# Simple DP problem solved in compile time

The problem is taken from one of recent SO questions:

Finding max sum of matrix elements with following constraints:

1. Exactly one row element has to be included in the sum
2. If element at (i, j) is selected, then (i + 1, j) is not.
3. Values in the matrix are nonnegative integers.

Below is my working (as far as I can tell) solution. It is based on the following observation: given a solution for a (N x M) matrix (in terms of taken path), the path is either the solution for the matrix without the last row (that is N -1 x M), or there exists at most one better path for the submatrix.
I've aimed to provide code testable at compile time, for that purpose I've also written a lightweight wrapper of std::array, with interface being a subset of e.g. eigen::Matrix classes.

Compiles fine with both gcc and clang (-std=c++17 -Werror -Wall -Wpedantic).

Any thoughts on how to improve it?

#include <array>

/*
* Small access wrapper for an array.
* Underlying storage kept puclic to allow efficient construction as an aggregate
*/
template <typename T, std::size_t rows_, std::size_t cols_>
struct Array2d {
T& operator()(std::size_t row, std::size_t col) {
return storage[index(row, col)];
}

constexpr T const & operator()(std::size_t row, std::size_t col) const {
return storage[index(row, col)];
}

constexpr std::size_t rows() const { return rows_; }

constexpr std::size_t cols() const { return cols_; }

std::array<T, rows_ * cols_> storage;

private:
constexpr std::size_t index (std::size_t row, std::size_t col) const {
return cols_ * row + col;
}
};

template<std::size_t rows, std::size_t cols>
using Problem = Array2d<int, rows, cols>;

namespace MaxSum {
namespace Details {

struct IndexedRowValue {
int val;
std::size_t index;
};

template<std::size_t count>
using TopElements = std::array<IndexedRowValue, count>;

constexpr TopElements<2> sorted_first_two (int v1, int v2) {
auto m1 = IndexedRowValue{v1, 0};
auto m2 = IndexedRowValue{v2, 1};
return v1 >= v2
? TopElements<2>{m1, m2}
: TopElements<2>{m2, m1};
}

constexpr void update_top (TopElements<2> & top_paths, int path_value,
std::size_t col) {
if (path_value > top_paths[0].val) {
top_paths[1] = top_paths[0];
top_paths[0] = IndexedRowValue{path_value, col};
}
else if (path_value > top_paths[1].val)
top_paths[1] = IndexedRowValue{path_value, col};
}

template<typename Matrix>
constexpr TopElements<2> find_top2_in_first_row (Matrix const &input) {
auto result = sorted_first_two(input(0, 0), input(0, 1));
for (auto i = 2u; i < input.cols(); ++i)
update_top(result,input(0, i), i);
return result;
}

constexpr int best_path_value_through_element(TopElements<2> const & top_last_row,
int val,
std::size_t col){
? top_last_row[0].val + val
: top_last_row[1].val + val;
}

template<typename Matrix>
constexpr TopElements<2> find_best_paths_for_row(TopElements<2> const & top_last_row,
std::size_t row,
Matrix const & input) {
auto path_0 = best_path_value_through_element(top_last_row, input(row, 0), 0u);
auto path_1 = best_path_value_through_element(top_last_row, input(row, 1), 1u);
auto top_paths = sorted_first_two(path_0, path_1);
for (auto i = 2u; i < input.cols(); ++i) {
auto path_i = best_path_value_through_element(top_last_row, input(row, i), i);
update_top(top_paths, path_i, i);
}
}

template<typename Matrix>
constexpr int solve_non_trivial(Matrix const & input) {
auto top_paths = find_top2_in_first_row(input);
for (auto i = 1u; i < input.rows(); ++i)
top_paths = find_best_paths_for_row(top_paths, i, input);
// key observation: optimal path at row i is either best or second best at i - 1
}
} // namespace Details

/*
* Finds max sum of elements of input Matrix, with following constraints:
* Exactly one element from each row can be selected
* If element at (i, j) has been selected, then (i + 1, j) can't be selected
*
* Matrix elements are required to be nonnegative integers.
*/
template<typename Matrix>
constexpr int solve (Matrix const & input) {
int result = 0; // reasonable answer for cases where rows > cols
// special case for 1x1 matrices
if (input.rows() == 1 && input.cols() == 1)
result = input(0, 0);
else if (input.rows() <=  input.cols()){
result = Details::solve_non_trivial(input);
}
return result;
}

} // namespace MaxSum

int main() {
constexpr auto trivial = Problem<1u, 1u>{{1}};
static_assert(MaxSum::solve(trivial) == 1);

constexpr auto problem2x2_0 = Problem<2u, 2u>{{1, 0, 0, 1}};
static_assert(MaxSum::solve(problem2x2_0) == 2);

constexpr auto problem2x2_1 = Problem<2u, 2u>{{10, 0, 9, 0}};
static_assert(MaxSum::solve(problem2x2_1) == 10);

constexpr auto problem2x2_2 = Problem<2u, 2u>{{10, 2, 9, 0}};
static_assert(MaxSum::solve(problem2x2_2) == 11);

constexpr auto problem1x5 = Problem<1u, 5u>{{10, 2, 9, 7, 6}};
static_assert(MaxSum::solve(problem1x5) == 10);

constexpr auto problem1x7 = Problem<1u, 7u>{{10, 2, 9, 7, 6, 12, 11}};
static_assert(MaxSum::solve(problem1x7) == 12);

constexpr auto problem3x3 = Problem<3u, 3u>{{1, 2, 3,
5, 6, 4,
3, 2, 4}};
static_assert(MaxSum::solve(problem3x3) == 13);

constexpr auto problem4x4 = Problem<4u, 4u>{{1, 2, 3, 4,
5, 6, 7, 8,
9, 1, 4, 2,
6, 3, 5, 7}};
static_assert(MaxSum::solve(problem4x4) == 27);
}

• Since you are aiming C++17, you can make the non-const operator[]s constexpr too: constexpr T& operator()(std::size_t row, std::size_t col) { /*...*/ } Feb 15, 2019 at 0:53

Given that the result can be computed as simply as this:

#include <array>

template <std::size_t MN>
constexpr int maximum_sum(std::array<int, MN> array, int M, int N, int current_row, int skip) {
int max = 0;
for (int x = 0; x < N; ++x) {
if (x == skip) continue;
int value = array[current_row * N + x];
max = std::max(max, current_row + 1 == M ? value : value + maximum_sum(array, M, N, current_row + 1, x));
}
return max;
}


I find your code really complicated.

The wrapper looks like a half-measure to me. If you only want a convenient way to handle an array as a 2-dimensional matrix for the task at hand, I'd say that inheritance is more powerful and concise:

template <std::size_t M, std::size_t N>
struct Array_2d : public std::array<int, M * N> {
constexpr int at(std::size_t m, std::size_t n) const { return (*this)[m * N + n]; }
}


Dimensions can then be deduced in the function call:

template <std::size_t M, std::size_t N>
constexpr int maximum_sum(Array_2d<M, N> array, std::size_t current_row = 0, std::size_t skip = N) {
// same as before
}


And if you want a solid, re-usable, constepxr matrix class then write it, but that goes far beyond the functionalities of your wrapper.

Burying types isn't a good thing either: your Problem is fundamentally an array, but there are two "indirection" levels before you can ascertain it: an alias, and a composition. Idem for TopElements, which is a simple pair the reader needs some work to recognize under an other alias and a custom class. One good way to look at your program is to understand that you are creating a language: do not create new words when you can use the ones everyone knows.

Your algorithm probably works, but it isn't clearly described (what is a "matrix, in terms of a taken path"?), and relies on a long chain of function calls which doesn't make it any easier to understand. Moving away from the simplest algorithm isn't justified unless you can prove it's faster (and that speed matters); but you don't study your algorithm complexity.

• Sorry if the description was unclear: it was "solution in terms of taken path" -> that is path maximizing the sum. The observation made allows the algorithm to run in O(M x N) (that is pretty much optimal, given that each matrix element has to be inpected) time with O(1) auxillary space (again can't do better than that I guess). Feb 15, 2019 at 12:10
• As to inheriting from std::array -> this seems wrong, as std::array does not have a virtual destructor. And if you decide to go for private / protected inheritance then you no longer deal with aggregate type. Feb 15, 2019 at 12:35
• @paler123: that's clearer now, but you should try to go the extra mile and describe your algorithm, not only the observation it's based on. It makes reading the code a lot easier. / As to inheritance, as I said, don't go for half-measures: if you want to write a library-grade matrix class, then go ahead, but your Array2d isn't up to the task; if you only want a quick prototype, then by all means, we're consenting adults around there, we won't use base pointers to allocate those arrays on the heap behind your back, so go for what's concise and expressive. Feb 15, 2019 at 15:13