算法计算的依赖关系图的偏序 [英] Algorithm for computing partial orderings of dependency graphs

问题描述

我试图计算依赖关系图，这实际上是一个DAG（有向无环图）的部分拓扑排序是precise;从而不平行依赖性冲突而执行任务。

我想出了这个简单的算法，因为我发现在谷歌是不是所有的有用的（我一直发现只有自己在并行运行的算法来计算一个正常的拓扑排序）。

 访问（节点）
{
    maxdist = 0;
    的foreach（E：in_e​​dge（节点））
    {
        maxdist = MAX（maxdist，1 +访问（源（E）））
    }
    距离[节点] = maxdist;
    返回距离[节点]
}

make_partial_ordering（节点）
{
    将所有节点的距离到+无穷大;
    num_levels =访问（节点，0）;
    返回排序（节点，通过增加距离），其中距离＆LT; +无限的;
}
 

（注意，这是唯一的伪code和有一定会是一些可能的小优化）

至于正确性，似乎pretty的明显的：对于叶子（：它们没有进一步依赖=节点）的最大距离 - 薄片始终为0（正确的，因为环被跳过由于0边缘）。 对于连接到节点n1的任何节点，...，nk的最大距离 - 薄片是1 +最大距离{[N1]，..，距离[NK]}

我没有找到这篇文章后，我写下来的算法：<一href="http://msdn.microsoft.com/en-us/magazine/dd569760.aspx">http://msdn.microsoft.com/en-us/magazine/dd569760.aspx 但说实话，他们为什么这样做所有的列表复制等特点，它只是似乎真的如此低效..？

无论如何，我不知道我的算法是否正确，如果是的话它叫什么，这样我可以读了一些东西吧。

更新：我实现的算法在我的计划，这似乎是伟大的工作，一切我测试。 code-明智的，它看起来是这样的：

 的typedef的boost :: graph_traits＆LT; my_graph＆GT; my_graph_traits;
  的typedef my_graph_traits :: vertices_size_type vertices_size_type;
  的typedef my_graph_traits ::顶点描述的顶点;
  的typedef my_graph_traits :: edge_descriptor边缘;

  vertices_size_type find_partial_ordering（顶点v，
      的std ::地图＆LT; vertices_size_type，性病::集＆LT;点＆GT; ＆GT;＆安培; distance_map）
  {
      vertices_size_type D = 0;

      BOOST_FOREACH（边e，in_edges（V））
      {
          D =的std ::最大（D，1 + find_partial_ordering（源（E），distance_map））;
      }

      distance_map并[d] .insert（ⅴ）;

      返回D组;
  }
 

解决方案

您算法（C ++）的作品，但它有非常坏的最坏情况下的时间复杂度。它计算 find_partial_ordering 尽可能多路径的节点上，因为有该节点。在树的情况下，路径的数目是1，但在一般的向无环图的路径的数量可以是指数的。

您可以通过修复此 memoizing 你的 find_partial_ordering 结果没有返回时，已经计算了一个特定节点的值递归。然而，这仍然给你留下一堆克星递归解决方案。

一个有效的（线性）算法的拓扑排序，给出维基百科。这是否符合你的需要？

编辑：啊，我明白了，你想知道的深度界限，让您可以并行的一切在给定的深度。您仍然可以使用该算法在维基百科上的这个（从而避免递归）。

首先，做一个拓扑排序与维基百科的算法。的现在的访问拓扑节点顺序计算深度：

 深度：数组1..N
因为我在1..N
    深度[I] = 0
    对于j的i的子女
        深度[I] =最大值（深度[I]中，深度[j]的+1）
回报深处
 

请注意，上面没有递归，只是一个普通的 0的（| V | + | E |）算法。这有相同的复杂性，在维基百科上的算法。

I'm trying to compute a partial "topological sort" of a dependency graph, which is actually a DAG (Directed Acyclic Graph) to be precise; so as to execute tasks without conflicting dependencies in parallel.

I came up with this simple algorithm because what I found on Google wasn't all that helpful (I keep finding only algorithms that themselves run in parallel to compute a normal topological sort).

visit(node)
{
    maxdist = 0;
    foreach (e : in_edge(node))
    {
        maxdist = max(maxdist, 1 + visit(source(e)))
    }
    distance[node] = maxdist;
    return distance[node];
}

make_partial_ordering(node)
{
    set all node distances to +infinite;
    num_levels = visit(node, 0);
    return sort(nodes, by increasing distance) where distances < +infinite;
}

(Note that this is only pseudocode and there would certainly be a few possible small optimizations)

As for the correctness, it seems pretty obvious: For the leafs (:= nodes which have no further dependency) the maximum distance-to-leaf is always 0 (correct because the loop gets skipped due to 0 edges). For any node connected to nodes n1,..,nk the maximum distance-to-leaf is 1 + max{distance[n1],..,distance[nk]}.

I did find this article after I had written down the algorithm: http://msdn.microsoft.com/en-us/magazine/dd569760.aspx But honestly, why are they doing all that list copying and so on, it just seems so really inefficient..?

Anyway I was wondering whether my algorithm is correct, and if so what it is called so that I can read up some stuff about it.

Update: I implemented the algorithm in my program and it seems to be working great for everything I tested. Code-wise it looks like this:

  typedef boost::graph_traits<my_graph> my_graph_traits;
  typedef my_graph_traits::vertices_size_type vertices_size_type;
  typedef my_graph_traits::vertex_descriptor vertex;
  typedef my_graph_traits::edge_descriptor edge;

  vertices_size_type find_partial_ordering(vertex v,
      std::map<vertices_size_type, std::set<vertex> >& distance_map)
  {
      vertices_size_type d = 0;

      BOOST_FOREACH(edge e, in_edges(v))
      {
          d = std::max(d, 1 + find_partial_ordering(source(e), distance_map));
      }

      distance_map[d].insert(v);

      return d;
  }

解决方案

Your algorithm (C++) works, but it has very bad worst-case time complexity. It computes find_partial_ordering on a node for as many paths as there are to that node. In the case of a tree, the number of paths is 1, but in a general directed acyclic graph the number of paths can be exponential.

You can fix this by memoizing your find_partial_ordering results and returning without recursing when you have already computed the value for a particular node. However, this still leaves you with a stack-busting recursive solution.

An efficient (linear) algorithm for topological sorting is given on Wikipedia. Doesn't this fit your needs?

Edit: ah, I see, you want to know where the depth boundaries are so that you can parallelize everything at a given depth. You can still use the algorithm on Wikipedia for this (and so avoid recursion).

First, Do a topological sort with the algorithm on Wikipedia. Now compute depths by visiting nodes in topological order:

depths : array 1..n
for i in 1..n
    depths[i] = 0
    for j in children of i
        depths[i] = max(depths[i], depths[j] + 1)
return depths

Notice that there's no recursion above, just a plain O(|V| + |E|) algorithm. This has the same complexity as the algorithm on Wikipedia.

这篇关于算法计算的依赖关系图的偏序的文章就介绍到这了，希望我们推荐的答案对大家有所帮助，也希望大家多多支持IT屋！