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

By : Dan
Source: Stackoverflow.com

I am working on a project that requires the manipulation of enormous matrices, particularly 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 (greater in number than the number of particles in the universe :p ), I need to store only the few non-zero elements. This could be several million entries, I currently work on a system that uses a binary search tree (b-tree) to store entries.

Does anyone know of a better system?

EDIT: Speed is a huge priority.

EDIT 2 : I like that solution. How would I go about dynamically choosing the number of variables in the class at runtime? [edit by MH: good question, updated in the answer]

By : Dan


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.

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

class triple {
    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;

For multiple variables:

The easiest way is to make the index a string, and then make the index strings look like "23,55" (2 vars) or "34,45,56" (3 vars):

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;

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.

Here is a relatively simple implementation that should provide a reasonable fast lookup (using a hash table) as well as fast iteration over non-zero elements in a row/column.

// Copyright 2014 Leo Osvald
// Licensed under the Apache License, Version 2.0 (the "License");
// you may not use this file except in compliance with the License.
// You may obtain a copy of the License at
//     http://www.apache.org/licenses/LICENSE-2.0
// Unless required by applicable law or agreed to in writing, software
// distributed under the License is distributed on an "AS IS" BASIS,
// See the License for the specific language governing permissions and
// limitations under the License.


#include <algorithm>
#include <limits>
#include <map>
#include <type_traits>
#include <unordered_map>
#include <utility>
#include <vector>

// A simple time-efficient implementation of an immutable sparse matrix
// Provides efficient iteration of non-zero elements by rows/cols,
// e.g. to iterate over a range [row_from, row_to) x [col_from, col_to):
//   for (int row = row_from; row < row_to; ++row) {
//     for (auto col_range = sm.nonzero_col_range(row, col_from, col_to);
//          col_range.first != col_range.second; ++col_range.first) {
//       int col = *col_range.first;
//       // use sm(row, col)
//       ...
//     }
template<typename T = double, class Coord = int>
class SparseMatrix {
  struct PointHasher;
  typedef std::map< Coord, std::vector<Coord> > NonZeroList;
  typedef std::pair<Coord, Coord> Point;

  typedef T ValueType;
  typedef Coord CoordType;
  typedef typename NonZeroList::mapped_type::const_iterator CoordIter;
  typedef std::pair<CoordIter, CoordIter> CoordIterRange;

  SparseMatrix() = default;

  // Reads a matrix stored in MatrixMarket-like format, i.e.:
  // <num_rows> <num_cols> <num_entries>
  // <row_1> <col_1> <val_1>
  // ...
  // Note: the header (lines starting with '%' are ignored).
  template<class InputStream, size_t max_line_length = 1024>
  void Init(InputStream& is) {
    rows_.clear(), cols_.clear();

    // skip the header (lines beginning with '%', if any)
    decltype(is.tellg()) offset = 0;
    for (char buf[max_line_length + 1];
         is.getline(buf, sizeof(buf)) && buf[0] == '%'; )
      offset = is.tellg();

    size_t n;
    is >> row_count_ >> col_count_ >> n;
    while (n--) {
      Coord row, col;
      typename std::remove_cv<T>::type val;
      is >> row >> col >> val;
      values_[Point(--row, --col)] = val;

  const T& operator()(const Coord& row, const Coord& col) const {
    static const T kZero = T();
    auto it = values_.find(Point(row, col));
    if (it != values_.end())
      return it->second;
    return kZero;

  nonzero_col_range(Coord row, Coord col_from, Coord col_to) const {
    CoordIterRange r;
    GetRange(cols_, row, col_from, col_to, &r);
    return r;

  nonzero_row_range(Coord col, Coord row_from, Coord row_to) const {
    CoordIterRange r;
    GetRange(rows_, col, row_from, row_to, &r);
    return r;

  Coord row_count() const { return row_count_; }
  Coord col_count() const { return col_count_; }
  size_t nonzero_count() const { return values_.size(); }
  size_t element_count() const { return size_t(row_count_) * col_count_; }

  typedef std::unordered_map<Point,
                             typename std::remove_cv<T>::type,
                             PointHasher> ValueMap;

  struct PointHasher {
    size_t operator()(const Point& p) const {
      return p.first << (std::numeric_limits<Coord>::digits >> 1) ^ p.second;

  static void SortAndShrink(NonZeroList& list) {
    for (auto& it : list) {
      auto& indices = it.second;
      std::sort(indices.begin(), indices.end());

    // insert a sentinel vector to handle the case of all zeroes
    if (list.empty())
      list.emplace(Coord(), std::vector<Coord>(Coord()));

  static void GetRange(const NonZeroList& list, Coord i, Coord from, Coord to,
                       CoordIterRange* r) {
    auto lr = list.equal_range(i);
    if (lr.first == lr.second) {
      r->first = r->second = list.begin()->second.end();

    auto begin = lr.first->second.begin(), end = lr.first->second.end();
    r->first = lower_bound(begin, end, from);
    r->second = lower_bound(r->first, end, to);

  ValueMap values_;
  NonZeroList rows_, cols_;
  Coord row_count_, col_count_;


For simplicity, it's immutable, but you can can make it mutable; be sure to change std::vector to std::set if you want a reasonable efficient "insertions" (changing a zero to a non-zero).

By : eold

This video can help you solving your question :)
By: admin