What is the best way to create a sparse array in C++?

I am working on a project that requires the manipulation of enormous matrices, specifically pyramidal summation for a copula calculation.

In short, I need to keep track of a relatively small number of values (usually a value of 1, and in rare cases more than 1) in a sea of zeros in the matrix (multidimensional array).

A sparse array allows the user to store a small number of values, and assume all undefined records to be a preset value. Since it is not physically possibly to store all values in memory, I need to store only the few non-zero elements. This could be several million entries.

Speed is a huge priority, and I would also like to dynamically choose the number of variables in the class at runtime.

I currently work on a system that uses a binary search tree (b-tree) to store entries. Does anyone know of a better system?

For C++, a map works well. Several million objects won't be a problem. 10 million items took about 4.4 seconds and about 57 meg on my computer.

My test application is as follows:

#include <stdio.h>
#include <stdlib.h>
#include <map>

class triple {
public:
    int x;
    int y;
    int z;
    bool operator<(const triple &other) const {
        if (x < other.x) return true;
        if (other.x < x) return false;
        if (y < other.y) return true;
        if (other.y < y) return false;
        return z < other.z;
    }
};

int main(int, char**)
{
    std::map<triple,int> data;
    triple point;
    int i;

    for (i = 0; i < 10000000; ++i) {
        point.x = rand();
        point.y = rand();
        point.z = rand();
        //printf("%d %d %d %d\n", i, point.x, point.y, point.z);
        data[point] = i;
    }
    return 0;
}

Now to dynamically choose the number of variables, the easiest solution is to represent index as a string, and then use string as a key for the map. For instance, an item located at [23][55] can be represented via "23,55" string. We can also extend this solution for higher dimensions; such as for three dimensions an arbitrary index will look like "34,45,56". A simple implementation of this technique is as follows:

std::map data<string,int> data;
char ix[100];

sprintf(ix, "%d,%d", x, y); // 2 vars
data[ix] = i;

sprintf(ix, "%d,%d,%d", x, y, z); // 3 vars
data[ix] = i;

The accepted answer recommends using strings to represent multi-dimensional indices.

However, constructing strings is needlessly wasteful for this. If the size isn’t known at compile time (and thus std::tuple doesn’t work), std::vector works well as an index, both with hash maps and ordered trees. For std::map, this is almost trivial:

#include <vector>
#include <map>

using index_type = std::vector<int>;

template <typename T>
using sparse_array = std::map<index_type, T>;

For std::unordered_map (or similar hash table-based dictionaries) it’s slightly more work, since std::vector does not specialise std::hash:

#include <vector>
#include <unordered_map>
#include <numeric>

using index_type = std::vector<int>;

struct index_hash {
    std::size_t operator()(index_type const& i) const noexcept {
        // Like boost::hash_combine; there might be some caveats, see
        // <https://stackoverflow.com/a/50978188/1968>
        auto const hash_combine = [](auto seed, auto x) {
            return std::hash<int>()(x) + 0x9e3779b9 + (seed << 6) + (seed >> 2);
        };
        return std::accumulate(i.begin() + 1, i.end(), i[0], hash_combine);
    }
};

template <typename T>
using sparse_array = std::unordered_map<index_type, T, index_hash>;

Either way, the usage is the same:

int main() {
    using i = index_type;

    auto x = sparse_array<int>();
    x[i{1, 2, 3}] = 42;
    x[i{4, 3, 2}] = 23;

    std::cout << x[i{1, 2, 3}] + x[i{4, 3, 2}] << '\n'; // 65
}

Boost has a templated implementation of BLAS called uBLAS that contains a sparse matrix.

https://www.boost.org/doc/libs/release/libs/numeric/ublas/doc/index.htm

Eigen is a C++ linear algebra library that has an implementation of a sparse matrix. It even supports matrix operations and solvers (LU factorization etc) that are optimized for sparse matrices.