TOPOLOGICAL SORT

Definition

A topological sort of a Directed Acylic Graph (DAG) is a linear ordering of all its vertices such that if there is a path from vi to vk then vi appears before vk in the ordering.

Algorithm for finding a topological sort

Initialize TSArray to null

for (i=1; i<=numofvertices; i++)

v = find vertex of degree zero

if v is null

Graph has a cycle. Exit

else

add v to TSArray

remove v and all its outgoing vertices

Print TSArray

Algorithm for enumerating all topological sorts

TopologicalSorts()

compute the indegree count array, Indegree[ ]

initialize TSArray[ ] to null

for each vertex v

if indegree[v] == zero

push Indegree[ ], TSArray[ ], v

while (stack not empty)

pop Indegree[ ], TSArray[ ], v

add v to TSArray

for each x adjacent to v, Indegree[x]--

for each vertex v

if indegree[v] == zero

push Indegree[ ], TSArray[ ], v

Find out the ordering

1 2

4

6 7

53

Solution

•1 2 5 4 3 7 6

•1 2 5 4 7 3 6

•Find out other possibilities.

                                                                                             

How many squares can you create in this figure by connecting any 4 dots (the corners of a square must lie upon a grid dot?

TRIANGLES: 

How many triangles are located in the image below?

There are 11 squares total; 5 small, 4 medium, and 2 large.

27 triangles.  There are 16 one-cell triangles, 7 four-cell triangles, 3 nine-cell triangles, and 1 sixteen-cell triangle.