bzoj1084

深度探讨：子矩阵求和优化算法

最新推荐文章于 2019-06-12 17:21:12 发布

原创最新推荐文章于 2019-06-12 17:21:12 发布 · 1.2k 阅读

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#div #output #input

本文深入分析了一种解决子矩阵求和问题的优化算法，通过状态压缩技术，实现快速计算多个子矩阵的最大和。适用于矩阵操作场景，特别针对矩阵元素分值不超过32767的情况。

Description

这里有一个n*m的矩阵，请你选出其中k个子矩阵，使得这个k个子矩阵分值之和最大。注意：选出的k个子矩阵不能相互重叠。

Input

第一行为n,m,k（1≤n≤100,1≤m≤2,1≤k≤10），接下来n行描述矩阵每行中的每个元素的分值(每个元素的分值的绝对值不超过32767)。

Output

只有一行为k个子矩阵分值之和最大为多少。

Sample Input

3 2 2
1 -3
2 3
-2 3

Sample Output

其实可以很简单的写的……心血来潮写了个状压的……

状态表示程序里注释写了……

program bzoj1084;
var
    n,m,i,j,k,ans:longint;
    map:array [0..101,0..3] of longint;
    f:array [0..101,0..4,0..10] of longint;

begin
    read(n,m,k);
    for i:=1 to n do
        for j:=1 to m do
            read(map[i,j]);
    if m=1 then
        begin
            for i:=0 to k do
                begin
                    f[0,1,i]:=-maxlongint div 10;
                    f[0,0,i]:=-maxlongint div 10;
                end;
            f[0,0,0]:=0;
            for i:=1 to n do
                for j:=0 to k do
                    begin
                        if f[i-1,1,j]>f[i-1,0,j] then
                            f[i,0,j]:=f[i-1,1,j]
                                                        else
                            f[i,0,j]:=f[i-1,0,j];
                        if j=0 then f[i,1,j]:=-maxlongint div 10
                                else
                            begin
                                f[i,1,j]:=f[i-1,1,j]+map[i,1];
                                if f[i,1,j]<f[i-1,0,j-1]+map[i,1] then
                                    f[i,1,j]:=f[i-1,0,j-1]+map[i,1];
                                if (j>0)and(f[i,1,j]<f[i-1,1,j-1]+map[i,1]) then 
                                    f[i,1,j]:=f[i-1,1,j-1]+map[i,1];
                            end;
                    end;
            if f[n,0,k]>f[n,1,k] then writeln(f[n,0,k])
                                        else writeln(f[n,1,k]);
            halt;
        end;
    for i:=0 to k do
        for j:=0 to 4 do
            f[0,j,i]:=-maxlongint div 10;
    f[0,0,0]:=0;
    for i:=1 to n do
        for j:=0 to k do
            begin
                ////00
                f[i,0,j]:=f[i-1,0,j];
                if f[i,0,j]<f[i-1,1,j] then f[i,0,j]:=f[i-1,1,j];
                if f[i,0,j]<f[i-1,2,j] then f[i,0,j]:=f[i-1,2,j];
                if f[i,0,j]<f[i-1,3,j] then f[i,0,j]:=f[i-1,3,j];
                if f[i,0,j]<f[i-1,4,j] then f[i,0,j]:=f[i-1,4,j];
                ////10
                f[i,1,j]:=f[i-1,1,j]+map[i,1];
                if (j>0)and(f[i,1,j]<f[i-1,0,j-1]+map[i,1]) then f[i,1,j]:=f[i-1,0,j-1]+map[i,1];
                if (j>0)and(f[i,1,j]<f[i-1,1,j-1]+map[i,1]) then f[i,1,j]:=f[i-1,1,j-1]+map[i,1];
                if (j>0)and(f[i,1,j]<f[i-1,2,j-1]+map[i,1]) then f[i,1,j]:=f[i-1,2,j-1]+map[i,1];
                if (j>0)and(f[i,1,j]<f[i-1,3,j-1]+map[i,1]) then f[i,1,j]:=f[i-1,3,j-1]+map[i,1];
                if (j>0)and(f[i,1,j]<f[i-1,4,j-1]+map[i,1]) then f[i,1,j]:=f[i-1,4,j-1]+map[i,1];
                if f[i,1,j]<f[i-1,3,j]+map[i,1] then f[i,1,j]:=f[i-1,3,j]+map[i,1];
                ////01
                f[i,2,j]:=f[i-1,2,j]+map[i,2];
                if (j>0)and(f[i,2,j]<f[i-1,0,j-1]+map[i,2]) then f[i,2,j]:=f[i-1,0,j-1]+map[i,2];
                if (j>0)and(f[i,2,j]<f[i-1,1,j-1]+map[i,2]) then f[i,2,j]:=f[i-1,1,j-1]+map[i,2];
                if (j>0)and(f[i,2,j]<f[i-1,2,j-1]+map[i,2]) then f[i,2,j]:=f[i-1,2,j-1]+map[i,2];
                if (j>0)and(f[i,2,j]<f[i-1,3,j-1]+map[i,2]) then f[i,2,j]:=f[i-1,3,j-1]+map[i,2];
                if (j>0)and(f[i,2,j]<f[i-1,4,j-1]+map[i,2]) then f[i,2,j]:=f[i-1,4,j-1]+map[i,2];
                if f[i,2,j]<f[i-1,3,j]+map[i,2] then f[i,2,j]:=f[i-1,3,j]+map[i,2];
                ////12
                f[i,3,j]:=f[i-1,3,j]+map[i,1]+map[i,2];
                if (j>=2)and(f[i,3,j]<f[i-1,0,j-2]+map[i,1]+map[i,2]) then f[i,3,j]:=f[i-1,0,j-2]+map[i,1]+map[i,2];
                if (j>=2)and(f[i,3,j]<f[i-1,1,j-2]+map[i,1]+map[i,2]) then f[i,3,j]:=f[i-1,1,j-2]+map[i,1]+map[i,2];
                if (j>=2)and(f[i,3,j]<f[i-1,2,j-2]+map[i,1]+map[i,2]) then f[i,3,j]:=f[i-1,2,j-2]+map[i,1]+map[i,2];
                if (j>=2)and(f[i,3,j]<f[i-1,3,j-2]+map[i,1]+map[i,2]) then f[i,3,j]:=f[i-1,3,j-2]+map[i,1]+map[i,2];
                if (j>=2)and(f[i,3,j]<f[i-1,4,j-2]+map[i,1]+map[i,2]) then f[i,3,j]:=f[i-1,4,j-2]+map[i,1]+map[i,2];
                if (j>=1)and(f[i,3,j]<f[i-1,1,j-1]+map[i,1]+map[i,2]) then f[i,3,j]:=f[i-1,1,j-1]+map[i,1]+map[i,2];
                if (j>=1)and(f[i,3,j]<f[i-1,2,j-1]+map[i,1]+map[i,2]) then f[i,3,j]:=f[i-1,2,j-1]+map[i,1]+map[i,2];
                if (j>=1)and(f[i,3,j]<f[i-1,3,j-1]+map[i,1]+map[i,2]) then f[i,3,j]:=f[i-1,3,j-1]+map[i,1]+map[i,2];
                ////11
                f[i,4,j]:=f[i-1,4,j]+map[i,1]+map[i,2];
                if (j>=1)and(f[i,4,j]<f[i-1,0,j-1]+map[i,1]+map[i,2]) then f[i,4,j]:=f[i-1,0,j-1]+map[i,1]+map[i,2];
                if (j>=1)and(f[i,4,j]<f[i-1,1,j-1]+map[i,1]+map[i,2]) then f[i,4,j]:=f[i-1,1,j-1]+map[i,1]+map[i,2];
                if (j>=1)and(f[i,4,j]<f[i-1,2,j-1]+map[i,1]+map[i,2]) then f[i,4,j]:=f[i-1,2,j-1]+map[i,1]+map[i,2];
                if (j>=1)and(f[i,4,j]<f[i-1,3,j-1]+map[i,1]+map[i,2]) then f[i,4,j]:=f[i-1,3,j-1]+map[i,1]+map[i,2];
                if (j>=1)and(f[i,4,j]<f[i-1,4,j-1]+map[i,1]+map[i,2]) then f[i,4,j]:=f[i-1,4,j-1]+map[i,1]+map[i,2];
            end;
    ans:=-maxlongint div 10;
    if ans<f[n,0,k] then ans:=f[n,0,k];
    if ans<f[n,1,k] then ans:=f[n,1,k];
    if ans<f[n,2,k] then ans:=f[n,2,k];
    if ans<f[n,3,k] then ans:=f[n,3,k];
    if ans<f[n,4,k] then ans:=f[n,4,k];
    writeln(ans);
end.