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Topological Sorting

In graph theory, a topological sort or topological ordering of a directed acyclic graph (DAG) is a linear ordering of its nodes in which each node comes before all nodes to which it has outbound edges. Every DAG has one or more topological sorts.

More formally, define the partial order relation R over the nodes of the DAG such that xRy if and only if there is a directed path from x to y. Then, a topological sort is a linear extension of this partial order, that is, a total order compatible with the partial order.
 
Examples
The canonical application of topological sorting (topological order) is in scheduling a sequence of jobs or tasks; topological sorting algorithms were first studied in the early 1960s in the context of the PERT technique for scheduling in project management. The jobs are represented by vertices, and there is an edge from x to y if job x must be completed before job y can be started (for example, when washing clothes, the washing machine must finish before we put the clothes to dry). Then, a topological sort gives an order in which to perform the jobs.
In computer science, applications of this type arise in instruction scheduling, ordering of formula cell evaluation when recomputing formula values in spreadsheets , logic synthesis, determining the order of compilation tasks to perform in makefiles, and resolving symbol dependencies in linkers.
The graph shown to the left has many valid topological sorts, including:
  • 7, 5, 3, 11, 8, 2, 9, 10 (visual left-to-right)
  • 3, 5, 7, 8, 11, 2, 9, 10 (smallest-numbered available vertex first)
  • 3, 7, 8, 5, 11, 10, 2, 9
  • 5, 7, 3, 8, 11, 10, 9, 2 (least number of edges first)
  • 7, 5, 11, 3, 10, 8, 9, 2 (largest-numbered available vertex first)
  • 7, 5, 11, 2, 3, 8, 9, 10 (visual top-to-bottom)

 

Algorithms
The usual algorithms for topological sorting have running time linear in the number of nodes plus the number of edges (O(|V|+|E|)).
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" which have no incoming edges and insert them into a set S; at least one such node must exist if graph is acyclic. Then:

L ← Empty list that will contain the sorted elements
S ← Set of all nodes with no incoming edges
 
while S is non-empty
    do remove a node n from S insert n into L   
        for each node m with an edge e from n to m 
            do remove edge e from the graph        
            if m has no other incoming edges 
                then insert m into S

if graph has edges then output error message (graph has at least one cycle)
else output message (proposed topologically sorted order: L)

If the graph was a DAG, a solution is contained in the list L (the solution is not unique). Otherwise, the graph has at least one cycle and therefore a topological sorting is impossible.

Note that, 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.

An alternative algorithm for topological sorting is based on depth-first-search . For this algorithm, edges point in the opposite direction as the previous algorithm (and the opposite direction to that shown in the diagram in the Examples section above). There is an edge from x to y if job x depends on job y (in other words, if job y must be completed before job x can be started). 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:

L ← Empty list that will contain the sorted nodes
S ← Set of all nodes
 
function visit(node n)  
if n has not been visited yet
    then mark n as visited
for each node m with an edge from n to m
    do visit(m)
    add n to L
for each node n in S
    do visit(n)
 
Note that each node n gets added to the output list L only after considering all other nodes on which n depends (all descendant nodes of n in the graph). Specifically, when the algorithm adds node n, we are guaranteed that all nodes on which n depends are already in the output list L: they were added to L either by the preceding recursive call to visit(), or by an earlier call to visit(). Since each edge and node is visited once, the algorithm runs in linear time. Note that the simple pseudocode above cannot detect the error case where the input graph contains cycles. The algorithm can be refined to detect cycles by watching for nodes which are visited more than once during any nested sequence of recursive calls to visit() (e.g., by passing a list down as an extra argument to visit(), indicating which nodes have already been visited in the current call stack). This depth-first-search-based algorithm is the one described by Cormen; it seems to have been first described in print by Tarjan.
 
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 Hamilton 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 polynomial 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.
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