普林斯頓算法課Part2第三週做業_BaseballElimination

做業地址：http://coursera.cs.princeton.edu/algs4/assignments/baseball.html

做業難點：

一、如何計算非平凡淘汰（Nontrivial elimination）？

　　關鍵在於理解題目的這句話「If all edges in the maxflow that are pointing from s are full, then this corresponds to assigning winners to all of the remaining games in such a way that no team wins more games than x. If some edges pointing from s are not full, then there is no scenario in which team x can win the division.」

　　1）平凡淘汰：從全部隊伍中直接淘汰w[i] + r[i] < max{w[i]}的隊伍；

　　2）非平凡淘汰的算法中心思想：若是一支隊伍完成的勝場不能被其餘隊伍超越，則這支隊伍不會被淘汰。極端狀況下，第i支隊伍完成w[i] + r[i]勝場時，剩下的隊伍輸掉全部與平凡淘汰的隊伍的對局其勝場最少。以其中一支未被平凡淘汰的隊伍的w[i] + r[i]做爲全部隊伍須要完成的勝場次，其餘隊伍須要完成w[i] + r[i] - w[j]個勝場可與第i支隊伍戰平，即[team vertices] -> [sink vertice]的capacity爲：w[i] + r[i] - w[j]；

　　3）根據剩餘隊伍之間的game[][]構建[source vertice] -> [game vertices]邊；

　　4）而後按題目提示構建FlowNetWork：

　　　　　　　　　　

from	to	capacity
[source vertice]	[game vertices]	game[i][j]
[game vertices]	[team vertices]	infinity
[team vertices]	[sink vertice]	w[i] + r[i] - w[j]

　　5）經過FordFulkerson算法求解每支隊伍source端的流量，肯定[source vertice] -> [game vertices]是否達到最大流量，若是沒有達到最大流量，說明FlowNetWork中的比賽沒有進行完某支隊伍的勝場已達到w[i]+r[i]，亦即存在最終獲勝的隊伍勝場大於w[i]+r[i]，說明這第i支隊伍須要被淘汰；

　　6）重複2）-5）步，直到遍歷求解完全部未被平凡淘汰的隊伍。

二、如何計算隊伍是被哪些隊伍淘汰的？

　　1）平凡淘汰：隊伍是被目前領先的隊伍直接淘汰；

　　2）非平凡淘汰：根據最大流最小切定理（以及題目提示：What the min cut tells us.），隊伍被過source的最小切上的頂點所淘汰。

容易扣分點：

本題目的特徵是要麼沒有理解題意難以求解，要麼很順利得出最終結果。

部分代碼：

一、數據結構：

    private int[] w, l, r;
    private int[][] games;
    private Map<String, Integer> teamMap;
    private int maxWins = -1;
    private String leadingTeam;    
    private static final double MAX_EDGE = Double.POSITIVE_INFINITY; 

    private class spGraph {
        FordFulkerson ff;
        FlowNetwork flowNetwork;
        int s;

        public spGraph(FordFulkerson ff, FlowNetwork network, int source, int sink) {
            super();
            this.ff = ff;
            this.flowNetwork = network;
            this.s = source;            
        }
    }

 

二、構造網絡流圖：

    private spGraph buildGraph(int id) {
        int n = numberOfTeams();
        int source = n;
        int sink = n + 1;
        int gameVertice = n + 2;
        int curMaxWins = w[id] + r[id];
        Set<FlowEdge> edges = new HashSet<FlowEdge>();
        for (int i = 0; i < n; i++) {
            if (i == id || trvialEnd(i)) 
                continue;            
            for (int j = 0; j < i; j++) {
                if (j == id || trvialEnd(j) || games[i][j] == 0)
                    continue;                
                edges.add(new FlowEdge(source, gameVertice, games[i][j]));
                edges.add(new FlowEdge(gameVertice, i, MAX_EDGE));
                edges.add(new FlowEdge(gameVertice, j, MAX_EDGE));
                gameVertice++;
            }
            edges.add(new FlowEdge(i, sink, curMaxWins - w[i]));
        }
        FlowNetwork flowNetwork = new FlowNetwork(gameVertice);
        for (FlowEdge edge : edges) 
            flowNetwork.addEdge(edge);        
        FordFulkerson ff = new FordFulkerson(flowNetwork, source, sink);
        return new spGraph(ff, flowNetwork, source, sink);
    }

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