src/flows/k-ary-optimization.hpp

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Last update: 2025-07-19 14:29:34+09:00
Include: #include "src/flows/k-ary-optimization.hpp"

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src/flows/dinic.hpp

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#pragma once
#include "src/flows/dinic.hpp"

/**
 * Author: Teetat T.
 * Date: 2024-07-16
 * Description: k-ary Optimization.
 * minimize $\kappa + \sum_i \theta_i(x_i) + \sum_{i<j} \phi_{ij}(x_i,x_j)$
 * where $x_i \in \{0,1,\ldots,k-1\}$ and $\phi_{i,j}$ is monge.
 * A function $f$ is monge if $f(a,c)+f(b,d) \leq f(a,d)+f(b,c)$ for all $a < b$ and $c < d$.
 * $\phi_{ij}(x-1,y)+\phi_{ij}(x,y+1) \leq \phi_{ij}(x-1,y+1)+\phi_{ij}(x,y)$.
 * $\phi_{ij}(x,y)+\phi_{ij}(x-1,y+1)-\phi_{ij}(x-1,y)-\phi_{ij}(x,y+1) \geq 0$.
 */

template<class T,bool minimize=true>
struct K_aryOptimization{
    static constexpr T INF=numeric_limits<T>::max()/2;
    int n,s,t,buf;
    T base;
    vector<int> ks;
    vector<vector<int>> id;
    map<pair<int,int>,T> edges;
    K_aryOptimization(int n,int k){init(vector<int>(n,k));}
    K_aryOptimization(const vector<int> &_ks){init(_ks);}
    void init(const vector<int> &_ks){
        ks=_ks;
        n=ks.size();
        s=0,t=1,buf=2;
        base=0;
        id.clear();
        edges.clear();
        for(auto &k:ks){
            assert(k>=1);
            vector<int> a(k+1);
            a[0]=s,a[k]=t;
            for(int i=1;i<k;i++)a[i]=buf++;
            id.emplace_back(a);
            for(int i=2;i<k;i++)add_edge(a[i],a[i-1],INF);
        }
    }
    void add_edge(int u,int v,T w){
        assert(w>=0);
        if(u==v||w==0)return;
        auto &e=edges[{u,v}];
        e=min(e+w,INF);
    }
    void add0(T w){
        base+=w;
    }
    void _add1(int i,vector<T> cost){
        add0(cost[0]);
        for(int j=1;j<ks[i];j++){
            T x=cost[j]-cost[j-1];
            if(x>0)add_edge(id[i][j],t,x);
            if(x<0)add0(x),add_edge(s,id[i][j],-x);
        }
    }
    void add1(int i,vector<T> cost){
        assert(0<=i&&i<n&&(int)cost.size()==ks[i]);
        if(!minimize)for(auto &x:cost)x=-x;
        _add1(i,cost);
    }
    void _add2(int i,int j,vector<vector<T>> cost){
        int h=ks[i],w=ks[j];
        _add1(j,cost[0]);
        for(int x=h-1;x>=0;x--)for(int y=0;y<w;y++)cost[x][y]-=cost[0][y];
        vector<T> a(h);
        for(int x=0;x<h;x++)a[x]=cost[x][w-1];
        _add1(i,a);
        for(int x=0;x<h;x++)for(int y=0;y<w;y++)cost[x][y]-=a[x];
        for(int x=1;x<h;x++){
            for(int y=0;y<w-1;y++){
                T w=cost[x][y]+cost[x-1][y+1]-cost[x-1][y]-cost[x][y+1];
                assert(w>=0); // monge
                add_edge(id[i][x],id[j][y+1],w);
            }
        }
    }
    void add2(int i,int j,vector<vector<T>> cost){
        assert(0<=i&&i<n&&0<=j&&j<n&&i!=j);
        assert((int)cost.size()==ks[i]);
        for(auto &v:cost)assert((int)v.size()==ks[j]);
        if(!minimize)for(auto &v:cost)for(auto &x:v)x=-x;
        _add2(i,j,cost);
    }
    pair<T,vector<int>> solve(){
        Dinic<T> dinic(buf,s,t);
        for(auto &[p,w]:edges){
            auto [u,v]=p;
            dinic.add_edge(u,v,w);
        }
        auto [val,cut]=dinic.cut();
        val+=base;
        if(!minimize)val=-val;
        vector<int> ans(n);
        for(int i=0;i<n;i++){
            ans[i]=ks[i]-1;
            for(int j=1;j<ks[i];j++)ans[i]-=cut[id[i][j]];
        }
        return {val,ans};
    }
};

#line 2 "src/flows/dinic.hpp"

/**
 * Author: Teetat T.
 * Date: 2024-07-15
 * Description: Dinic's Algorithm for finding the maximum flow.
 * Time: O(V E \log U) where U is the maximum flow.
 */

template<class T,bool directed=true,bool scaling=true>
struct Dinic{
    static constexpr T INF=numeric_limits<T>::max()/2;
    struct Edge{
        int to;
        T flow,cap;
        Edge(int _to,T _cap):to(_to),flow(0),cap(_cap){}
        T remain(){return cap-flow;}
    };
    int n,s,t;
    T U;
    vector<Edge> e;
    vector<vector<int>> g;
    vector<int> ptr,lv;
    bool calculated;
    T max_flow;
    Dinic(){}
    Dinic(int n,int s,int t){init(n,s,t);}
    void init(int _n,int _s,int _t){
        n=_n,s=_s,t=_t;
        U=0;
        e.clear();
        g.assign(n,{});
        calculated=false;
    }
    void add_edge(int from,int to,T cap){
        assert(0<=from&&from<n&&0<=to&&to<n);
        g[from].emplace_back(e.size());
        e.emplace_back(to,cap);
        g[to].emplace_back(e.size());
        e.emplace_back(from,directed?0:cap);
        U=max(U,cap);
    }
    bool bfs(T scale){
        lv.assign(n,-1);
        vector<int> q{s};
        lv[s]=0;
        for(int i=0;i<(int)q.size();i++){
            int u=q[i];
            for(int j:g[u]){
                int v=e[j].to;
                if(lv[v]==-1&&e[j].remain()>=scale){
                    q.emplace_back(v);
                    lv[v]=lv[u]+1;
                }
            }
        }
        return lv[t]!=-1;
    }
    T dfs(int u,int t,T f){
        if(u==t||f==0)return f;
        for(int &i=ptr[u];i<(int)g[u].size();i++){
            int j=g[u][i];
            int v=e[j].to;
            if(lv[v]==lv[u]+1){
                T res=dfs(v,t,min(f,e[j].remain()));
                if(res>0){
                    e[j].flow+=res;
                    e[j^1].flow-=res;
                    return res;
                }
            }
        }
        return 0;
    }
    T flow(){
        if(calculated)return max_flow;
        calculated=true;
        max_flow=0;
        for(T scale=scaling?1LL<<(63-__builtin_clzll(U)):1LL;scale>0;scale>>=1){
            while(bfs(scale)){
                ptr.assign(n,0);
                while(true){
                    T f=dfs(s,t,INF);
                    if(f==0)break;
                    max_flow+=f;
                }
            }
        }
        return max_flow;
    }
    pair<T,vector<int>> cut(){
        flow();
        vector<int> res(n);
        for(int i=0;i<n;i++)res[i]=(lv[i]==-1);
        return {max_flow,res};
    }
};
#line 3 "src/flows/k-ary-optimization.hpp"

/**
 * Author: Teetat T.
 * Date: 2024-07-16
 * Description: k-ary Optimization.
 * minimize $\kappa + \sum_i \theta_i(x_i) + \sum_{i<j} \phi_{ij}(x_i,x_j)$
 * where $x_i \in \{0,1,\ldots,k-1\}$ and $\phi_{i,j}$ is monge.
 * A function $f$ is monge if $f(a,c)+f(b,d) \leq f(a,d)+f(b,c)$ for all $a < b$ and $c < d$.
 * $\phi_{ij}(x-1,y)+\phi_{ij}(x,y+1) \leq \phi_{ij}(x-1,y+1)+\phi_{ij}(x,y)$.
 * $\phi_{ij}(x,y)+\phi_{ij}(x-1,y+1)-\phi_{ij}(x-1,y)-\phi_{ij}(x,y+1) \geq 0$.
 */

template<class T,bool minimize=true>
struct K_aryOptimization{
    static constexpr T INF=numeric_limits<T>::max()/2;
    int n,s,t,buf;
    T base;
    vector<int> ks;
    vector<vector<int>> id;
    map<pair<int,int>,T> edges;
    K_aryOptimization(int n,int k){init(vector<int>(n,k));}
    K_aryOptimization(const vector<int> &_ks){init(_ks);}
    void init(const vector<int> &_ks){
        ks=_ks;
        n=ks.size();
        s=0,t=1,buf=2;
        base=0;
        id.clear();
        edges.clear();
        for(auto &k:ks){
            assert(k>=1);
            vector<int> a(k+1);
            a[0]=s,a[k]=t;
            for(int i=1;i<k;i++)a[i]=buf++;
            id.emplace_back(a);
            for(int i=2;i<k;i++)add_edge(a[i],a[i-1],INF);
        }
    }
    void add_edge(int u,int v,T w){
        assert(w>=0);
        if(u==v||w==0)return;
        auto &e=edges[{u,v}];
        e=min(e+w,INF);
    }
    void add0(T w){
        base+=w;
    }
    void _add1(int i,vector<T> cost){
        add0(cost[0]);
        for(int j=1;j<ks[i];j++){
            T x=cost[j]-cost[j-1];
            if(x>0)add_edge(id[i][j],t,x);
            if(x<0)add0(x),add_edge(s,id[i][j],-x);
        }
    }
    void add1(int i,vector<T> cost){
        assert(0<=i&&i<n&&(int)cost.size()==ks[i]);
        if(!minimize)for(auto &x:cost)x=-x;
        _add1(i,cost);
    }
    void _add2(int i,int j,vector<vector<T>> cost){
        int h=ks[i],w=ks[j];
        _add1(j,cost[0]);
        for(int x=h-1;x>=0;x--)for(int y=0;y<w;y++)cost[x][y]-=cost[0][y];
        vector<T> a(h);
        for(int x=0;x<h;x++)a[x]=cost[x][w-1];
        _add1(i,a);
        for(int x=0;x<h;x++)for(int y=0;y<w;y++)cost[x][y]-=a[x];
        for(int x=1;x<h;x++){
            for(int y=0;y<w-1;y++){
                T w=cost[x][y]+cost[x-1][y+1]-cost[x-1][y]-cost[x][y+1];
                assert(w>=0); // monge
                add_edge(id[i][x],id[j][y+1],w);
            }
        }
    }
    void add2(int i,int j,vector<vector<T>> cost){
        assert(0<=i&&i<n&&0<=j&&j<n&&i!=j);
        assert((int)cost.size()==ks[i]);
        for(auto &v:cost)assert((int)v.size()==ks[j]);
        if(!minimize)for(auto &v:cost)for(auto &x:v)x=-x;
        _add2(i,j,cost);
    }
    pair<T,vector<int>> solve(){
        Dinic<T> dinic(buf,s,t);
        for(auto &[p,w]:edges){
            auto [u,v]=p;
            dinic.add_edge(u,v,w);
        }
        auto [val,cut]=dinic.cut();
        val+=base;
        if(!minimize)val=-val;
        vector<int> ans(n);
        for(int i=0;i<n;i++){
            ans[i]=ks[i]-1;
            for(int j=1;j<ks[i];j++)ans[i]-=cut[id[i][j]];
        }
        return {val,ans};
    }
};

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src/flows/k-ary-optimization.hpp

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