General Algorithms - Graph

BFS

Red Knight’s Shortest Path - World CodeSprint 12 -

While loop 和 Recusive function可以互相用来替换; 0 为任意随机起始点， 4 为任意终点， 1，2，3为过程中步骤

DFS

Even Tree

• 由于已知node范围在2-100，把adjacency list(vector)的范围直接写成 ”>100” 大一点的可以帮助更快确定位置。Sparse 的这种不确定分布的图，使用matrix会增加storage的负担，最大情况会需要一个100x100的matrix， 然而不是每个matrix 的cell都会用到。

• 整个DFS里，不用一个一个node来回测试相连为不为even的tree，只用计算出DFS 遍历的nodes的和能不能被%2。使用公共的Variable来 counts 可以砍掉的edges的数量。

• 最要小心是return的时候，nodes的总和计算要加上当前的这一个node （1）, 即：dfs(当前node)+1。dfs(当前node) recursion （back tracking）的作用是累计当前node的 子nodes 的个数。

• 用 num_vertex+=num_nodes 累计到当前node，需要经过的子nodes.

• Visit[] 的boolean判断在DFS非要不可，是为了track node.

Roads and Libraries

• 此题已知edge 和 node统一Weights，则BFS没有优势但也可以用

• 除了BFS， DFS这些能够确保每个点和边都会经过的，还有Minimum Spanning Tree 可以用. MST运用了Union-Find的Data structure，比如根据Weight来Swap

MST解法 1

import java.io.BufferedReader;
import java.io.InputStreamReader;
import java.util.*;

class MinSpanningTree {

    public static final class Edge implements Comparable<Edge> {
        final int srcId;
        final int trgId;
        final long value;

        Edge(int srcId, int trgId, long value) {
            this.srcId = srcId;
            this.trgId = trgId;
            this.value = value;
        }

        public int compareTo(Edge o) {
            return (value < o.value) ? -1 : (value > o.value) ? 1 : 0;
        }
    }

    private static final class NodeEntry {
        NodesSet owner;
        NodeEntry next;     
    }

    private static final class NodesSet {
        int size;
        NodeEntry first;
        NodeEntry last;

        NodesSet(NodeEntry node) {
            size = 1;
            first = node;
            last = node;
        }

        static void merge(NodesSet set1, NodesSet set2) {
            if (set1.size < set2.size) {
                merge(set2, set1);
                return;
            }
            set1.size += set2.size;
            set1.last.next = set2.first;
            set1.last = set2.last;
            NodeEntry node = set2.first;
            while (node != null) {
                node.owner = set1;
                node = node.next;
            }
        }
    }

    private int nodesCount;
    private int edgesCount;
    private Edge[] edges;

    public MinSpanningTree(int nodes) {
        this.nodesCount = nodes;
        this.edgesCount = 0;
        this.edges = new Edge[nodesCount];
    }

    public void addEdge(int srcId, int trgId, long value) {
        if (edgesCount == edges.length) {
            edges = Arrays.copyOf(edges, 2 * edges.length);
        }
        edges[edgesCount++] = new Edge(srcId, trgId, value);
    }

    public Edge[] getMinSpanningTree() {
        NodeEntry[] nodes = new NodeEntry[nodesCount];
        for (int i = 0; i < nodesCount; i++) {
            nodes[i] = new NodeEntry();
            nodes[i].owner = new NodesSet(nodes[i]);
        }
        edges = Arrays.copyOf(edges, edgesCount);
        Arrays.sort(edges, 0, edgesCount);
        Edge[] spanningTree = new Edge[nodesCount-1];
        int k = 0;
        for (int i = 0; i < edgesCount & k+1 < nodesCount; i++) {
            NodesSet set1 = nodes[edges[i].srcId].owner;
            NodesSet set2 = nodes[edges[i].trgId].owner;
            if (set1 != set2) {
                NodesSet.merge(set1, set2);
                spanningTree[k++] = edges[i];
            }
        }
        return (k+1 < nodesCount) ? null : spanningTree;
    }

    public long getMinSpanningTreeSize() {
        Edge[] spanningTree = getMinSpanningTree();
        if (spanningTree == null) return -1;
        long treeSize = 0;
        for (Edge edge : spanningTree) treeSize += edge.value;
        return treeSize;
    }
}

public class Solution {
    public static void main(String[] args) {
        Scanner in = new Scanner(new BufferedReader(new InputStreamReader(System.in), 256 << 10));
        int testsNumber = in.nextInt();
        for (int test = 0; test < testsNumber; test++) {
            int n = in.nextInt();
            int m = in.nextInt();
            int costOfLib = in.nextInt();
            int costOfRoad = in.nextInt();
            MinSpanningTree tree = new MinSpanningTree(n+1);
            for (int i = 0; i < n; i++) {
                tree.addEdge(0, i+1, costOfLib);
            }
            for (int i = 0; i < m; i++) {
                int v1 = in.nextInt();
                int v2 = in.nextInt();
                tree.addEdge(v1, v2, costOfRoad);
            }
            System.out.println(tree.getMinSpanningTreeSize());
        }
        in.close();
    }
}

MST解法2

import java.io.*;
import java.util.*;
import java.text.*;
import java.math.*;
import java.util.regex.*;

public class Solution {

    public static class Edge {
        public int u, v;
        public long w;
        public Edge(int u, int v, long w) {
            this.u = u;
            this.v = v;
            this.w = w;
        }
    }

    public static class UnionFind {
        private int parent[];
        private int rank[];

        public UnionFind(int n) {
            parent = new int[n];
            rank = new int[n];
            for (int i = 0; i < n; i++) {
                parent[i] = i;
            }
        }

        public int findSet(int x) {
            while (parent[x] != x) {
                parent[x] = parent[parent[x]];
                x = parent[x];
            }
            return x;
        }

        public void union(int a, int b) {
            int setA = findSet(a);
            int setB = findSet(b);

            if (rank[setA] > rank[setB]) {
                parent[setB] = setA;
            } else {
                parent[setA] = setB;
                if (rank[setA] == rank[setB]) {
                    rank[setB] += 1;
                }
            }
        }

        public boolean connected(int a, int b) {
            return (findSet(a) == findSet(b));
        }
    }

    public static void main(String[] args) {
        Scanner in = new Scanner(System.in);
        int q = in.nextInt();
        for(int a0 = 0; a0 < q; a0++){
            long n = in.nextLong();
            long m = in.nextLong();
            long lib = in.nextLong();
            long road = in.nextLong();

            List edges = new ArrayList<>();
            for(int a1 = 0; a1 < m; a1++){
                int city_1 = in.nextInt();
                int city_2 = in.nextInt();
                edges.add(new Edge(city_1, city_2, road));
            }

            UnionFind uf = new UnionFind((int)(n + 1));
            long minCost = n * lib;
            long roadCosts = 0;
            long libCosts = n * lib;

            for (Edge edge : edges) {
                if (!uf.connected(edge.u, edge.v)) {
                    uf.union(edge.u, edge.v);
                    roadCosts += road;
                    libCosts -= lib;
                    minCost = Math.min(minCost, roadCosts + libCosts);
                }
            }

            System.out.println(minCost);
        }
    }
}

MST

Kruskal (MST): Really Special Subtree

按照算法，右图的大括号里的已经是所有edges从小到大的排序。已知从最小的那个开始直到所有nodes都包含在一个Minimum Spanning Tree里：先是Edge(1,3), 到Edge(2,3),当到Edge(1,2)时，由于前面的两个Edges已经包含了1 与 2 这两个node,所以忽略Edge(1,2)；到Edge(3,4), 然后到Edge(1,4),由于前面也已经包含了Node 1 和 4，所以忽略Edge(1,4), 到Edge(2,4)同理，也被忽略。则最后只留有：

Edge(1,3) +Edge(2,3)+Edge(3,4) = 3+4+5 = 12

• 这个算法的time complexity 由Sort来决定。因为在找MST时的For loop只为Edge的数量级，很少很简单。
  
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