Topological sorting

Algorithms

The usual algorithms for topological sorting have running time linear in the number of nodes plus the number of edges, asymptotically, ${\displaystyle O(\left|V\right|+\left|E\right|).}$

Kahn's algorithm

One of these algorithms, first described by Kahn (1962), works by choosing vertices in the same order as the eventual topological sort. First, find a list of "start nodes" that have no incoming edges and insert them into a set S; at least one such node must exist in a non-empty (finite) acyclic graph. Then:

<i id="mwQw">L</i> ← Empty list that will contain the sorted elements <i id="mwRA">S</i> ← Set of all nodes with no incoming edge <b id="mwRQ">while</b> <i id="mwRg">S</i> <b id="mwRw">is not</b> empty <b id="mwSA">do</b> remove a node <i id="mwSQ">n</i> from <i id="mwSg">S</i> add <i id="mwSw">n</i> to <i id="mwTA">L</i> <b id="mwTQ">for each</b> node <i id="mwTg">m</i> with an edge <i id="mwTw">e</i> from <i id="mwUA">n</i> to <i id="mwUQ">m</i> <b id="mwUg">do</b> remove edge <i id="mwUw">e</i> from the graph <b id="mwVA">if</b> <i id="mwVQ">m</i> has no other incoming edges <b id="mwVg">then</b> insert <i id="mwVw">m</i> into <i id="mwWA">S</i> <b id="mwWQ">if</b> <i id="mwWg">graph</i> has edges <b id="mwWw">then</b> <b id="mwXA">return</b> error <i id="mwXQ">(graph has at least one cycle)</i> <b id="mwXg">else</b> <b id="mwXw">return</b> <i id="mwYA">L</i> <i id="mwYQ">(a topologically sorted order)</i>

If the graph is a DAG, a solution will be contained in the list L (although the solution is not necessarily unique). Otherwise, the graph must have at least one cycle and therefore a topological sort is impossible.

Reflecting the non-uniqueness of the resulting sort, the structure S can be simply a set or a queue or a stack. Depending on the order that nodes n are removed from set S, a different solution is created. A variation of Kahn's algorithm that breaks ties lexicographically forms a key component of the Coffman–Graham algorithm for parallel scheduling and layered graph drawing.

Depth-first search

An alternative algorithm for topological sorting is based on depth-first search. The algorithm loops through each node of the graph, in an arbitrary order, initiating a depth-first search that terminates when it hits any node that has already been visited since the beginning of the topological sort or the node has no outgoing edges (i.e., a leaf node):

<i id="mwbA">L</i> ← Empty list that will contain the sorted nodes <b id="mwbQ">while</b> exists nodes without a permanent mark <b id="mwbg">do</b> select an unmarked node <i id="mwbw">n</i> visit(<i id="mwcA">n</i>) <b id="mwcQ">function</b> visit(node <i id="mwcg">n</i>) <b id="mwcw">if</b> <i id="mwdA">n</i> has a permanent mark <b id="mwdQ">then</b> <b id="mwdg">return</b> <b id="mwdw">if</b> <i id="mweA">n</i> has a temporary mark <b id="mweQ">then</b> <b id="mweg">stop</b> <i id="mwew">(graph has at least one cycle)</i> mark <i id="mwfA">n</i> with a temporary mark <b id="mwfQ">for each</b> node <i id="mwfg">m</i> with an edge from <i id="mwfw">n</i> to <i id="mwgA">m</i> <b id="mwgQ">do</b> visit(<i id="mwgg">m</i>) mark <i id="mwgw">n</i> with a permanent mark add <i id="mwhA">n</i> to head of <i id="mwhQ">L</i>

Each node n gets prepended to the output list L only after considering all other nodes that depend on n (all descendants of n in the graph). Specifically, when the algorithm adds node n, we are guaranteed that all nodes that depend on n are already in the output list L: they were added to L either by the recursive call to visit() that ended before the call to visit n, or by a call to visit() that started even before the call to visit n. Since each edge and node is visited once, the algorithm runs in linear time. This depth-first-search-based algorithm is the one described by Cormen et al. (2001); it seems to have been first described in print by Tarjan in 1976.

Parallel algorithms

On a parallel random-access machine, a topological ordering can be constructed in O((log n)²) time using a polynomial number of processors, putting the problem into the complexity class NC². One method for doing this is to repeatedly square the adjacency matrix of the given graph, logarithmically many times, using min-plus matrix multiplication with maximization in place of minimization. The resulting matrix describes the longest path distances in the graph. Sorting the vertices by the lengths of their longest incoming paths produces a topological ordering.

An algorithm for parallel topological sorting on distributed memory machines parallelizes the algorithm of Kahn for a DAG ${\displaystyle G=(V,E)}$. On a high level, the algorithm of Kahn repeatedly removes the vertices of indegree 0 and adds them to the topological sorting in the order in which they were removed. Since the outgoing edges of the removed vertices are also removed, there will be a new set of vertices of indegree 0, where the procedure is repeated until no vertices are left. This algorithm performs ${\displaystyle D+1}$ iterations, where D is the longest path in G. Each iteration can be parallelized, which is the idea of the following algorithm.

In the following, it is assumed that the graph partition is stored on p processing elements (PE), which are labeled ${\displaystyle 0,\dots ,p-1}$. Each PE i initializes a set of local vertices ${\displaystyle Q_i^1}$ with indegree 0, where the upper index represents the current iteration. Since all vertices in the local sets ${\displaystyle Q_0^1,\dots ,Q_{p-1}^1}$ have indegree 0, i.e., they are not adjacent, they can be given in an arbitrary order for a valid topological sorting. To assign a global index to each vertex, a prefix sum is calculated over the sizes of ${\displaystyle Q_0^1,\dots ,Q_{p-1}^1}$. So, each step, there are $\sum _{i=0}^{p-1}|Q_i|$ vertices added to the topological sorting.

In the first step, PE j assigns the indices $\sum _{i=0}^{j-1}|Q_i^1|,\dots ,\left(\sum _{i=0}^j|Q_i^1|\right)-1$ to the local vertices in ${\displaystyle Q_j^1}$. These vertices in ${\displaystyle Q_j^1}$ are removed, together with their corresponding outgoing edges. For each outgoing edge ${\displaystyle (u,v)}$ with endpoint v in another PE ${\displaystyle l,j\ne l}$, the message ${\displaystyle (u,v)}$ is posted to PE l. After all vertices in ${\displaystyle Q_j^1}$ are removed, the posted messages are sent to their corresponding PE. Each message ${\displaystyle (u,v)}$ received updates the indegree of the local vertex v. If the indegree drops to zero, v is added to ${\displaystyle Q_j^2}$. Then the next iteration starts.

In step k, PE j assigns the indices $a_{k-1}+\sum _{i=0}^{j-1}|Q_i^k|,\dots ,a_{k-1}+\left(\sum _{i=0}^j|Q_i^k|\right)-1$, where ${\displaystyle a_{k-1}}$ is the total number of processed vertices after step ⁠${\displaystyle k-1}$⁠. This procedure repeats until there are no vertices left to process, hence $\sum _{i=0}^{p-1}|Q_i^{D+1}|=0$. Below is a high level, single program, multiple data pseudo-code overview of this algorithm.

Note that the prefix sum for the local offsets $a_{k-1}+\sum _{i=0}^{j-1}|Q_i^k|,\dots ,a_{k-1}+\left(\sum _{i=0}^j|Q_i^k|\right)-1$ can be efficiently calculated in parallel.

<b id="mw4A">p</b> processing elements with IDs from 0 to <i id="mw4Q">p</i>-1 <b id="mw4g">Input:</b> G = (V, E) DAG, distributed to PEs, PE index j = 0, ..., p - 1 <b id="mw4w">Output:</b> topological sorting of G <b id="mw5A">function</b> traverseDAGDistributed δ incoming degree of local vertices <i id="mw5Q">V</i> <span class="texhtml " about="#mwt64" typeof="mw:Transclusion" data-mw='{"parts":[{"template":{"target":{"wt":"math","href":"./Template:Math"},"params":{"1":{"wt":"''Q'' = {{mset|''v'' ∈ ''V'' | δ[''v''] {{=}} 0}}"}},"i":0}}]}' id="mw5g"><i>Q</i> = {<i>v</i> ∈ <i>V</i> | δ[<i>v</i>] = 0}</span> // All vertices with indegree 0 nrOfVerticesProcessed = 0 <b id="mw5w">do</b> <b id="mw6A">global</b> build prefix sum over size of <i id="mw6Q">Q</i> // get offsets and total number of vertices in this step offset = nrOfVerticesProcessed + sum(Q<sub id="mw6g">i</sub>, i = 0 to j - 1) // <i id="mw6w">j</i> is the processor index <b id="mw7A">foreach</b> u <b id="mw7Q">in</b> Q localOrder[u] = index++; <b id="mw7g">foreach</b> (u,v) in E <b id="mw7w">do</b> post message (<i id="mw8A">u, v</i>) to PE owning vertex <i id="mw8Q">v</i> nrOfVerticesProcessed += sum(|Q<sub id="mw8g">i</sub>|, i = 0 to p - 1) deliver all messages to neighbors of vertices in Q receive messages for local vertices V remove all vertices in Q <b id="mw8w">foreach</b> message (<i id="mw9A">u, v</i>) received: <b id="mw9Q">if</b> --δ[v] = 0 add <i id="mw9g">v</i> to <i id="mw9w">Q</i> <b id="mw-A">while</b> global size of <i id="mw-Q">Q</i> > 0 <b id="mw-g">return</b> localOrder

The communication cost depends heavily on the given graph partition. As for runtime, on a CRCW-PRAM model that allows fetch-and-decrement in constant time, this algorithm runs in $\mathcal{O}\left(\frac{m+n}{p}+D(\mathrm{\Delta }+\log n)\right)$, where D is again the longest path in G and Δ the maximum degree.

Application to shortest path finding

The topological ordering can also be used to quickly compute shortest paths through a weighted directed acyclic graph. Let V be the list of vertices in such a graph, in topological order. Then the following algorithm computes the shortest path from some source vertex s to all other vertices:

• Let d be an array of the same length as V; this will hold the shortest-path distances from s. Set d[s] = 0, all other d[u] = ∞.
• Let p be an array of the same length as V, with all elements initialized to nil. Each p[u] will hold the predecessor of u in the shortest path from s to u.
• Loop over the vertices u as ordered in V, starting from s:
  • For each vertex v directly following u (i.e., there exists an edge from u to v):
    • Let w be the weight of the edge from u to v.
    • Relax the edge: if d[v] > d[u] + w, set
      • d[v] ← d[u] + w,
      • p[v] ← u.

Equivalently:

• Let d be an array of the same length as V; this will hold the shortest-path distances from s. Set d[s] = 0, all other d[u] = ∞.
• Let p be an array of the same length as V, with all elements initialized to nil. Each p[u] will hold the predecessor of u in the shortest path from s to u.
• Loop over the vertices u as ordered in V, starting from s:
  • For each vertex v into u (i.e., there exists an edge from v to u):
    • Let w be the weight of the edge from v to u.
    • Relax the edge: if d[u] > d[v] + w, set
      • d[u] ← d[v] + w,
      • p[u] ← v.

On a graph of n vertices and m edges, this algorithm takes Θ(n + m), i.e., linear, time.

Uniqueness

If a topological sort has the property that all pairs of consecutive vertices in the sorted order are connected by edges, then these edges form a directed Hamiltonian path in the DAG. If a Hamiltonian path exists, the topological sort order is unique; no other order respects the edges of the path. Conversely, if a topological sort does not form a Hamiltonian path, the DAG will have two or more valid topological orderings, for in this case it is always possible to form a second valid ordering by swapping two consecutive vertices that are not connected by an edge to each other. Therefore, it is possible to test in linear time whether a unique ordering exists, and whether a Hamiltonian path exists, despite the NP-hardness of the Hamiltonian path problem for more general directed graphs (i.e., cyclic directed graphs).

Relation to partial orders

Topological orderings are also closely related to the concept of a linear extension of a partial order in mathematics. A partially ordered set is just a set of objects together with a definition of the "≤" inequality relation, satisfying the axioms of reflexivity (x ≤ x), antisymmetry (if x ≤ y and y ≤ x then x = y) and transitivity (if x ≤ y and y ≤ z, then x ≤ z). A total order is a partial order in which, for every two objects x and y in the set, either x ≤ y or y ≤ x. Total orders are familiar in computer science as the comparison operators needed to perform comparison sorting algorithms. For finite sets, total orders may be identified with linear sequences of objects, where the "≤" relation is true whenever the first object precedes the second object in the order; a comparison sorting algorithm may be used to convert a total order into a sequence in this way. A linear extension of a partial order is a total order that is compatible with it, in the sense that, if x ≤ y in the partial order, then x ≤ y in the total order as well.

One can define a partial ordering from any DAG by letting the set of objects be the vertices of the DAG, and defining x ≤ y to be true, for any two vertices x and y, whenever there exists a directed path from x to y; that is, whenever y is reachable from x. With these definitions, a topological ordering of the DAG is the same thing as a linear extension of this partial order. Conversely, any partial ordering may be defined as the reachability relation in a DAG. One way of doing this is to define a DAG that has a vertex for every object in the partially ordered set, and an edge xy for every pair of objects for which x ≤ y. An alternative way of doing this is to use the transitive reduction of the partial ordering; in general, this produces DAGs with fewer edges, but the reachability relation in these DAGs is still the same partial order. By using these constructions, one can use topological ordering algorithms to find linear extensions of partial orders.

Relation to scheduling optimisation

By definition, the solution of a scheduling problem that includes a precedence graph is a valid solution to topological sort (irrespective of the number of machines), however, topological sort in itself is not enough to optimally solve a scheduling optimisation problem. Hu's algorithm is a popular method used to solve scheduling problems that require a precedence graph and involve processing times (where the goal is to minimise the largest completion time amongst all the jobs). Like topological sort, Hu's algorithm is not unique and can be solved using DFS (by finding the largest path length and then assigning the jobs).