<< problem 343 - Fractional Sequences Strong Repunits - problem 346 >>

# Problem 345: Matrix Sum

We define the Matrix Sum of a matrix as the maximum sum of matrix elements with each element being the only one in his row and column. For example, the Matrix Sum of the matrix below equals 3315 ( = 863 + 383 + 343 + 959 + 767):

 7 53 183 439 863
497 383 563 79 973
287 63 343 169 583
627 343 773 959 943
767 473 103 699 303

Find the Matrix Sum of:

 7 53 183 439 863 497 383 563 79 973 287 63 343 169 583
627 343 773 959 943 767 473 103 699 303 957 703 583 639 913
447 283 463 29 23 487 463 993 119 883 327 493 423 159 743
217 623 3 399 853 407 103 983 89 463 290 516 212 462 350
960 376 682 962 300 780 486 502 912 800 250 346 172 812 350
870 456 192 162 593 473 915 45 989 873 823 965 425 329 803
973 965 905 919 133 673 665 235 509 613 673 815 165 992 326
322 148 972 962 286 255 941 541 265 323 925 281 601 95 973
445 721 11 525 473 65 511 164 138 672 18 428 154 448 848
414 456 310 312 798 104 566 520 302 248 694 976 430 392 198
184 829 373 181 631 101 969 613 840 740 778 458 284 760 390
821 461 843 513 17 901 711 993 293 157 274 94 192 156 574
 34 124 4 878 450 476 712 914 838 669 875 299 823 329 699
815 559 813 459 522 788 168 586 966 232 308 833 251 631 107
813 883 451 509 615 77 281 613 459 205 380 274 302 35 805

# My Algorithm

I wrote a simple Dynamic Programming solution:
to avoid processing all 15! combinations (about 1.3 * 10^12) it keeps track of the best solution so far and aborts a search if the current
path in the search tree can't exceed that solution.
Thus only 2100 path are fully evaluated (make it to if (row == Size)) and the correct solution is printed in about 0.02 seconds.

## Alternative Approaches

I learnt that a so-called Hungarian algorithm (see en.wikipedia.org/wiki/Hungarian_algorithm) can solve this problem even faster.

## Note

Pretty sure you can create a "bad" matrix where my algorithm needs ages to find the correct solution.
Even flipping the matrix along the main diagonal (swap(matrix[row][column], matrix[column][row])) make it about 70% slower.

# Interactive test

This feature is not available for the current problem.

# My code

… was written in C++11 and can be compiled with G++, Clang++, Visual C++. You can download it, too. Or just jump to my GitHub repository.

       #include <iostream>
#include <vector>

// 15x15 grid
const unsigned int Size = 15;
// matrix from problem statement
unsigned short matrix[Size][Size] =
{
{   7,  53, 183, 439, 863, 497, 383, 563,  79, 973, 287,  63, 343, 169, 583 },
{ 627, 343, 773, 959, 943, 767, 473, 103, 699, 303, 957, 703, 583, 639, 913 },
{ 447, 283, 463,  29,  23, 487, 463, 993, 119, 883, 327, 493, 423, 159, 743 },
{ 217, 623,   3, 399, 853, 407, 103, 983,  89, 463, 290, 516, 212, 462, 350 },
{ 960, 376, 682, 962, 300, 780, 486, 502, 912, 800, 250, 346, 172, 812, 350 },
{ 870, 456, 192, 162, 593, 473, 915,  45, 989, 873, 823, 965, 425, 329, 803 },
{ 973, 965, 905, 919, 133, 673, 665, 235, 509, 613, 673, 815, 165, 992, 326 },
{ 322, 148, 972, 962, 286, 255, 941, 541, 265, 323, 925, 281, 601,  95, 973 },
{ 445, 721,  11, 525, 473,  65, 511, 164, 138, 672,  18, 428, 154, 448, 848 },
{ 414, 456, 310, 312, 798, 104, 566, 520, 302, 248, 694, 976, 430, 392, 198 },
{ 184, 829, 373, 181, 631, 101, 969, 613, 840, 740, 778, 458, 284, 760, 390 },
{ 821, 461, 843, 513,  17, 901, 711, 993, 293, 157, 274,  94, 192, 156, 574 },
{  34, 124,   4, 878, 450, 476, 712, 914, 838, 669, 875, 299, 823, 329, 699 },
{ 815, 559, 813, 459, 522, 788, 168, 586, 966, 232, 308, 833, 251, 631, 107 },
{ 813, 883, 451, 509, 615,  77, 281, 613, 459, 205, 380, 274, 302,  35, 805 }
};

// sum of the highest value of the current row and all further rows (sum[current..15])
unsigned int maxRemaining[Size];

// try to find the highest value for "optimum"
// row        - current row, start at top-most row 0
// columnMask - bitmask of all used columns (1 - used, 0 - available)
// sum        - current sum
// atLeast    - best solution so far
unsigned int search(unsigned int row = 0, unsigned int columnMask = 0, unsigned int sum = 0, unsigned int atLeast = 0)
{
// done ?
if (row == Size)
return sum;

// even if choosing the highest value of each of the next rows:
// is it possible hat this combination exceeds the previously highest sum ?
if (sum + maxRemaining[row] <= atLeast)
return 0;

// look at all values of the current row
for (unsigned int column = 0; column < Size; column++)
{
auto mask = 1 << column;
continue;

// "occupy" column and continue with next row
auto current = search(row + 1, columnMask | mask, sum + matrix[row][column], atLeast);
if (atLeast < current)
atLeast = current;
}

return atLeast;
}

int main()
{
// find highest element of each row
unsigned int maxValuePerRow[Size];
for (unsigned int row = 0; row < Size; row++)
{
maxValuePerRow[row] = matrix[0][row];
for (unsigned int column = 1; column < Size; column++)
if (maxValuePerRow[row] < matrix[column][row])
maxValuePerRow[row] = matrix[column][row];
}
// compute the maximum sum of the last rows, ignoring collisions (invalid choices)
maxRemaining[Size - 1] = maxValuePerRow[Size - 1];
for (auto row = Size - 1; row > 0; row--)
maxRemaining[row - 1] = maxRemaining[row] + maxValuePerRow[row - 1];

// let's go !
std::cout << search() << std::endl;
return 0;
}


This solution contains 9 empty lines, 17 comments and 2 preprocessor commands.

# Benchmark

The correct solution to the original Project Euler problem was found in 0.01 seconds on an Intel® Core™ i7-2600K CPU @ 3.40GHz.
(compiled for x86_64 / Linux, GCC flags: -O3 -march=native -fno-exceptions -fno-rtti -std=gnu++11 -DORIGINAL)

See here for a comparison of all solutions.

Note: interactive tests run on a weaker (=slower) computer. Some interactive tests are compiled without -DORIGINAL.

# Changelog

July 18, 2017 submitted solution

# Difficulty

Project Euler ranks this problem at 15% (out of 100%).

# Heatmap

Please click on a problem's number to open my solution to that problem:

 green solutions solve the original Project Euler problem and have a perfect score of 100% at Hackerrank, too yellow solutions score less than 100% at Hackerrank (but still solve the original problem easily) gray problems are already solved but I haven't published my solution yet blue solutions are relevant for Project Euler only: there wasn't a Hackerrank version of it (at the time I solved it) or it differed too much orange problems are solved but exceed the time limit of one minute or the memory limit of 256 MByte red problems are not solved yet but I wrote a simulation to approximate the result or verified at least the given example - usually I sketched a few ideas, too black problems are solved but access to the solution is blocked for a few days until the next problem is published [new] the flashing problem is the one I solved most recently

I stopped working on Project Euler problems around the time they released 617.
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The 310 solved problems (that's level 12) had an average difficulty of 32.6% at Project Euler and
I scored 13526 points (out of 15700 possible points, top rank was 17 out of ≈60000 in August 2017) at Hackerrank's Project Euler+.

My username at Project Euler is stephanbrumme while it's stbrumme at Hackerrank.

Look at my progress and performance pages to get more details.

 << problem 343 - Fractional Sequences Strong Repunits - problem 346 >>
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