<< problem 66 - Diophantine equation | Magic 5-gon ring - problem 68 >> |

# Problem 67: Maximum path sum II

(see projecteuler.net/problem=67)

By starting at the top of the triangle below and moving to adjacent numbers on the row below, the maximum total from top to bottom is 23.

` 3`

` 7 4`

` 2 4 6`

`8 5 9 3`

That is, 3 + 7 + 4 + 9 = 23.

Find the maximum total from top to bottom in triangle.txt (right click and 'Save Link/Target As...'), a 15K text file containing a triangle with one-hundred rows.

*NOTE:* This is a much more difficult version of Problem 18. It is not possible to try every route to solve this problem, as there are 299 altogether!

If you could check one trillion (10^12) routes every second it would take over twenty billion years to check them all. There is an efficient algorithm to solve it. ;o)

# Algorithm

The algorithm and code were copied from problem 18:

The main idea is to build a data structure similar to the input data:

but instead of just storing the raw input we store the biggest sum up to this point.

All data is processed row-by-row

Of course, the first row consists of a single number and it has no "parents", that means no rows above it.

Therefore the "sum" is the number itself.

This row now becomes my "parent row" called `last`

.

For each element of the next rows I have to find its parents (some have one, some have two),

figure out which parent is bigger and then add the current input to it.

This sum is stored in `current`

.

When a row is fully processed, `current`

becomes `last`

.

When all rows are processed, the largest element in `last`

is the result of the algorithm.

Example:

` 1`

` 2 3`

`4 5 6`

initialize:

`last[0] = 1;`

read second line:

`current[0] = 2 + last[0] = 3`

`current[1] = 3 + last[0] = 4`

copy current to last (which becomes { 3, 4 })

read third line:

`current[0] = 4 + last[0] = 7`

`current[1] = 5 + max(last[0], last[1]) = 9`

`current[2] = 6 + last[1] = 10`

copy current to last (which becomes { 7, 9, 10 })

finally:

print max(last) = 10

# 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 <iostream>
#include <vector>
#include <algorithm>
int main()
{
unsigned int tests = 1;
unsigned int numRows = 100;
//#define ORIGINAL

#ifndef ORIGINAL
std::cin >> tests;
#endif
while (tests--)
{
#ifndef ORIGINAL
std::cin >> numRows;
#endif
// process input row-by-row
// each time a number is read we add it to the two numbers above it
// choose the bigger sum and store it
// if all rows are finished, find the largest number in the last row
// read first line, just one number
std::vector<unsigned int> last(1);
std::cin >> last[0];
// read the remaining lines
for (unsigned int row = 1; row < numRows; row++)
{
// prepare array for new row
unsigned int numElements = row + 1;
std::vector<unsigned int> current;
// read all numbers of current row
for (unsigned int column = 0; column < numElements; column++)
{
unsigned int x;
std::cin >> x;
// find sum of elements in row above (going a half step to the left)
unsigned int leftParent = 0;
// only if left parent is available
if (column > 0)
leftParent = last[column - 1];
// find sum of elements in row above (going a half step to the right)
unsigned int rightParent = 0;
// only if right parent is available
if (column < last.size())
rightParent = last[column];
// add larger parent to current input
unsigned int sum = x + std::max(leftParent, rightParent);
// and store this sum
current.push_back(sum);
}
// row is finished, it become the "parent" row
last = current;
}
// find largest sum in final row
std::cout << *std::max_element(last.begin(), last.end()) << std::endl;
}
return 0;
}

This solution contains 13 empty lines, 17 comments and 7 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 "" | ./67`

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

February 24, 2017 submitted solution

April 26, 2017 added comments

# Hackerrank

see https://www.hackerrank.com/contests/projecteuler/challenges/euler067

My code solved **20** out of **20** test cases (score: **100%**)

# Difficulty

Project Euler ranks this problem at **5%** (out of 100%).

Hackerrank describes this problem as **medium**.

*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=67 - **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-67-efficient-algorithm-triangle/ (written by Kristian Edlund)

Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p067.java (written by Nayuki)

Go: github.com/frrad/project-euler/blob/master/golang/Problem067.go (written by Frederick Robinson)

Scala: github.com/samskivert/euler-scala/blob/master/Euler067.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.

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

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