必会算法总结4—迪杰斯特拉算法

必会算法总结(4) - 迪杰斯特拉算法

​ 迪杰斯特拉算法的核心在于贪心+BFS。它和上一篇介绍到的拓扑排序一样属于图论算法，是大学计算机专业必学的算法之一，也是笔试经常出的算法。那么下面让我们一起来学习迪杰斯特拉算法的实现。

算法思想

迪杰斯特拉算法需要维护以下两张列表：

• visited：记录已经计算出最短路径的节点
• shortest：记录到达每个节点的最短路径

每次选取shortest中的权值最小的节点加入visited中，并实时维护shortest列表。

图的存储实现

在本次实例中我选用图的邻接表存储结构：

• Node类

  public class Node {
      int value;
      Map<Node, Integer> neighbors;
      
      public Node() {}
      
      public Node(int value) {
          this.value = value;
      }
      
      public Node(int value, Map<Node, Integer> neighbors) {
          this.value = value;
          neighbors = this.neighbors;
      }
  }
  

• Graph类

  public class Graph {
      List<Node> nodes;
      
      public Graph() {}
      
      public Graph(List<Node> nodes) {
          this.nodes = nodes;
      }
  }
  

算法核心

下列代码是迪杰斯特拉算法的核心，我们使用Integer.MAX_VALUE来表示无穷大：

public Map<Node, Integer> shortestPath(Graph graph, Node source) {
    // 创建列表shortest和visited
    Map<Node, Integer> shortest = new HashMap<>();
    Set<Node> visited = new HashSet<>();
    // 初始化每个node的值为Integer.MAX_VALUE
    for (Node node : graph.nodes) {
        shortest.put(node, Integer.MAX_VALUE);
    }
    // 设置源点source
    shortest.put(source, 0);
    // 选择shortest中的最小值加入visited中，并更新列表shortest
    while (true) {
        Node node = null;
        int min = Integer.MAX_VALUE;
        for (Map.Entry<Node, Integer> entry : shortest.entrySet()) {
            if (!visited.contains(entry.getKey()) && min > entry.getValue()) {
                min = entry.getValue();
                node = entry.getKey();
            }
        }
        if (node == null) {
            break;
        }
        visited.add(node);
        for (Map.Entry<Node, Integer> entry : node.neighbors.entrySet()) {
            if (!visited.contains(entry.getKey())) {
                shortest.put(entry.getKey(), Math.min(
                    shortest.get(entry.getKey()), min + entry.getValue()));
            }
        }
    }
    return shortest;
}

总结

下面的代码是上述所有代码的汇总，并附加了一个例子：

public class Main {
    
    static class Node {
        int value;
        Map<Node, Integer> neighbors;

        public Node() {}

        public Node(int value) {
            this.value = value;
        }

        public Node(int value, Map<Node, Integer> neighbors) {
            this.value = value;
            neighbors = this.neighbors;
        }
	}
    
    static class Graph {
        List<Node> nodes;

        public Graph() {}

        public Graph(List<Node> nodes) {
            this.nodes = nodes;
        }
	}
    
    public Map<Node, Integer> shortestPath(Graph graph, Node source) {
        // 创建列表shortest和visited
        Map<Node, Integer> shortest = new HashMap<>();
        Set<Node> visited = new HashSet<>();
        // 初始化每个node的值为Integer.MAX_VALUE
        for (Node node : graph.nodes) {
            shortest.put(node, Integer.MAX_VALUE);
        }
        // 设置源点source
        shortest.put(source, 0);
        // 选择shortest中的最小值加入visited中，并更新列表shortest
        while (true) {
            Node node = null;
            int min = Integer.MAX_VALUE;
            for (Map.Entry<Node, Integer> entry : shortest.entrySet()) {
                if (!visited.contains(entry.getKey()) && min > entry.getValue()) {
                    min = entry.getValue();
                    node = entry.getKey();
                }
            }
            if (node == null) {
                break;
            }
            visited.add(node);
            for (Map.Entry<Node, Integer> entry : node.neighbors.entrySet()) {
                if (!visited.contains(entry.getKey())) {
                    shortest.put(entry.getKey(), Math.min(
                        shortest.get(entry.getKey()), min + entry.getValue()));
                }
            }
        }
        return shortest;
	}
    
	public static void main(String[] args) {
        Node node1 = new Node(1);
        Node node2 = new Node(2);
        Node node3 = new Node(3);
        Node node4 = new Node(4);
        Node node5 = new Node(5);
        Node node6 = new Node(6);
        Node node7 = new Node(7);

        node1.neighbors = new HashMap<>();
        node2.neighbors = new HashMap<>();
        node3.neighbors = new HashMap<>();
        node4.neighbors = new HashMap<>();
        node5.neighbors = new HashMap<>();
        node6.neighbors = new HashMap<>();
        node7.neighbors = new HashMap<>();

        node1.neighbors.put(node2, 4);
        node1.neighbors.put(node3, 6);
        node1.neighbors.put(node4, 6);
        node2.neighbors.put(node3, 1);
        node2.neighbors.put(node5, 7);
        node3.neighbors.put(node5, 6);
        node3.neighbors.put(node6, 4);
        node4.neighbors.put(node3, 2);
        node4.neighbors.put(node6, 5);
        node5.neighbors.put(node7, 6);
        node6.neighbors.put(node5, 1);
        node6.neighbors.put(node7, 8);

        Graph graph = new Graph(new ArrayList<>(
            Arrays.asList(node1, node2, node3, node4, node5, node6, node7)));

        System.out.println(shortestPath(graph, node1));
    }
    
}