Operator< and strict weak ordering

Question

How to define operator< on n-tuple (for example on 3-tuple) so that it satisfy strict weak ordering concept ? I know that boost library has tuple class with correctly defined operator< but for some reasons I can't use it.

Answer 1

strict weak ordering

This is a mathematical term to define a relationship between two objects.
Its definition is:

Two objects x and y are equivalent if both f(x, y) and f(y, x) are false. Note that an object is always (by the irreflexivity invariant) equivalent to itself.

In terms of C++ this means if you have two objects of a given type, you should return the following values when compared with the operator <.

X    a;
X    b;

Condition:                  Test:     Result
a is equivalent to b:       a < b     false
a is equivalent to b        b < a     false

a is less than b            a < b     true
a is less than b            b < a     false

b is less than a            a < b     false
b is less than a            b < a     true

How you define equivalent/less is totally dependent on the type of your object.

Formal Definition:
Strict Weak ordering

Computer Science:
Strict Weak Ordering

How it relates to operators:
Comparator

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As a side note we can implement strict weak ordering manually. But we can do it simply using the std::tuple which has implemented it for you. You simply need to create a tuple without copying the objects.

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

Note: This assumes that thingA and thingB already implement strict weak ordering themselves.

We can also implement equality the same way:

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

Note again: This assumes that thingA and thingB already implement equality.