Optimized way to find M largest elements in an NxN array using C++

I need a blazing fast way to find the 2D positions and values of the M largest elements in an NxN array.

right now I'm doing this:

struct SourcePoint {
    Point point;
    float value;
}

SourcePoint* maxValues = new SourcePoint[ M ];

maxCoefficients = new SourcePoint*[
for (int j = 0; j < rows; j++) {
    for (int i = 0; i < cols; i++) {
        float sample = arr[i][j];
        if (sample > maxValues[0].value) {
            int q = 1;
            while ( sample > maxValues[q].value && q < M ) {
                maxValues[q-1] = maxValues[q];      // shuffle the values back
                 q++;
            }
            maxValues[q-1].value = sample;
            maxValues[q-1].point = Point(i,j);
        }
    }
}

A Point struct is just two ints - x and y.

This code basically does an insertion sort of the values coming in. maxVa开发者_开发问答lues[0] always contains the SourcePoint with the lowest value that still keeps it within the top M values encoutered so far. This gives us a quick and easy bailout if sample <= maxValues, we don't do anything. The issue I'm having is the shuffling every time a new better value is found. It works its way all the way down maxValues until it finds it's spot, shuffling all the elements in maxValues to make room for itself.

I'm getting to the point where I'm ready to look into SIMD solutions, or cache optimisations, since it looks like there's a fair bit of cache thrashing happening. Cutting the cost of this operation down will dramatically affect the performance of my overall algorithm since this is called many many times and accounts for 60-80% of my overall cost.

I've tried using a std::vector and make_heap, but I think the overhead for creating the heap outweighed the savings of the heap operations. This is likely because M and N generally aren't large. M is typically 10-20 and N 10-30 (NxN 100 - 900). The issue is this operation is called repeatedly, and it can't be precomputed.

I just had a thought to pre-load the first M elements of maxValues which may provide some small savings. In the current algorithm, the first M elements are guaranteed to shuffle themselves all the way down just to initially fill maxValues.

Any help from optimization gurus would be much appreciated :)

A few ideas you can try. In some quick tests with N=100 and M=15 I was able to get it around 25% faster in VC++ 2010 but test it yourself to see whether any of them help in your case. Some of these changes may have no or even a negative effect depending on the actual usage/data and compiler optimizations.

• Don't allocate a new maxValues array each time unless you need to. Using a stack variable instead of dynamic allocation gets me +5%.
• Changing g_Source[i][j] to g_Source[j][i] gains you a very little bit (not as much as I'd thought there would be).
• Using the structure SourcePoint1 listed at the bottom gets me another few percent.
• The biggest gain of around +15% was to replace the local variable sample with g_Source[j][i]. The compiler is likely smart enough to optimize out the multiple reads to the array which it can't do if you use a local variable.
• Trying a simple binary search netted me a small loss of a few percent. For larger M/Ns you'd likely see a benefit.
• If possible try to keep the source data in arr[][] sorted, even if only partially. Ideally you'd want to generate maxValues[] at the same time the source data is created.
• Look at how the data is created/stored/organized may give you patterns or information to reduce the amount of time to generate your maxValues[] array. For example, in the best case you could come up with a formula that gives you the top M coordinates without needing to iterate and sort.

Code for above:

struct SourcePoint1 {
     int x;
     int y;
     float value;
     int test;       //Play with manual/compiler padding if needed
};

If you want to go into micro-optimizations at this point, the a simple first step should be to get rid of the Points and just stuff both dimensions into a single int. That reduces the amount of data you need to shift around, and gets SourcePoint down to being a power of two long, which simplifies indexing into it.

Also, are you sure that keeping the list sorted is better than simply recomputing which element is the new lowest after each time you shift the old lowest out?

(Updated 22:37 UTC 2011-08-20)

I propose a binary min-heap of fixed size holding the M largest elements (but still in min-heap order!). It probably won't be faster in practice, as I think OPs insertion sort probably has decent real world performance (at least when the recommendations of the other posteres in this thread are taken into account).

Look-up in the case of failure should be constant time: If the current element is less than the minimum element of the heap (containing the max M elements) we can reject it outright.

If it turns out that we have an element bigger than the current minimum of the heap (the Mth biggest element) we extract (discard) the previous min and insert the new element.

If the elements are needed in sorted order the heap can be sorted afterwards.

First attempt at a minimal C++ implementation:

template<unsigned size, typename T>
class m_heap {
private: 
    T nodes[size];
    static const unsigned last = size - 1;

    static unsigned parent(unsigned i) { return (i - 1) / 2; }
    static unsigned left(unsigned i) { return i * 2; }
    static unsigned right(unsigned i) { return i * 2 + 1; }

    void bubble_down(unsigned int i) {
        for (;;) { 
            unsigned j = i;
            if (left(i) < size && nodes[left(i)] < nodes[i])
                j = left(i);
            if (right(i) < size && nodes[right(i)] < nodes[j])
                j = right(i);
            if (i != j) {
                swap(nodes[i], nodes[j]);
                i = j;
            } else {
                break;
            }
        }
    }

    void bubble_up(unsigned i) {
        while (i > 0 && nodes[i] < nodes[parent(i)]) {
            swap(nodes[parent(i)], nodes[i]);
            i = parent(i);
        }
    }

public:
    m_heap() {
        for (unsigned i = 0; i < size; i++) {
            nodes[i] = numeric_limits<T>::min();
        }
    }

    void add(const T& x) {
        if (x < nodes[0]) {
            // reject outright 
            return;
        }
        nodes[0] = x;
        swap(nodes[0], nodes[last]);
        bubble_down(0);
    }
};

Small test/usage case:

#include <iostream>
#include <limits>
#include <algorithm>
#include <vector>
#include <stdlib.h>
#include <assert.h>
#include <math.h>

using namespace std;

// INCLUDE TEMPLATED CLASS FROM ABOVE

typedef vector<float> vf;
bool compare(float a, float b) { return a > b; }

int main()
{
    int N = 2000;
    vf v;
    for (int i = 0; i < N; i++) v.push_back( rand()*1e6 / RAND_MAX);

    static const int M = 50;
    m_heap<M, float> h;
    for (int i = 0; i < N; i++) h.add( v[i] );

    sort(v.begin(), v.end(), compare);

    vf heap(h.get(), h.get() + M); // assume public in m_heap: T* get() { return nodes; }
    sort(heap.begin(), heap.end(), compare);

    cout << "Real\tFake" << endl;
    for (int i = 0; i < M; i++) {
        cout << v[i] << "\t" << heap[i] << endl;
        if (fabs(v[i] - heap[i]) > 1e-5) abort();
    }
}

You're looking for a priority queue:

template < class T, class Container = vector<T>,
       class Compare = less<typename Container::value_type> > 
       class priority_queue;

You'll need to figure out the best underlying container to use, and probably define a Compare function to deal with your Point type.

If you want to optimize it, you could run a queue on each row of your matrix in its own worker thread, then run an algorithm to pick the largest item of the queue fronts until you have your M elements.

A quick optimization would be to add a sentinel value to yourmaxValues array. If you have maxValues[M].value equal to std::numeric_limits<float>::max() then you can eliminate the q < M test in your while loop condition.

One idea would be to use the std::partial_sort algorithm on a plain one-dimensional sequence of references into your NxN array. You could probably also cache this sequence of references for subsequent calls. I don't know how well it performs, but it's worth a try - if it works good enough, you don't have as much "magic". In particular, you don't resort to micro optimizations.

Consider this showcase:

#include <algorithm>
#include <iostream>
#include <vector>

#include <stddef.h>

static const int M = 15;
static const int N = 20;

// Represents a reference to a sample of some two-dimensional array
class Sample
{
public:
    Sample( float *arr, size_t row, size_t col )
        : m_arr( arr ),
        m_row( row ),
        m_col( col )
    {
    }

    inline operator float() const {
        return m_arr[m_row * N + m_col];
    }

    bool operator<( const Sample &rhs ) const {
        return (float)other < (float)*this;
    }

    int row() const {
        return m_row;
    }

    int col() const {
        return m_col;
    }

private:
    float *m_arr;
    size_t m_row;
    size_t m_col;
};

int main()
{
    // Setup a demo array
    float arr[N][N];
    memset( arr, 0, sizeof( arr ) );

    // Put in some sample values
    arr[2][1] = 5.0;
    arr[9][11] = 2.0;
    arr[5][4] = 4.0;
    arr[15][7] = 3.0;
    arr[12][19] = 1.0;

    //  Setup the sequence of references into this array; you could keep
    // a copy of this sequence around to reuse it later, I think.
    std::vector<Sample> samples;
    samples.reserve( N * N );
    for ( size_t row = 0; row < N; ++row ) {
        for ( size_t col = 0; col < N; ++col ) {
            samples.push_back( Sample( (float *)arr, row, col ) );
        }
    }

    // Let partial_sort find the M largest entry
    std::partial_sort( samples.begin(), samples.begin() + M, samples.end() );

    // Print out the row/column of the M largest entries.
    for ( std::vector<Sample>::size_type i = 0; i < M; ++i ) {
        std::cout << "#" << (i + 1) << " is " << (float)samples[i] << " at " << samples[i].row() << "/" << samples[i].col() << std::endl;
    }
}

First of all, you are marching through the array in the wrong order!

You always, always, always want to scan through memory linearly. That means the last index of your array needs to be changing fastest. So instead of this:

for (int j = 0; j < rows; j++) {
    for (int i = 0; i < cols; i++) {
        float sample = arr[i][j];

Try this:

for (int i = 0; i < cols; i++) {
    for (int j = 0; j < rows; j++) {
        float sample = arr[i][j];

I predict this will make a bigger difference than any other single change.

Next, I would use a heap instead of a sorted array. The standard <algorithm> header already has push_heap and pop_heap functions to use a vector as a heap. (This will probably not help all that much, though, unless M is fairly large. For small M and a randomized array, you do not wind up doing all that many insertions on average... Something like O(log N) I believe.)

Next after that is to use SSE2. But that is peanuts compared to marching through memory in the right order.

You should be able to get nearly linear speedup with parallel processing.

With N CPUs, you can process a band of rows/N rows (and all columns) with each CPU, finding the top M entries in each band. And then do a selection sort to find the overall top M.

You could probably do that with SIMD as well (but here you'd divide up the task by interleaving columns instead of banding the rows). Don't try to make SIMD do your insertion sort faster, make it do more insertion sorts at once, which you combine at the end using a single very fast step.

Naturally you could do both multi-threading and SIMD, but on a problem which is only 30x30, that's not likely to be worthwhile.

I tried replacing float by double, and interestingly that gave me a speed improvement of about 20% (using VC++ 2008). That's a bit counterintuitive, but it seems modern processors or compilers are optimized for double value processing.

Use a linked list to store the best yet M values. You'll still have to iterate over it to find the right spot, but the insertion is O(1). It would probably even be better than binary search and insertion O(N)+O(1) vs O(lg(n))+O(N). Interchange the fors, so you're not accessing every N element in memory and trashing the cache.

LE: Throwing another idea that might work for uniformly distributed values.
Find the min, max in 3/2*O(N^2) comparisons.
Create anywhere from N to N^2 uniformly distributed buckets, preferably closer to N^2 than N.
For every element in the NxN matrix place it in bucket[(int)(value-min)/range], range=max-min.
Finally create a set starting from the highest bucket to the lowest, add elements from other buckets to it while |current set| + |next bucket| <=M.
If you get M elements you're done. You'll likely get less elements than M, let's say P.
Apply your algorithm for the remaining bucket and get biggest M-P elements out of it.
If elements are uniform and you use N^2 buckets it's complexity is about 3.5*(N^2) vs your current solution which is about O(N^2)*ln(M).