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Здравствуйте! Прошу помощи в следующем вопросе: 1. Решить задачу, методом поиска с возвратом, используя рекурсивный или нерекурсивный вариант. 2. Решить те же задачи с помощью жадного алгоритма (можно вручную). 3. Обосновать возможность решения данных задач методом динамического программирования.
Задача о ладьях. На клетках шахматной доски размера 9*9 написаны числа. Расставить на доске 9 ладей, чтобы они не били друг друга, а сумма накрытых ими чисел была максимальной.
1. Вот текст программы с комментариями. [code h=200]const coin:array [1..9,1..9]of integer=( (27,12,23,20,22,21,34,25,13), (30,12,19,26,21,35,34,12,32), (27,12,28,17,25,16,36,35,13), (18,23,24,33,20,22,20,34,17), (14,18,29,36,20,26,36,25,20), (12,15,19,23,13,15,33,34,25), (29,27,32,19,33,21,16,15,14), (33,33,17,18,13,22,17,31,23), (27,12,29,35,25,15,23,33,27) ); var rook,max:array[1..9]of integer;{массивы горизонталей тур (текущий и наилучший)} rows:array[1..9]of boolean; {признак незанятости горизонтали} sum_max:integer; {сумма накрытых чисел для наилучшей расстановки}
procedure p(a,sum:integer); var s:integer; begin for s:=1 to 9 do {перебираем горизонтали} if rows[s] then {если она незанята} begin rook[a]:=s; {занимаем} rows[s]:=false; if a<9 then p(a+1,sum+coin[a,rook[a]]) {если непоследняя, то переходим к следующей} { (увеличивая сумму)} else if sum+coin[a,rook[a]]>sum_max then {иначе сравниваем текущую расстановку } begin {c наилучшей} sum_max:=sum+coin[a,rook[a]]; max:=rook; end; rook[a]:=0; {освобождаем горизонталь} rows[s]:=true; end; end;
var i:integer;
begin{main} for i:=1 to 9 do begin rook[i]:=0; rows[i]:=true; end; sum_max:=0; p(1,0); for i:=1 to 9 do write(i:3,':',max[i],'=',coin[i,max[i]]); writeln; writeln('Sum:',sum_max); readln; end.{main}[/code]
2. Здесь жадный алгоритм, вероятно, реализуется так: а) находим на доске ячейку с самым большим числом и ставим ладью; б) вычёркиваем горизонталь и вертикаль; с) в оставшихся ячейках находим самое большое число; и т.д.
В прикреплённом файле весь процесс показан в excel таблице. Сумма составила 277. Т.е. меньше, чем в рекурсивном алгоритме, в котором происходит полный перебор всех вариантов.
Кстати, на первой же ладье имеется нексолько ячеек с максимальным значением 36. Однако, перебор вариантов тут уже не делается. Идея жадного алгоритма -- накидать всё побыстрее и попроще. Результат будет не самый-самый лучший, но достаточно большой в любой практической задаче.
Всем: [code h=200]const coin:array [1..9,1..9]of integer=( (27,12,23,20,22,21,34,25,13), (30,12,19,26,21,35,34,12,32), (27,12,28,17,25,16,36,35,13), (18,23,24,33,20,22,20,34,17), (14,18,29,36,20,26,36,25,20), (12,15,19,23,13,15,33,34,25), (29,27,32,19,33,21,16,15,14), (33,33,17,18,13,22,17,31,23), (27,12,29,35,25,15,23,33,27) ); var m_x,m_y,w,x,y,z:integer; lin,row:array[1..9]of boolean; begin {A} writeln('row 1:9;col 1:9'); for z:=1 to 9 do begin lin[z]:=true; row[z]:=true; end; w:=0; for z:=1 to 9 do begin m_x:=0;repeat inc(m_x) until lin[m_x]; m_y:=0;repeat inc(m_y) until lin[m_y]; for y:=1 to 9 do if row[y] then for x:=1 to 9 do if lin[x] then if coin[x,y]>coin[m_x,m_y]then begin m_y:=y; m_x:=x; end; row[m_y]:=false; lin[m_x]:=false; write(m_x:3,':',m_y,'=',coin[m_x,m_y]); w:=w+coin[m_x,m_y]; end; writeln(' s:=',w); {B} writeln('col 1:9;row 1:9'); for z:=1 to 9 do begin lin[z]:=true; row[z]:=true; end; w:=0; for z:=1 to 9 do begin m_x:=0;repeat inc(m_x) until lin[m_x]; m_y:=0;repeat inc(m_y) until row[m_y]; for x:=1 to 9 do if lin[x] then for y:=1 to 9 do if row[y] then if coin[x,y]>coin[m_x,m_y]then begin m_y:=y; m_x:=x; end; row[m_y]:=false; lin[m_x]:=false; write(m_x:3,':',m_y,'=',coin[m_x,m_y]); w:=w+coin[m_x,m_y]; end; writeln(' s:=',w);
{C} writeln('row 9:1;col 9:1'); for z:=1 to 9 do begin lin[z]:=true; row[z]:=true; end; w:=0; for z:=1 to 9 do begin m_x:=0;repeat inc(m_x) until lin[m_x]; m_y:=0;repeat inc(m_y) until lin[m_y]; for y:=9 downto 1 do if row[y] then for x:=9 downto 1 do if lin[x] then if coin[x,y]>coin[m_x,m_y]then begin m_y:=y; m_x:=x; end; row[m_y]:=false; lin[m_x]:=false; write(m_x:3,':',m_y,'=',coin[m_x,m_y]); w:=w+coin[m_x,m_y]; end; writeln(' s:=',w); {D} writeln('col 9:1; row 9:1'); for z:=1 to 9 do begin lin[z]:=true; row[z]:=true; end; w:=0; for z:=1 to 9 do begin m_x:=0;repeat inc(m_x) until lin[m_x]; m_y:=0;repeat inc(m_y) until row[m_y]; for x:=9 downto 1 do if lin[x] then for y:=9 downto 1 do if row[y] then if coin[x,y]>coin[m_x,m_y]then begin m_y:=y; m_x:=x; end; row[m_y]:=false; lin[m_x]:=false; write(m_x:3,':',m_y,'=',coin[m_x,m_y]); w:=w+coin[m_x,m_y]; end; writeln(' s:=',w); readln; end.{}[/code] жадный алгоритм(который предложил Сергей) с 4-мя стратегиями поиска максимальных элементов: по строкам, по столбцам, с возрастанием индекса поиска и его убыванием.
Судя по всему и полный перебор не дает полной гарантии.
нашел ошибку в своем коде:
Код:
m_x:=0;repeat inc(m_x) until lin[m_x]; m_y:=0;repeat inc(m_y) until lin[m_y];
из-за этого учитывались у же вычеркнутые линии.
Об авторе:
Мне безразлично, что Вы думаете о обо мне, но я рад за Вас - Вы начали думать.
Неизвестный
26.01.2012, 16:02
общий
большое спасибо и за ваш вариант
Форма ответа
Отправка постов/ответов доступна только зарегистрированным и подтвержденным пользователям.
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