How to Implement a Strict Weak Ordering `

How to Define the Operator< on an n-Tuple for Strict Weak Ordering

Introduction

Strict weak ordering is a mathematical concept that defines a relationship between two objects. It specifies that if objects x and y are equivalent, then both f(x, y) and f(y, x) are false; and if a is less than b, then f(a, b) is true and f(b, a) is false.

Definition for n-Tuples

For an n-tuple, the operator< can be defined to satisfy strict weak ordering by following these conditions:

• Equivalence: Two n-tuples are equivalent if they are equal in all their components.
• Less Than: An n-tuple a is less than another n-tuple b if any component of a is less than the corresponding component of b.

Implementation

The definition above can be implemented in C using the following steps:

1. Create a comparison function that returns false for equivalence and true for less than, as per the conditions above.
2. Provide operator overloading for the < operator using the comparison function.

Example

For a 3-tuple, the operator< can be defined as follows:

struct Triple {
    int x;
    int y;
    int z;
    bool operator<(const Triple& other) const {
        return (x < other.x) || (y < other.y) || (z < other.z);
    }
};

Using this definition, two triples would be equivalent if they have identical values for x, y, and z. A triple would be considered less than another if any of its components are less than those of the other triple.

std::tuple

Alternatively, the std::tuple class provided by the C Standard Library already implements strict weak ordering for tuples. To take advantage of this, tuples can be used without copying the objects they contain.

struct S {
    int a;
    int b;
};
bool operator<(const S& lhs, const S& rhs) {
    return std::tie(lhs.a, lhs.b) < std::tie(rhs.a, rhs.b);
}

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