How to resolve the algorithm Tropical algebra overloading step by step in the C++ programming language

Published on 7 June 2024 03:52 AM
#cpp

How to resolve the algorithm Tropical algebra overloading step by step in the C++ programming language

Table of Contents

Problem Statement

In algebra, a max tropical semiring (also called a max-plus algebra) is the semiring (ℝ ∪ -Inf, ⊕, ⊗) containing the ring of real numbers ℝ augmented by negative infinity, the max function (returns the greater of two real numbers), and addition. In max tropical algebra, x ⊕ y = max(x, y) and x ⊗ y = x + y. The identity for ⊕ is -Inf (the max of any number with -infinity is that number), and the identity for ⊗ is 0. Show that 2 ⊗ -2 is 0, -0.001 ⊕ -Inf is -0.001, 0 ⊗ -Inf is -Inf, 1.5 ⊕ -1 is 1.5, and -0.5 ⊗ 0 is -0.5. where ⊗ has precedence over ⊕. Demonstrate that 5 ⊗ (8 ⊕ 7) equals 5 ⊗ 8 ⊕ 5 ⊗ 7.

Let's start with the solution:

Step by Step solution about How to resolve the algorithm Tropical algebra overloading step by step in the C++ programming language

The provided C++ code defines a TropicalAlgebra class and various operators to perform operations on values in a tropical algebra setting. Tropical algebra is a mathematical structure that uses the operations of maximum and addition instead of the usual multiplication and addition operations. In this code, negative infinity (-Inf) is represented using an unset std::optional<double>.

Here's a breakdown of the code:

  • TropicalAlgebra Class:

    • This class represents a point in tropical algebra, where the value can be initialized to a finite double or left unset to represent negative infinity.
    • It defines overloaded operators for addition (+=), multiplication (*=), and exponentiation (pow).

  • Operator Overloads:

    • operator+: This operator adds two TropicalAlgebra values using the += operator defined in the TropicalAlgebra class.
    • operator*: This operator multiplies two TropicalAlgebra values using the *= operator defined in the TropicalAlgebra class.
    • operator<<: This operator overloads the insertion operator (<<) to print the TropicalAlgebra value. It prints "-Inf" if the value is unset or the actual value otherwise.

  • pow Function:

    • This function exponentiates a TropicalAlgebra value to a given exponent using successive multiplication.

  • main Function:

    • This function demonstrates the use of the TropicalAlgebra class and its operators:
      • It creates several TropicalAlgebra objects with different values, including negative infinity.
      • It performs various operations such as addition, multiplication, and exponentiation on these values.
      • It prints the results of these operations, which can be either finite values or negative infinity.

Overall, this code showcases the use of C++ classes and operator overloading to implement a tropical algebra, where negative infinity is handled using an unset std::optional<double>. It demonstrates how tropical algebra operations (maximum and addition) can be applied to values, including negative infinity, to obtain meaningful results.

Source code in the cpp programming language

#include <iostream>
#include <optional>

using namespace std;

class TropicalAlgebra
{
    // use an unset std::optional to represent -infinity
    optional<double> m_value;
    
public:
    friend std::ostream& operator<<(std::ostream&, const TropicalAlgebra&);
    friend TropicalAlgebra pow(const TropicalAlgebra& base, unsigned int exponent) noexcept;
    
    // create a point that is initialized to -infinity
    TropicalAlgebra() = default;

    // construct with a value
    explicit TropicalAlgebra(double value) noexcept
        : m_value{value} {}

    // add a value to this one ( p+=q ).  it is common to also overload 
    // the += operator when overloading +
    TropicalAlgebra& operator+=(const TropicalAlgebra& rhs) noexcept
    {
        if(!m_value)
        {
            // this point is -infinity so the other point is max
            *this = rhs;
        }
        else if (!rhs.m_value)
        {
            // since rhs is -infinity this point is max
        }
        else
        {
            // both values are valid, find the max
            *m_value = max(*rhs.m_value, *m_value);
        }

        return *this;
    }
    
    // multiply this value by another (p *= q)
    TropicalAlgebra& operator*=(const TropicalAlgebra& rhs) noexcept
    {
        if(!m_value)
        {
            // since this value is -infinity this point does not need to be
            // modified
        }
        else if (!rhs.m_value)
        {
            // the other point is -infinity, make this -infinity too
            *this = rhs;
        }
        else
        {
            *m_value += *rhs.m_value;
        }

        return *this;
    }
};

// add values (p + q)
inline TropicalAlgebra operator+(TropicalAlgebra lhs, const TropicalAlgebra& rhs) noexcept
{
    // implemented using the += operator defined above
    lhs += rhs;
    return lhs;
}

// multiply values (p * q)
inline TropicalAlgebra operator*(TropicalAlgebra lhs, const TropicalAlgebra&  rhs) noexcept
{
    lhs *= rhs;
    return lhs;
}

// pow is the idomatic way for exponentiation in C++
inline TropicalAlgebra pow(const TropicalAlgebra& base, unsigned int exponent) noexcept
{
    auto result = base;
    for(unsigned int i = 1; i < exponent; i++)
    {
        // compute the power by successive multiplication 
        result *= base;
    }
    return result;
}

// print the point
ostream& operator<<(ostream& os, const TropicalAlgebra& pt)
{
    if(!pt.m_value) cout << "-Inf\n";
    else cout << *pt.m_value << "\n";
    return os;
}

int main(void) {
    const TropicalAlgebra a(-2);
    const TropicalAlgebra b(-1);
    const TropicalAlgebra c(-0.5);
    const TropicalAlgebra d(-0.001);
    const TropicalAlgebra e(0);
    const TropicalAlgebra h(1.5);
    const TropicalAlgebra i(2);
    const TropicalAlgebra j(5);
    const TropicalAlgebra k(7);
    const TropicalAlgebra l(8);
    const TropicalAlgebra m; // -Inf
    
    cout << "2 * -2 == " << i * a;
    cout << "-0.001 + -Inf == " << d + m;
    cout << "0 * -Inf == " << e * m;
    cout << "1.5 + -1 == " << h + b;
    cout << "-0.5 * 0 == " << c * e;
    cout << "pow(5, 7) == " << pow(j, 7);
    cout << "5 * (8 + 7)) == " << j * (l + k);
    cout << "5 * 8 + 5 * 7 == " << j * l + j * k;
}

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