<< problem 17 - Number letter counts | Counting Sundays - problem 19 >> |

# Problem 18: Maximum path sum I

(see projecteuler.net/problem=18)

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 of the triangle below:

` 75`

` 95 64`

` 17 47 82`

` 18 35 87 10`

` 20 04 82 47 65`

` 19 01 23 75 03 34`

` 88 02 77 73 07 63 67`

` 99 65 04 28 06 16 70 92`

` 41 41 26 56 83 40 80 70 33`

` 41 48 72 33 47 32 37 16 94 29`

` 53 71 44 65 25 43 91 52 97 51 14`

` 70 11 33 28 77 73 17 78 39 68 17 57`

` 91 71 52 38 17 14 91 43 58 50 27 29 48`

` 63 66 04 68 89 53 67 30 73 16 69 87 40 31`

`04 62 98 27 23 09 70 98 73 93 38 53 60 04 23`

*NOTE:* As there are only 16384 routes, it is possible to solve this problem by trying every route.

However, Problem 67, is the same challenge with a triangle containing one-hundred rows; it cannot be solved by brute force, and requires a clever method! ;o)

# Algorithm

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

## Note

Exactly the same algorithm is used for problem 67.

# My code

… was written in C++ 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;
//#define ORIGINAL

#ifndef ORIGINAL
std::cin >> tests;
#endif
while (tests--)
{
unsigned int numRows = 15;
#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 "1 4 3 7 4 2 4 6 8 5 9 3" | ./18`

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 3, 2017 added comments

# Hackerrank

see https://www.hackerrank.com/contests/projecteuler/challenges/euler018

My code solved **6** out of **6** test cases (score: **100%**)

# Difficulty

Project Euler ranks this problem at **5%** (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=18 - **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-18/ (written by Kristian Edlund)

Haskell: github.com/nayuki/Project-Euler-solutions/blob/master/haskell/p018.hs (written by Nayuki)

Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p018.java (written by Nayuki)

Mathematica: github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p018.mathematica (written by Nayuki)

C: github.com/eagletmt/project-euler-c/blob/master/10-19/problem18.c (written by eagletmt)

Go: github.com/frrad/project-euler/blob/master/golang/Problem018.go (written by Frederick Robinson)

Javascript: github.com/dsernst/ProjectEuler/blob/master/18 Maximum path sum I.js (written by David Ernst)

Scala: github.com/samskivert/euler-scala/blob/master/Euler018.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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