Data Structures Using C++ 2E 1

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

Data Structures Using C++ 2E

Data Structures Using C++ 2E

Data Structures Using C++ 2E

topologicalOrder[n-1]

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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

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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 240024013610 4040 …

EE1024…

• Breadth-first: CS 2400EE1024 Chem 1510 Math 2301 CS 2401 …

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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).

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How to initialize preCount?

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How to do breadth-first traversal?

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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.

Data Structures Using C++ 2E

Breadth First Topological Ordering

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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

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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

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FIGURE 12-20 Arrays predCount, topologicalOrder, and queue after the second iteration of Step 3

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FIGURE 12-21 Arrays predCount, topologicalOrder, and queue after the third iteration of Step 3

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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

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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]… ..

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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.

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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: