Минимальные пути (маршруты) в нагруженных ориентированных графах (неориентированных графах)
Алгоритм Форда-Беллмана нахождения минимального пути в нагруженном графе D из v1 в vi1 (i1≠l):
Шаг 1.
Если
,
то вершина
не достижима из v1
(предполагаем,
что все величины l(х),
х
Х,
конечны). В этом случае работа алгоритма
заканчивается.
Шаг 2.
Пусть
.
Тогда число
выражает длину любого минимального
пути из v1
в vi1
в нагруженном
орграфе D.
Определим минимальное число k1>l,
при котором выполняется равенство
.
По определению чисел
получаем, что k1
— минимальное число дуг в пути среди
всех минимальных путей из V1
в
в нагруженном орграфе D.
Шаг 3. Последовательно определяем номера i2,...,ik1+1 такие, что
Складывая равенства
(9.14) и учитывая (9.15), имеем l(v1,vk1
,,..,vi2,vi1)=
.
Заметим, что в этом пути ровно k1
дуг. Следовательно, мы определили путь
с минимальным числом дуг среди всех
минимальных путей из v1
в vi1
в нагруженном орграфе D.
Номера i2,i3,...,ik1, удовлетворяющие (9.14), вообще говоря, могут быть выделены неоднозначно. Эта неоднозначность соответствует случаям, когда существует несколько различных путей из v1 в vi1 в нагруженном орграфе D.
Пример. Определим минимальный путь из v1 в v6 в нагруженном орграфе D, изображенном на рис., около каждой дуги которого указана ее длина.
Рис.
Таблица
Составим матрицу
C(D)
длин дуг нагруженного графа D.
Эта матрица представлена в таблице.
Справа от матрицы C(D)
припишем шесть столбцов, которые будем
определять, используя рекуррентное
соотношение. Величина
выражает
длину минимального пути из v1
в v6
в нагруженном орграфе D.
Найдем минимальное число k1>i,
при котором выполняется равенство
.
Из таблицы получаем, что k1=4.
Таким образом, минимальное число дуг в
пути среди всех минимальных путей из
v1
в v6
в нагруженном орграфе D
равняется 4. Определим теперь
последовательность номеров,
удовлетворяющих
(9.14). Из таблицы получаем, что в качестве
такой последовательности надо взять
номера 6, 2, 3, 5, 1, так как
Тогда v1v5v3v2v6 - искомый минимальный путь из v1 в v6 в нагруженном орграфе D, причем он содержит минимальное число дуг среди всех возможных минимальных путей из v1 в v6.
Практические задания
1) Определить путь из v1 в v7 минимальной длины в нагруженном орграфе, заданной матрицей длин дуг.
2) С помощью алгоритма определить минимальное остовное дерево нагруженного графа.
3) Найти маршрут из v1 в v7 с минимальным числом ребер в ориентированном графе.
Вариант № 1
|
|
1 |
|
4 |
|
5 |
13 |
|
0 |
2 |
0 |
0 |
3 |
4 |
0 |
|
0 |
0 |
0 |
1 |
1 |
0 |
0 |
|
|
|
|
|
9 |
3 |
11 |
|
2 |
0 |
6 |
4 |
5 |
0 |
3 |
|
1 |
0 |
1 |
0 |
0 |
1 |
1 |
|
|
7 |
|
|
4 |
|
6 |
|
0 |
6 |
0 |
9 |
11 |
0 |
0 |
|
1 |
1 |
1 |
1 |
0 |
0 |
1 |
1) |
|
|
|
|
|
2 |
|
2) |
0 |
4 |
9 |
0 |
0 |
5 |
7 |
3) |
0 |
0 |
0 |
1 |
1 |
1 |
0 |
|
7 |
|
|
2 |
|
5 |
1 |
|
3 |
5 |
11 |
0 |
0 |
1 |
7 |
|
1 |
0 |
0 |
1 |
0 |
1 |
0 |
|
|
|
1 |
|
|
|
|
|
4 |
0 |
0 |
5 |
1 |
0 |
0 |
|
0 |
1 |
1 |
1 |
0 |
0 |
0 |
|
|
|
|
6 |
|
8 |
|
|
0 |
3 |
0 |
7 |
7 |
0 |
0 |
|
1 |
1 |
0 |
0 |
1 |
1 |
0 |
Вариант № 2
|
|
7 |
|
5 |
1 |
|
|
|
0 |
0 |
8 |
7 |
3 |
0 |
5 |
|
0 |
0 |
0 |
0 |
1 |
1 |
0 |
|
|
|
4 |
1 |
4 |
|
|
|
0 |
0 |
0 |
3 |
6 |
0 |
0 |
|
1 |
0 |
1 |
0 |
1 |
0 |
1 |
|
|
|
|
|
3 |
1 |
2 |
|
8 |
0 |
0 |
2 |
7 |
0 |
5 |
|
1 |
1 |
0 |
1 |
1 |
0 |
0 |
1) |
|
|
4 |
|
|
|
|
2) |
7 |
3 |
2 |
0 |
11 |
9 |
0 |
3) |
0 |
0 |
1 |
0 |
1 |
1 |
0 |
|
|
|
|
|
|
1 |
|
|
3 |
6 |
7 |
11 |
0 |
0 |
6 |
|
1 |
0 |
0 |
1 |
0 |
1 |
0 |
|
|
|
|
2 |
|
|
9 |
|
0 |
0 |
0 |
9 |
0 |
0 |
4 |
|
0 |
0 |
1 |
0 |
1 |
0 |
0 |
|
6 |
|
|
|
4 |
|
|
|
5 |
0 |
5 |
0 |
6 |
4 |
0 |
|
1 |
0 |
1 |
1 |
0 |
0 |
1 |
Вариант № 3
|
|
5 |
1 |
|
8 |
|
|
|
0 |
5 |
0 |
3 |
11 |
0 |
0 |
|
0 |
0 |
0 |
1 |
0 |
1 |
0 |
|
|
|
|
|
2 |
|
|
|
5 |
0 |
7 |
0 |
3 |
0 |
0 |
|
0 |
1 |
1 |
0 |
1 |
0 |
1 |
|
|
3 |
|
|
|
|
11 |
|
0 |
7 |
0 |
0 |
8 |
5 |
9 |
|
1 |
1 |
0 |
1 |
1 |
1 |
0 |
1) |
|
|
1 |
|
|
|
2 |
2) |
3 |
0 |
0 |
0 |
2 |
0 |
0 |
3) |
1 |
0 |
0 |
0 |
1 |
0 |
0 |
|
|
|
|
2 |
|
3 |
|
|
11 |
3 |
8 |
2 |
0 |
6 |
0 |
|
1 |
0 |
1 |
1 |
0 |
1 |
0 |
|
7 |
|
3 |
4 |
|
|
6 |
|
0 |
0 |
5 |
0 |
6 |
0 |
4 |
|
0 |
0 |
0 |
1 |
1 |
0 |
0 |
|
6 |
5 |
|
|
|
|
|
|
0 |
0 |
9 |
0 |
0 |
4 |
0 |
|
0 |
1 |
1 |
1 |
1 |
0 |
0 |
Вариант № 4
|
|
1 |
|
4 |
|
6 |
12 |
|
0 |
4 |
7 |
12 |
3 |
11 |
0 |
|
0 |
1 |
0 |
1 |
1 |
0 |
0 |
|
|
|
|
|
2 |
7 |
11 |
|
4 |
0 |
5 |
9 |
14 |
0 |
0 |
|
1 |
0 |
1 |
1 |
1 |
0 |
0 |
|
|
7 |
|
|
4 |
|
5 |
|
7 |
5 |
0 |
6 |
5 |
0 |
0 |
|
1 |
0 |
0 |
1 |
0 |
0 |
1 |
1) |
|
|
|
|
|
2 |
|
2) |
12 |
9 |
6 |
0 |
4 |
0 |
5 |
3) |
0 |
1 |
0 |
0 |
1 |
1 |
0 |
|
7 |
|
|
2 |
|
2 |
9 |
|
3 |
14 |
5 |
4 |
0 |
2 |
1 |
|
1 |
0 |
1 |
0 |
1 |
1 |
0 |
|
|
|
1 |
|
|
|
|
|
11 |
0 |
0 |
0 |
2 |
0 |
8 |
|
0 |
1 |
1 |
0 |
0 |
0 |
1 |
|
|
|
|
6 |
|
8 |
|
|
0 |
0 |
0 |
5 |
1 |
8 |
0 |
|
0 |
1 |
0 |
1 |
0 |
0 |
1 |
Вариант № 5
|
¥ |
4 |
¥ |
11 |
8 |
¥ |
¥ |
|
0 |
3 |
8 |
9 |
0 |
0 |
0 |
|
0 |
0 |
0 |
1 |
0 |
0 |
0 |
|
¥ |
¥ |
9 |
7 |
3 |
¥ |
¥ |
|
3 |
0 |
5 |
0 |
4 |
0 |
0 |
|
1 |
0 |
1 |
0 |
1 |
0 |
0 |
|
¥ |
¥ |
¥ |
¥ |
3 |
1 |
2 |
|
8 |
5 |
0 |
7 |
11 |
0 |
4 |
|
1 |
0 |
0 |
1 |
0 |
1 |
0 |
1) |
¥ |
¥ |
1 |
¥ |
¥ |
¥ |
¥ |
2) |
9 |
0 |
7 |
0 |
0 |
0 |
5 |
3) |
0 |
1 |
1 |
0 |
0 |
0 |
0 |
|
¥ |
¥ |
¥ |
¥ |
¥ |
2 |
¥ |
|
0 |
4 |
11 |
0 |
0 |
7 |
9 |
|
0 |
0 |
1 |
0 |
1 |
0 |
1 |
|
¥ |
¥ |
¥ |
1 |
¥ |
¥ |
9 |
|
0 |
0 |
0 |
0 |
7 |
0 |
6 |
|
1 |
0 |
0 |
0 |
0 |
0 |
1 |
|
6 |
¥ |
¥ |
¥ |
4 |
¥ |
¥ |
|
0 |
0 |
4 |
5 |
9 |
6 |
0 |
|
1 |
1 |
0 |
1 |
0 |
0 |
0 |
Вариант № 6
|
¥ |
5 |
1 |
¥ |
8 |
¥ |
¥ |
|
0 |
4 |
0 |
0 |
5 |
4 |
0 |
|
1 |
1 |
1 |
0 |
0 |
1 |
0 |
|
¥ |
¥ |
¥ |
¥ |
3 |
¥ |
¥ |
|
4 |
0 |
8 |
6 |
5 |
0 |
4 |
|
0 |
0 |
1 |
0 |
0 |
1 |
0 |
|
¥ |
2 |
¥ |
¥ |
¥ |
¥ |
14 |
|
0 |
8 |
0 |
7 |
9 |
0 |
0 |
|
1 |
1 |
0 |
0 |
0 |
1 |
0 |
1) |
¥ |
¥ |
1 |
¥ |
¥ |
¥ |
2 |
2) |
0 |
6 |
7 |
0 |
0 |
3 |
2 |
3) |
1 |
0 |
1 |
0 |
1 |
0 |
1 |
|
¥ |
¥ |
¥ |
7 |
¥ |
1 |
¥ |
|
5 |
5 |
9 |
0 |
0 |
11 |
8 |
|
0 |
1 |
1 |
1 |
0 |
0 |
1 |
|
7 |
¥ |
3 |
5 |
¥ |
¥ |
9 |
|
4 |
0 |
0 |
3 |
11 |
0 |
0 |
|
0 |
0 |
1 |
1 |
1 |
1 |
0 |
|
6 |
5 |
¥ |
¥ |
¥ |
¥ |
¥ |
|
0 |
4 |
0 |
2 |
8 |
0 |
0 |
|
1 |
0 |
0 |
0 |
1 |
1 |
0 |
Вариант № 7
|
|
7 |
|
3 |
|
6 |
14 |
|
0 |
0 |
5 |
4 |
8 |
0 |
6 |
|
1 |
0 |
0 |
1 |
0 |
0 |
0 |
|
|
|
|
|
2 |
7 |
11 |
|
0 |
0 |
0 |
11 |
10 |
0 |
0 |
|
0 |
0 |
1 |
1 |
1 |
0 |
0 |
|
|
7 |
|
|
3 |
|
9 |
|
5 |
0 |
0 |
2 |
8 |
0 |
5 |
|
1 |
1 |
0 |
1 |
1 |
1 |
0 |
1) |
|
|
|
|
|
1 |
|
2) |
4 |
11 |
2 |
0 |
7 |
5 |
0 |
3) |
1 |
1 |
0 |
0 |
0 |
1 |
0 |
|
7 |
|
|
2 |
|
2 |
5 |
|
8 |
10 |
8 |
7 |
0 |
0 |
6 |
|
0 |
0 |
1 |
1 |
1 |
0 |
1 |
|
|
|
1 |
|
|
|
|
|
0 |
0 |
0 |
5 |
0 |
0 |
3 |
|
1 |
0 |
1 |
0 |
1 |
0 |
0 |
|
|
|
|
6 |
|
8 |
|
|
6 |
0 |
5 |
0 |
6 |
3 |
0 |
|
0 |
1 |
1 |
1 |
0 |
1 |
0 |
Вариант № 8
|
¥ |
3 |
¥ |
8 |
8 |
¥ |
¥ |
|
0 |
4 |
0 |
1 |
9 |
0 |
0 |
|
0 |
0 |
0 |
0 |
1 |
1 |
0 |
|
¥ |
¥ |
9 |
1 |
3 |
¥ |
¥ |
|
4 |
0 |
5 |
0 |
2 |
0 |
0 |
|
0 |
0 |
1 |
1 |
1 |
0 |
0 |
|
¥ |
¥ |
¥ |
¥ |
3 |
1 |
6 |
|
0 |
5 |
0 |
0 |
6 |
4 |
7 |
|
1 |
0 |
1 |
0 |
0 |
1 |
1 |
1) |
¥ |
¥ |
2 |
¥ |
¥ |
¥ |
¥ |
2) |
1 |
0 |
0 |
0 |
5 |
0 |
0 |
3) |
0 |
0 |
1 |
1 |
0 |
1 |
1 |
|
¥ |
¥ |
¥ |
¥ |
¥ |
2 |
¥ |
|
9 |
2 |
6 |
5 |
0 |
8 |
0 |
|
1 |
1 |
0 |
0 |
0 |
1 |
0 |
|
¥ |
¥ |
¥ |
1 |
¥ |
¥ |
4 |
|
0 |
0 |
4 |
0 |
8 |
0 |
4 |
|
1 |
1 |
0 |
0 |
1 |
1 |
0 |
|
6 |
¥ |
¥ |
¥ |
4 |
¥ |
¥ |
|
0 |
0 |
7 |
0 |
0 |
4 |
0 |
|
1 |
0 |
0 |
1 |
1 |
0 |
0 |
Вариант № 9
|
¥ |
5 |
2 |
¥ |
9 |
¥ |
¥ |
|
0 |
3 |
2 |
11 |
5 |
9 |
0 |
|
0 |
1 |
0 |
0 |
0 |
0 |
0 |
|
¥ |
¥ |
¥ |
¥ |
4 |
¥ |
¥ |
|
3 |
0 |
5 |
8 |
1 |
0 |
0 |
|
1 |
0 |
1 |
0 |
1 |
0 |
0 |
|
¥ |
2 |
¥ |
¥ |
¥ |
¥ |
11 |
|
2 |
5 |
0 |
6 |
5 |
0 |
0 |
|
0 |
1 |
0 |
1 |
0 |
1 |
0 |
1) |
¥ |
¥ |
1 |
¥ |
¥ |
¥ |
2 |
2) |
11 |
8 |
6 |
0 |
2 |
0 |
4 |
3) |
1 |
0 |
1 |
0 |
0 |
0 |
1 |
|
¥ |
¥ |
¥ |
7 |
¥ |
1 |
¥ |
|
5 |
1 |
5 |
2 |
0 |
3 |
7 |
|
1 |
1 |
0 |
1 |
1 |
0 |
0 |
|
3 |
¥ |
3 |
5 |
¥ |
¥ |
3 |
|
9 |
0 |
0 |
0 |
3 |
0 |
6 |
|
0 |
1 |
0 |
0 |
1 |
1 |
1 |
|
7 |
1 |
¥ |
¥ |
¥ |
¥ |
¥ |
|
0 |
0 |
0 |
4 |
7 |
6 |
0 |
|
0 |
0 |
1 |
1 |
0 |
1 |
0 |
Вариант № 10
|
|
11 |
|
2 |
|
6 |
17 |
|
0 |
3 |
6 |
9 |
0 |
0 |
0 |
|
0 |
0 |
1 |
1 |
0 |
1 |
0 |
|
|
|
|
|
7 |
7 |
6 |
|
3 |
0 |
6 |
0 |
7 |
0 |
0 |
|
1 |
1 |
0 |
0 |
1 |
0 |
1 |
|
|
5 |
|
|
8 |
|
12 |
|
6 |
6 |
0 |
1 |
4 |
0 |
5 |
|
0 |
0 |
1 |
0 |
1 |
1 |
0 |
1) |
|
|
|
|
|
1 |
|
2) |
9 |
0 |
1 |
0 |
0 |
0 |
2 |
3) |
1 |
0 |
0 |
1 |
0 |
1 |
0 |
|
7 |
|
|
2 |
|
2 |
5 |
|
0 |
7 |
4 |
0 |
0 |
2 |
7 |
|
1 |
1 |
0 |
1 |
0 |
0 |
0 |
|
|
|
1 |
|
|
|
|
|
0 |
0 |
0 |
0 |
2 |
0 |
3 |
|
0 |
0 |
1 |
0 |
1 |
1 |
0 |
|
|
|
|
6 |
|
8 |
|
|
0 |
0 |
5 |
2 |
7 |
3 |
0 |
|
1 |
0 |
1 |
1 |
0 |
0 |
0 |
Вариант № 11
|
¥ |
2 |
¥ |
8 |
8 |
¥ |
¥ |
|
0 |
2 |
0 |
0 |
3 |
4 |
0 |
|
0 |
0 |
0 |
1 |
1 |
0 |
0 |
|
¥ |
¥ |
4 |
3 |
6 |
¥ |
¥ |
|
2 |
0 |
6 |
4 |
5 |
0 |
3 |
|
1 |
0 |
1 |
0 |
0 |
1 |
1 |
|
¥ |
¥ |
¥ |
¥ |
1 |
7 |
6 |
|
0 |
6 |
0 |
9 |
11 |
0 |
0 |
|
1 |
1 |
1 |
1 |
0 |
0 |
1 |
1) |
¥ |
¥ |
3 |
¥ |
¥ |
¥ |
¥ |
2) |
0 |
4 |
9 |
0 |
0 |
5 |
7 |
3) |
0 |
0 |
0 |
1 |
1 |
1 |
0 |
|
¥ |
¥ |
¥ |
¥ |
¥ |
2 |
¥ |
|
3 |
5 |
11 |
0 |
0 |
1 |
7 |
|
1 |
0 |
0 |
1 |
0 |
1 |
0 |
|
¥ |
¥ |
¥ |
1 |
¥ |
¥ |
2 |
|
4 |
0 |
0 |
5 |
1 |
0 |
0 |
|
0 |
1 |
1 |
1 |
0 |
0 |
0 |
|
6 |
¥ |
¥ |
¥ |
4 |
¥ |
¥ |
|
0 |
3 |
0 |
7 |
7 |
0 |
0 |
|
1 |
1 |
0 |
0 |
1 |
1 |
0 |
Вариант № 12
|
¥ |
4 |
9 |
¥ |
9 |
¥ |
¥ |
|
0 |
0 |
8 |
7 |
3 |
0 |
5 |
|
0 |
0 |
0 |
0 |
1 |
1 |
0 |
|
¥ |
¥ |
¥ |
¥ |
2 |
¥ |
¥ |
|
0 |
0 |
0 |
3 |
6 |
0 |
0 |
|
1 |
0 |
1 |
0 |
1 |
0 |
1 |
|
¥ |
2 |
¥ |
¥ |
¥ |
¥ |
3 |
|
8 |
0 |
0 |
2 |
7 |
0 |
5 |
|
1 |
1 |
0 |
1 |
1 |
0 |
0 |
1) |
¥ |
¥ |
1 |
¥ |
¥ |
¥ |
5 |
2) |
7 |
3 |
2 |
0 |
11 |
9 |
0 |
3) |
0 |
0 |
1 |
0 |
1 |
1 |
0 |
|
¥ |
¥ |
¥ |
1 |
¥ |
4 |
¥ |
|
3 |
6 |
7 |
11 |
0 |
0 |
6 |
|
1 |
0 |
0 |
1 |
0 |
1 |
0 |
|
2 |
¥ |
3 |
5 |
¥ |
¥ |
3 |
|
0 |
0 |
0 |
9 |
0 |
0 |
4 |
|
0 |
0 |
1 |
0 |
1 |
0 |
0 |
|
9 |
4 |
¥ |
¥ |
¥ |
¥ |
¥ |
|
5 |
0 |
5 |
0 |
6 |
4 |
0 |
|
1 |
0 |
1 |
1 |
0 |
0 |
1 |
Вариант № 13
|
|
7 |
|
2 |
|
6 |
11 |
|
0 |
5 |
0 |
3 |
11 |
0 |
0 |
|
0 |
0 |
0 |
1 |
0 |
1 |
0 |
|
|
|
|
|
2 |
7 |
6 |
|
5 |
0 |
7 |
0 |
3 |
0 |
0 |
|
0 |
1 |
1 |
0 |
1 |
0 |
1 |
|
|
2 |
|
|
8 |
|
8 |
|
0 |
7 |
0 |
0 |
8 |
5 |
9 |
|
1 |
1 |
0 |
1 |
1 |
1 |
0 |
1) |
|
|
|
|
|
1 |
|
2) |
3 |
0 |
0 |
0 |
2 |
0 |
0 |
3) |
1 |
0 |
0 |
0 |
1 |
0 |
0 |
|
7 |
|
|
2 |
|
2 |
1 |
|
11 |
3 |
8 |
2 |
0 |
6 |
0 |
|
1 |
0 |
1 |
1 |
0 |
1 |
0 |
|
|
|
1 |
|
|
|
|
|
0 |
0 |
5 |
0 |
6 |
0 |
4 |
|
0 |
0 |
0 |
1 |
1 |
0 |
0 |
|
|
|
|
6 |
|
8 |
|
|
0 |
0 |
9 |
0 |
0 |
4 |
0 |
|
0 |
1 |
1 |
1 |
1 |
0 |
0 |
Вариант № 14
|
¥ |
3 |
¥ |
3 |
7 |
¥ |
¥ |
|
0 |
4 |
7 |
12 |
3 |
11 |
0 |
|
0 |
1 |
0 |
1 |
1 |
0 |
0 |
|
¥ |
¥ |
4 |
3 |
6 |
¥ |
¥ |
|
4 |
0 |
5 |
9 |
14 |
0 |
0 |
|
1 |
0 |
1 |
1 |
1 |
0 |
0 |
|
¥ |
¥ |
¥ |
¥ |
1 |
5 |
6 |
|
7 |
5 |
0 |
6 |
5 |
0 |
0 |
|
1 |
0 |
0 |
1 |
0 |
0 |
1 |
1) |
¥ |
¥ |
2 |
¥ |
¥ |
¥ |
¥ |
2) |
12 |
9 |
6 |
0 |
4 |
0 |
5 |
3) |
0 |
1 |
0 |
0 |
1 |
1 |
0 |
|
¥ |
¥ |
¥ |
¥ |
¥ |
2 |
¥ |
|
3 |
14 |
5 |
4 |
0 |
2 |
1 |
|
1 |
0 |
1 |
0 |
1 |
1 |
0 |
|
¥ |
¥ |
¥ |
1 |
¥ |
¥ |
1 |
|
11 |
0 |
0 |
0 |
2 |
0 |
8 |
|
0 |
1 |
1 |
0 |
0 |
0 |
1 |
|
6 |
¥ |
¥ |
¥ |
4 |
¥ |
¥ |
|
0 |
0 |
0 |
5 |
1 |
8 |
0 |
|
0 |
1 |
0 |
1 |
0 |
0 |
1 |
Вариант № 15
|
¥ |
1 |
12 |
¥ |
9 |
¥ |
¥ |
|
0 |
3 |
8 |
9 |
0 |
0 |
0 |
|
0 |
0 |
0 |
1 |
0 |
0 |
0 |
|
¥ |
¥ |
¥ |
¥ |
2 |
¥ |
¥ |
|
3 |
0 |
5 |
0 |
4 |
0 |
0 |
|
1 |
0 |
1 |
0 |
1 |
0 |
0 |
|
¥ |
2 |
¥ |
¥ |
¥ |
¥ |
2 |
|
8 |
5 |
0 |
7 |
11 |
0 |
4 |
|
1 |
0 |
0 |
1 |
0 |
1 |
0 |
1) |
¥ |
¥ |
1 |
¥ |
¥ |
¥ |
6 |
2) |
9 |
0 |
7 |
0 |
0 |
0 |
5 |
3) |
0 |
1 |
1 |
0 |
0 |
0 |
0 |
|
¥ |
¥ |
¥ |
7 |
¥ |
5 |
¥ |
|
0 |
4 |
11 |
0 |
0 |
7 |
9 |
|
0 |
0 |
1 |
0 |
1 |
0 |
1 |
|
5 |
¥ |
2 |
5 |
¥ |
¥ |
5 |
|
0 |
0 |
0 |
0 |
7 |
0 |
6 |
|
1 |
0 |
0 |
0 |
0 |
0 |
1 |
|
11 |
12 |
¥ |
¥ |
¥ |
¥ |
¥ |
|
0 |
0 |
4 |
5 |
9 |
6 |
0 |
|
1 |
1 |
0 |
1 |
0 |
0 |
0 |
Вариант № 16
|
¥ |
8 |
¥ |
4 |
¥ |
4 |
13 |
|
0 |
4 |
0 |
0 |
5 |
4 |
0 |
|
1 |
1 |
1 |
0 |
0 |
1 |
0 |
|
¥ |
¥ |
¥ |
¥ |
3 |
3 |
11 |
|
4 |
0 |
8 |
6 |
5 |
0 |
4 |
|
0 |
0 |
1 |
0 |
0 |
1 |
0 |
|
¥ |
1 |
¥ |
¥ |
7 |
¥ |
6 |
|
0 |
8 |
0 |
7 |
9 |
0 |
0 |
|
1 |
1 |
0 |
0 |
0 |
1 |
0 |
1) |
¥ |
¥ |
¥ |
¥ |
¥ |
2 |
¥ |
2) |
0 |
6 |
7 |
0 |
0 |
3 |
2 |
3) |
1 |
0 |
1 |
0 |
1 |
0 |
1 |
|
7 |
¥ |
¥ |
2 |
¥ |
5 |
1 |
|
5 |
5 |
9 |
0 |
0 |
11 |
8 |
|
0 |
1 |
1 |
1 |
0 |
0 |
1 |
|
¥ |
¥ |
2 |
¥ |
¥ |
¥ |
¥ |
|
4 |
0 |
0 |
3 |
11 |
0 |
0 |
|
0 |
0 |
1 |
1 |
1 |
1 |
0 |
|
¥ |
¥ |
¥ |
6 |
¥ |
8 |
¥ |
|
0 |
4 |
0 |
2 |
8 |
0 |
0 |
|
1 |
0 |
0 |
0 |
1 |
1 |
0 |
Вариант № 17
|
¥ |
3 |
14 |
¥ |
8 |
¥ |
¥ |
|
0 |
0 |
5 |
4 |
8 |
0 |
6 |
|
1 |
0 |
0 |
1 |
0 |
0 |
0 |
|
¥ |
¥ |
¥ |
¥ |
4 |
¥ |
¥ |
|
0 |
0 |
0 |
11 |
10 |
0 |
0 |
|
0 |
0 |
1 |
1 |
1 |
0 |
0 |
|
¥ |
2 |
¥ |
¥ |
¥ |
¥ |
2 |
|
5 |
0 |
0 |
2 |
8 |
0 |
5 |
|
1 |
1 |
0 |
1 |
1 |
1 |
0 |
1) |
¥ |
¥ |
2 |
¥ |
¥ |
¥ |
7 |
2) |
4 |
11 |
2 |
0 |
7 |
5 |
0 |
3) |
1 |
1 |
0 |
0 |
0 |
1 |
0 |
|
¥ |
¥ |
¥ |
6 |
¥ |
1 |
¥ |
|
8 |
10 |
8 |
7 |
0 |
0 |
6 |
|
0 |
0 |
1 |
1 |
1 |
0 |
1 |
|
3 |
¥ |
5 |
2 |
¥ |
¥ |
8 |
|
0 |
0 |
0 |
5 |
0 |
0 |
3 |
|
1 |
0 |
1 |
0 |
1 |
0 |
0 |
|
3 |
4 |
¥ |
¥ |
¥ |
¥ |
¥ |
|
6 |
0 |
5 |
0 |
6 |
3 |
0 |
|
0 |
1 |
1 |
1 |
0 |
1 |
0 |
Вариант № 18
|
¥ |
2 |
8 |
¥ |
7 |
¥ |
¥ |
|
0 |
4 |
0 |
1 |
9 |
0 |
0 |
|
0 |
0 |
0 |
0 |
1 |
1 |
0 |
|
¥ |
¥ |
¥ |
¥ |
1 |
¥ |
¥ |
|
4 |
0 |
5 |
0 |
2 |
0 |
0 |
|
0 |
0 |
1 |
1 |
1 |
0 |
0 |
|
¥ |
2 |
¥ |
¥ |
¥ |
¥ |
2 |
|
0 |
5 |
0 |
0 |
6 |
4 |
7 |
|
1 |
0 |
1 |
0 |
0 |
1 |
1 |
1) |
¥ |
¥ |
1 |
¥ |
¥ |
¥ |
2 |
2) |
1 |
0 |
0 |
0 |
5 |
0 |
0 |
3) |
0 |
0 |
1 |
1 |
0 |
1 |
1 |
|
¥ |
¥ |
¥ |
6 |
¥ |
3 |
¥ |
|
9 |
2 |
6 |
5 |
0 |
8 |
0 |
|
1 |
1 |
0 |
0 |
0 |
1 |
0 |
|
1 |
¥ |
2 |
1 |
¥ |
¥ |
5 |
|
0 |
0 |
4 |
0 |
8 |
0 |
4 |
|
1 |
1 |
0 |
0 |
1 |
1 |
0 |
|
3 |
7 |
¥ |
¥ |
¥ |
¥ |
¥ |
|
0 |
0 |
7 |
0 |
0 |
4 |
0 |
|
1 |
0 |
0 |
1 |
1 |
0 |
0 |
Вариант № 19
|
¥ |
3 |
¥ |
11 |
¥ |
11 |
17 |
|
0 |
3 |
2 |
11 |
5 |
9 |
0 |
|
0 |
1 |
0 |
0 |
0 |
0 |
0 |
|
¥ |
¥ |
¥ |
¥ |
2 |
8 |
14 |
|
3 |
0 |
5 |
8 |
1 |
0 |
0 |
|
1 |
0 |
1 |
0 |
1 |
0 |
0 |
|
¥ |
1 |
¥ |
¥ |
7 |
¥ |
3 |
|
2 |
5 |
0 |
6 |
5 |
0 |
0 |
|
0 |
1 |
0 |
1 |
0 |
1 |
0 |
1) |
¥ |
¥ |
¥ |
¥ |
¥ |
1 |
¥ |
2) |
11 |
8 |
6 |
0 |
2 |
0 |
4 |
3) |
1 |
0 |
1 |
0 |
0 |
0 |
1 |
|
7 |
¥ |
¥ |
4 |
¥ |
6 |
12 |
|
5 |
1 |
5 |
2 |
0 |
3 |
7 |
|
1 |
1 |
0 |
1 |
1 |
0 |
0 |
|
¥ |
¥ |
2 |
¥ |
¥ |
¥ |
¥ |
|
9 |
0 |
0 |
0 |
3 |
0 |
6 |
|
0 |
1 |
0 |
0 |
1 |
1 |
1 |
|
¥ |
¥ |
¥ |
3 |
¥ |
1 |
¥ |
|
0 |
0 |
0 |
4 |
7 |
6 |
0 |
|
0 |
0 |
1 |
1 |
0 |
1 |
0 |
Вариант № 20
|
¥ |
3 |
13 |
¥ |
5 |
¥ |
¥ |
|
0 |
3 |
6 |
9 |
0 |
0 |
0 |
|
0 |
0 |
1 |
1 |
0 |
1 |
0 |
|
¥ |
¥ |
¥ |
¥ |
4 |
¥ |
¥ |
|
3 |
0 |
6 |
0 |
7 |
0 |
0 |
|
1 |
1 |
0 |
0 |
1 |
0 |
1 |
|
¥ |
2 |
¥ |
¥ |
¥ |
¥ |
4 |
|
6 |
6 |
0 |
1 |
4 |
0 |
5 |
|
0 |
0 |
1 |
0 |
1 |
1 |
0 |
1) |
¥ |
¥ |
2 |
¥ |
¥ |
¥ |
7 |
2) |
9 |
0 |
1 |
0 |
0 |
0 |
2 |
3) |
1 |
0 |
0 |
1 |
0 |
1 |
0 |
|
¥ |
¥ |
¥ |
6 |
¥ |
2 |
¥ |
|
0 |
7 |
4 |
0 |
0 |
2 |
7 |
|
1 |
1 |
0 |
1 |
0 |
0 |
0 |
|
11 |
¥ |
6 |
3 |
¥ |
¥ |
11 |
|
0 |
0 |
0 |
0 |
2 |
0 |
3 |
|
0 |
0 |
1 |
0 |
1 |
1 |
0 |
|
6 |
4 |
¥ |
¥ |
¥ |
¥ |
¥ |
|
0 |
0 |
5 |
2 |
7 |
3 |
0 |
|
1 |
0 |
1 |
1 |
0 |
0 |
0 |
Вариант № 21
|
|
1 |
|
4 |
|
5 |
13 |
|
0 |
3 |
6 |
9 |
0 |
0 |
0 |
|
0 |
0 |
0 |
1 |
1 |
0 |
0 |
|
|
|
|
|
9 |
3 |
11 |
|
3 |
0 |
6 |
0 |
7 |
0 |
0 |
|
1 |
0 |
1 |
0 |
0 |
1 |
1 |
|
|
7 |
|
|
4 |
|
6 |
|
6 |
6 |
0 |
1 |
4 |
0 |
5 |
|
1 |
1 |
1 |
1 |
0 |
0 |
1 |
1) |
|
|
|
|
|
2 |
|
2) |
9 |
0 |
1 |
0 |
0 |
0 |
2 |
3) |
0 |
0 |
0 |
1 |
1 |
1 |
0 |
|
7 |
|
|
2 |
|
5 |
1 |
|
0 |
7 |
4 |
0 |
0 |
2 |
7 |
|
1 |
0 |
0 |
1 |
0 |
1 |
0 |
|
|
|
1 |
|
|
|
|
|
0 |
0 |
0 |
0 |
2 |
0 |
3 |
|
0 |
1 |
1 |
1 |
0 |
0 |
0 |
|
|
|
|
6 |
|
8 |
|
|
0 |
0 |
5 |
2 |
7 |
3 |
0 |
|
1 |
1 |
0 |
0 |
1 |
1 |
0 |
Вариант № 22
|
|
7 |
|
5 |
1 |
|
|
|
0 |
3 |
2 |
11 |
5 |
9 |
0 |
|
0 |
0 |
0 |
0 |
1 |
1 |
0 |
|
|
|
4 |
1 |
4 |
|
|
|
3 |
0 |
5 |
8 |
1 |
0 |
0 |
|
1 |
0 |
1 |
0 |
1 |
0 |
1 |
|
|
|
|
|
3 |
1 |
2 |
|
2 |
5 |
0 |
6 |
5 |
0 |
0 |
|
1 |
1 |
0 |
1 |
1 |
0 |
0 |
1) |
|
|
4 |
|
|
|
|
2) |
11 |
8 |
6 |
0 |
2 |
0 |
4 |
3) |
0 |
0 |
1 |
0 |
1 |
1 |
0 |
|
|
|
|
|
|
1 |
|
|
5 |
1 |
5 |
2 |
0 |
3 |
7 |
|
1 |
0 |
0 |
1 |
0 |
1 |
0 |
|
|
|
|
2 |
|
|
9 |
|
9 |
0 |
0 |
0 |
3 |
0 |
6 |
|
0 |
0 |
1 |
0 |
1 |
0 |
0 |
|
6 |
|
|
|
4 |
|
|
|
0 |
0 |
0 |
4 |
7 |
6 |
0 |
|
1 |
0 |
1 |
1 |
0 |
0 |
1 |
Вариант № 23
|
|
5 |
1 |
|
8 |
|
|
|
0 |
4 |
0 |
1 |
9 |
0 |
0 |
|
0 |
0 |
0 |
1 |
0 |
1 |
0 |
|
|
|
|
|
2 |
|
|
|
4 |
0 |
5 |
0 |
2 |
0 |
0 |
|
0 |
1 |
1 |
0 |
1 |
0 |
1 |
|
|
3 |
|
|
|
|
11 |
|
0 |
5 |
0 |
0 |
6 |
4 |
7 |
|
1 |
1 |
0 |
1 |
1 |
1 |
0 |
1) |
|
|
1 |
|
|
|
2 |
2) |
1 |
0 |
0 |
0 |
5 |
0 |
0 |
3) |
1 |
0 |
0 |
0 |
1 |
0 |
0 |
|
|
|
|
2 |
|
3 |
|
|
9 |
2 |
6 |
5 |
0 |
8 |
0 |
|
1 |
0 |
1 |
1 |
0 |
1 |
0 |
|
7 |
|
3 |
4 |
|
|
6 |
|
0 |
0 |
4 |
0 |
8 |
0 |
4 |
|
0 |
0 |
0 |
1 |
1 |
0 |
0 |
|
6 |
5 |
|
|
|
|
|
|
0 |
0 |
7 |
0 |
0 |
4 |
0 |
|
0 |
1 |
1 |
1 |
1 |
0 |
0 |
Вариант № 24
|
|
1 |
|
4 |
|
6 |
12 |
|
0 |
0 |
5 |
4 |
8 |
0 |
6 |
|
0 |
1 |
0 |
1 |
1 |
0 |
0 |
|
|
|
|
|
2 |
7 |
11 |
|
0 |
0 |
0 |
11 |
10 |
0 |
0 |
|
1 |
0 |
1 |
1 |
1 |
0 |
0 |
|
|
7 |
|
|
4 |
|
5 |
|
5 |
0 |
0 |
2 |
8 |
0 |
5 |
|
1 |
0 |
0 |
1 |
0 |
0 |
1 |
1) |
|
|
|
|
|
2 |
|
2) |
4 |
11 |
2 |
0 |
7 |
5 |
0 |
3) |
0 |
1 |
0 |
0 |
1 |
1 |
0 |
|
7 |
|
|
2 |
|
2 |
9 |
|
8 |
10 |
8 |
7 |
0 |
0 |
6 |
|
1 |
0 |
1 |
0 |
1 |
1 |
0 |
|
|
|
1 |
|
|
|
|
|
0 |
0 |
0 |
5 |
0 |
0 |
3 |
|
0 |
1 |
1 |
0 |
0 |
0 |
1 |
|
|
|
|
6 |
|
8 |
|
|
6 |
0 |
5 |
0 |
6 |
3 |
0 |
|
0 |
1 |
0 |
1 |
0 |
0 |
1 |
Вариант № 25
|
¥ |
4 |
¥ |
11 |
8 |
¥ |
¥ |
|
0 |
4 |
0 |
0 |
5 |
4 |
0 |
|
0 |
0 |
0 |
1 |
0 |
0 |
0 |
|
¥ |
¥ |
9 |
7 |
3 |
¥ |
¥ |
|
4 |
0 |
8 |
6 |
5 |
0 |
4 |
|
1 |
0 |
1 |
0 |
1 |
0 |
0 |
|
¥ |
¥ |
¥ |
¥ |
3 |
1 |
2 |
|
0 |
8 |
0 |
7 |
9 |
0 |
0 |
|
1 |
0 |
0 |
1 |
0 |
1 |
0 |
1) |
¥ |
¥ |
1 |
¥ |
¥ |
¥ |
¥ |
2) |
0 |
6 |
7 |
0 |
0 |
3 |
2 |
3) |
0 |
1 |
1 |
0 |
0 |
0 |
0 |
|
¥ |
¥ |
¥ |
¥ |
¥ |
2 |
¥ |
|
5 |
5 |
9 |
0 |
0 |
11 |
8 |
|
0 |
0 |
1 |
0 |
1 |
0 |
1 |
|
¥ |
¥ |
¥ |
1 |
¥ |
¥ |
9 |
|
4 |
0 |
0 |
3 |
11 |
0 |
0 |
|
1 |
0 |
0 |
0 |
0 |
0 |
1 |
|
6 |
¥ |
¥ |
¥ |
4 |
¥ |
¥ |
|
0 |
4 |
0 |
2 |
8 |
0 |
0 |
|
1 |
1 |
0 |
1 |
0 |
0 |
0 |