<< problem 81 - Path sum: two ways | Path sum: four ways - problem 83 >> |

# Problem 82: Path sum: three ways

(see projecteuler.net/problem=82)

The minimal path sum in the 5 by 5 matrix below, by starting in any cell in the left column and finishing in any cell in the right column,

and only moving up, down, and right, is indicated in bold; the sum is equal to 994.

`131 673 `

*234* *103* *18*

*201* *96* *342* 965 150

`630 803 746 422 111`

`537 699 497 121 956`

`805 732 524 37 331`

Find the minimal path sum, in matrix.txt (right click and "Save Link/Target As..."), a 31K text file containing a 80 by 80 matrix, from the left column to the right column.

# Algorithm

My code is almost identical to problem 81.

Relevant changes:

- "seed" all values of left-most column instead of just the upper-left cell

- stop whenever we reach any cell of the right-most column

- add the upper neighbor to the priority queue, too

- and that's it !

# My code

… was written in C++11 and can be compiled with G++, Clang++, Visual C++. You can download it, as well as the input data, too.

The code contains `#ifdef`

s to switch between the original problem and the Hackerrank version.

Enable `#ifdef ORIGINAL`

to produce the result for the original problem (default setting for most problems).

#include <queue>
#include <vector>
#include <iostream>
// 2D matrix: unfortunately x and y are swapped, so we need to write matrix[y][x]
// instead of the more common matrix[x][y]

typedef std::vector<std::vector<unsigned int>> Matrix;
// use a priority queue to find the next cell to process

struct Cell
{
// position
unsigned int x, y;
// sum of shortest route so far
unsigned long long weight;
Cell(unsigned int x_, unsigned int y_, unsigned long long weight_)
: x(x_), y(y_), weight(weight_) {}
// std::priority_queue returns the LARGEST element, therefore I implement this function "the other way 'round"
bool operator<(const Cell& cell) const
{
return weight > cell.weight; // ">" is not a typo !
}
};
// breadth-search

unsigned long long search(const Matrix& matrix)
{
// matrix height/width
const auto size = matrix.size();
// store already processed cells
std::vector<std::vector<bool>> processed(matrix.size());
for (auto& row : processed)
row.resize(matrix.size(), false);
// process cells in increasing distance from starting point
std::priority_queue<Cell> next;
// add starting points (left column)
for (unsigned int i = 0; i < size; i++)
next.push(Cell(0, i, matrix[i][0]));
while (!next.empty())
{
// get cell with the smallest weight
Cell cell = next.top();
// and remove it from the queue
next.pop();
// we have been here before ?
// must have been on a shorter route, hence discard current cell
if (processed[cell.y][cell.x])
continue;
processed[cell.y][cell.x] = true;
// finished ?
if (cell.x == size - 1)
return cell.weight;
// one step right
if (cell.x + 1 < size)
next.push(Cell(cell.x + 1, cell.y, cell.weight + matrix[cell.y][cell.x + 1]));
// one step down
if (cell.y + 1 < size)
next.push(Cell(cell.x, cell.y + 1, cell.weight + matrix[cell.y + 1][cell.x]));
// one step up
if (cell.y > 0)
next.push(Cell(cell.x, cell.y - 1, cell.weight + matrix[cell.y - 1][cell.x]));
}
return -1; // failed
}
int main()
{
unsigned int size = 80;
//#define ORIGINAL

#ifndef ORIGINAL
std::cin >> size;
#endif
Matrix matrix(size);
for (auto& row : matrix)
{
row.resize(size);
for (auto& cell : row)
{
#ifdef ORIGINAL
// unfortunately, Project Euler used a CSV format which is a bit tricky to parse in C++
cell = 0;
// read until the number is complete or we run out of input
while (std::cin)
{
char c;
std::cin.get(c);
// number complete ?
if (c < '0' || c > '9')
break;
// add digit to current number
cell *= 10;
cell += c - '0';
}
#else
std::cin >> cell;
#endif
}
}
// go !
std::cout << search(matrix) << std::endl;
return 0;
}

This solution contains 18 empty lines, 25 comments and 8 preprocessor commands.

# Interactive test

You can submit your own input to my program and it will be instantly processed at my server:

This live test is based on the Hackerrank problem.

This is equivalent to`echo "" | ./82`

Output:

*(this interactive test is still under development, computations will be aborted after one second)*

# Benchmark

The correct solution to the original Project Euler problem was found in **less than 0.01** seconds on a Intel® Core™ i7-2600K CPU @ 3.40GHz.

(compiled for x86_64 / Linux, GCC flags: `-O3 -march=native -fno-exceptions -fno-rtti -std=c++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

March 12, 2017 submitted solution

May 4, 2017 added comments

# Hackerrank

see https://www.hackerrank.com/contests/projecteuler/challenges/euler082

My code solved **7** out of **7** test cases (score: **100%**)

# Difficulty

Project Euler ranks this problem at **20%** (out of 100%).

Hackerrank describes this problem as **easy**.

*Note:*

Hackerrank has strict execution time limits (typically 2 seconds for C++ code) and often a much wider input range than the original problem.

In my opinion, Hackerrank's modified problems are usually a lot harder to solve. As a rule thumb: brute-force is never an option.

# Links

projecteuler.net/thread=82 - **the** best forum on the subject (*note:* you have to submit the correct solution first)

Code in various languages:

Python: www.mathblog.dk/project-euler-82-find-the-minimal-path-sum-from-the-left-column-to-the-right-column/ (written by Kristian Edlund)

Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p082.java (written by Nayuki)

Scala: github.com/samskivert/euler-scala/blob/master/Euler082.scala (written by Michael Bayne)

# Heatmap

green problems solve the original Project Euler problem and have a perfect score of 100% at Hackerrank, too.

yellow problems score less than 100% at Hackerrank (but still solve the original problem).

gray problems are already solved but I haven't published my solution yet.

blue problems are already solved and there wasn't a Hackerrank version of it (at the time I solved it) or I didn't care about it because it differed too much.

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<< problem 81 - Path sum: two ways | Path sum: four ways - problem 83 >> |