題意:給一段子序列,定義密度:子序列中的逆序對數/子序列的長度
求這個序列的對大密度.
分析:將序列中的每個位置視作點,逆序對\(<i,j>\)之間表示點i與點j之間有一條無向邊.所以就轉化為了最大密度子圖的模型.

#include<bits/stdc++.h>
using namespace std;
#define eps 1e-7
#define INF 0x3f3f3f3f
const int MAXN=1010;//點數的最大值
const int MAXM=400010;//邊數的最大值
#define captype double
struct Edge{
    int from,to,next;
    captype cap;
};

struct SAP_MaxFlow{
    Edge edges[MAXM];
    int tot,head[MAXN];
    int gap[MAXN];
    int dis[MAXN];
    int cur[MAXN];
    int pre[MAXN];

    void init(){
        tot=0;
        memset(head,-1,sizeof(head));
    }
    void AddEdge(int u,int v,captype c,captype rc=0){
        edges[tot] = (Edge){u,v,head[u],c};  head[u]=tot++;
        edges[tot] = (Edge){v,u,head[v],rc}; head[v]=tot++;
    }

    captype maxFlow_sap(int sNode,int eNode, int n){//n是包括源點和匯點的總點個數，這個一定要註意
        memset(gap,0,sizeof(gap));
        memset(dis,0,sizeof(dis));
        memcpy(cur,head,sizeof(head));
        pre[sNode] = -1;
        gap[0]=n;
        captype ans=0;
        int u=sNode;
        while(dis[sNode]<n){
            if(u==eNode){
                captype Min=INF ;
                int inser;
                for(int i=pre[u]; i!=-1; i=pre[edges[i^1].to])
                if(Min>edges[i].cap){
                    Min=edges[i].cap;
                    inser=i;
                }
                for(int i=pre[u]; i!=-1; i=pre[edges[i^1].to]){
                    edges[i].cap-=Min;
                    edges[i^1].cap+=Min;
                }
                ans+=Min;
                u=edges[inser^1].to;
                continue;
            }
            bool flag = false;
            int v;
            for(int i=cur[u]; i!=-1; i=edges[i].next){
                v=edges[i].to;
                if(edges[i].cap>0 && dis[u]==dis[v]+1){
                    flag=true;
                    cur[u]=pre[v]=i;
                    break;
                }
            }
            if(flag){
                u=v;
                continue;
            }
            int Mind= n;
            for(int i=head[u]; i!=-1; i=edges[i].next)
            if(edges[i].cap>0 && Mind>dis[edges[i].to]){
                Mind=dis[edges[i].to];
                cur[u]=i;
            }
            gap[dis[u]]--;
            if(gap[dis[u]]==0) return ans;
            dis[u]=Mind+1;
            gap[dis[u]]++;
            if(u!=sNode)  u=edges[pre[u]^1].to;  //退一條邊
        }
        return ans;
    }
}F;

int N, M;
int S,T;
int deg[MAXN];
int ed[MAXM][2];

bool check(double mid)
{
    F.init();
    S = 0, T = N+1;
    for(int i=1;i<=N;++i){
        F.AddEdge(S,i,M*1.0);
        F.AddEdge(i,T,1.0*M + 2*mid - deg[i]);
    }
    for(int i=1;i<=M;++i){
        F.AddEdge(ed[i][0],ed[i][1], 1.0);
        F.AddEdge(ed[i][1],ed[i][0], 1.0);
    }
    double hg = ( 1.0 * M * N - F.maxFlow_sap(S,T,T+1)) *0.5;
    return hg>eps;
}

int p[MAXN];

int main()
{
    #ifndef ONLINE_JUDGE
        freopen("in.txt", "r", stdin);
        freopen("out.txt", "w", stdout);
    #endif
    int u,v;
    int T,cas=1; scanf("%d",&T);
    while(T--){
        printf("Case #%d: ",cas++);
        scanf("%d",&N);
        for(int i=1;i<=N;++i) scanf("%d",&p[i]);
        M = 0 ;
        memset(deg,0,sizeof(deg));
        for(int i=1;i<=N;++i){
            for(int j=i+1;j<=N;++j){
                if(p[i]>p[j]){
                    M ++;
                    ed[M][0] = i, ed[M][1] = j;
                    deg[i]++, deg[j]++;
                }
            }
        }
        double L = 0.0, R = M*1.0;

        while(R-L>= eps){
            double mid = (R+L)*0.5;
            if(check(mid)){
                L = mid;
            }else{
                R = mid;
            }
        }
        printf("%.7f\n",L);
    }
    return 0;
}
 

Uvalive 7037 The Problem Needs 3D Arrays（最大密度子圖）