Блог пользователя Corei13

Автор Corei13, 10 лет назад, По-английски

For those who are interested, a C++11 implementation of highest-label push relabel maximum flow algorithm. Style and format is taken from here.

Uses just gap relabeling heuristic. Global relabeling heuristic was implemented but removed because of poor performance.

/*
    Implementation of highest-label push-relabel maximum flow
    with gap relabeling heuristic.

    Running time:
        O(|V|^2|E|^{1/2})

    Usage:
        - add edges by AddEdge()
        - GetMaxFlow(s, t) returns the maximum flow from s to t
    
    Input:
        - graph, constructed using AddEdge()
        - (s, t), (source, sink)

    Output:
        - maximum flow value

    Todo:
        - implement Phase II (flow network from preflow network)
        - implement GetMinCut()
*/

template <class T> struct Edge {
    int from, to, index;
    T cap, flow;

    Edge(int from, int to, T cap, T flow, int index): from(from), to(to), cap(cap), flow(flow), index(index) {}
};

template <class T> struct PushRelabel {
    int n;
    vector <vector <Edge <T>>> adj;
    vector <T> excess;
    vector <int> dist, count;
    vector <bool> active;
    vector <vector <int>> B;
    int b;
    queue <int> Q;

    PushRelabel (int n): n(n), adj(n) {}

    void AddEdge (int from, int to, int cap) {
        adj[from].push_back(Edge <T>(from, to, cap, 0, adj[to].size()));
        if (from == to) {
            adj[from].back().index++;
        }
        adj[to].push_back(Edge <T>(to, from, 0, 0, adj[from].size() - 1));

    }

    void Enqueue (int v) {
        if (!active[v] && excess[v] > 0 && dist[v] < n) {
            active[v] = true;
            B[dist[v]].push_back(v);
            b = max(b, dist[v]);
        }
    }

    void Push (Edge <T> &e) {
        T amt = min(excess[e.from], e.cap - e.flow);
        if (dist[e.from] == dist[e.to] + 1 && amt > T(0)) {
            e.flow += amt;
            adj[e.to][e.index].flow -= amt;
            excess[e.to] += amt;    
            excess[e.from] -= amt;
            Enqueue(e.to);
        }
    }

    void Gap (int k) {
        for (int v = 0; v < n; v++) if (dist[v] >= k) {
            count[dist[v]]--;
            dist[v] = max(dist[v], n);
            count[dist[v]]++;
            Enqueue(v);
        }
    }

    void Relabel (int v) {
        count[dist[v]]--;
        dist[v] = n;
        for (auto e: adj[v]) if (e.cap - e.flow > 0) {
            dist[v] = min(dist[v], dist[e.to] + 1);
        }
        count[dist[v]]++;
        Enqueue(v);
    }

    void Discharge(int v) {
        for (auto &e: adj[v]) {
            if (excess[v] > 0) {
                Push(e);
            } else {
                break;
            }
        }

        if (excess[v] > 0) {
            if (count[dist[v]] == 1) {
                Gap(dist[v]); 
            } else {
                Relabel(v);
            }
        }
    }

    T GetMaxFlow (int s, int t) {
        dist = vector <int>(n, 0), excess = vector<T>(n, 0), count = vector <int>(n + 1, 0), active = vector <bool>(n, false), B = vector <vector <int>>(n), b = 0;
        
        for (auto &e: adj[s]) {
            excess[s] += e.cap;
        }

        count[0] = n;
        Enqueue(s);
        active[t] = true;
        
        while (b >= 0) {
            if (!B[b].empty()) {
                int v = B[b].back();
                B[b].pop_back();
                active[v] = false;
                Discharge(v);
            } else {
                b--;
            }
        }
        return excess[t];
    }

    T GetMinCut (int s, int t, vector <int> &cut);
};

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10 лет назад, # |
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I read a lot of articles about push-relabel algorithm, I'm still not able to understand it :(

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    10 лет назад, # ^ |
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    The TopCoder one is pretty good save for one or two of the proof explanations later on. The basis for the algorithm is the operations themselves (push, relabel) and a few key properties that hold throughout the entire algorithm, the most important of which being that a residual edge implies a height restriction. For me, I had trouble understanding it on the large scale, so I worked through each individual part and understood how they fit together for the bigger algorithm. Although I still have some trouble understanding why the whole algorithm works intuitively, I understand each individual part, which is helpful when actually implementing the algorithm. If you have any questions regarding the individual parts, I'm sure I or someone else in the community could explain it. Also, if someone could post a larger scale explanation or motivation for the algorithm that would be very beneficial.

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10 лет назад, # |
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Although Ford-Fulkerson method with shortest augment implementation has complexity O(V2E), it is in practice faster than push-relabel method, and also easy to implement(about 15 lines for a recursive version, and 25 lines for a non-recursive one). That bound looks like not possible to reach, however complexity of push-relabel might be a tight bound.

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    10 лет назад, # ^ |
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    I think FF method will get TLE if the graph of some data is special. I used to choose Dinic algorithm to solve network flow.

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    9 лет назад, # ^ |
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    Ford-Fulkerson with shortest augment

    It is usually called Edmonds-Karp algorithm.

    ...it is in practice faster than push-relabel method

    That's wrong, there are many problems even on the programming contests about networks with unit capacities where E-K works significantly slower than Dinic algorithm or preflow-push algorithm.

    however complexity of push-relabel might be a tight bound

    That's also wrong, there are algorithms that are faster. The fastest known is Orlin's algorithm, it has a running time of O(nm) that is believed to be a tight bound for the maximum flow problem in general case (though the tightness is not proven).

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8 лет назад, # |
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if you want to check if an edge from node u to node v is part of the minimum cut you can add this function

bool is_cut(int u,int v) {
    return (dist[u] >= n) && (dist[v] < n);
}
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8 лет назад, # |
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I'd be curious about how you concluded that the global relabeling heuristics exhibits poor performance.