Lecture 14: Graph Traversals
CS6507– Python Programming and Data Science Applications
Summary
Breadth-first and depth-first search. Applications.
Dr Kieran T. Herley
Department of Computer Science
University College Cork
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Depth First Search–Pseduocode
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DFS cont’d
Noteworthy features
Algorithm Explore(G, v):
Mark v visited
for each edge (v, w) touching v do
if w is not marked visited then
Explore(G, w)
Algorithm Explore(G, v ):
Mark v visited
for each edge (v, w) touching v do
if w is not marked visited then
Explore(G, w)
Algorithm DFS(G, v):
Mark every node unvisited
Explore(G, v)
Algorithm DFS(G, v):
Mark every node unvisited
Explore(G, v)
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Recursion: Explore(G , v ) may
call Explore for v ’s neighbours
Node marking: nodes
visited/unvisited
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DFS in action
Illustration 2 – execution trace
Explore(G , x)
Mark x visited
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Start:
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Is y visited? (No)
Explore(G, 1)
Explore(G , y )
→ Explore(G, 5)
→ Explore(G, 9)
y
· · · → Explore(G, 2)
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For edge (x, y )
x
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z
Algorithm Explore(G, v ):
Mark v visited
for each edge (v, w) touching v do
if w is not marked visited then
Explore(G, w)
→ Explore(G, 6)
Algorithm Explore(G, v ):
Mark v visited
for each edge (v, w) touching v do
if w is not marked visited then
Explore(G, w)
Mark y visited
For edge (y , x)
Is x visited? (Yes)
For edge (y , z)
Is z visited ? (No)
Explore(G , z)
(Backtrack to Explore(G, 3))
→ Explore(G, 7)
Mark z visited
For edge (z, x)
Is x visited? (Yes)
For edge (z, y )
Is y visited? (Yes)
Algorithm DFS(G, v):
Mark every node unvisited
Explore(G, v)
→ Explore(G, 10)
Unwind recursion
For edge (x, z)
Is z visited? (Yes)
All done!
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Observations 1
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Observation 2
Algorithm Explore(G, v ):
Mark v visited
for each edge (v, w) touching v do
if w is not marked visited then
Explore(G, w)
Observation
No node is marked visited twice
Algorithm Explore(G, v ):
Mark v visited
for each edge (v, w) touching v do
if w is not marked visited then
Explore(G, w)
Why?
Node x marked visited only at
beginning of Explore(G, x)
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Observation
Once a node is marked visited, so
too will all its neighbours
(eventually)
Why?
Explore(G, x) marks x visited
Explore(G, x) only called if x
is unmarked . . .
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Calls Explore(G, y) for each
unmarked neighbour y
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Observation 3
Observation
Observation
Every node reachable from the
starting point is eventually visited
Algorithm Explore(G, v ):
Mark v visited
for each edge (v, w) touching v do
if w is not marked visited then
Explore(G, w)
Why?
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During traversal all graph edge categorized a either as
Tree edges (red) reflecting pattern of calls to Explore
Node y reachable from x
implies path between them
adjacent members of which
are neighbours
Result follows from previous
observation
Back edges (blue) reflecting
edges leading to nodes already
visited
a
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Usefulness of DFS
Algorithm in dfs.c does not explicitly mark edges as tree/back but can be
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modified
do so.
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Path Discovery
Depth-first search forms basis for solving lots of problems:
Reachability can node y be
reached from node y ?
Objective Given x and y , find path from x to y
Observation Explore(G , x) visits y , if it is reachable from x
Connectedness can every node
be reached from every other?
Idea Build up path info. as Explore executes
Acyclicity does graph contain
any cycles?
Planarity can graph be drawn
without edges crossing (hard)
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A Useful Observation
Path Discovery cont’d
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Pattern of calls to
Explore(G , x) defines a tree
(red edges) rooted at x (here
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Tree “spans” (i.e. touches) all
nodes reachable from x
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Note: to find x-to-y path
follow tree edges from y to x
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Previous Explore augmented
Note array parent []
Algorithm Explore(G, v ):
Mark v visited
for each edge (v, w) touching v do
if w is not marked visited then
parent [w] ←tree1v
Explore(G, w)
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Notes and Acknowledgements
Reading
Code
Acknowledgements
Lecture 14: Graph Traversals
Explore(G , x) calls
Explore(G , y ) implies
x = parent[y]
Encodes a spanning tree
rooted at y
Back Material
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Records recursive calling
pattern
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