Dijkstra 算法是求解有向图中单源最短距离（Single Source Shortest Path，简称为 SSSP）的经典算法。

最短距离：对一个有权重的有向图 G=(V,E)，从一个源点 s 到汇点 v 有很多路径，其中边权和最小的路径，称从 s 到 v 的最短距离。

算法基本原理，如下所示：
  • 初始化：源点 s 到 s 自身的距离（d[s]=0），其他点 u 到 s 的距离为无穷（d[u]=∞）。
  • 迭代：若存在一条从 u 到 v 的边，那么从 s 到 v 的最短距离更新为：d[v]=min(d[v], d[u]+weight(u, v))，直到所有的点到 s 的距离不再发生变化时，迭代结束。

由算法基本原理可以看出，此算法非常适合使用 MaxCompute Graph 程序进行求解：每个点维护到源点的当前最短距离值，当这个值变化时，将新值加上边的权值发送消息通知其邻接点，下一轮迭代时，邻接点根据收到的消息更新其当前最短距离，当所有点当前最短距离不再变化时，迭代结束。

代码示例

单源最短距离的代码，如下所示： 
import java.io.IOException;

import com.aliyun.odps.io.WritableRecord;
import com.aliyun.odps.graph.Combiner;
import com.aliyun.odps.graph.ComputeContext;
import com.aliyun.odps.graph.Edge;
import com.aliyun.odps.graph.GraphJob;
import com.aliyun.odps.graph.GraphLoader;
import com.aliyun.odps.graph.MutationContext;
import com.aliyun.odps.graph.Vertex;
import com.aliyun.odps.graph.WorkerContext;
import com.aliyun.odps.io.LongWritable;
import com.aliyun.odps.data.TableInfo;

public class SSSP {

  public static final String START_VERTEX = "sssp.start.vertex.id";

  public static class SSSPVertex extends
      Vertex<LongWritable, LongWritable, LongWritable, LongWritable> {

    private static long startVertexId = -1;

    public SSSPVertex() {
      this.setValue(new LongWritable(Long.MAX_VALUE));
    }

    public boolean isStartVertex(
        ComputeContext<LongWritable, LongWritable, LongWritable, LongWritable> context) {
      if (startVertexId == -1) {
        String s = context.getConfiguration().get(START_VERTEX);
        startVertexId = Long.parseLong(s);
      }
      return getId().get() == startVertexId;
    }

    @Override
    public void compute(
        ComputeContext<LongWritable, LongWritable, LongWritable, LongWritable> context,
        Iterable<LongWritable> messages) throws IOException {
      long minDist = isStartVertex(context) ? 0 : Integer.MAX_VALUE;
      for (LongWritable msg : messages) {
        if (msg.get() < minDist) {
          minDist = msg.get();
        }
      }

      if (minDist < this.getValue().get()) {
        this.setValue(new LongWritable(minDist));
        if (hasEdges()) {
          for (Edge<LongWritable, LongWritable> e : this.getEdges()) {
            context.sendMessage(e.getDestVertexId(), new LongWritable(minDist
                + e.getValue().get()));
          }
        }
      } else {
        voteToHalt();
      }
    }

    @Override
    public void cleanup(
        WorkerContext<LongWritable, LongWritable, LongWritable, LongWritable> context)
        throws IOException {
      context.write(getId(), getValue());
    }
  }

  public static class MinLongCombiner extends
      Combiner<LongWritable, LongWritable> {

    @Override
    public void combine(LongWritable vertexId, LongWritable combinedMessage,
        LongWritable messageToCombine) throws IOException {
      if (combinedMessage.get() > messageToCombine.get()) {
        combinedMessage.set(messageToCombine.get());
      }
    }

  }

  public static class SSSPVertexReader extends
      GraphLoader<LongWritable, LongWritable, LongWritable, LongWritable> {

    @Override
    public void load(
        LongWritable recordNum,
        WritableRecord record,
        MutationContext<LongWritable, LongWritable, LongWritable, LongWritable> context)
        throws IOException {
      SSSPVertex vertex = new SSSPVertex();
      vertex.setId((LongWritable) record.get(0));
      String[] edges = record.get(1).toString().split(",");
      for (int i = 0; i < edges.length; i++) {
        String[] ss = edges[i].split(":");
        vertex.addEdge(new LongWritable(Long.parseLong(ss[0])),
            new LongWritable(Long.parseLong(ss[1])));
      }

      context.addVertexRequest(vertex);
    }

  }

  public static void main(String[] args) throws IOException {
    if (args.length < 2) {
      System.out.println("Usage: <startnode> <input> <output>");
      System.exit(-1);
    }

    GraphJob job = new GraphJob();
    job.setGraphLoaderClass(SSSPVertexReader.class);
    job.setVertexClass(SSSPVertex.class);
    job.setCombinerClass(MinLongCombiner.class);

    job.set(START_VERTEX, args[0]);
    job.addInput(TableInfo.builder().tableName(args[1]).build());
    job.addOutput(TableInfo.builder().tableName(args[2]).build());

    long startTime = System.currentTimeMillis();
    job.run();
    System.out.println("Job Finished in "
        + (System.currentTimeMillis() - startTime) / 1000.0 + " seconds");
  }
}
上述代码，说明如下：
  • 第 19 行：定义 SSSPVertex ，其中：
    • 点值表示该点到源点 startVertexId 的当前最短距离。
    • compute() 方法使用迭代公式：d[v]=min(d[v], d[u]+weight(u, v)) 更新点值。
    • cleanup() 方法把点及其到源点的最短距离写到结果表中。
  • 第 58 行：当点值没发生变化时，调用 voteToHalt() 告诉框架该点进入 halt 状态，当所有点都进入 halt 状态时，计算结束。
  • 第 70 行：定义 MinLongCombiner，对发送给同一个点的消息进行合并，优化性能，减少内存占用。
  • 第 83 行：定义 SSSPVertexReader 类，加载图，将表中每一条记录解析为一个点，记录的第一列是点标识，第二列存储该点起始的所有的边集，内容如：2:2，3:1，4:4。
  • 第 106 行：主程序（main 函数），定义 GraphJob，指定 Vertex/GraphLoader/Combiner 等的实现，指定输入输出表。