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std::lexicographical_compare - cppreference.com
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std::lexicographical_compare

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Definido en el archivo de encabezado <algorithm>
template< class InputIt1, class InputIt2 >
bool lexicographical_compare( InputIt1 first1, InputIt1 last1,
                              InputIt2 first2, InputIt2 last2 );
(1)
template< class InputIt1, class InputIt2, class Compare >
bool lexicographical_compare( InputIt1 first1, InputIt1 last1,
                              InputIt2 first2, InputIt2 last2,
                              Compare comp );
(2)
Comprueba si el primer intervalo es [first1, last1) lexicogrfico' menor que el [first2, last2) segundo rango. La primera versin utiliza operator< para comparar los elementos, la segunda versin utiliza la funcin de comparacin dado comp .
Original:
Checks if the first range [first1, last1) is lexicographically less than the second range [first2, last2). The first version uses operator< to compare the elements, the second version uses the given comparison function comp.
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Comparacin lexicogrfica es una operacin con las siguientes propiedades:
Original:
Lexicographical comparison is a operation with the following properties:
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  • Dos cadenas se comparan elemento por elemento .
    Original:
    Two ranges are compared element by element.
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  • El elemento descalce primera define qu rango es menos lexicogrfico''' o mayor que el otro .
    Original:
    The first mismatching element defines which range is lexicographically less or greater than the other.
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  • Si una serie es un prefijo de otro, el rango ms corto es lexicogrfico' menos que el otro .
    Original:
    If one range is a prefix of another, the shorter range is lexicographically less than the other.
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  • Si dos gamas tienen elementos equivalentes y son de la misma longitud, entonces los rangos son iguales lexicogrficamente' .
    Original:
    If two ranges have equivalent elements and are of the same length, then the ranges are lexicographically equal.
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  • Un rango de vaco es lexicogrfico' menos de cualquier rango que no est vaca .
    Original:
    An empty range is lexicographically less than any non-empty range.
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  • Dos cadenas vacas son lexicogrfico' igual .
    Original:
    Two empty ranges are lexicographically equal.
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Parmetros

first1, last1 -
la primera gama de elementos a examinar
Original:
the first range of elements to examine
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first2, last2 -
la segunda gama de elementos a examinar
Original:
the second range of elements to examine
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comp - objeto funcin de comparacin (es decir, un objeto que satisface los requerimientos de Compare) que devuelve true si el primer argumento es menor que el segundo.

La signatura de la funcin de comparacin deber ser equivalente a lo siguiente:

bool cmp(const Type1 &a, const Type2 &b);

Mientras que la signatura no necesita ser const &, la funcin no debe modificar los objetos que se le pasaron y debe admitir todos los valores de los tipos (posiblemente const) Type1 y Type2 a pesar de la categora de valor (por consiguiente, no se permite a Type1 & , ni tampoco a Type1 a menos que para Type1 un movimiento sea equivalente a una copia (desde C++11)).
Los tipos Type1 y Type2 deben ser tales que objetos de tipo InputIt1 y InputIt2 pueden ser desreferenciados y luego convertidos implcitamente a Type1 and Type2 respectively.

Requisitos de tipo
-
InputIt1, InputIt2 debe reunir los requerimientos de InputIterator.

Valor de retorno

true si el primer intervalo es lexicogrficamente' menos que el segundo .
Original:
true if the first range is lexicographically less than the second.
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Complejidad

En la mayora de aplicaciones 2·min(N1, N2) de la operacin de comparacin, donde N1 = std::distance(first1, last1) y N2 = std::distance(first2, last2) .
Original:
At most 2·min(N1, N2) applications of the comparison operation, where N1 = std::distance(first1, last1) and N2 = std::distance(first2, last2).
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Posible implementacin

Primera versin
template<class InputIt1, class InputIt2>
bool lexicographical_compare(InputIt1 first1, InputIt1 last1,
                             InputIt2 first2, InputIt2 last2)
{
    for ( ; (first1 != last1) && (first2 != last2); first1++, first2++ ) {
        if (*first1 < *first2) return true;
        if (*first2 < *first1) return false;
    }
    return (first1 == last1) && (first2 != last2);
}
Segunda versin
template<class InputIt1, class InputIt2, class Compare>
bool lexicographical_compare(InputIt1 first1, InputIt1 last1,
                             InputIt2 first2, InputIt2 last2,
                             Compare comp)
{
    for ( ; (first1 != last1) && (first2 != last2); first1++, first2++ ) {
        if (comp(*first1, *first2)) return true;
        if (comp(*first2, *first1)) return false;
    }
    return (first1 == last1) && (first2 != last2);
}

Ejemplo

#include <algorithm>
#include <iostream>
#include <vector>
#include <cstdlib>
#include <ctime>

int main()
{
    std::vector<char> v1 {'a', 'b', 'c', 'd'};
    std::vector<char> v2 {'a', 'b', 'c', 'd'};

    std::srand(std::time(0));
    while (!std::lexicographical_compare(v1.begin(), v1.end(),
                                         v2.begin(), v2.end())) {
        for (auto c : v1) std::cout << c << ' ';
        std::cout << ">= ";
        for (auto c : v2) std::cout << c << ' ';
        std::cout << '\n';

        std::random_shuffle(v1.begin(), v1.end());
        std::random_shuffle(v2.begin(), v2.end());
    }

    for (auto c : v1) std::cout << c << ' ';
    std::cout << "< ";
    for (auto c : v2) std::cout << c << ' ';
    std::cout << '\n';
}

Posible salida:

a b c d >= a b c d 
d a b c >= c b d a 
b d a c >= a d c b 
a c d b < c d a b

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