图的存储方式

• 临接表
• 临接矩阵

表达

• 点集/边集
• 有向图/无向图

Graph（大结构就是图）（包含点集合和边集合）

import java.util.HashMap;
import java.util.HashSet;
 
public class Graph {
    public HashMap<Integer, Node> nodes;//点集合
    public HashSet<Edge> edges;//边集合
 
    public Graph() {
        nodes = new HashMap<>();
        edges = new HashSet<>();
    }
}

Node（点集合）

import java.util.ArrayList;
 
public class Node {
    public int value;//数值
    public int in;//入度
    public int out;//出度
    public ArrayList<Node> nexts;//从当前点出发，连接的数据点（出度连接的点）
    public ArrayList<Edge> edges;//从当前点出发，连接的边（出度的边）
 
    public Node(int value) {
        this.value = value;
        in = 0;
        out = 0;
        nexts = new ArrayList<>();
        edges = new ArrayList<>();
    }
}

Edge（边集合）

 public class Edge {
    public int weight;//边的权重
    public Node from;//有向边
    public Node to;//有向边
 
    public Edge(int weight, Node from, Node to) {
        this.weight = weight;
        this.from = from;
        this.to = to;
    }
 
}

接口函数（将未知的题目类型转化为我们所熟知的类型，如上图所示）

public class GraphGenerator {

	// matrix 所有的边
	// N*3 的矩阵
	// [weight, from节点上面的值，to节点上面的值]

	//example
	//【7，0，1】第一个代表from，第二个代表to，第三个代表weight
	//【6，1，2】
	//【4，2，0】
	public static Graph createGraph(Integer[][] matrix) {
		Graph graph = new Graph();
		for (int i = 0; i < matrix.length; i++) { // from：matrix[0][0], to：matrix[0][1]  权重（weight）：matrix[0][2]
			Integer from = matrix[i][0];
			Integer to = matrix[i][1];
			Integer weight = matrix[i][2];
			if (!graph.nodes.containsKey(from)) {//将未出现的from放入点集合
				graph.nodes.put(from, new Node(from));
			}
			if (!graph.nodes.containsKey(to)) {//将未出现的to放入点集合
				graph.nodes.put(to, new Node(to));
			}
			Node fromNode = graph.nodes.get(from);//获取from点
			Node toNode = graph.nodes.get(to);//获取to点
			Edge newEdge = new Edge(weight, fromNode, toNode);//创建边
			fromNode.nexts.add(toNode);//from邻居
			fromNode.out++;//出度++
			toNode.in++;//入度++
			fromNode.edges.add(newEdge);//将边放到我所拥有的边集合里面
			graph.edges.add(newEdge);//放入边值
		}
		return graph;
	}

}

图的遍历

宽度优先遍历

代码

import java.util.HashSet;
import java.util.LinkedList;
import java.util.Queue;

public class Code01_BFS {

	// 从node出发，进行宽度优先遍历
	public static void bfs(Node node) {
		if (node == null) {
			return;
		}
		Queue<Node> queue = new LinkedList<>();
		HashSet<Node> set = new HashSet<>();//防止数据重复
		queue.add(node);
		set.add(node);
		while (!queue.isEmpty()) {
			Node cur = queue.poll();
			System.out.println(cur.value);
			for (Node next : cur.nexts) {
				if (!set.contains(next)) {
					set.add(next);
					queue.add(next);
				}
			}
		}
	}

}

深度优先遍历

代码

import java.util.HashSet;
import java.util.Stack;

public class Code02_DFS {

	public static void dfs(Node node) {
		if (node == null) {
			return;
		}
		Stack<Node> stack = new Stack<>();//保证深度的路径
		HashSet<Node> set = new HashSet<>();
		stack.add(node);
		set.add(node);
		System.out.println(node.value);
		while (!stack.isEmpty()) {
			Node cur = stack.pop();
			for (Node next : cur.nexts) {
				if (!set.contains(next)) {
					stack.push(cur);
					stack.push(next);
					set.add(next);
					System.out.println(next.value);
					break;
				}
			}
		}
	}

}

拓扑排序算法

• 要求是有向图，入度为0的节点，且没有环

代码

package class06;

import java.util.ArrayList;
import java.util.HashMap;
import java.util.LinkedList;
import java.util.List;
import java.util.Queue;

public class Code03_TopologySort {

	// directed graph and no loop
	public static List<Node> sortedTopology(Graph graph) {
		// key：某一个node
		// value：剩余的入度
		HashMap<Node, Integer> inMap = new HashMap<>();
		// 入度为0的点，才能进这个队列
		Queue<Node> zeroInQueue = new LinkedList<>();
		for (Node node : graph.nodes.values()) {
			inMap.put(node, node.in);
			if (node.in == 0) {
				zeroInQueue.add(node);
			}
		}
		// 拓扑排序的结果，依次加入result
		List<Node> result = new ArrayList<>();
		while (!zeroInQueue.isEmpty()) {
			Node cur = zeroInQueue.poll();
			result.add(cur);
			for (Node next : cur.nexts) {
				inMap.put(next, inMap.get(next) - 1);
				if (inMap.get(next) == 0) {
					zeroInQueue.add(next);
				}
			}
		}
		return result;
	}
}

Kruskal算法（并查集）

无向图

• 加最小的边，只要不形成环，就加上，否则不加入

代码

package class06;

import java.util.ArrayList;
import java.util.Collection;
import java.util.Comparator;
import java.util.HashMap;
import java.util.HashSet;
import java.util.List;
import java.util.PriorityQueue;
import java.util.Set;
import java.util.Stack;

//undirected graph only
public class Code04_Kruskal {
	
	public static class MySets{
		public HashMap<Node, List<Node>>  setMap;
		public MySets(List<Node> nodes) {
			for(Node cur : nodes) {
				List<Node> set = new ArrayList<Node>();
				set.add(cur);
				setMap.put(cur, set);
			}
		}
		
		
		public boolean isSameSet(Node from, Node to) {//判断from和to是否在一个集合
			List<Node> fromSet  = setMap.get(from);
			List<Node> toSet = setMap.get(to);
			return fromSet == toSet;//内存地址
		}
		
		
		public void union(Node from, Node to) {//
			List<Node> fromSet  = setMap.get(from);
			List<Node> toSet = setMap.get(to);
			for(Node toNode : toSet) {
				fromSet.add(toNode);
				setMap.put(toNode, fromSet);
			}
		}
	}
	
	

	
	
	// Union-Find Set
	public static class UnionFind {
		// key 某一个节点， value key节点往上的节点
		private HashMap<Node, Node> fatherMap;
		// key 某一个集合的代表节点, value key所在集合的节点个数
		private HashMap<Node, Integer> sizeMap;

		public UnionFind() {
			fatherMap = new HashMap<Node, Node>();
			sizeMap = new HashMap<Node, Integer>();
		}
		
		public void makeSets(Collection<Node> nodes) {
			fatherMap.clear();
			sizeMap.clear();
			for (Node node : nodes) {
				fatherMap.put(node, node);
				sizeMap.put(node, 1);
			}
		}

		private Node findFather(Node n) {
			Stack<Node> path = new Stack<>();
			while(n != fatherMap.get(n)) {
				path.add(n);
				n = fatherMap.get(n);
			}
			while(!path.isEmpty()) {
				fatherMap.put(path.pop(), n);
			}
			return n;
		}

		public boolean isSameSet(Node a, Node b) {
			return findFather(a) == findFather(b);
		}

		public void union(Node a, Node b) {
			if (a == null || b == null) {
				return;
			}
			Node aDai = findFather(a);
			Node bDai = findFather(b);
			if (aDai != bDai) {
				int aSetSize = sizeMap.get(aDai);
				int bSetSize = sizeMap.get(bDai);
				if (aSetSize <= bSetSize) {
					fatherMap.put(aDai, bDai);
					sizeMap.put(bDai, aSetSize + bSetSize);
					sizeMap.remove(aDai);
				} else {
					fatherMap.put(bDai, aDai);
					sizeMap.put(aDai, aSetSize + bSetSize);
					sizeMap.remove(bDai);
				}
			}
		}
	}
	

	public static class EdgeComparator implements Comparator<Edge> {

		@Override
		public int compare(Edge o1, Edge o2) {
			return o1.weight - o2.weight;
		}

	}

	public static Set<Edge> kruskalMST(Graph graph) {
		UnionFind unionFind = new UnionFind();
		unionFind.makeSets(graph.nodes.values());
		PriorityQueue<Edge> priorityQueue = new PriorityQueue<>(new EdgeComparator());
		for (Edge edge : graph.edges) { // M 条边
			priorityQueue.add(edge);  // O(logM)
		}
		Set<Edge> result = new HashSet<>();
		while (!priorityQueue.isEmpty()) { // M 条边
			Edge edge = priorityQueue.poll(); // O(logM)
			if (!unionFind.isSameSet(edge.from, edge.to)) { // O(1)
				result.add(edge);
				unionFind.union(edge.from, edge.to);
			}
		}
		return result;
	}
}

prim算法

无向图

代码

package class06;

import java.util.Comparator;
import java.util.HashSet;
import java.util.PriorityQueue;
import java.util.Set;

// undirected graph only
public class Code05_Prim {

	public static class EdgeComparator implements Comparator<Edge> {

		@Override
		public int compare(Edge o1, Edge o2) {
			return o1.weight - o2.weight;
		}

	}

	public static Set<Edge> primMST(Graph graph) {
		// 解锁的边进入小根堆
		PriorityQueue<Edge> priorityQueue = new PriorityQueue<>(
				new EdgeComparator());
		HashSet<Node> set = new HashSet<>();
		Set<Edge> result = new HashSet<>(); // 依次挑选的的边在result里
		for (Node node : graph.nodes.values()) { // 随便挑了一个点
			// node 是开始点
			if (!set.contains(node)) {
				set.add(node);
				for (Edge edge : node.edges) { // 由一个点，解锁所有相连的边
					priorityQueue.add(edge);
				}
				while (!priorityQueue.isEmpty()) {
					Edge edge = priorityQueue.poll(); // 弹出解锁的边中，最小的边
					Node toNode = edge.to; // 可能的一个新的点
					if (!set.contains(toNode)) { // 不含有的时候，就是新的点
						set.add(toNode);
						result.add(edge);
						for (Edge nextEdge : toNode.edges) {
							priorityQueue.add(nextEdge);
						}
					}
				}
			}
			//break;
		}
		return result;
	}

	// 请保证graph是连通图
	// graph[i][j]表示点i到点j的距离，如果是系统最大值代表无路
	// 返回值是最小连通图的路径之和
	public static int prim(int[][] graph) {
		int size = graph.length;
		int[] distances = new int[size];
		boolean[] visit = new boolean[size];
		visit[0] = true;
		for (int i = 0; i < size; i++) {
			distances[i] = graph[0][i];
		}
		int sum = 0;
		for (int i = 1; i < size; i++) {
			int minPath = Integer.MAX_VALUE;
			int minIndex = -1;
			for (int j = 0; j < size; j++) {
				if (!visit[j] && distances[j] < minPath) {
					minPath = distances[j];
					minIndex = j;
				}
			}
			if (minIndex == -1) {
				return sum;
			}
			visit[minIndex] = true;
			sum += minPath;
			for (int j = 0; j < size; j++) {
				if (!visit[j] && distances[j] > graph[minIndex][j]) {
					distances[j] = graph[minIndex][j];
				}
			}
		}
		return sum;
	}

	public static void main(String[] args) {
		System.out.println("hello world!");
	}

}

Dijkstra算法（单元最远路径算法）

• 要求没有权值为负的边

代码

package class06;

import java.util.HashMap;
import java.util.HashSet;
import java.util.Map.Entry;

// no negative weight
public class Code06_Dijkstra {

	public static HashMap<Node, Integer> dijkstra1(Node head) {
		// 从head出发到所有点的最小距离
		// key : 从head出发到达key
		// value : 从head出发到达key的最小距离
		// 如果在表中，没有T的记录，含义是从head出发到T这个点的距离为正无穷
		HashMap<Node, Integer> distanceMap = new HashMap<>();
		distanceMap.put(head, 0);
		// 已经求过距离的节点，存在selectedNodes中，以后再也不碰
		HashSet<Node> selectedNodes = new HashSet<>();
		Node minNode = getMinDistanceAndUnselectedNode(distanceMap, selectedNodes);
		while (minNode != null) {
			int distance = distanceMap.get(minNode);
			for (Edge edge : minNode.edges) {
				Node toNode = edge.to;
				if (!distanceMap.containsKey(toNode)) {
					distanceMap.put(toNode, distance + edge.weight);
				} else {
					distanceMap.put(edge.to, Math.min(distanceMap.get(toNode), distance + edge.weight));
				}
			}
			selectedNodes.add(minNode);
			minNode = getMinDistanceAndUnselectedNode(distanceMap, selectedNodes);
		}
		return distanceMap;
	}

	public static Node getMinDistanceAndUnselectedNode(HashMap<Node, Integer> distanceMap, HashSet<Node> touchedNodes) {
		Node minNode = null;
		int minDistance = Integer.MAX_VALUE;
		for (Entry<Node, Integer> entry : distanceMap.entrySet()) {
			Node node = entry.getKey();
			int distance = entry.getValue();
			if (!touchedNodes.contains(node) && distance < minDistance) {
				minNode = node;
				minDistance = distance;
			}
		}
		return minNode;
	}

	public static class NodeRecord {
		public Node node;
		public int distance;

		public NodeRecord(Node node, int distance) {
			this.node = node;
			this.distance = distance;
		}
	}

	public static class NodeHeap {
		private Node[] nodes; // 实际的堆结构
		// key 某一个node， value 上面数组中的位置
		private HashMap<Node, Integer> heapIndexMap;
		// key 某一个节点， value 从源节点出发到该节点的目前最小距离
		private HashMap<Node, Integer> distanceMap;
		private int size; // 堆上有多少个点

		public NodeHeap(int size) {
			nodes = new Node[size];
			heapIndexMap = new HashMap<>();
			distanceMap = new HashMap<>();
			size = 0;
		}

		public boolean isEmpty() {
			return size == 0;
		}

		// 有一个点叫node，现在发现了一个从源节点出发到达node的距离为distance
		// 判断要不要更新，如果需要的话，就更新
		public void addOrUpdateOrIgnore(Node node, int distance) {
			if (inHeap(node)) {
				distanceMap.put(node, Math.min(distanceMap.get(node), distance));
				insertHeapify(node, heapIndexMap.get(node));
			}
			if (!isEntered(node)) {
				nodes[size] = node;
				heapIndexMap.put(node, size);
				distanceMap.put(node, distance);
				insertHeapify(node, size++);
			}
		}

		public NodeRecord pop() {
			NodeRecord nodeRecord = new NodeRecord(nodes[0], distanceMap.get(nodes[0]));
			swap(0, size - 1);
			heapIndexMap.put(nodes[size - 1], -1);
			distanceMap.remove(nodes[size - 1]);
			// free C++同学还要把原本堆顶节点析构，对java同学不必
			nodes[size - 1] = null;
			heapify(0, --size);
			return nodeRecord;
		}

		private void insertHeapify(Node node, int index) {
			while (distanceMap.get(nodes[index]) < distanceMap.get(nodes[(index - 1) / 2])) {
				swap(index, (index - 1) / 2);
				index = (index - 1) / 2;
			}
		}

		private void heapify(int index, int size) {
			int left = index * 2 + 1;
			while (left < size) {
				int smallest = left + 1 < size && distanceMap.get(nodes[left + 1]) < distanceMap.get(nodes[left])
						? left + 1
						: left;
				smallest = distanceMap.get(nodes[smallest]) < distanceMap.get(nodes[index]) ? smallest : index;
				if (smallest == index) {
					break;
				}
				swap(smallest, index);
				index = smallest;
				left = index * 2 + 1;
			}
		}

		private boolean isEntered(Node node) {
			return heapIndexMap.containsKey(node);
		}

		private boolean inHeap(Node node) {
			return isEntered(node) && heapIndexMap.get(node) != -1;
		}

		private void swap(int index1, int index2) {
			heapIndexMap.put(nodes[index1], index2);
			heapIndexMap.put(nodes[index2], index1);
			Node tmp = nodes[index1];
			nodes[index1] = nodes[index2];
			nodes[index2] = tmp;
		}
	}

	// 改进后的dijkstra算法
	// 从head出发，所有head能到达的节点，生成到达每个节点的最小路径记录并返回
	public static HashMap<Node, Integer> dijkstra2(Node head, int size) {
		NodeHeap nodeHeap = new NodeHeap(size);
		nodeHeap.addOrUpdateOrIgnore(head, 0);
		HashMap<Node, Integer> result = new HashMap<>();
		while (!nodeHeap.isEmpty()) {
			NodeRecord record = nodeHeap.pop();
			Node cur = record.node;
			int distance = record.distance;
			for (Edge edge : cur.edges) {
				nodeHeap.addOrUpdateOrIgnore(edge.to, edge.weight + distance);
			}
			result.put(cur, distance);
		}
		return result;
	}

}