Graph traversal Topological sort Breadth-first search Cinda Heeren / Andy Roth / Will Evans / March 27, 2020 1 Geoffrey Tien
Announcements • All CPSC 221 deadlines postponed to 23:59, Apr.09 • TA evaluations and instructor/course evaluations will be done online – Check your e-mail for links/instructions soon Cinda Heeren / Andy Roth / Will Evans / March 27, 2020 2 Geoffrey Tien
Total order 1 2 5 4 7 6 3 means A must go before B A B Cinda Heeren / Andy Roth / Will Evans / March 27, 2020 3 Geoffrey Tien
Partial order Getting dressed pants shirt socks coat undies belt watch pocket boots knife Cinda Heeren / Andy Roth / Will Evans / March 27, 2020 4 Geoffrey Tien
Topological sort • Given a graph, 𝐻 = 𝑊, 𝐹 , output all vertices in V such that no vertex is output before any other vertex with an edge to it • Classic application: course prerequisites MATH 100 MATH 101 CPSC 110 CPSC 121 MATH 200 MATH 221 CPSC 213 CPSC 221 CPSC 210 STAT 241 CPSC 310 CPSC 313 CPSC 320 Cinda Heeren / Andy Roth / Will Evans / March 27, 2020 5 Geoffrey Tien
Topological sort • Label each vertex's in-degree (# of inbound edges) • Initialize a queue to contain all vertices with in-degree zero • While there are vertices remaining in the queue – Pick a vertex 𝑤 from the queue and output it – Reduce the in-degree of all vertices adjacent to 𝑤 – Put any of these with updated zero in-degree on the queue – Remove 𝑤 from the queue Cinda Heeren / Andy Roth / Will Evans / March 27, 2020 6 Geoffrey Tien
Topological sort Queue: 1 6 Vertex In-degree 1 0 2 2 1 3 3 1 1 6 4 2 5 2 5 4 6 0 Cinda Heeren / Andy Roth / Will Evans / March 27, 2020 7 Geoffrey Tien
Topological sort Queue: 1 6 2 Vertex In-degree 1 0 2 2 0 1 3 3 1 1 6 4 1 2 5 2 5 4 6 0 1 6 Cinda Heeren / Andy Roth / Will Evans / March 27, 2020 8 Geoffrey Tien
Topological sort Queue: 2 3 Vertex In-degree 1 0 2 2 0 3 0 3 1 1 6 4 1 5 2 5 4 6 0 1 6 2 Cinda Heeren / Andy Roth / Will Evans / March 27, 2020 9 Geoffrey Tien
Topological sort Queue: 3 4 Vertex In-degree 1 0 2 2 0 3 3 0 1 6 0 4 1 1 5 2 5 4 6 0 1 6 2 3 Cinda Heeren / Andy Roth / Will Evans / March 27, 2020 10 Geoffrey Tien
Topological sort Queue: 4 5 Vertex In-degree 1 0 2 2 0 3 3 0 1 6 4 0 0 5 1 5 4 6 0 1 6 2 3 4 5 Cinda Heeren / Andy Roth / Will Evans / March 27, 2020 11 Geoffrey Tien
How are algorithms affected? Adjacency list vs. adjacency matrix • Label each vertex's in-degree (# of inbound edges) 𝑃 𝑛 + 𝑜 vs 𝑃 𝑜 2 • Initialize a queue to contain all vertices with in-degree zero 𝑃 𝑜 • While there are vertices remaining in the queue 𝑜 times... – Pick a vertex 𝑤 from the queue and output it – Reduce the in-degree of all vertices adjacent to 𝑤 𝑃 𝑒𝑓 𝑤 vs 𝑃 𝑜 – Put any of these with updated zero in-degree on the queue – Remove 𝑤 from the queue For the adjacency list, how 2 many vertices/edges have 3 been processed in total? 1 6 4 5 Cinda Heeren / Andy Roth / Will Evans / March 27, 2020 12 Geoffrey Tien
Breadth-first search • Visits all vertices within 𝑒 "hops" away from starting vertex, before visiting vertices within 𝑒 + 1 hops, where 𝑒 begins at 0 start 2 hops 1 hop 4 hops 3 hops This looks very similar to a level-order traversal in a tree! How to control it? Use a queue! Cinda Heeren / Andy Roth / Will Evans / March 27, 2020 13 Geoffrey Tien
Breadth-first search • Need to be able to identify the neighbours of a given node, and to check if a vertex has been added to the queue already – Adjacency matrix / list, and a Boolean array of size 𝑊 5 0 1, 3 1 0, 2, 4, 6, 7 0 4 2 1, 3, 8, 9 3 0, 2 3 1 7 4 1, 5, 6, 7 5 4 2 6 6 1, 4, 7 7 1, 4, 6 Identified: 9 8 0 1 2 3 4 5 6 7 8 9 8 2 9 2 Cinda Heeren / Andy Roth / Will Evans / March 27, 2020 14 Geoffrey Tien
Breadth-first search 5 0 1, 3 1 0, 2, 4, 6, 7 0 4 2 1, 3, 8, 9 3 0, 2 3 1 7 4 1, 5, 6, 7 5 4 2 6 6 1, 4, 7 7 1, 4, 6 T T T T T T T T T T Identified: 9 8 0 1 2 3 4 5 6 7 8 9 8 2 9 2 0 1 3 2 4 6 7 8 9 5 Queue: 0 1 3 2 4 6 7 8 9 5 Visited: 1 hop 2 hops 3 hops Cinda Heeren / Andy Roth / Will Evans / March 27, 2020 15 Geoffrey Tien
Breadth-first search Algorithm Algorithm BFS 𝐻, 𝑤 { mark 𝑤 as identified enqueue( 𝑤 ) while (queue is not empty) { 𝑣 = dequeue() for all vertices 𝑥 adjacent to 𝑣 , do { if ( 𝑥 has not been identified or visited) { mark 𝑥 as identified enqueue( 𝑥 ) } } output 𝑣 to visited collection } } A for loop nested inside a while loop. Running time? Cinda Heeren / Andy Roth / Will Evans / March 27, 2020 16 Geoffrey Tien
Readings for this lesson • Carrano & Henry – Chapter 20.3 (Graph traversals) • Next class – Carrano & Henry, Chapter 20.4.2 (Spanning trees) – Carrano & Henry, Chapter 20.4.3 (Minimum spanning trees) Cinda Heeren / Andy Roth / Will Evans / March 27, 2020 17 Geoffrey Tien
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