<< problem 88 - Product-sum numbers | Cube digit pairs - problem 90 >> |

# Problem 89: Roman numerals

(see projecteuler.net/problem=89)

For a number written in Roman numerals to be considered valid there are basic rules which must be followed.

Even though the rules allow some numbers to be expressed in more than one way there is always a "best" way of writing a particular number.

For example, it would appear that there are at least six ways of writing the number sixteen:

`IIIIIIIIIIIIIIII`

`VIIIIIIIIIII`

`VVIIIIII`

`XIIIIII`

`VVVI`

`XVI`

However, according to the rules only `XIIIIII`

and `XVI`

are valid, and the last example is considered to be the most efficient, as it uses the least number of numerals.

The 11K text file, roman.txt (right click and 'Save Link/Target As...'), contains one thousand numbers written in valid, but not necessarily minimal, Roman numerals;

see About... Roman Numerals for the definitive rules for this problem.

Find the number of characters saved by writing each of these in their minimal form.

Note: You can assume that all the Roman numerals in the file contain no more than four consecutive identical units.

# Algorithm

There are two functions:

- `roman2number`

takes a valid Roman number and converts it to an integer

- `number2roman`

converts an integer to a minimal ("optimal") Roman number

My program reads all Roman numbers, converts them to integers and back to an optimal Roman number.

The difference of the strings' lengths is what I'm looking for.

`roman2number`

reads the Roman numbers backwards:

- if the current letter is smaller than the previous (its right neighbor) then the current letter must be subtracted else added

- my code can subtract multiple identical letters, too, e.g. 8 = `IIX`

which is shorter than `VIII`

`number2roman`

has a list of conversion rules to convert an integer to a Roman number.

Each rule is identified by a number and that rule is applied as long as the integer is larger or equal.

Whenever `rules[i]`

applies, one or two letters `action[i]`

are added to the `result`

:

`i`

`rules[i]`

`action[i]`

01000`M`

1900`CM`

2500`D`

3400`CD`

4100`C`

590`XC`

650`L`

740`XL`

810`X`

99`IX`

105`V`

114`IV`

121`I`

## Modifications by HackerRank

I have to print the optimized Roman numbers instead of finding the sum of the length differences.

## Note

The problem can be solved by simple search'n'replace, too.

However, I liked the challenge to write a basic parser for Roman numbers.

# 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 <string>
// convert valid roman numbers to binary numbers

unsigned int roman2number(const std::string& roman)
{
unsigned int result = 0;
// remember the value of the previous Roman letter
unsigned int last = 0;
// true, if the current letter is subtracted (and the next identical letters)
bool subtract = false;
// walk through the whole string from the end to the beginning ...
for (auto i = roman.rbegin(); i != roman.rend(); i++)
{
unsigned int current = 0;
switch (*i)
{
case 'M': current = 1000; break;
case 'D': current = 500; break;
case 'C': current = 100; break;
case 'L': current = 50; break;
case 'X': current = 10; break;
case 'V': current = 5; break;
case 'I': current = 1; break;
}
// smaller than its right neighbor ? => we must subtract
if (current < last)
{
subtract = true;
last = current;
}
// bigger than its right neighbor ? => we must add
else if (current > last)
{
subtract = false;
last = current;
}
// note: if current == last then we keep the variables "subtract" and "last" in their current state
// add/subtract accordingly
if (subtract)
result -= current;
else
result += current;
}
return result;
}
std::string number2roman(unsigned int number)
{
// apply these rules in the presented order:
// - as long as number >= steps[i] add roman[i] to result
const unsigned int NumRules = 13;
const unsigned int rules[NumRules] =
{ 1000, 900, 500, 400, 100, 90, 50, 40, 10, 9, 5, 4, 1 };
const char* action[NumRules] =
{ "M", "CM", "D", "CD", "C", "XC", "L", "XL", "X", "IX", "V", "IV", "I" };
// apply all rules ...
std::string result;
for (unsigned int i = 0; i < NumRules; i++)
// ... as often as needed
while (number >= rules[i])
{
// reduce integer
number -= rules[i];
// add letter(s)
result += action[i];
}
return result;
}
int main()
{
// letters saved by optimization
unsigned int saved = 0;
unsigned int tests = 1000;
//#define ORIGINAL

#ifndef ORIGINAL
std::cin >> tests;
#endif
while (tests--)
{
// read Roman number
std::string roman;
std::cin >> roman;
// convert it to an integer and back to an optimal Roman number
auto number = roman2number(roman);
auto optimized = number2roman(number);
// count how many character were saved
saved += roman.size() - optimized.size();
#ifndef ORIGINAL
// print Roman number
std::cout << optimized << std::endl;
#endif
}
#ifdef ORIGINAL
std::cout << saved << std::endl;
#endif
return 0;
}

This solution contains 17 empty lines, 20 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 "1 " | ./89`

Output:

*Note:* the original problem's input `50000000`

__cannot__ be entered

because just copying results is a soft skill reserved for idiots.

*(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 15, 2017 submitted solution

May 6, 2017 added comments

# Hackerrank

see https://www.hackerrank.com/contests/projecteuler/challenges/euler089

My code solves **4** out of **4** 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=89 - **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-89-develop-a-method-to-express-roman-numerals-in-minimal-form/ (written by Kristian Edlund)

Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p089.java (written by Nayuki)

Scala: github.com/samskivert/euler-scala/blob/master/Euler089.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 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:*

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 |

26 | 27 | 28 | 29 | 30 | 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 | 41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 | 49 | 50 |

51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | 59 | 60 | 61 | 62 | 63 | 64 | 65 | 66 | 67 | 68 | 69 | 70 | 71 | 72 | 73 | 74 | 75 |

76 | 77 | 78 | 79 | 80 | 81 | 82 | 83 | 84 | 85 | 86 | 87 | 88 | 89 | 90 | 91 | 92 | 93 | 94 | 95 | 96 | 97 | 98 | 99 | 100 |

101 | 102 | 103 | 104 | 105 | 106 | 107 | 108 | 109 | 110 | 111 | 112 | 113 | 114 | 115 | 116 | 117 | 118 | 119 | 120 | 121 | 122 | 123 | 124 | 125 |

126 | 127 | 128 | 129 | 130 | 131 | 132 | 133 | 134 | 135 | 136 | 137 | 138 | 139 | 140 | 141 | 142 | 143 | 144 | 145 | 146 | 147 | 148 | 149 | 150 |

151 | 152 | 153 | 154 | 155 | 156 | 157 | 158 | 159 | 160 | 161 | 162 | 163 | 164 | 165 | 166 | 167 | 168 | 169 | 170 | 171 | 172 | 173 | 174 | 175 |

176 | 177 | 178 | 179 | 180 | 181 | 182 | 183 | 184 | 185 | 186 | 187 | 188 | 189 | 190 | 191 | 192 | 193 | 194 | 195 | 196 | 197 | 198 | 199 | 200 |

201 | 202 | 203 | 204 | 205 | 206 | 207 | 208 | 209 | 210 | 211 | 212 | 213 | 214 | 215 | 216 | 217 | 218 | 219 | 220 | 221 | 222 | 223 | 224 | 225 |

226 | 227 | 228 | 229 | 230 | 231 | 232 | 233 | 234 | 235 | 236 | 237 | 238 | 239 | 240 | 241 | 242 | 243 | 244 | 245 | 246 | 247 | 248 | 249 | 250 |

My username at Project Euler is

**stephanbrumme**while it's stbrumme at Hackerrank.

<< problem 88 - Product-sum numbers | Cube digit pairs - problem 90 >> |