2-SAT【模板】

摘自http://www.cnblogs.com/kuangbin/archive/2012/10/05/2712429.html 
现有一个由N个布尔值组成的序列A,给出一些限制关系,比如A[x] && A[y] = 0、A[x] || A[y] || A[z]=1等,要确定A[0..N-1]的值,使得其满足所有限制关系。这个称为SAT问题,特别的,若每种限制关系中最多只对两个元素进行限制,则称为2-SAT问题。

由于在2-SAT问题中,最多只对两个元素进行限制,所以可能的限制关系共有11种: 
A[x] 
NOT A[x] 
A[x] AND A[y] 
A[x] AND NOT A[y] 
A[x] OR A[y] 
A[x] OR NOT A[y] 
NOT (A[x] AND A[y]) 
NOT (A[x] OR A[y]) 
A[x] XOR A[y] 
NOT (A[x] XOR A[y]) 
A[x] XOR NOT A[y] 
进一步,A[x] AND A[y]相当于(A[x]) AND (A[y])(也就是可以拆分成A[x]与A[y]两个限制关系),NOT(A[x] OR A[y])相当于NOT A[x] AND NOT A[y](也就是可以拆分成NOT A[x]与NOT A[y]两个限制关系)。因此,可能的限制关系最多只有9种。

2-SAT问题在大多数时候表现成以下形式:有N对物品,每对物品中必须选取一个,也只能选取一个,并且它们之间存在某些限制关系(如某两个物品不能都选,某两个物品不能都不选,某两个物品必须且只能选一个,某个物品必选)等,这时,可以将每对物品当成一个布尔值(选取第一个物品相当于0,选取第二个相当于1),如果所有的限制关系最多只对两个物品进行限制,则它们都可以转化成9种基本限制关系,从而转化为2-SAT模型。

【建模】 
可以构造有向图G,G中包含2*N个顶点,前N个顶点(1~N)表示第i个元素能被选择,后N个顶点(N+1~2*N)表示第i个元素不能被选择。Ai和A(i+N)不能同时被选择。同理Ai + Bj = ~( A(i+N) + B(j+N) ),A(i+N)和B(j+N)不能同时被选。选中~A,必须选择B;如果选择~B,必须选择A。 
若图中i到j有路径,则若i选,则j也要选;或者说,若j不选,则i也不能选。

<code class="hljs cpp has-numbering" style="display: block; padding: 0px; color: inherit; box-sizing: border-box; font-family: 'Source Code Pro', monospace;font-size:undefined; white-space: pre; border-radius: 0px; word-wrap: normal; background: transparent;"><span class="hljs-preprocessor" style="color: rgb(68, 68, 68); box-sizing: border-box;">#include<iostream></span>
<span class="hljs-preprocessor" style="color: rgb(68, 68, 68); box-sizing: border-box;">#include<algorithm></span>
<span class="hljs-preprocessor" style="color: rgb(68, 68, 68); box-sizing: border-box;">#include<cstdio></span>
<span class="hljs-preprocessor" style="color: rgb(68, 68, 68); box-sizing: border-box;">#include<cstring></span>
<span class="hljs-preprocessor" style="color: rgb(68, 68, 68); box-sizing: border-box;">#include<cmath></span>
<span class="hljs-preprocessor" style="color: rgb(68, 68, 68); box-sizing: border-box;">#include<queue></span>
<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">using</span> <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">namespace</span> <span class="hljs-built_in" style="color: rgb(102, 0, 102); box-sizing: border-box;">std</span>;
<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">const</span> <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> MAXN = <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">2200</span>;
<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">const</span> <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> MAXM = MAXN*MAXN;

<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">struct</span> EdgeNode
{
    <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> to;
    <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> next;
}Edges[MAXM];

<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> Head[MAXN];
<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> dfn[MAXN],low[MAXN],belong[MAXN],Stack[MAXN],vis[MAXN];
<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> m,id,lay,scc,N,M;
<span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//belong[]来判断i和i+N是否在一个强连通分量里</span>
<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">void</span> AddEdges(<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> u,<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> v)
{
    Edges[id].to = v;
    Edges[id].next = Head[u];
    Head[u] = id++;
}

<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">void</span> TarBFS(<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> pos)
{
    dfn[pos] = low[pos] = ++lay;
    Stack[m++] = pos;
    vis[pos] = <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">1</span>;

    <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">for</span>(<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> i = Head[pos]; i != -<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">1</span>; i = Edges[i].next)
    {
        <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> v = Edges[i].to;
        <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">if</span>( !dfn[v] )
        {
            TarBFS(v);
            low[pos] = min(low[pos],low[v]);
        }
        <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">else</span> <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">if</span>(vis[v])
            low[pos] = min(low[pos],low[v]);
    }
    <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> v;
    <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">if</span>(dfn[pos] == low[pos])
    {
        ++scc;
        <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">do</span>
        {
            v = Stack[--m];
            belong[v] = scc;
            vis[v] = <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>;
        }<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">while</span>(v != pos);
    }
}

<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> main()
{
    <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> u,v;
    <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">while</span>(~<span class="hljs-built_in" style="color: rgb(102, 0, 102); box-sizing: border-box;">scanf</span>(<span class="hljs-string" style="color: rgb(0, 136, 0); box-sizing: border-box;">"%d%d"</span>,&N,&M))
    {
        id = m = scc = lay = <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>;
        <span class="hljs-built_in" style="color: rgb(102, 0, 102); box-sizing: border-box;">memset</span>(Head,-<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">1</span>,<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">sizeof</span>(Head));
        <span class="hljs-built_in" style="color: rgb(102, 0, 102); box-sizing: border-box;">memset</span>(vis,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>,<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">sizeof</span>(vis));
        <span class="hljs-built_in" style="color: rgb(102, 0, 102); box-sizing: border-box;">memset</span>(low,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>,<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">sizeof</span>(low));
        <span class="hljs-built_in" style="color: rgb(102, 0, 102); box-sizing: border-box;">memset</span>(dfn,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>,<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">sizeof</span>(dfn));
        <span class="hljs-built_in" style="color: rgb(102, 0, 102); box-sizing: border-box;">memset</span>(belong,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>,<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">sizeof</span>(belong));
        <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">for</span>(<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> i = <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>; i < M; ++i)
        {
            <span class="hljs-built_in" style="color: rgb(102, 0, 102); box-sizing: border-box;">scanf</span>(<span class="hljs-string" style="color: rgb(0, 136, 0); box-sizing: border-box;">"%d%d"</span>,&u,&v);
            <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> a = <span class="hljs-built_in" style="color: rgb(102, 0, 102); box-sizing: border-box;">abs</span>(u);
            <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> b = <span class="hljs-built_in" style="color: rgb(102, 0, 102); box-sizing: border-box;">abs</span>(v);

            <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">if</span>(u > <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span> && v > <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>)
            {   <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//a、b至少一个被选中</span>
                AddEdges(a+N,b);      <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//如果a不被选,b就必须被选</span>
                AddEdges(b+N,a);      <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//如果b不被选,a就必须被选</span>
            }

            <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">if</span>(u < <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span> && v < <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>)
            {   <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//a、b至少有一个不被选中</span>
                AddEdges(a,b+N);      <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//如果a被选,b就必须不被选</span>
                AddEdges(b,a+N);      <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//如果b被选,a就必须不被选</span>
            }

            <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">if</span>(u > <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span> && v < <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>)
            {   <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//a被选中和b不被选中两件事至少发生一件</span>
                AddEdges(a+N,b+N);    <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//如果a不被选中,b必须不被选中</span>
                AddEdges(b,a);        <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//如果b被选中,那么a必须被选中</span>
            }

            <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">if</span>(u < <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span> && v > <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>)
            {   <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//a不被选中和b被选中至少发生一件</span>
                AddEdges(a,b);        <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//如果a被选中,b必须被选中</span>
                AddEdges(b+N,a+N);    <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//如果b不被选中,a必须不被选中</span>
            }
        }

        <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">for</span>(<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> i = <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">1</span>; i <= <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">2</span>*N; ++i)
            <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">if</span>( !dfn[i] )
                TarBFS(i);
        <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> ans = <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">1</span>;
        <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">for</span>(<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> i = <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">1</span>; i <= N; ++i)
        {
            <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">if</span>(belong[i] == belong[i+N])    <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//如果i和i+N同在一个连通分量里,则2-SAT不满足</span>
            {
                ans = <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>;
                <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">break</span>;
            }
        }

        <span class="hljs-built_in" style="color: rgb(102, 0, 102); box-sizing: border-box;">printf</span>(<span class="hljs-string" style="color: rgb(0, 136, 0); box-sizing: border-box;">"%d\n"</span>,ans); <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//不存在就满足2-SAT</span>
    }

    <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">return</span> <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>;
}
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padding: 0px 5px;">63</li><li style="box-sizing: border-box; padding: 0px 5px;">64</li><li style="box-sizing: border-box; padding: 0px 5px;">65</li><li style="box-sizing: border-box; padding: 0px 5px;">66</li><li style="box-sizing: border-box; padding: 0px 5px;">67</li><li style="box-sizing: border-box; padding: 0px 5px;">68</li><li style="box-sizing: border-box; padding: 0px 5px;">69</li><li style="box-sizing: border-box; padding: 0px 5px;">70</li><li style="box-sizing: border-box; padding: 0px 5px;">71</li><li style="box-sizing: border-box; padding: 0px 5px;">72</li><li style="box-sizing: border-box; padding: 0px 5px;">73</li><li style="box-sizing: border-box; padding: 0px 5px;">74</li><li style="box-sizing: border-box; padding: 0px 5px;">75</li><li style="box-sizing: border-box; padding: 0px 5px;">76</li><li style="box-sizing: border-box; padding: 0px 5px;">77</li><li style="box-sizing: border-box; padding: 0px 5px;">78</li><li style="box-sizing: border-box; padding: 0px 5px;">79</li><li style="box-sizing: border-box; padding: 0px 5px;">80</li><li style="box-sizing: border-box; padding: 0px 5px;">81</li><li style="box-sizing: border-box; padding: 0px 5px;">82</li><li style="box-sizing: border-box; padding: 0px 5px;">83</li><li style="box-sizing: border-box; padding: 0px 5px;">84</li><li style="box-sizing: border-box; padding: 0px 5px;">85</li><li style="box-sizing: border-box; padding: 0px 5px;">86</li><li style="box-sizing: border-box; padding: 0px 5px;">87</li><li style="box-sizing: border-box; padding: 0px 5px;">88</li><li style="box-sizing: border-box; padding: 0px 5px;">89</li><li style="box-sizing: border-box; padding: 0px 5px;">90</li><li style="box-sizing: border-box; padding: 0px 5px;">91</li><li style="box-sizing: border-box; padding: 0px 5px;">92</li><li style="box-sizing: border-box; padding: 0px 5px;">93</li><li style="box-sizing: border-box; padding: 0px 5px;">94</li><li style="box-sizing: border-box; padding: 0px 5px;">95</li><li style="box-sizing: border-box; padding: 0px 5px;">96</li><li style="box-sizing: border-box; padding: 0px 5px;">97</li><li style="box-sizing: border-box; padding: 0px 5px;">98</li><li style="box-sizing: border-box; padding: 0px 5px;">99</li><li style="box-sizing: border-box; padding: 0px 5px;">100</li><li style="box-sizing: border-box; padding: 0px 5px;">101</li><li style="box-sizing: border-box; padding: 0px 5px;">102</li><li style="box-sizing: border-box; padding: 0px 5px;">103</li><li style="box-sizing: border-box; padding: 0px 5px;">104</li><li style="box-sizing: border-box; padding: 0px 5px;">105</li><li style="box-sizing: border-box; padding: 0px 5px;">106</li><li style="box-sizing: border-box; padding: 0px 5px;">107</li><li style="box-sizing: border-box; padding: 0px 5px;">108</li><li style="box-sizing: border-box; padding: 0px 5px;">109</li><li style="box-sizing: border-box; padding: 0px 5px;">110</li><li style="box-sizing: border-box; padding: 0px 5px;">111</li><li style="box-sizing: border-box; padding: 0px 5px;">112</li><li style="box-sizing: border-box; padding: 0px 5px;">113</li><li style="box-sizing: border-box; padding: 0px 5px;">114</li><li style="box-sizing: border-box; padding: 0px 5px;">115</li><li style="box-sizing: border-box; padding: 0px 5px;">116</li><li style="box-sizing: border-box; padding: 0px 5px;">117</li><li style="box-sizing: border-box; padding: 0px 5px;">118</li></ul>
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