普林斯顿算法课Part2第三周作业_BaseballElimination

作业地址：http://coursera.cs.princeton.edu/algs4/assignments/baseball.html

作业难点：

1、如何计算非平凡淘汰（Nontrivial elimination）？

　　关键在于理解题目的这句话“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：从所有队伍中直接淘汰w[i] + r[i] < max{w[i]}的队伍；

　　2）非平凡淘汰的算法中心思想：如果一支队伍完成的胜场不能被其他队伍超越，则这支队伍不会被淘汰。极端情况下，第i支队伍完成w[i] + r[i]胜场时，剩下的队伍输掉所有与平凡淘汰的队伍的对局其胜场最少。以其中一支未被平凡淘汰的队伍的w[i] + r[i]作为所有队伍需要完成的胜场次，其他队伍需要完成w[i] + r[i] - w[j]个胜场可与第i支队伍战平，即[team vertices] -> [sink vertice]的capacity为：w[i] + r[i] - w[j]；

　　3）根据剩余队伍之间的game[][]构建[source vertice] -> [game vertices]边；

　　4）然后按题目提示构建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）通过FordFulkerson算法求解每支队伍source端的流量，确定[source vertice] -> [game vertices]是否达到最大流量，如果没有达到最大流量，说明FlowNetWork中的比赛没有进行完某支队伍的胜场已达到w[i]+r[i]，亦即存在最终获胜的队伍胜场大于w[i]+r[i]，说明这第i支队伍需要被淘汰；

　　6）重复2）-5）步，直到遍历求解完所有未被平凡淘汰的队伍。

2、如何计算队伍是被哪些队伍淘汰的？

　　1）平凡淘汰：队伍是被目前领先的队伍直接淘汰；

　　2）非平凡淘汰：根据最大流最小切定理（以及题目提示：What the min cut tells us.），队伍被过source的最小切上的顶点所淘汰。

容易扣分点：

本题目的特征是要么没有理解题意难以求解，要么很顺利得出最终结果。

部分代码：

1、数据结构：

    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;            
        }
    }

2、构造网络流图：

    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);
    }