1. Lecture 7 Graph Traversal
Undirected graph
Directed graph
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2. Overview Graph Undirected graph DFS, BFS, Application Directed graph
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3. Graph theory The Königsberg Bridge problem (Source from Wikipedia)
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4. Graph everywhere
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5. Graph terminology Undirected graph Directed graph 4/28/2017
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6. Adjacency Matrix v.s. Adjacency List
G=<V, E>
Directed: n + m
Undirected: n + 2m
Directed: n2
Undirected: n2
For every edge connected with v ...
Is u and v connected with an edge?
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7. Important graph problems
Path. Is there a directed path from s to t ? Shortest path. What is the shortest directed path from s to t ? Topological sort. Can you draw the digraph so that all edges point upwards? Strong connectivity. Is there a directed path between all pairs of vertices? Transitive closure. For each vertices v and w, is there a path from v to w ?
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8. Where are we? Graph Undirected graph DFS, BFS, Application
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9. DFS Depth-first-search. Unroll a ball of string behind you.
忒修斯
克里特岛迷宫
米诺斯
阿里阿德涅公主
米诺牛
剑和线团
Shannon的故事
Depth-first-search.
Unroll a ball of string behind you.
Mark each visited intersection and each visited passage.
Retrace steps when no unvisited options.
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10. Maze exploration
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11. Depth-first search pre/post = 0/0 u 1/ 6 u v w v 2/ 5 y x y 3/ 4 z w z4/ 1 5/ 3 w x y 6/ 2 z v
DFS tree u
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12. Depth-first search time = 0 pre/post u 1/ 12 u v w v 2/ 11 y x y 3/ 10z w z x 4/ 5 6/ 9 w x y 7/ 8 z v
DFS tree u
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13. DFS tree: undirected How to figure out back edges? u 1/ 6 v 2/ 5
tree edge:
back edge: y 3/ 4 w x 4/ 1 5/ 3 6/ 2 z
DFS tree
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14. 4/28/2017
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15. time <-- 0 v.pre <-- infinity v.post <-- infinity
time <-- time + 1
v.pre <-- time
time <-- time + 1
v.post <-- time
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16. Quiz: Complexity Time Space Undirected Adj. Matrix ? Adj. List
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17. Complexity Time Space Undirected Adj. Matrix O(|V|2) O(|V|) Adj. List
O(|V| + 2|E|)
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18. DFS correctness Proposition
DFS marks all vertices connected to s in time proportional to the sum of their degrees.
Pf.
Correctness:
if w marked, then w connected to s (why?)
if w connected to s, then w marked (if w unmarked, then consider last edge on a path from s to w that goes from a marked vertex to an unmarked one)
Running time: each vertex connected to s is visited once
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19. DFS application?
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20. Breadth-first search u v w x y z a b u 1 x 2 v 2 y 3 w z 4 u v x y w z5 a 5
BFS tree
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21. 4/28/2017
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22. Quiz: Complexity Time Space Undirected Adj. Matrix ? Adj. List
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23. Complexity Time Space Undirected Adj. Matrix O(|V|2) O(|V|) Adj. List
O(|V| + 2|E|)
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24. BFS Application: The shortest pathu x v y a b w z u v w x y z a b
Discussion: How to record the shortest path?
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25. Breadth-first search u v w x y z a b u 1 x 2 v 2 y 3 w z 4 u v x y w z5 a 5 u v x y w z a b
BFS tree u u v y y
parent w z
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26. 4/28/2017
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27. w.parent <-- v
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28. Connectivity
#trees == #connected components
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29. Graph acyclicity
NO back edges!
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30. Where are we? Graph Undirected graph DFS, BFS, Application
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31. DFS tree: directed time = 0 u w w 1/ 8 9/ 12 u v v 10/ 11 2/ 7 z x y3/ 6 z y x 4/ 5 x
DFS trees y z v w u
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32. DFS tree u w tree edge: back edge: forward edge: cross edge: 1/ 8 9/12 v
10/ 11 2/ 7 z
type
edge(u,v)
tree
[u [v ]v ]u
back
[v [u ]u ]v
forward
cross
[v ]v [u ]u 3/ 6 y x 4/ 5
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33. Quiz
Run DFS on the following graph. Which type are the following edges:
CB, DC, FC
(Whenever you have a choice of vertices to explore, always pick the one that is alphabetically first.)
A. tree edge B. back edge
C. forward edge D. cross edge

34. Application: Garbage collector
Mark-sweep algorithm. [McCarthy, 1960]
Mark: mark all reachable objects.
Sweep: if object is unmarked, it is garbage (so add to free list).
Memory cost. Uses 1 extra mark bit per object (plus DFS stack).
roots
DFS needs O(|V|) space.
How to do Mark-sweep with O(1) space?
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35. A directed acyclic graph is usually called a DAG.
Graph acyclicity
NO back edges!
A directed acyclic graph is usually called a DAG.
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36. An edge (u,v) is a back edge iff
Graph acyclicity u w
tree edge:
back edge:
forward edge:
cross edge: 1/ 8 9/ 12 v
10/ 11 2/ 7 z
type
edge(u,v)
tree
[u [v ]v ]u
back
[v [u ]u ]v
forward
cross
[v ]v [u ]u 3/ 6 y x 4/ 5
An edge (u,v) is a back edge iff
post(u) < post(v)
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37. Applications
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38. DAG: Topological sort Problem: TopoSort
Input: A DAG (directed acyclic graph) G = (V , E )
Output: A linear ordering of its vertices in such a way that if (v,w) ∈ E, then v appears before w in the ordering.
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39. Topological order
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40. Topological sort by DFS
Proposition
In DAG, every edge (u,v) yields post[v] < post[u]. u v x y w z 1/ 2/ 3/ 4/ 5 6 7 8 9/
10/ 11 12
tree edge:
back edge:
forward edge:
cross edge:
type
edge(u,v)
tree
[u [v ]v ]u
back
[v [u ]u ]v
forward
cross
[v ]v [u ]u
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41. DAG: Topological sort Θ(|E|+|V|) TOPOLOGICAL-SORT(G)
Call DFS(G) to compute finishing times of each vertex v
As each vertex is finished, insert it onto the front of a linked list
Return the linked list of vertices.
Θ(|E|+|V|)
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42. Quiz
Give the topological-sort algorithm on the following graph, give the topological order.
(Whenever you have a choice of vertices to explore, always pick the one that is alphabetically first.) ♣
A. BACEDFGH
B. ABCDEFGH
C. ABCEDFHG
D. None of above

43. ? Connectivity #trees == #connected components u v w x y z 4/28/2017
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44. Strong connected components
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45. Ecological food webs 4/28/2017 Xiaojuan Cai
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46. Strong connected components
Lemma Every directed graph is a DAG of its strongly connected component.
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47. Some properties Property 1
If dfs is started at node u, then it will terminate precisely when all nodes reachable from u have been visited.
Property 2
The node that receives the highest post number in DFS must lie in a source strongly connected component.
Property 3
If C and C′ are scc, and there is an edge from a node in C to a node in C′, then the highest post number in C is bigger than the highest post number in C′.
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48. Kosaraju-Sharir algorithm
Reversed graph
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49. Kosaraju-Sharir algorithm
Run DFS on GR.
Run the undirected connected components algorithm on G, and during the DFS, process the vertices in decreasing order of their post numbers from step 1.
Θ(|E|+|V|)
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50. Tarjan’s algorithm (briefly)
//color[u] = 0: unvisited; 1: visited and in Stack, 2: visited and not in Stack
try all vertex u, if color[u] = 0, DFS(u)
DFS(u):
Push u into stack, color[u] = 1, pre[u], low[u] = ++time
try all neighbor v of u
if color[v] = 0, DFS(v), low[u] = min{low[u],low[v]}
else if color[v] = 1, low[u] = min{low[u],pre[v]}
if low[u]==pre[u]
Pop v from stack, color[v] = 2, until v=u;
One Dfs, More efficient than Kosaraju-Sharir

51. Articulation points Problem: ArticulationPoints
Input: An undirected graph G = (V , E )
Output: Articulation points. (A point v is articulate iff any path between other two points must pass through v).
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52. Conclusion Graph Undirected graph DFS, BFS, Application Directed graph
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53. BFS, DFS applications
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54. BFS, DFS applications BFS. Choose root web page as source s.
Maintain a Queue of websites to explore.
Maintain a SET of discovered websites.
Dequeue the next website and enqueue websites to which it links (provided you haven't done so before).
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55. BFS, DFS applications Vertex: pixel.
Edge: between two adjacent gray pixels.
Blob: all pixels connected to given pixel.
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56. BFS, DFS applications Every data structure is a digraph.
Vertex = object.
Edge = reference.
Roots. Objects known to be directly accessible by program (e.g., stack).
Reachable objects. Objects indirectly accessible by program
(starting at a root and following a chain of pointers).
roots
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57. BFS, DFS applications Every program is a digraph.
Vertex = basic block of instructions
Edge = jump.
Dead-code elimination. Find (and remove) unreachable code.
Infinite-loop detection. Determine whether exit is unreachable.
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58. Quiz Which of the following statements are true for undirected graph?
A. If u is connected to v and v is connected to w, then u is connected to w.
B. Removing any edge from a connected graph G breaks the graph into two connected components.
C. An acyclic graph G is connected if and only if it has V-1 edges.
D. If you add two edges to a graph G, its number of components can decrease by at most 2.
♠ ♣ ♦

59. Quiz Which of the following statements are true for directed graph?
A. A digraph on V vertices with fewer than V edges contains more than one strong component.
B. If we modify the Kosaraju-Sharir algorithm to replace the second DFS with BFS, then it will still find the strong components.
C. If u is strongly connected to v and v is strongly connected to w, then u is strongly connected to w.
D. If you add two edges to a digraph, its number of strong components can decrease by at most 2.

60. Challenges
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61. Challenge 1 Problem. Is a graph bipartite? How difficult?
0-1
0-2
0-5
0-6
1-3
2-3
2-4
4-5
4-6
Problem. Is a graph bipartite?
How difficult?
Any programmer could do it.
Need to be a typical diligent SE222 student.
Hire an expert.
Intractable.
No one knows.
Impossible. 1 2 6 3 4 5
0-1
0-2
0-5
0-6
1-3
2-3
2-4
4-5
4-6 1 2 6 3 4 5
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62. Challenge 1 Problem. Is a graph bipartite? How difficult?
0-1
0-2
0-5
0-6
1-3
2-3
2-4
4-5
4-6
Problem. Is a graph bipartite?
How difficult?
Any programmer could do it.
Need to be a typical diligent SE222 student.
Hire an expert.
Intractable.
No one knows.
Impossible. 1 2 6 3 4 5 ✓
0-1
0-2
0-5
0-6
1-3
2-3
2-4
4-5
4-6 1 2 6 3 4 5
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63. Challenge 2 Problem. Find a cycle. How difficult?
0-1
0-2
0-5
0-6
1-3
2-3
2-4
4-5
4-6
Problem. Find a cycle.
How difficult?
Any programmer could do it.
Need to be a typical diligent SE222 student.
Hire an expert.
Intractable.
No one knows.
Impossible. 1 2 6 3 4 5 ✓ 1 2 6 3 4 5
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64. Challenge 3
Problem. Find a cycle that uses every edge exactly once. (Eulerian tour)
How difficult?
Any programmer could do it.
Need to be a typical diligent SE222 student.
Hire an expert.
Intractable.
No one knows.
Impossible.
0-1
0-2
0-5
0-6
1-2
2-3
2-4
3-4
4-5
4-6 6 4 2 1 5 3 ✓
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65. Challenge 4
Problem. Find a cycle that uses every vertex exactly once. (Hamiltonian tour)
How difficult?
Any programmer could do it.
Need to be a typical diligent SE222 student.
Hire an expert.
Intractable.
No one knows.
Impossible.
0-1
0-2
0-5
0-6
1-2
2-6
3-4
3-5
4-5
4-6 1 2 6 3 4 5 ✓
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66. Challenge 5
Problem. Are two graphs identical except for vertex names? (Graph isomorphism)
How difficult?
Any programmer could do it.
Need to be a typical diligent SE222 student.
Hire an expert.
Intractable.
No one knows.
Impossible.
0-1
0-2
0-5
0-6
3-4
3-5
4-5
4-6 6 4 2 1 5 3
0↔4, 1↔3, 2↔2, 3↔6, 4↔5, 5↔0, 6↔1
0-4
1-4
1-5
2-4
5-6 ✓
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67. Challenge 6
Problem. Lay out a graph in the plane without crossing edges?
How difficult?
Any programmer could do it.
Need to be a typical diligent SE222 student.
Hire an expert.
Intractable.
No one knows.
Impossible. 1
0-1
0-2
0-5
0-6
3-4
3-5
4-5
4-6 2 6 3 4 5 ✓ 1 2 6 3 4 5
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Xiaojuan Cai