[PPT] - EE 355 Unit 18 DFS and Topological Sort Mark Redekopp 2 PowerPoint Presentation

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EE 355 Unit 18

DFS and Topological Sort Mark Redekopp

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Topological Sort

Given a graph of dependencies

(tasks, prerequisities, etc.) topological sort creates a consistent ordering of tasks (vertices) where no dependencies are violated

Many possible valid topological
rderings exist

– EE 109L, EE 209, EE 354L, EE 454L, EE 457, MATH 245, PHYS 152, EE 202L,… – MATH 245, EE 109L, PHYS 152L, EE 154,… EE 109L EE 209 EE 354L MA 245 EE 202L EE 457 EE 454L EE 338 EE 348 PHYS 152 EE 447 EE 448

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Topological Sort

Another example

– Getting dressed

More Examples:

– Project management scheduling – Build order in a Makefile or

ther compile project

– Cooking using a recipe – Instruction execution on an

ut-of-order pipelined CPU

– Production of output values in a simulation of a combinational gate network

Underwear Pants Belt Undershirt Shoes Tie Shirt Socks

http://www.personal.kent.edu/~rmuhamma/Algorithms/MyAlgorith ms/GraphAlgor/topoSort.htm

Jacket

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Topological Sort

Does breadth-first search

work?

–

No. What if we started at EE200…

– We'd go to EE 202L before PHYS 152

All parent nodes need to be

completed before any child node

BFS only guarantees some

parent has completed before child

Turns out a Depth-First Search

will be part of our solution

EE 109L EE 209 EE 354L MA 245 EE 202L EE 457 EE 454L EE 338 EE 348 PHYS 152 EE 447 EE 448

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Depth First Search

Explores ALL children

before completing a parent

– Note BFS completes a parent before ANY children

For DFS let us assign:

– A start time when the node is first found – A finish time when a node is completed

If we look at our nodes in

reverse order of finish time (i.e. last one to finish back to first

ne to finish) we arrive at a…

– Topological ordering!!!

EE 109L EE 209 EE 354L MA 245 EE 202L EE 457 EE 454L EE 338 EE 348 PHYS 152 EE 447 EE 448

10 1 2 3 4 6 5 7 8 9 11 12 13 15 16 18 14 17 19 20 21 22 24 23 1 2 Start Time Finish Time PHYS 152, MATH 245, EE 202L, EE 348, EE 448, EE 447, EE 338, EE 109L, EE 209, EE 354L, EE 454L, EE 457 Reverse Finish Time Order

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DFS & Topological Sort

Imagine we flipped our

dependencies (edge direction) and performed a DFS starting at nodes with no incoming edges

Now as nodes

“finish/complete” we could enter them in a list and that list would provide a valid topological ordering

EE 109L EE 209 EE 354L MA 245 EE 202L EE 457 EE 454L EE 338 EE 348 PHYS 152 EE 447 EE 448

1 4 5 3 2 12 10 11 9 6 7 8 1st DFS 2nd DFS 3rd DFS 4th DFS 5th DFS

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DFS Algorithm

Visit a node

– Mark as visited – For each visited neighbor, visit it and perform DFS on all of their children – Only then, mark as finished

DFS is recursive!!
If cycles in the graph, ensure we

don’t get caught visiting neighbors endlessly

– Color them as we go – White = unvisited, – Gray = visited but not finished – Black = finished

DFS (G) 1 for each vertex u 2 u.color = WHITE 3 u.pred = nil 4 time = 0 5 finish_list = empty_list 6 for each vertex u do 7 if u.color == WHITE then 8 DFS-Visit (G, u, finish_list) DFS-Visit (G, u) 1 u.color = GRAY 2 u.start = ++time 3 for each vertex v in Adj(u) do 5 if v.color = WHITE then 6 v.pred = u 7 DFS-Visit (G, v) 8 u.color = BLACK 9 finish_list.append(u)

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Simplified DFS for Topo-sort

For topological sort, all we care

about is finish time

If in line 4 of toposort() we

simply iterate through goal/output nodes as our start locations then the ordering in the finish_list will be a correct toposort with input nodes first…

This is a recursive solution

– We can have an iterative solution shown later…

Toposort(G) 1 for each vertex u 2 u.color = WHITE 3 finish_list = empty_list 4 for each vertex u do 5 if u.color == WHITE then 6 DFS-Visit (G, u, finish_list) DFS-Visit (G, u) 1 u.color = GRAY 2 for each vertex v in Adj(u) do 3 if v.color = WHITE then 4 DFS-Visit (G, v) 5 u.color = BLACK 6 finish_list.append(u)

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Depth First-Search

Toposort(G) 1 for each vertex u 2 u.color = WHITE 3 finish_list = empty_list 4 for each vertex u do 5 if u.color == WHITE then 6 DFS-Visit (G, u, finish_list) DFS-Visit (G, u) 1 u.color = GRAY 2 for each vertex v in Adj(u) do 3 if v.color = WHITE then 4 DFS-Visit (G, v) 5 u.color = BLACK 6 finish_list.append(u) a b c d e f g h

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Depth First-Search

Toposort(G) 1 for each vertex u 2 u.color = WHITE 3 finish_list = empty_list 4 for each vertex u do 5 if u.color == WHITE then 6 DFS-Visit (G, u, finish_list) DFS-Visit (G, u) 1 u.color = GRAY 2 for each vertex v in Adj(u) do 3 if v.color = WHITE then 4 DFS-Visit (G, v) 5 u.color = BLACK 6 finish_list.append(u) a b c d e f g h

DFS-Visit(G,a): u v

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Depth First-Search

Toposort(G) 1 for each vertex u 2 u.color = WHITE 3 finish_list = empty_list 4 for each vertex u do 5 if u.color == WHITE then 6 DFS-Visit (G, u, finish_list) DFS-Visit (G, u) 1 u.color = GRAY 2 for each vertex v in Adj(u) do 3 if v.color = WHITE then 4 DFS-Visit (G, v) 5 u.color = BLACK 6 finish_list.append(u) a b c d e f g h

DFS-Visit(G,a): u v DFS-Visit(G,d):

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Depth First-Search

Toposort(G) 1 for each vertex u 2 u.color = WHITE 3 finish_list = empty_list 4 for each vertex u do 5 if u.color == WHITE then 6 DFS-Visit (G, u, finish_list) DFS-Visit (G, u) 1 u.color = GRAY 2 for each vertex v in Adj(u) do 3 if v.color = WHITE then 4 DFS-Visit (G, v) 5 u.color = BLACK 6 finish_list.append(u) a b c d e f g h

DFS-Visit(G,a): u v DFS-Visit(G,d): DFS-Visit(G,f):

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Depth First-Search

Toposort(G) 1 for each vertex u 2 u.color = WHITE 3 finish_list = empty_list 4 for each vertex u do 5 if u.color == WHITE then 6 DFS-Visit (G, u, finish_list) DFS-Visit (G, u) 1 u.color = GRAY 2 for each vertex v in Adj(u) do 3 if v.color = WHITE then 4 DFS-Visit (G, v) 5 u.color = BLACK 6 finish_list.append(u) a b c d e f g h

DFS-Visit(G,a): u DFS-Visit(G,d): DFS-Visit(G,f): DFS-Visit(G,h):

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Depth First-Search

Toposort(G) 1 for each vertex u 2 u.color = WHITE 3 finish_list = empty_list 4 for each vertex u do 5 if u.color == WHITE then 6 DFS-Visit (G, u, finish_list) DFS-Visit (G, u) 1 u.color = GRAY 2 for each vertex v in Adj(u) do 3 if v.color = WHITE then 4 DFS-Visit (G, v) 5 u.color = BLACK 6 finish_list.append(u) a b c d e f g h

DFS-Visit(G,a): u DFS-Visit(G,d): DFS-Visit(G,f): DFS-Visit(G,h): Finish_list: h

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Depth First-Search

Toposort(G) 1 for each vertex u 2 u.color = WHITE 3 finish_list = empty_list 4 for each vertex u do 5 if u.color == WHITE then 6 DFS-Visit (G, u, finish_list) DFS-Visit (G, u) 1 u.color = GRAY 2 for each vertex v in Adj(u) do 3 if v.color = WHITE then 4 DFS-Visit (G, v) 5 u.color = BLACK 6 finish_list.append(u) a b c d e f g h

DFS-Visit(G,a): u DFS-Visit(G,d): DFS-Visit(G,f): Finish_list: h v

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Depth First-Search

Toposort(G) 1 for each vertex u 2 u.color = WHITE 3 finish_list = empty_list 4 for each vertex u do 5 if u.color == WHITE then 6 DFS-Visit (G, u, finish_list) DFS-Visit (G, u) 1 u.color = GRAY 2 for each vertex v in Adj(u) do 3 if v.color = WHITE then 4 DFS-Visit (G, v) 5 u.color = BLACK 6 finish_list.append(u) a b c d e f g h

DFS-Visit(G,a): DFS-Visit(G,d): DFS-Visit(G,f): Finish_list: h u DFS-Visit(G,g):

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Depth First-Search

Toposort(G) 1 for each vertex u 2 u.color = WHITE 3 finish_list = empty_list 4 for each vertex u do 5 if u.color == WHITE then 6 DFS-Visit (G, u, finish_list) DFS-Visit (G, u) 1 u.color = GRAY 2 for each vertex v in Adj(u) do 3 if v.color = WHITE then 4 DFS-Visit (G, v) 5 u.color = BLACK 6 finish_list.append(u) a b c d e f g h

DFS-Visit(G,a): DFS-Visit(G,d): DFS-Visit(G,f): Finish_list: h, g u DFS-Visit(G,g):

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Depth First-Search

Toposort(G) 1 for each vertex u 2 u.color = WHITE 3 finish_list = empty_list 4 for each vertex u do 5 if u.color == WHITE then 6 DFS-Visit (G, u, finish_list) DFS-Visit (G, u) 1 u.color = GRAY 2 for each vertex v in Adj(u) do 3 if v.color = WHITE then 4 DFS-Visit (G, v) 5 u.color = BLACK 6 finish_list.append(u) a b c d e f g h

DFS-Visit(G,a): DFS-Visit(G,d): DFS-Visit(G,f): Finish_list: h, g, f u

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Depth First-Search

Toposort(G) 1 for each vertex u 2 u.color = WHITE 3 finish_list = empty_list 4 for each vertex u do 5 if u.color == WHITE then 6 DFS-Visit (G, u, finish_list) DFS-Visit (G, u) 1 u.color = GRAY 2 for each vertex v in Adj(u) do 3 if v.color = WHITE then 4 DFS-Visit (G, v) 5 u.color = BLACK 6 finish_list.append(u) a b c d e f g h

DFS-Visit(G,a): DFS-Visit(G,d): Finish_list: h, g, f, d u

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Depth First-Search

Toposort(G) 1 for each vertex u 2 u.color = WHITE 3 finish_list = empty_list 4 for each vertex u do 5 if u.color == WHITE then 6 DFS-Visit (G, u, finish_list) DFS-Visit (G, u) 1 u.color = GRAY 2 for each vertex v in Adj(u) do 3 if v.color = WHITE then 4 DFS-Visit (G, v) 5 u.color = BLACK 6 finish_list.append(u) a b c d e f g h

DFS-Visit(G,a): Finish_list: h, g, f, d u v

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Depth First-Search

Toposort(G) 1 for each vertex u 2 u.color = WHITE 3 finish_list = empty_list 4 for each vertex u do 5 if u.color == WHITE then 6 DFS-Visit (G, u, finish_list) DFS-Visit (G, u) 1 u.color = GRAY 2 for each vertex v in Adj(u) do 3 if v.color = WHITE then 4 DFS-Visit (G, v) 5 u.color = BLACK 6 finish_list.append(u) a b c d e f g h

DFS-Visit(G,a): Finish_list: h, g, f, d u v DFS-Visit(G,c):

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Depth First-Search

Toposort(G) 1 for each vertex u 2 u.color = WHITE 3 finish_list = empty_list 4 for each vertex u do 5 if u.color == WHITE then 6 DFS-Visit (G, u, finish_list) DFS-Visit (G, u) 1 u.color = GRAY 2 for each vertex v in Adj(u) do 3 if v.color = WHITE then 4 DFS-Visit (G, v) 5 u.color = BLACK 6 finish_list.append(u) a b c d e f g h

DFS-Visit(G,a): Finish_list: h, g, f, d u DFS-Visit(G,c): DFS-Visit(G,e):

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Depth First-Search

Toposort(G) 1 for each vertex u 2 u.color = WHITE 3 finish_list = empty_list 4 for each vertex u do 5 if u.color == WHITE then 6 DFS-Visit (G, u, finish_list) DFS-Visit (G, u) 1 u.color = GRAY 2 for each vertex v in Adj(u) do 3 if v.color = WHITE then 4 DFS-Visit (G, v) 5 u.color = BLACK 6 finish_list.append(u) a b c d e f g h

DFS-Visit(G,a): Finish_list: h, g, f, d. e u DFS-Visit(G,c): DFS-Visit(G,e):

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Depth First-Search

Toposort(G) 1 for each vertex u 2 u.color = WHITE 3 finish_list = empty_list 4 for each vertex u do 5 if u.color == WHITE then 6 DFS-Visit (G, u, finish_list) DFS-Visit (G, u) 1 u.color = GRAY 2 for each vertex v in Adj(u) do 3 if v.color = WHITE then 4 DFS-Visit (G, v) 5 u.color = BLACK 6 finish_list.append(u) a b c d e f g h

DFS-Visit(G,a): Finish_list: h, g, f, d. e, c u DFS-Visit(G,c):

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Depth First-Search

Toposort(G) 1 for each vertex u 2 u.color = WHITE 3 finish_list = empty_list 4 for each vertex u do 5 if u.color == WHITE then 6 DFS-Visit (G, u, finish_list) DFS-Visit (G, u) 1 u.color = GRAY 2 for each vertex v in Adj(u) do 3 if v.color = WHITE then 4 DFS-Visit (G, v) 5 u.color = BLACK 6 finish_list.append(u) a b c d e f g h

DFS-Visit(G,a): Finish_list: h, g, f, d. e, c, a u

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Depth First-Search

Toposort(G) 1 for each vertex u 2 u.color = WHITE 3 finish_list = empty_list 4 for each vertex u do 5 if u.color == WHITE then 6 DFS-Visit (G, u, finish_list) DFS-Visit (G, u) 1 u.color = GRAY 2 for each vertex v in Adj(u) do 3 if v.color = WHITE then 4 DFS-Visit (G, v) 5 u.color = BLACK 6 finish_list.append(u) a b c d e f g h

DFS-Visit(G,b): Finish_list: h, g, f, d. e, c, a u

May iterate through many complete vertices before finding b to launch a new search from

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Depth First-Search

Toposort(G) 1 for each vertex u 2 u.color = WHITE 3 finish_list = empty_list 4 for each vertex u do 5 if u.color == WHITE then 6 DFS-Visit (G, u, finish_list) DFS-Visit (G, u) 1 u.color = GRAY 2 for each vertex v in Adj(u) do 3 if v.color = WHITE then 4 DFS-Visit (G, v) 5 u.color = BLACK 6 finish_list.append(u) a b c d e f g h

DFS-Visit(G,b): Finish_list: h, g, f, d. e, c, a, b u

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Depth First-Search

Toposort(G) 1 for each vertex u 2 u.color = WHITE 3 finish_list = empty_list 4 for each vertex u do 5 if u.color == WHITE then 6 DFS-Visit (G, u, finish_list) DFS-Visit (G, u) 1 u.color = GRAY 2 for each vertex v in Adj(u) do 3 if v.color = WHITE then 4 DFS-Visit (G, v) 5 u.color = BLACK 6 finish_list.append(u) a b c d e f g h

Finish_list: h, g, f, d. e, c, a, b u

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ANOTHER EXAMPLE

With Cycles in the graph

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Depth First-Search

Toposort(G) 1 for each vertex u 2 u.color = WHITE 3 finish_list = empty_list 4 for each vertex u do 5 if u.color == WHITE then 6 DFS-Visit (G, u, finish_list) DFS-Visit (G, u) 1 u.color = GRAY 2 for each vertex v in Adj(u) do 3 if v.color = WHITE then 4 DFS-Visit (G, v) 5 u.color = BLACK 6 finish_list.append(u) a b d c h e f g

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Depth First-Search

a b d c h e f g

DFS-Visit(G,a):

Toposort(G) 1 for each vertex u 2 u.color = WHITE 3 finish_list = empty_list 4 for each vertex u do 5 if u.color == WHITE then 6 DFS-Visit (G, u, finish_list) DFS-Visit (G, u) 1 u.color = GRAY 2 for each vertex v in Adj(u) do 3 if v.color = WHITE then 4 DFS-Visit (G, v) 5 u.color = BLACK 6 finish_list.append(u)

u v

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Depth First-Search

a b d c h e f g

DFS-Visit(G,a):

Toposort(G) 1 for each vertex u 2 u.color = WHITE 3 finish_list = empty_list 4 for each vertex u do 5 if u.color == WHITE then 6 DFS-Visit (G, u, finish_list) DFS-Visit (G, u) 1 u.color = GRAY 2 for each vertex v in Adj(u) do 3 if v.color = WHITE then 4 DFS-Visit (G, v) 5 u.color = BLACK 6 finish_list.append(u)

DFS-Visit(G,c): v u

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Depth First-Search

a b d c h e f g

DFS-Visit(G,a):

Toposort(G) 1 for each vertex u 2 u.color = WHITE 3 finish_list = empty_list 4 for each vertex u do 5 if u.color == WHITE then 6 DFS-Visit (G, u, finish_list) DFS-Visit (G, u) 1 u.color = GRAY 2 for each vertex v in Adj(u) do 3 if v.color = WHITE then 4 DFS-Visit (G, v) 5 u.color = BLACK 6 finish_list.append(u)

DFS-Visit(G,c): DFS-Visit(G,b): u v

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Depth First-Search

a b d c h e f g

DFS-Visit(G,a):

Toposort(G) 1 for each vertex u 2 u.color = WHITE 3 finish_list = empty_list 4 for each vertex u do 5 if u.color == WHITE then 6 DFS-Visit (G, u, finish_list) DFS-Visit (G, u) 1 u.color = GRAY 2 for each vertex v in Adj(u) do 3 if v.color = WHITE then 4 DFS-Visit (G, v) 5 u.color = BLACK 6 finish_list.append(u)

DFS-Visit(G,c): DFS-Visit(G,b): DFS-Visit(G,h): v u

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Depth First-Search

a b d c h e f g

DFS-Visit(G,a):

Toposort(G) 1 for each vertex u 2 u.color = WHITE 3 finish_list = empty_list 4 for each vertex u do 5 if u.color == WHITE then 6 DFS-Visit (G, u, finish_list) DFS-Visit (G, u) 1 u.color = GRAY 2 for each vertex v in Adj(u) do 3 if v.color = WHITE then 4 DFS-Visit (G, v) 5 u.color = BLACK 6 finish_list.append(u)

DFS-Visit(G,c): DFS-Visit(G,b): DFS-Visit(G,h): DFS-Visit(G,g): u v

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Depth First-Search

a b d c h e f g

DFS-Visit(G,a):

Toposort(G) 1 for each vertex u 2 u.color = WHITE 3 finish_list = empty_list 4 for each vertex u do 5 if u.color == WHITE then 6 DFS-Visit (G, u, finish_list) DFS-Visit (G, u) 1 u.color = GRAY 2 for each vertex v in Adj(u) do 3 if v.color = WHITE then 4 DFS-Visit (G, v) 5 u.color = BLACK 6 finish_list.append(u)

DFS-Visit(G,c): DFS-Visit(G,b): DFS-Visit(G,h): DFS-Visit(G,g): u v

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Depth First-Search

a b d c h e f g

DFS-Visit(G,a):

Toposort(G) 1 for each vertex u 2 u.color = WHITE 3 finish_list = empty_list 4 for each vertex u do 5 if u.color == WHITE then 6 DFS-Visit (G, u, finish_list) DFS-Visit (G, u) 1 u.color = GRAY 2 for each vertex v in Adj(u) do 3 if v.color = WHITE then 4 DFS-Visit (G, v) 5 u.color = BLACK 6 finish_list.append(u)

DFS-Visit(G,c): DFS-Visit(G,b): DFS-Visit(G,h): DFS-Visit(G,g): DFS-Visit(G,f): u v

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Depth First-Search

a b d c h e f g

DFS-Visit(G,a):

Toposort(G) 1 for each vertex u 2 u.color = WHITE 3 finish_list = empty_list 4 for each vertex u do 5 if u.color == WHITE then 6 DFS-Visit (G, u, finish_list) DFS-Visit (G, u) 1 u.color = GRAY 2 for each vertex v in Adj(u) do 3 if v.color = WHITE then 4 DFS-Visit (G, v) 5 u.color = BLACK 6 finish_list.append(u)

DFS-Visit(G,c): DFS-Visit(G,b): DFS-Visit(G,h): DFS-Visit(G,g): DFS-Visit(G,f): DFS-Visit(G,d): u v

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Depth First-Search

a b d c h e f g

DFS-Visit(G,a):

Toposort(G) 1 for each vertex u 2 u.color = WHITE 3 finish_list = empty_list 4 for each vertex u do 5 if u.color == WHITE then 6 DFS-Visit (G, u, finish_list) DFS-Visit (G, u) 1 u.color = GRAY 2 for each vertex v in Adj(u) do 3 if v.color = WHITE then 4 DFS-Visit (G, v) 5 u.color = BLACK 6 finish_list.append(u)

DFS-Visit(G,c): DFS-Visit(G,b): DFS-Visit(G,h): DFS-Visit(G,g): DFS-Visit(G,f): DFS-Visit(G,d): DFSQ: d u

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Depth First-Search

a b d c h e f g

DFS-Visit(G,a):

Toposort(G) 1 for each vertex u 2 u.color = WHITE 3 finish_list = empty_list 4 for each vertex u do 5 if u.color == WHITE then 6 DFS-Visit (G, u, finish_list) DFS-Visit (G, u) 1 u.color = GRAY 2 for each vertex v in Adj(u) do 3 if v.color = WHITE then 4 DFS-Visit (G, v) 5 u.color = BLACK 6 finish_list.append(u)

DFS-Visit(G,c): DFS-Visit(G,b): DFS-Visit(G,h): DFS-Visit(G,g): DFS-Visit(G,f): DFSQ: d u v

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Depth First-Search

a b d c h e f g

DFS-Visit(G,a):

Toposort(G) 1 for each vertex u 2 u.color = WHITE 3 finish_list = empty_list 4 for each vertex u do 5 if u.color == WHITE then 6 DFS-Visit (G, u, finish_list) DFS-Visit (G, u) 1 u.color = GRAY 2 for each vertex v in Adj(u) do 3 if v.color = WHITE then 4 DFS-Visit (G, v) 5 u.color = BLACK 6 finish_list.append(u)

DFS-Visit(G,c): DFS-Visit(G,b): DFS-Visit(G,h): DFS-Visit(G,g): DFS-Visit(G,f): DFSQ: d DFS-Visit(G,e): u v v

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Depth First-Search

a b d c h e f g

DFS-Visit(G,a):

Toposort(G) 1 for each vertex u 2 u.color = WHITE 3 finish_list = empty_list 4 for each vertex u do 5 if u.color == WHITE then 6 DFS-Visit (G, u, finish_list) DFS-Visit (G, u) 1 u.color = GRAY 2 for each vertex v in Adj(u) do 3 if v.color = WHITE then 4 DFS-Visit (G, v) 5 u.color = BLACK 6 finish_list.append(u)

DFS-Visit(G,c): DFS-Visit(G,b): DFS-Visit(G,h): DFS-Visit(G,g): DFS-Visit(G,f): DFSQ: d e DFS-Visit(G,e): u

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Depth First-Search

a b d c h e f g

DFS-Visit(G,a):

Toposort(G) 1 for each vertex u 2 u.color = WHITE 3 finish_list = empty_list 4 for each vertex u do 5 if u.color == WHITE then 6 DFS-Visit (G, u, finish_list) DFS-Visit (G, u) 1 u.color = GRAY 2 for each vertex v in Adj(u) do 3 if v.color = WHITE then 4 DFS-Visit (G, v) 5 u.color = BLACK 6 finish_list.append(u)

DFS-Visit(G,c): DFS-Visit(G,b): DFS-Visit(G,h): DFS-Visit(G,g): DFSQ: d e f DFS-Visit(G,f): u

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Depth First-Search

a b d c h e f g

DFS-Visit(G,a):

Toposort(G) 1 for each vertex u 2 u.color = WHITE 3 finish_list = empty_list 4 for each vertex u do 5 if u.color == WHITE then 6 DFS-Visit (G, u, finish_list) DFS-Visit (G, u) 1 u.color = GRAY 2 for each vertex v in Adj(u) do 3 if v.color = WHITE then 4 DFS-Visit (G, v) 5 u.color = BLACK 6 finish_list.append(u)

DFSQ: d e f g h b c u

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Depth First-Search

a b d c h e f g Toposort(G) 1 for each vertex u 2 u.color = WHITE 3 finish_list = empty_list 4 for each vertex u do 5 if u.color == WHITE then 6 DFS-Visit (G, u, finish_list) DFS-Visit (G, u) 1 u.color = GRAY 2 for each vertex v in Adj(u) do 3 if v.color = WHITE then 4 DFS-Visit (G, v) 5 u.color = BLACK 6 finish_list.append(u)

DFSQ: d e f g h b c a

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BFS vs. DFS Algorithm

BFS and DFS are more similar than you think

– Do we use a FIFO/Queue (BFS) or LIFO/Stack (DFS) to store vertices as we find them

BFS-Visit (G, start_node) 1 for each vertex u 2 u.color = WHITE 3 u.pred = nil 4 bfsq = new deque 5 bfsq.push_back(start_node) 6 while bfsq not empty 7 u = bfsq.pop_front() 8 if u.color == WHITE 9 u.color = GRAY 10 foreach vertex v in Adj(u) do 11 bfsq.push_back(v) DFS-Visit (G, start_node) 1 for each vertex u 2 u.color = WHITE 3 u.pred = nil 4 dfsq = new deque 5 dfsq.push_back(start_node) 6 while dfsq not empty 7 u = dfsq.pop_back() 8 if u.color == WHITE 9 u.color = GRAY 10 foreach vertex v in Adj(u) do 11 dfsq.push_back(v)

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ITERATIVE VERSION

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48

Depth First-Search

DFS (G,s) 1 for each vertex u 2 u.color = WHITE 3 dfsq = new deque 4 dfsq.push_back(s) 5 while dfsq not empty 6 u = dfsq.back() 7 if u.color == WHITE then 8 u.color = GRAY 9 foreach vertex v in Adj(u) do 10 if v.color == WHITE 11 dfsq.push_back(v) 12 else if u.color == GRAY 13 u.color = BLACK 14 dfsq.pop_back() a b d c h e f g

DFSQ: a

SLIDE 49

49

Depth First-Search

DFS (G,s) 1 for each vertex u 2 u.color = WHITE 3 dfsq = new deque 4 dfsq.push_back(s) 5 while dfsq not empty 6 u = dfsq.back() 7 if u.color == WHITE then 8 u.color = GRAY 9 foreach vertex v in Adj(u) do 10 if v.color == WHITE 11 dfsq.push_back(v) 12 else if u.color == GRAY 13 u.color = BLACK 14 dfsq.pop_back() a b d c h e f g

DFSQ: a c e

SLIDE 50

50

Depth First-Search

DFS (G,s) 1 for each vertex u 2 u.color = WHITE 3 dfsq = new deque 4 dfsq.push_back(s) 5 while dfsq not empty 6 u = dfsq.back() 7 if u.color == WHITE then 8 u.color = GRAY 9 foreach vertex v in Adj(u) do 10 if v.color == WHITE 11 dfsq.push_back(v) 12 else if u.color == GRAY 13 u.color = BLACK 14 dfsq.pop_back() a b d c h e f g

DFSQ: a c e c f

SLIDE 51

51

Depth First-Search

DFS (G,s) 1 for each vertex u 2 u.color = WHITE 3 dfsq = new deque 4 dfsq.push_back(s) 5 while dfsq not empty 6 u = dfsq.back() 7 if u.color == WHITE then 8 u.color = GRAY 9 foreach vertex v in Adj(u) do 10 if v.color == WHITE 11 dfsq.push_back(v) 12 else if u.color == GRAY 13 u.color = BLACK 14 dfsq.pop_back() a b d c h e f g

DFSQ: a c e c f d g

SLIDE 52

52

Depth First-Search

DFS (G,s) 1 for each vertex u 2 u.color = WHITE 3 dfsq = new deque 4 dfsq.push_back(s) 5 while dfsq not empty 6 u = dfsq.back() 7 if u.color == WHITE then 8 u.color = GRAY 9 foreach vertex v in Adj(u) do 10 if v.color == WHITE 11 dfsq.push_back(v) 12 else if u.color == GRAY 13 u.color = BLACK 14 dfsq.pop_back() a b d c h e f g

DFSQ: a c e c f d g c h

SLIDE 53

53

Depth First-Search

DFS (G,s) 1 for each vertex u 2 u.color = WHITE 3 dfsq = new deque 4 dfsq.push_back(s) 5 while dfsq not empty 6 u = dfsq.back() 7 if u.color == WHITE then 8 u.color = GRAY 9 foreach vertex v in Adj(u) do 10 if v.color == WHITE 11 dfsq.push_back(v) 12 else if u.color == GRAY 13 u.color = BLACK 14 dfsq.pop_back() a b d c h e f g

DFSQ: a c e c f d g c h b

SLIDE 54

54

Depth First-Search

DFS (G,s) 1 for each vertex u 2 u.color = WHITE 3 dfsq = new deque 4 dfsq.push_back(s) 5 while dfsq not empty 6 u = dfsq.back() 7 if u.color == WHITE then 8 u.color = GRAY 9 foreach vertex v in Adj(u) do 10 if v.color == WHITE 11 dfsq.push_back(v) 12 else if u.color == GRAY 13 u.color = BLACK 14 dfsq.pop_back() a b d c h e f g

DFSQ: a c e c f d g c h b c

SLIDE 55

55

Depth First-Search

DFS (G,s) 1 for each vertex u 2 u.color = WHITE 3 dfsq = new deque 4 dfsq.push_back(s) 5 while dfsq not empty 6 u = dfsq.back() 7 if u.color == WHITE then 8 u.color = GRAY 9 foreach vertex v in Adj(u) do 10 if v.color == WHITE 11 dfsq.push_back(v) 12 else if u.color == GRAY 13 u.color = BLACK 14 dfsq.pop_back() a b d c h e f g

DFSQ: a c e c f d g c h b c d

SLIDE 56

56

Depth First-Search

DFS (G,s) 1 for each vertex u 2 u.color = WHITE 3 dfsq = new deque 4 dfsq.push_back(s) 5 while dfsq not empty 6 u = dfsq.back() 7 if u.color == WHITE then 8 u.color = GRAY 9 foreach vertex v in Adj(u) do 10 if v.color == WHITE 11 dfsq.push_back(v) 12 else if u.color == GRAY 13 u.color = BLACK 14 dfsq.pop_back() a b d c h e f g

DFSQ: a c e c f d g c h b c d

SLIDE 57

57

Depth First-Search

DFS (G,s) 1 for each vertex u 2 u.color = WHITE 3 dfsq = new deque 4 dfsq.push_back(s) 5 while dfsq not empty 6 u = dfsq.back() 7 if u.color == WHITE then 8 u.color = GRAY 9 foreach vertex v in Adj(u) do 10 if v.color == WHITE 11 dfsq.push_back(v) 12 else if u.color == GRAY 13 u.color = BLACK 14 dfsq.pop_back() a b d c h e f g

DFSQ: a c e c f d g c h b c d

SLIDE 58

58

Depth First-Search

DFS (G,s) 1 for each vertex u 2 u.color = WHITE 3 dfsq = new deque 4 dfsq.push_back(s) 5 while dfsq not empty 6 u = dfsq.back() 7 if u.color == WHITE then 8 u.color = GRAY 9 foreach vertex v in Adj(u) do 10 if v.color == WHITE 11 dfsq.push_back(v) 12 else if u.color == GRAY 13 u.color = BLACK 14 dfsq.pop_back() a b d c h e f g

DFSQ: a c e c f d g c h b c

SLIDE 59

59

Depth First-Search

DFS (G,s) 1 for each vertex u 2 u.color = WHITE 3 dfsq = new deque 4 dfsq.push_back(s) 5 while dfsq not empty 6 u = dfsq.back() 7 if u.color == WHITE then 8 u.color = GRAY 9 foreach vertex v in Adj(u) do 10 if v.color == WHITE 11 dfsq.push_back(v) 12 else if u.color == GRAY 13 u.color = BLACK 14 dfsq.pop_back() a b d c h e f g

DFSQ: a c e c f d g c h b

SLIDE 60

60

Depth First-Search

DFS (G,s) 1 for each vertex u 2 u.color = WHITE 3 dfsq = new deque 4 dfsq.push_back(s) 5 while dfsq not empty 6 u = dfsq.back() 7 if u.color == WHITE then 8 u.color = GRAY 9 foreach vertex v in Adj(u) do 10 if v.color == WHITE 11 dfsq.push_back(v) 12 else if u.color == GRAY 13 u.color = BLACK 14 dfsq.pop_back() a b d c h e f g

DFSQ: a