

from __future__ import annotations

import random

# Adjacency list representation of this graph:
# https://en.wikipedia.org/wiki/File:Single_run_of_Karger%E2%80%99s_Mincut_algorithm.svg
A_						:     Union[str, Any]			  =				{
    '1': ['2', '3', '4', '5'],
    '2': ['1', '3', '4', '5'],
    '3': ['1', '2', '4', '5', '10'],
    '4': ['1', '2', '3', '5', '6'],
    '5': ['1', '2', '3', '4', '7'],
    '6': ['7', '8', '9', '10', '4'],
    '7': ['6', '8', '9', '10', '5'],
    '8': ['6', '7', '9', '10'],
    '9': ['6', '7', '8', '10'],
    '10': ['6', '7', '8', '9', '3'],
}


def  __a    (				SCREAMING_SNAKE_CASE ) ->       set[tuple[str, str]]:






    '''simple docstring'''
    __UpperCAmelCase			   =     {node: {node} for node in graph}

    __UpperCAmelCase			   =     {node: graph[node][:] for node in graph}

    while len(SCREAMING_SNAKE_CASE ) > 2:
        # Choose a random edge.
        __UpperCAmelCase			   =     random.choice(list(graph_copy.keys() ) )
        __UpperCAmelCase			   =     random.choice(graph_copy[u] )

        # Contract edge (u, v) to new node uv
        __UpperCAmelCase			   =     u + v
        __UpperCAmelCase			   =     list(set(graph_copy[u] + graph_copy[v] ) )
        uv_neighbors.remove(SCREAMING_SNAKE_CASE )
        uv_neighbors.remove(SCREAMING_SNAKE_CASE )
        __UpperCAmelCase			   =     uv_neighbors
        for neighbor in uv_neighbors:
            graph_copy[neighbor].append(SCREAMING_SNAKE_CASE )

        __UpperCAmelCase			   =     set(contracted_nodes[u].union(contracted_nodes[v] ) )

        # Remove nodes u and v.
        del graph_copy[u]
        del graph_copy[v]
        for neighbor in uv_neighbors:
            if u in graph_copy[neighbor]:
                graph_copy[neighbor].remove(SCREAMING_SNAKE_CASE )
            if v in graph_copy[neighbor]:
                graph_copy[neighbor].remove(SCREAMING_SNAKE_CASE )

    # Find cutset.
    __UpperCAmelCase			   =     [contracted_nodes[node] for node in graph_copy]
    return {
        (node, neighbor)
        for node in groups[0]
        for neighbor in graph[node]
        if neighbor in groups[1]
    }


if __name__ == "__main__":
       print(partition_graph(TEST_GRAPH))







from __future__ import annotations


def  __a    (				SCREAMING_SNAKE_CASE ) ->       list[int]:






    '''simple docstring'''
    if not list_of_ints:
        return []

    if min(SCREAMING_SNAKE_CASE ) < 0:
        raise ValueError('''All numbers must be positive''' )

    __UpperCAmelCase			   =     max(len(bin(SCREAMING_SNAKE_CASE )[2:] ) for x in list_of_ints )
    return _msd_radix_sort(SCREAMING_SNAKE_CASE    ,				SCREAMING_SNAKE_CASE )


def  __a    (				SCREAMING_SNAKE_CASE    ,				SCREAMING_SNAKE_CASE ) ->       list[int]:






    '''simple docstring'''
    if bit_position == 0 or len(SCREAMING_SNAKE_CASE ) in [0, 1]:
        return list_of_ints

    __UpperCAmelCase			   =     []
    __UpperCAmelCase			   =     []
    # Split numbers based on bit at bit_position from the right
    for number in list_of_ints:
        if (number >> (bit_position - 1)) & 1:
            # number has a one at bit bit_position
            ones.append(SCREAMING_SNAKE_CASE )
        else:
            # number has a zero at bit bit_position
            zeros.append(SCREAMING_SNAKE_CASE )

    # recursively split both lists further
    __UpperCAmelCase			   =     _msd_radix_sort(SCREAMING_SNAKE_CASE    ,				bit_position - 1 )
    __UpperCAmelCase			   =     _msd_radix_sort(SCREAMING_SNAKE_CASE    ,				bit_position - 1 )

    # recombine lists
    __UpperCAmelCase			   =     zeros
    res.extend(SCREAMING_SNAKE_CASE )

    return res


def  __a    (				SCREAMING_SNAKE_CASE ) ->       Optional[int]:






    '''simple docstring'''
    __UpperCAmelCase			   =     len(SCREAMING_SNAKE_CASE )
    if not list_of_ints or length == 1:
        return

    if min(SCREAMING_SNAKE_CASE ) < 0:
        raise ValueError('''All numbers must be positive''' )

    __UpperCAmelCase			   =     max(len(bin(SCREAMING_SNAKE_CASE )[2:] ) for x in list_of_ints )
    _msd_radix_sort_inplace(SCREAMING_SNAKE_CASE    ,				SCREAMING_SNAKE_CASE    ,				0    ,				SCREAMING_SNAKE_CASE )


def  __a    (				SCREAMING_SNAKE_CASE    ,				SCREAMING_SNAKE_CASE    ,				SCREAMING_SNAKE_CASE    ,				SCREAMING_SNAKE_CASE ) ->       int:






    '''simple docstring'''
    if bit_position == 0 or end_index - begin_index <= 1:
        return

    bit_position -= 1

    __UpperCAmelCase			   =     begin_index
    __UpperCAmelCase			   =     end_index - 1
    while i <= j:
        __UpperCAmelCase			   =     False
        if not (list_of_ints[i] >> bit_position) & 1:
            # found zero at the beginning
            i += 1
            __UpperCAmelCase			   =     True
        if (list_of_ints[j] >> bit_position) & 1:
            # found one at the end
            j -= 1
            __UpperCAmelCase			   =     True

        if changed:
            continue

        __UpperCAmelCase   ,       __UpperCAmelCase			   =     list_of_ints[j], list_of_ints[i]
        j -= 1
        if j != i:
            i += 1

    _msd_radix_sort_inplace(SCREAMING_SNAKE_CASE    ,				SCREAMING_SNAKE_CASE    ,				SCREAMING_SNAKE_CASE    ,				SCREAMING_SNAKE_CASE )
    _msd_radix_sort_inplace(SCREAMING_SNAKE_CASE    ,				SCREAMING_SNAKE_CASE    ,				SCREAMING_SNAKE_CASE    ,				SCREAMING_SNAKE_CASE )


if __name__ == "__main__":
       import doctest

       doctest.testmod()




