<< problem 100 - Arranged probability | Triangle containment - problem 102 >> |

# Problem 101: Optimum polynomial

(see projecteuler.net/problem=101)

If we are presented with the first k terms of a sequence it is impossible to say with certainty the value of the next term,

as there are infinitely many polynomial functions that can model the sequence.

As an example, let us consider the sequence of cube numbers. This is defined by the generating function,

u_n = n^3: 1, 8, 27, 64, 125, 216, ...

Suppose we were only given the first two terms of this sequence. Working on the principle that "simple is best" we should assume a linear relationship

and predict the next term to be 15 (common difference 7). Even if we were presented with the first three terms, by the same principle of simplicity, a quadratic relationship should be assumed.

We shall define OP(k, n) to be the nth term of the optimum polynomial generating function for the first k terms of a sequence.

It should be clear that OP(k, n) will accurately generate the terms of the sequence for n <= k, and potentially the first incorrect term (FIT) will be OP(k, k+1);

in which case we shall call it a bad OP (BOP).

As a basis, if we were only given the first term of sequence, it would be most sensible to assume constancy; that is, for n >= 2, OP(1, n) = u_1.

Hence we obtain the following OPs for the cubic sequence:

OP(1, n) = 1

1, *1*, 1, 1, ...

OP(2, n) = 7n-6

1, 8, *15*, ...

OP(3, n) = 6n^2-11n+6

1, 8, 27, *58*, ...

OP(4, n) = n^3

1, 8, 27, 64, 125, ...

Clearly no BOPs exist for k >= 4.

By considering the sum of FITs generated by the BOPs (indicated in *bold* above), we obtain 1 + 15 + 58 = 74.

Consider the following tenth degree polynomial generating function:

u_n = 1 - n + n^2 - n^3 + n^4 - n^5 + n^6 - n^7 + n^8 - n^9 + n^10

Find the sum of FITs for the BOPs.

# My Algorithm

When I read this problem, I didn't know anything about polynomial interpolation - and started reading the Wikipedia page: en.wikipedia.org/wiki/Polynomial_interpolation

The Lagrange polynomial caught my attention but my implementation just couldn't find the correct solution.

Then I tried the Newton polynomial and succeeded. Even more, I was able to fix the off-by-one error in my Lagrange code ...

Nevertheless, I kept both routines in my solution. They return the same result.

## Modifications by HackerRank

[TODO] I modified my code to accept arbitrary coefficients for the polynomial generating function (originally `{ +1, -1, +1, -1, +1, -1, +1, -1, +1, -1, +1 }`

)

but still fail to solve anything else than the default input.

# My code

… was written in C++11 and can be compiled with G++, Clang++, Visual C++. You can download it, 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).

//#define ORIGINAL

#include <iostream>
#include <vector>
// compute sum(coeff[i] * x^i)

template <typename T>
T sequence(T x, const std::vector<T>& coefficients)
{
T xx = 1;
T result = 0;
for (auto i : coefficients)
{
result += i * xx;
xx *= x;
}
return result;
}
// given f(1),f(2),f(3),...f(n) then find f(n+1)
// (Lagrange polynomials)

template <typename T>
T lagrange(const std::vector<T>& known)
{
T result = 0;
size_t next = known.size() + 1;
for (size_t i = 1; i < next; i++)
{
// build Lagrange polynomials
// n = numerator, d = denominator
T n = 1;
T d = 1;
for (size_t j = 1; j < next; j++)
{
if (i == j)
continue;
n *= next - j;
d *= i - j;
}
// evaluate
result += known[i - 1] * (n / d);
}
return result;
}
// given f(1),f(2),f(3),...f(n) then find f(n+1)
// (Newton divided differences)

template <typename T>
T newton(std::vector<T> known)
{
T result = known[0];
size_t j = 1;
size_t k = known.size();
for (size_t last = known.size() - 1; last > 0; last--)
{
for (size_t i = 0; i < last; i++)
known[i] = (known[i + 1] - known[i]) / j;
T multDiff = 1;
for (size_t i = 0; i < j; i++)
multDiff *= k - i;
result += known[0] * multDiff;
j++;
}
return result;
}
int main()
{
// read coefficients
#ifdef ORIGINAL
std::vector<long long> coefficients = { +1, -1, +1, -1, +1, -1, +1, -1, +1, -1, +1 };
#else
size_t numCoefficients;
std::cin >> numCoefficients;
std::vector<long long> coefficients(numCoefficients + 1);
for (auto& i : coefficients)
std::cin >> i;
#endif
long long sum = 0;
std::vector<long long> data;
// iterate over 10 points
for (long long x = 1; x < (long long)coefficients.size(); x++)
{
// add the next point
data.push_back(sequence(x, coefficients));
// estimate next point
long long next = lagrange(data);
//long long next = newton(data);
sum += next;
#ifndef ORIGINAL
std::cout << (next % 1000000007) << " ";
#endif
}
#ifdef ORIGINAL
std::cout << sum << std::endl;
#endif
return 0;
}

This solution contains 16 empty lines, 13 comments and 9 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 "0 0 0 1" | ./101`

Output:

*Note:* the original problem's input `11 1 -1 1 -1 1 -1 1 -1 1 -1 1`

__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

May 23, 2017 submitted solution

May 23, 2017 added comments

# Difficulty

Project Euler ranks this problem at **35%** (out of 100%).

# Links

projecteuler.net/thread=101 - **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-101-optimum-polynomial-function/ (written by Kristian Edlund)

Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p101.java (written by Nayuki)

# 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.

red problems are solved but exceed the time limit of one minute or the memory limit of 256 MByte.

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I scored 13,183 points (out of 15300 possible points, top rank was 17 out of ≈60000 in August 2017) at Hackerrank's Project Euler+.

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

My username at Project Euler is

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

# Copyright

I hope you enjoy my code and learn something - or give me feedback how I can improve my solutions.

All of my solutions can be used for any purpose and I am in no way liable for any damages caused.

You can even remove my name and claim it's yours. But then you shall burn in hell.

The problems and most of the problems' images were created by Project Euler.

Thanks for all their endless effort.

<< problem 100 - Arranged probability | Triangle containment - problem 102 >> |