Topological Sorting

In a construction project, some jobs must be done before others can begin.

We would like to list them so that each job precedes those that must be done after it. In a program library, a program a may call program h. Program h in turn may call programs d and e, and so on. We would like to order the pro-grams so that a program appears before all the propro-grams it calls. (The Unix program lorder does this.) These problems and others like them are instances of the problem of topological sorting: finding an ordering that satisfies a set of constraints of the form "x must come before y." In a topological sort any linear ordering that satisfies the partial order represented by the constraints is sufficient.

The constraints can be represented by a graph in which the nodes are labeled by the names, and there is an edge from node x to node y if x must come before y. The following graph is an example:

If a graph contains an edge from x toy, then x is called a predecessor of y, and y is a successor of x. Suppose the constraints come in the form of predecessor-successor pairs where each input line contains x and y representing an edge from node x to node y, as in this description of the graph above:

a h

b g

c f

c h

d i

e d

f b

f g

h d

h e

i b

If there is an edge from x toy, then x must appear before y in the output.

Given the input above, one possible output is the list a c f h e d i b g

There are many other linear orders that contain the partial order depicted in the graph; another is

c a h e d i f b g

The problem of topological sorting is that of ordering the nodes of a graph so

that all predecessors appear before their successors. Such an ordering is possi-ble if and only if the graph does not contain a cycle, which is a sequence of edges that leads from a node back to itself. If the input graph contains a cycle, then we must say so and indicate that no linear ordering exists.

Breadth-First Topological Sort

There are many algorithms that can be used to sort a graph topologically.

Perhaps the simplest is one that at each iteration removes from the graph a node with no predecessors. If all nodes can be removed from the graph this way, the sequence in which the nodes are removed is a topological sort of the graph. In the graph above, we could begin by removing node a and the edge that comes from it. Then we could remove node c, then nodes f and h in either order, and so on.

Our implementation uses a first-in, first-out data structure called a queue to sequence the processing of nodes with no predecessors in a "breadth-first"

manner. After all the data has been read in, a loop counts the nodes and places all nodes with no predecessors on the queue. A second loop removes the node at the front of the queue, prints its name, and decrements the predecessor count of each of its successors. If the predecessor count of any of its successors becomes zero, those successors are put on the back of the queue. When the front catches up to the back and all nodes have been considered, the job is done. But, if some nodes are never put on the queue, those nodes are involved in cycles and no topological sort is possible. When no cycles are present, the sequence of nodes printed is a topological sort.

The first three statements of tsort read the predecessor-successor pairs from the input and construct a successor-list data structure like this:

node pent sent slist

a 0 I h

b 2 I q

e 0 2 f, h

d 2 I i

e I I d

f 1 2 b, q

q 2 0

h 2 2 d, e

i I 1 b

The arrays pent and sent keep track of the number of predecessors and successors for each node; slist[x ,i] gives the node that is the i-th successor of node x. The first line creates an element of pent if it is not already present.

172 EXPERIMENTS WITH ALGORITHMS

# tsort - topological sort of a graph

# input: predecessor-successor pairs

# output: linear order, predecessors first

Consider the following graph. If it starts at node 1, a depth-first search will visit nodes 1, 2, 3, and 4. At that point, if it starts with another unvisited node such as 5, it will then visit nodes 5, 6, and 7. If it starts at a different place, however, a different sequence of visits will be made.

Depth-first search is useful for finding cycles. An edge like (3, 1) that goes from a node to a previously visited ancestor is called a back edge. Since a back edge identifies a cycle, to find cycles all we need to do is find back edges. The following function will test whether a graph, stored as a successor-list data structure like that in tsort, contains a cycle reachable from node:

# dfs - depth-first search for cycles function dfs(node, i, s) {

visited[node]

=

for (i = 1; i <= scnt[node]; i++)

if (visited[s

=

else if (visited[s]

==

print "cycle with back edge (" node " " s ")"

visited[node]

=

This function uses an array visited to determine whether a node has been traversed. Initially, visi ted[x] is 0 for all nodes. Entering a node x for the first time, dfs sets visited[x] to 1, and leaving x for the last time it sets visited[x] to 2. During the traversal, dfs uses visited to determine whether a node y is an ancestor of the current node (and hence previously visited), in which case visited[y] is 1, or whether y has been previously visited, in which case visited[y] is 2.

Depth-First Topological Sort

The function dfs can easily be turned into a node-sorting procedure. If it prints the name of each node once the search from that node is completed, it will generate a list of nodes that is a reverse topological sort, provided again there are no cycles in the graph. The program rtsort prints the reverse of a topological sort of a graph, given a sequence of predecessor-successor pairs as input. It applies depth-first search to every node with no predecessors. The

174 EXPERlMENTS WITH ALGORITHMS CHAPTER 7

data structure is the same as that in tsort.

# rtsort - reverse topological sort

# input: predecessor-successor pairs

# output: linear order, successors first

# put $1 in pent if ( l($1 in pent))

pcnt[$1] = 0 pcnt[$2]++

slist[$1, ++scnt[$1]] :::

# count predecessors of $2

$2 # add $2 to successors of $1 END for (node in pent)

nodecnt++

if (pcnt[node] 0) rtsort(node)

}

if (pncnt l= nodecnt)

print "error: input contains a cycle"

print£ ( "\n")

function rtsort(node, i, s) { visited[node] ::: 1

for (i ::: 1; i <= scnt[node]; i++)

if (visited[s ::: slist[node, ill 0) rtsort(s)

else if (visited[s] :::::: 1)

printf("error: nodes "s and "s are in a cycle\n", s, node)

visited[node] ::: 2 print£ ( " "s" , node)

pncnt++ # count nodes printed

Applied to the predecessor-successor pairs at the beginning of this section, rtsort would print

g b i d e h a f c

Notice that this algorithm detects some cycles explicitly by finding a back edge, while it detects other cycles only implicitly, by failing to print all the nodes, as in this graph:

Exercise 7-10. Modify rtsort to print its output in the usual order, predecessors first.

Can you achieve the same effect without modifying rtsort'? D