Topological Sort. Version of October 11, Version of October 11, 2014 Topological Sort 1 / 14

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1 Topological Sort Version of October 11, 2014 Version of October 11, 2014 Topological Sort 1 / 14

2 Directed Graph In a directed graph, we distinguish between edge (u, v) and edge (v, u) Version of October 11, 2014 Topological Sort 2 / 14

3 Directed Graph In a directed graph, we distinguish between edge (u, v) and edge (v, u) out-degree of a vertex is the number of edges leaving it Version of October 11, 2014 Topological Sort 2 / 14

4 Directed Graph In a directed graph, we distinguish between edge (u, v) and edge (v, u) out-degree of a vertex is the number of edges leaving it in-degree of a vertex is the number of edges entering it Version of October 11, 2014 Topological Sort 2 / 14

5 Directed Graph In a directed graph, we distinguish between edge (u, v) and edge (v, u) out-degree of a vertex is the number of edges leaving it in-degree of a vertex is the number of edges entering it Each edge (u, v) contributes one to the out-degree of u and one to the in-degree of v Version of October 11, 2014 Topological Sort 2 / 14

6 Directed Graph In a directed graph, we distinguish between edge (u, v) and edge (v, u) out-degree of a vertex is the number of edges leaving it in-degree of a vertex is the number of edges entering it Each edge (u, v) contributes one to the out-degree of u and one to the in-degree of v out-degree(v) = in-degree(v) = E v V v V Version of October 11, 2014 Topological Sort 2 / 14

7 Usage of Directed Graph Directed graphs are often used to represent order-dependent tasks Version of October 11, 2014 Topological Sort 3 / 14

8 Usage of Directed Graph Directed graphs are often used to represent order-dependent tasks That is, we cannot start a task before another task finishes Version of October 11, 2014 Topological Sort 3 / 14

9 Usage of Directed Graph Directed graphs are often used to represent order-dependent tasks That is, we cannot start a task before another task finishes Edge (u, v) denotes that task v cannot start until task u is finished u v Version of October 11, 2014 Topological Sort 3 / 14

10 Usage of Directed Graph Directed graphs are often used to represent order-dependent tasks That is, we cannot start a task before another task finishes Edge (u, v) denotes that task v cannot start until task u is finished u v Clearly, for the system not to hang, the graph must be acyclic Version of October 11, 2014 Topological Sort 3 / 14

11 Usage of Directed Graph Directed graphs are often used to represent order-dependent tasks That is, we cannot start a task before another task finishes Edge (u, v) denotes that task v cannot start until task u is finished u v Clearly, for the system not to hang, the graph must be acyclic It must be a directed acyclic graph (or DAG) Version of October 11, 2014 Topological Sort 3 / 14

12 Course dependence chart 09/10 Red: COMP/CSIE Core Green: COMP/CSIE Required Purple: CSIE (NW) Required Blue: CSIE (MC) Required (sum,s) 211 (F,S) 21 (F) M144 (F,S)/E210 (F) (F) 12 (F,S) 21 (F,S) (S) 221 M021 (F,S) M113 (F,S) 22 (S) 10 (F) M100/M10 (F,S) E101/E102 (F,S) 10 (S) 361 (F,S) 22 (F) , (s), 343 (F,S), 344 (F), 364 (F) 332, (F) E211 (Sum,F) E214 (S) F and S means offered in Fall and Spring respectively. Course offering schedule shown here is for reference only; the actual offering schedule may vary slightly from year to year. E314 (F) Version of October 11, 2014 Topological Sort 4 / 14

13 Topological Sort A Topological ordering of a graph is a linear ordering of the vertices of a DAG such that if (u, v) is in the graph, u appears before v in the linear ordering Version of October 11, 2014 Topological Sort / 14

14 Topological Sort A Topological ordering of a graph is a linear ordering of the vertices of a DAG such that if (u, v) is in the graph, u appears before v in the linear ordering e.g., order in which classes can be taken Version of October 11, 2014 Topological Sort / 14

15 Topological Sort A Topological ordering of a graph is a linear ordering of the vertices of a DAG such that if (u, v) is in the graph, u appears before v in the linear ordering e.g., order in which classes can be taken Version of October 11, 2014 Topological Sort / 14

16 Topological Sort A Topological ordering of a graph is a linear ordering of the vertices of a DAG such that if (u, v) is in the graph, u appears before v in the linear ordering e.g., order in which classes can be taken Topological ordering may not be unique as there are many equal elements! Version of October 11, 2014 Topological Sort / 14

17 Topological Sort A Topological ordering of a graph is a linear ordering of the vertices of a DAG such that if (u, v) is in the graph, u appears before v in the linear ordering e.g., order in which classes can be taken Topological ordering may not be unique as there are many equal elements! E.G., there are several topological orderings 0, 6, 1, 4, 3, 2,,,, 9 0, 4, 1, 6, 2,, 3,,, 9... Version of October 11, 2014 Topological Sort / 14

18 Topological Sort Algorithm Observations A DAG must contain at least one vertex with in-degree zero (why?) Version of October 11, 2014 Topological Sort 6 / 14

19 Topological Sort Algorithm Observations A DAG must contain at least one vertex with in-degree zero (why?) Algorithm: Topological Sort 1 Output a vertex u with in-degree zero in current graph. 2 Remove u and all edges (u, v) from current graph. 3 If graph is not empty, goto step 1. Version of October 11, 2014 Topological Sort 6 / 14

20 Topological Sort Algorithm Observations A DAG must contain at least one vertex with in-degree zero (why?) Algorithm: Topological Sort 1 Output a vertex u with in-degree zero in current graph. 2 Remove u and all edges (u, v) from current graph. 3 If graph is not empty, goto step 1. Correctness Version of October 11, 2014 Topological Sort 6 / 14

21 Topological Sort Algorithm Observations A DAG must contain at least one vertex with in-degree zero (why?) Algorithm: Topological Sort 1 Output a vertex u with in-degree zero in current graph. 2 Remove u and all edges (u, v) from current graph. 3 If graph is not empty, goto step 1. Correctness At every stage, current graph is a DAG (why?) Because current graph is always a DAG, algorithm can always output some vertex. So algorithm outputs all vertices. Suppose order output was not a topological order. Then there is some edge (u, v) such that v appears before u in the order. This is impossible, though, because v can not be output until edge (u, v) is removed! Version of October 11, 2014 Topological Sort 6 / 14

22 Topological Sort Algorithm Topological sort(g) Initialize Q to be an empty queue; foreach u in V do if in-degree(u) = 0 then // Find all starting vertices Enqueue(Q, u); end end while Q is not empty do u = Dequeue(Q); Output u; foreach v in Adj(u) do // remove u s outgoing edges in-degree(v) = in-degree(v) 1; if in-degree(v) = 0 then Enqueue(Q, v); end end end Version of October 11, 2014 Topological Sort / 14

23 Example Q = {} Q= {0} 4 Version of October 11, 2014 Topological Sort / 14

24 Example Q = {6, 1, 4} Output: Q= {1, 4, 3} Output: 0, 6 4 Version of October 11, 2014 Topological Sort 9 / 14

25 Example Q = {4, 3, 2} 4 Output: 0, 6, Q = {3, 2} Output: 0, 6, 1, 4 Version of October 11, 2014 Topological Sort 10 / 14

26 Example 2 9 Q = {2} Output: 0, 6, 1, 4, 3 9 Q = {, } Output: 0, 6, 1, 4, 3, 2 Version of October 11, 2014 Topological Sort 11 / 14

27 Example 9 Q = {, } Output: 0, 6, 1, 4, 3, 2, 9 Q = {} Output: 0, 6, 1, 4, 3, 2,, Version of October 11, 2014 Topological Sort 12 / 14

28 Example 9 Q = {9} Output: 0, 6, 1, 4, 3, 2,,, Q = {} Output: 0, 6, 1, 4, 3, 2,,,, 9 4 Done! Version of October 11, 2014 Topological Sort 13 / 14

29 Topological Sort: Complexity We never visit a vertex more than once Version of October 11, 2014 Topological Sort 14 / 14

30 Topological Sort: Complexity We never visit a vertex more than once For each vertex, we examine all outgoing edges v V out-degree(v) = E Version of October 11, 2014 Topological Sort 14 / 14

31 Topological Sort: Complexity We never visit a vertex more than once For each vertex, we examine all outgoing edges v V out-degree(v) = E Therefore, the running time is O(V + E) Version of October 11, 2014 Topological Sort 14 / 14

32 Topological Sort: Complexity We never visit a vertex more than once For each vertex, we examine all outgoing edges v V out-degree(v) = E Therefore, the running time is O(V + E) Question Can we use DFS to implement topological sort? Version of October 11, 2014 Topological Sort 14 / 14