1. Data Structures Using C++ 2E1 Topic of this lecture: Topological Order OU CS prerequisite bubble chart http://www.ohio.edu/engineering/eecs/undergradtools/ upload/CS-BUBBLE-CHART-October-2012.pdf

2. Data Structures Using C++ 2E

4. topologicalOrder[n-1]

5. Data Structures Using C++ 2E5 Topological Order Algorithm topological order –Outputs directed graph vertices in topological order –Assume graph has no cycles There exists a vertex v in G such that v has no successor There exists a vertex u in G such that u has no predecessor

6. Data Structures Using C++ 2E6 Topological Order (cont’d.) Topological sort algorithm –Implemented with the depth first traversal or the breadth first traversal Depth-first (stack) vs. Breadth-first (queue) Depth-first: CS 2400  2401  3610  4040  …  EE1024… Breadth-first: CS 2400  EE1024  Chem 1510  Math 2301  CS 2401 …

7. Data Structures Using C++ 2E7 Breadth First Topological Ordering General algorithm: 1.Use an array predCount to record the number of predecessors (prerequisites) for each node (course) 2.For a node (course) without any predecessor (prerequisite), it can be output (taken) 3.Use a queue for breadth-first traversal (maintain the prerequisite-cleared courses).

8. 8 How to initialize preCount?

9. 9 How to do breadth-first traversal?

10. Data Structures Using C++ 2E10 How to do Breadth First Topological Ordering? Similar to breadth-first traversal: using a queue, but only put prerequisite-cleared items ( predCount == 0) into the queue. When an item is removed from the queue, update the predCount values for the successors predCount[CS3610] = 2 After taking CS 2400, it should be reduced by 1. After taking CS 3000, it will be become 0.

11. Data Structures Using C++ 2E Breadth First Topological Ordering

12. Data Structures Using C++ 2E12 Breadth First Topological Ordering (cont’d.) Breadth First Topological order –0 9 1 7 2 5 4 6 3 8 10 FIGURE 12-18 Arrays predCount, topologicalOrder, and queue after Steps 1 and 2 execute

13. Data Structures Using C++ 2E13 Breadth First Topological Ordering (cont’d.) Breadth First Topological order –0 9 1 7 2 5 4 6 3 8 10 FIGURE 12-19 Arrays predCount, topologicalOrder, and queue after the first iteration of Step 3

14. Data Structures Using C++ 2E14 FIGURE 12-20 Arrays predCount, topologicalOrder, and queue after the second iteration of Step 3

15. Data Structures Using C++ 2E15 FIGURE 12-21 Arrays predCount, topologicalOrder, and queue after the third iteration of Step 3

16. Data Structures Using C++ 2E16 Breadth First Topological Ordering (cont’d.) See code on pages 718-719 –Function implementing breadth first topological ordering algorithm FIGURE 12-22 Arrays predCount, topologicalOrder, and queue after Step 3 executes

17. Data Structures Using C++ 2E17 How to do Depth First Topological Ordering? Project 5 question. Use depth-first traversal with recursive calls. Find a course without successor (e.g, CS 4561) through depth-first traversal and assign it into topologicalOrder[n- 1]. When you trace back, assign topologicalOrder[n-2], [n-3]…..

18. Data Structures Using C++ 2E18 How to do Depth First Topological Ordering? A course should NOT be output (taken) until the subtrees have been visited. Decrement topIndex each time when you output a course. Assign topologicalOrder[topIndex--] at the end of recursive calls. Pre-requisites are guaranteed to be taken first.

19. Data Structures Using C++ 2E19 Depth First Topological Ordering If start at 0, dft to 10, then 10 will be put as the last element in toplogicalOrder[]; 8 will be the second last. Output: