0,3,4,4},{0,2,1,1,3}}
We can now generate all windows:
{{0,0,3},{0,3,4},{3,4,4}},{{0,2,1},{2,1,1},{1,1,3}}
Using a per tuple prefix sum and applying the aforementioned representation of each tuple as a single base-4 number, we get:
{{0,0,48},{0,12,76},{3,19,83}},{{0,8,24},{2,6,22},{1,5,53}}
Now we reorder the values so we have the numbers which represent a subarray of a specific length next to each other:
{{0,0,3},{0,12,19},{48,76,83}},{0,2,1},{8,6,5},{24,22,53}}
We then sort within each group:
{{0,0,3},{0,12,19},{48,76,83}},{0,1,2},{5,6,8},{22,24,53}}
Now we can count how often a number occurs in each group:
2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
Applying the log-formula results in
0.60206,0,0,0,0,0
Now we fetch the minimum value per array:
0,0
#include <thrust/device_vector.h>
#include <thrust/copy.h>
#include <thrust/transform.h>
#include <thrust/iterator/counting_iterator.h>
#include <thrust/iterator/zip_iterator.h>
#include <thrust/functional.h>
#include <thrust/random.h>
#include <iostream>
#include <thrust/tuple.h>
#include <thrust/reduce.h>
#include <thrust/scan.h>
#include <thrust/gather.h>
#include <thrust/sort.h>
#include <math.h>
#include <chrono>
#ifdef PRINT_ENABLED
#define PRINTER(name) print(#name, (name))
#else
#define PRINTER(name)
#endif
template <template <typename...> class V, typename T, typename ...Args>
void print(const char* name, const V<T,Args...> & v)
{
std::cout << name << ":\t";
thrust::copy(v.begin(), v.end(), std::ostream_iterator<T>(std::cout, "\t"));
std::cout << std::endl;
}
template <typename Integer, Integer Min, Integer Max>
struct random_filler
{
__device__
Integer operator()(std::size_t index) const
{
thrust::default_random_engine rng;
thrust::uniform_int_distribution<Integer> dist(Min, Max);
rng.discard(index);
return dist(rng);
}
};
template <std::size_t ArraySize,
std::size_t ArrayCount,
std::size_t WindowSize,
typename T,
std::size_t WindowCount = ArraySize - (WindowSize-1),
std::size_t PerArrayCount = WindowSize * WindowCount>
__device__ __inline__
thrust::tuple<T,T,T,T> calc_indices(const T& i0)
{
const T i1 = i0 / PerArrayCount;
const T i2 = i0 % PerArrayCount;
const T i3 = i2 / WindowSize;
const T i4 = i2 % WindowSize;
return thrust::make_tuple(i1,i2,i3,i4);
}
template <typename Iterator,
std::size_t ArraySize,
std::size_t ArrayCount,
std::size_t WindowSize,
std::size_t WindowCount = ArraySize - (WindowSize-1),
std::size_t PerArrayCount = WindowSize * WindowCount,
std::size_t TotalCount = PerArrayCount * ArrayCount
>
class sliding_window
{
public:
typedef typename thrust::iterator_difference<Iterator>::type difference_type;
struct window_functor : public thrust::unary_function<difference_type,difference_type>
{
__host__ __device__
difference_type operator()(const difference_type& i0) const
{
auto t = calc_indices<ArraySize, ArrayCount,WindowSize>(i0);
return thrust::get<0>(t) * ArraySize + thrust::get<2>(t) + thrust::get<3>(t);
}
};
typedef typename thrust::counting_iterator<difference_type> CountingIterator;
typedef typename thrust::transform_iterator<window_functor, CountingIterator> TransformIterator;
typedef typename thrust::permutation_iterator<Iterator,TransformIterator> PermutationIterator;
typedef PermutationIterator iterator;
sliding_window(Iterator first) : first(first){}
iterator begin(void) const
{
return PermutationIterator(first, TransformIterator(CountingIterator(0), window_functor()));
}
iterator end(void) const
{
return begin() + TotalCount;
}
protected:
Iterator first;
};
template <std::size_t ArraySize,
std::size_t ArrayCount,
std::size_t WindowSize,
typename Iterator>
sliding_window<Iterator, ArraySize, ArrayCount, WindowSize>
make_sliding_window(Iterator first)
{
return sliding_window<Iterator, ArraySize, ArrayCount, WindowSize>(first);
}
template <typename KeyType,
std::size_t ArraySize,
std::size_t ArrayCount,
std::size_t WindowSize>
struct key_generator : thrust::unary_function<KeyType, thrust::tuple<KeyType,KeyType> >
{
__device__
thrust::tuple<KeyType,KeyType> operator()(std::size_t i0) const
{
auto t = calc_indices<ArraySize, ArrayCount,WindowSize>(i0);
return thrust::make_tuple(thrust::get<0>(t),thrust::get<2>(t));
}
};
template <typename Integer,
std::size_t Base,
std::size_t ArraySize,
std::size_t ArrayCount,
std::size_t WindowSize>
struct base_n : thrust::unary_function<thrust::tuple<Integer, Integer>, Integer>
{
__host__ __device__
Integer operator()(const thrust::tuple<Integer, Integer> t) const
{
const auto i = calc_indices<ArraySize, ArrayCount, WindowSize>(thrust::get<0>(t));
// ipow could be optimized by precomputing a lookup table at compile time
const auto result = thrust::get<1>(t)*ipow(Base, thrust::get<3>(i));
return result;
}
// taken from http://stackoverflow.com/a/101613/678093
__host__ __device__ __inline__
Integer ipow(Integer base, Integer exp) const
{
Integer result = 1;
while (exp)
{
if (exp & 1)
result *= base;
exp >>= 1;
base *= base;
}
return result;
}
};
template <std::size_t ArraySize,
std::size_t ArrayCount,
std::size_t WindowSize,
typename T,
std::size_t WindowCount = ArraySize - (WindowSize-1),
std::size_t PerArrayCount = WindowSize * WindowCount>
__device__ __inline__
thrust::tuple<T,T,T,T> calc_sort_indices(const T& i0)
{
const T i1 = i0 % PerArrayCount;
const T i2 = i0 / PerArrayCount;
const T i3 = i1 % WindowCount;
const T i4 = i1 / WindowCount;
return thrust::make_tuple(i1,i2,i3,i4);
}
template <typename Integer,
std::size_t ArraySize,
std::size_t ArrayCount,
std::size_t WindowSize,
std::size_t WindowCount = ArraySize - (WindowSize-1),
std::size_t PerArrayCount = WindowSize * WindowCount>
struct pre_sort : thrust::unary_function<Integer, Integer>
{
__device__
Integer operator()(Integer i0) const
{
auto t = calc_sort_indices<ArraySize, ArrayCount,WindowSize>(i0);
const Integer i_result = ( thrust::get<2>(t) * WindowSize + thrust::get<3>(t) ) + thrust::get<1>(t) * PerArrayCount;
return i_result;
}
};
template <typename Integer,
std::size_t ArraySize,
std::size_t ArrayCount,
std::size_t WindowSize,
std::size_t WindowCount = ArraySize - (WindowSize-1),
std::size_t PerArrayCount = WindowSize * WindowCount>
struct generate_sort_keys : thrust::unary_function<Integer, Integer>
{
__device__
thrust::tuple<Integer,Integer> operator()(Integer i0) const
{
auto t = calc_sort_indices<ArraySize, ArrayCount,WindowSize>(i0);
return thrust::make_tuple( thrust::get<1>(t), thrust::get<3>(t));
}
};
template<typename... Iterators>
__host__ __device__
thrust::zip_iterator<thrust::tuple<Iterators...>> zip(Iterators... its)
{
return thrust::make_zip_iterator(thrust::make_tuple(its...));
}
struct calculate_log : thrust::unary_function<std::size_t, float>
{
__host__ __device__
float operator()(std::size_t i) const
{
return i*log10f(i);
}
};
int main()
{
typedef int Integer;
typedef float Real;
const std::size_t array_count = ARRAY_COUNT;
const std::size_t array_size = ARRAY_SIZE;
const std::size_t window_size = WINDOW_SIZE;
const std::size_t window_count = array_size - (window_size-1);
const std::size_t input_size = array_count * array_size;
const std::size_t base = 4;
thrust::device_vector<Integer> input_arrays(input_size);
thrust::counting_iterator<Integer> counting_it(0);
thrust::transform(counting_it,
counting_it + input_size,
input_arrays.begin(),
random_filler<Integer,0,base>());
PRINTER(input_arrays);
const int runs = 100;
auto start = std::chrono::high_resolution_clock::now();
for (int k = 0 ; k < runs; ++k)
{
auto sw = make_sliding_window<array_size, array_count, window_size>(input_arrays.begin());
const std::size_t total_count = window_size * window_count * array_count;
thrust::device_vector<Integer> result(total_count);
thrust::copy(sw.begin(), sw.end(), result.begin());
PRINTER(result);
auto ti_begin = thrust::make_transform_iterator(counting_it, key_generator<Integer, array_size, array_count, window_size>());
auto base_4_ti = thrust::make_transform_iterator(zip(counting_it, sw.begin()), base_n<Integer, base, array_size, array_count, window_size>());
thrust::inclusive_scan_by_key(ti_begin, ti_begin+total_count, base_4_ti, result.begin());
PRINTER(result);
thrust::device_vector<Integer> result_2(total_count);
auto ti_pre_sort = thrust::make_transform_iterator(counting_it, pre_sort<Integer, array_size, array_count, window_size>());
thrust::gather(ti_pre_sort,
ti_pre_sort+total_count,
result.begin(),
result_2.begin());
PRINTER(result_2);
thrust::device_vector<Integer> sort_keys_1(total_count);
thrust::device_vector<Integer> sort_keys_2(total_count);
auto zip_begin = zip(sort_keys_1.begin(),sort_keys_2.begin());
thrust::transform(counting_it,
counting_it+total_count,
zip_begin,
generate_sort_keys<Integer, array_size, array_count, window_size>());
thrust::stable_sort_by_key(result_2.begin(), result_2.end(), zip_begin);
thrust::stable_sort_by_key(zip_begin, zip_begin+total_count, result_2.begin());
PRINTER(result_2);
thrust::device_vector<Integer> key_counts(total_count);
thrust::device_vector<Integer> sort_keys_1_reduced(total_count);
thrust::device_vector<Integer> sort_keys_2_reduced(total_count);
// count how often each sub array occurs
auto zip_count_begin = zip(sort_keys_1.begin(), sort_keys_2.begin(), result_2.begin());
auto new_end = thrust::reduce_by_key(zip_count_begin,
zip_count_begin + total_count,
thrust::constant_iterator<Integer>(1),
zip(sort_keys_1_reduced.begin(), sort_keys_2_reduced.begin(), thrust::make_discard_iterator()),
key_counts.begin()
);
std::size_t new_size = new_end.second - key_counts.begin();
key_counts.resize(new_size);
sort_keys_1_reduced.resize(new_size);
sort_keys_2_reduced.resize(new_size);
PRINTER(key_counts);
PRINTER(sort_keys_1_reduced);
PRINTER(sort_keys_2_reduced);
auto log_ti = thrust::make_transform_iterator (key_counts.begin(), calculate_log());
thrust::device_vector<Real> log_result(new_size);
auto zip_keys_reduced_begin = zip(sort_keys_1_reduced.begin(), sort_keys_2_reduced.begin());
auto log_end = thrust::reduce_by_key(zip_keys_reduced_begin,
zip_keys_reduced_begin + new_size,
log_ti,
zip(sort_keys_1.begin(),thrust::make_discard_iterator()),
log_result.begin()
);
std::size_t final_size = log_end.second - log_result.begin();
log_result.resize(final_size);
sort_keys_1.resize(final_size);
PRINTER(log_result);
thrust::device_vector<Real> final_result(final_size);
auto final_end = thrust::reduce_by_key(sort_keys_1.begin(),
sort_keys_1.begin() + final_size,
log_result.begin(),
thrust::make_discard_iterator(),
final_result.begin(),
thrust::equal_to<Integer>(),
thrust::minimum<Real>()
);
final_result.resize(final_end.second-final_result.begin());
PRINTER(final_result);
}
auto duration = std::chrono::duration_cast<std::chrono::milliseconds>(std::chrono::high_resolution_clock::now() - start);
std::cout << "took " << duration.count()/runs << "milliseconds" << std::endl;
return 0;
}
compile using
nvcc -std=c++11 conditional_entropy.cu -o benchmark -DARRAY_SIZE=1000 -DARRAY_COUNT=1000 -DWINDOW_SIZE=10 && ./benchmark
This configuration takes 133 milliseconds on my GPU (GTX 680), so around 0.1 milliseconds per array.
The implementation can definitely be optimized, e.g. using a precomputed lookup table for the base-4 conversion and maybe some of the thrust calls can be avoided.
来源：https://stackoverflow.com/questions/32386014/cuda-thrust-counting-matching-subarrays