1. Advanced DFS, BFS, Graph Modeling
19/2/2005

2. Introduction Depth-first search (DFS) Breadth-first search (BFS)
Graph Modeling
Model a graph from a problem, ie. transform a problem into a graph problem

3. What is a graph?
A set of vertices and edges
vertex
edge

4. Trees and related terms
root
siblings
descendents
children
ancestors
parent

5. Graph Traversal Given: a graph
Goal: visit all (or some) vertices and edges of the graph using some strategy (the order of visit is systematic)
DFS, BFS are examples of graph traversal algorithms
Some shortest path algorithms and spanning tree algorithms have specific visit order

6. Intuition of DFS of BFS This is a brief idea of DFS, BFS
DFS: continue visiting next vertex whenever there is a road, go back if no road (ie. visit to the depth of current path)
Example: a human want to visit a place, but do not know the path
BFS: go through all the adjacent vertices before going further (ie. spread among next vertices)
Example: set a house on fire, the fire will spread through the house

7. DFS (pseudo code) Initially all vertices are marked as unvisited
DFS (vertex u) {
mark u as visited
for each vertex v directly reachable from u
if v is unvisited
DFS (v) }
Initially all vertices are marked as unvisited

8. DFS (Demonstration) F A B C D E
unvisited
visited

9. “Advanced” DFS
Apart from just visiting the vertices, DFS can also provide us with valuable information
DFS can be enhanced by introducing:
birth time and death time of a vertex
birth time: when the vertex is first visited
death time: when we retreat from the vertex
DFS tree
parent of a vertex

10. DFS spanning tree / forest
A rooted tree
The root is the start vertex
If v is first visited from u, then u is the parent of v in the DFS tree
Edges are those in forward direction of DFS, ie. when visiting vertices that are not visited before
If some vertices are not reachable from the start vertex, those vertices will form other spanning trees (1 or more)
The collection of the trees are called forest

11. DFS (pseudo code) DFS (vertex u) { mark u as visited time  time+1
for each vertex v directly reachable from u
if v is unvisited
DFS (v) }

12. DFS forest (Demonstration)B C D E F G H
birth
death
parent A 1 2 3 13 10 4 14 6 D 12 9 8 16 11 5 15 7 B - A B - A C D C E G A F B C D E G H C F H
unvisited
visited
visited (dead)

13. Classification of edgesB C F E D G H
Tree edge
Forward edge
Back edge
Cross edge
Question: which type of edges is always absent in an undirected graph?

14. Determination of edge types
How to determine the type of an arbitrary edge (u, v) after DFS?
Tree edge
parent [v] = u
Forward edge
not a tree edge; and
birth [v] > birth [u]; and
death [v] < death [u]
How about back edge and cross edge?

15. Determination of edge types
Tree edge
Forward Edge
Back Edge
Cross Edge
parent [v] = u
not a tree edge
birth[v] > birth[u]
death[v] < death[u]
birth[v] < birth[u]
death[v] > death[u]
death[v] >death[u]

16. Applications of DFS Forests
Topological sorting (Tsort)
Strongly-connected components (SCC)
Some more “advanced” algorithms

17. Example: Tsort
Topological order: A numbering of the vertices of a directed acyclic graph such that every edge from a vertex numbered i to a vertex numbered j satisfies i<j
Tsort: Number the vertices in topological order 3 6 1 7 2 5 4

18. Tsort Algorithm
If the graph has more then one vertex that has indegree 0, add a vertice to connect to all indegree-0 vertices
Let the indegree 0 vertice be s
Use s as start vertice, and compute the DFS forest
The death time of the vertices represent the reverse of topological order

19. A B C G S D F E G C F B A E D  D E A B F C G S A B C D E F G birth 12 3 4 12 13 8 5
death 16 11 10 7 15 14 9 6 A B C G S D F E
G C F B A E D  D E A B F C G

20. Example: SCC A graph is strongly-connected if
for any pair of vertices u and v, one can go from u to v and from v to u.
Informally speaking, an SCC of a graph is a subset of vertices that
forms a strongly-connected subgraph
does not form a strongly-connected subgraph with the addition of any new vertex

21. SCC (Illustration)

22. SCC (Algorithm)
Compute the DFS forest of the graph G to get the death time of the vertices
Reverse all edges in G to form G’
Compute a DFS forest of G’, but always choose the vertex with the latest death time when choosing the root for a new tree
The SCCs of G are the DFS trees in the DFS forest of G’

23. SCC (Demonstration) E A F B C D G H F A E B H D A D G F B C C G H A B
birth
death
parent 1 2 3 13 10 4 14 6 12 9 8 16 11 5 15 7 - A B - A C D C E A F B C D G H F A E B H D A D G F B C C G H

24. SCC (Demonstration) A F B C D G H E D G A E B F C H

25. DFS Summary DFS spanning tree / forest
We can use birth time and death time in DFS spanning tree to do varies things, such as Tsort, SCC
Notice that in the previous slides, we related birth time and death time. But in the discussed applications, birth time and death time can be independent, ie. birth time and death time can use different time counter

26. DFS (vertex u) {
mark u as visited
birthtime  birthtime + 1
for each vertex v directly reachable from u
if v is unvisited
DFS (v)
deathtime  deathtime + 1 }
time  time+1

27. Breadth-first search (BFS)
Revised:
DFS: continue visiting next vertex whenever there is a road, go back if no road (ie. visit to the depth of current path)
BFS: go through all the adjacent vertices before going further (ie. spread among next vertices)
In order to “spread”, we need to makes use of a data structure, queue ,to remember just visited vertices

28. BFS (Pseudo code)
while queue not empty
dequeue the first vertex u from queue
for each vertex v directly reachable from u
if v is unvisited
enqueue v to queue
mark v as visited
Initially all vertices except the start vertex are marked as unvisited and the queue contains the start vertex only

29. BFS (Demonstration) Queue: A B C F D E H G J I A B C D E F G H I J
unvisited
visited
visited (dequeued)

30. Applications of BFS Shortest paths finding
Flood-fill (can also be handled by DFS)

31. Comparisons of DFS and BFS
Depth-first
Breadth-first
Stack
Queue
Does not guarantee shortest paths
Guarantees shortest paths

32. Bidirectional search (BDS)
Searches simultaneously from both the start vertex and goal vertex
Commonly implemented as bidirectional BFS
start
goal

33. BDS Example: Bomber Man (1 Bomb)
find the shortest path from the upper-left corner to the lower-right corner in a maze using a bomb. The bomb can destroy a wall. S E

34. Bomber Man (1 Bomb) S 1 2 3 4 E 13 12 11 12 4 10 8 7 6 5 6 7 8 9 9 5 10 11 12 13 4 3 2 1
Shortest Path length = 7
What will happen if we stop once we find a path?

35. Example S 1 2 3 4 5 6 7 8 9 10 21 11 20 12 19 13 18 14 17 16 15 E

36. Iterative deepening search (IDS)
Iteratively performs DFS with increasing depth bound
Shortest paths are guaranteed

37. IDS

38. IDS (pseudo code) DFS (vertex u, depth d) { mark u as visited
if (d>0)
for each vertex v directly reachable from u
if v is unvisited
DFS (v,d-1) }
i=0
Do {
DFS(start vertex,i)
Increment i
}While (target is not found)

39. IDS Complexity (the details can be skipped)
( )=bm
( ) =bm
- b is branching factor - td is the number of vertices visited for depth d

40. IDS
( )=bm
( ) =bm
- b is branching factor - td is the number of vertices visited for depth d
Conclusion: the complexity of IDS is the same as DFS

41. Summary of DFS, BFS We learned some variations of DFS and BFS
Bidirectional search (BDS)
Iterative deepening search (IDS)

42. What is graph modeling? Conversion of a problem into a graph problem
Sometimes a problem can be easily solved once its underlying graph model is recognized
Graph modeling appears almost every year in NOI or IOI

43. Basics of graph modeling
A few steps:
identify the vertices and the edges
identify the objective of the problem
state the objective in graph terms
implementation:
construct the graph from the input instance
run the suitable graph algorithms on the graph
convert the output to the required format

44. Simple examples (1)
Given a grid maze with obstacles, find a shortest path between two given points
start
goal

45. Simple examples (2)
A student has the phone numbers of some other students
Suppose you know all pairs (A, B) such that A has B’s number
Now you want to know Alan’s number, what is the minimum number of calls you need to make?

46. Simple examples (2) Vertex: student Edge: whether A has B’s number
Add an edge from A to B if A has B’s number
Problem: find a shortest path from your vertex to Alan’s vertex

47. Complex examples (1) Same settings as simple example 1
You know a trick – walking through an obstacle! However, it can be used for only once (Bomber Man with 1 bomb)
What should a vertex represent?
your position only?
your position + whether you have used the trick

48. Complex examples (1) A vertex is in the form (position, used)
The vertices are divided into two groups
trick used
trick not used

49. Complex examples (1)
unused
start
goal
start
goal
used
goal

50. Complex examples (1)
How about you can walk through obstacles for k times?

51. Complex examples (2) The famous 8-puzzle
Given a state, find the moves that bring it to the goal state 1 2 3 4 5 6 7 8

52. Complex examples (2) What does a vertex represent? What are the edges?
the position of the empty square?
the number of tiles that are in wrong positions?
the state (the positions of the eight tiles)
What are the edges?
What is the equivalent graph problem?

53. Complex examples (2) 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 2 3 4 6 7 5 8 1 2 3 4 5 7 8 6 1 2 4 5 3 7 8 6 1 2 3 4 5 6 7 8 1 2 3 4 5 7 8 6

54. Complex examples (3) Theseus and Minotaur
Extract:
Theseus must escape from a maze. There is also a mechanical Minotaur in the maze. For every turn that Theseus takes, the Minotaur takes two turns. The Minotaur follows this program for each of his two turns:
First he tests if he can move horizontally and get closer to Theseus. If he can, he will move one square horizontally. If he can’t, he will test if he could move vertically and get closer to Theseus. If he can, he will move one square vertically. If he can’t move either horizontally or vertically, then he just skips that turn.

55. Complex examples (3) What does a vertex represent? Theseus’ position
Minotaur’s position
Both

56. Some more examples How can the followings be modeled?
Tilt maze (Single-goal mazes only)
Double title maze
No-left-turn maze
Same as complex example 1, but you can use the trick for k times

57. Competition problems HKOI2000 S – Wormhole Labyrinth
HKOI2001 S – A Node Too Far
HKOI2004 S – Teacher’s Problem *
TFT2001 – OIMan *
TFT2002 – Bomber Man *
NOI2001 – cung1 ming4 dik7 daa2 zi6 jyun4
NOI2001 – Equation *
IOI2000 – Walls
IOI2002 – Troublesome Frog
IOI2003 – Amazing Robots

58. Teacher’s Problem
Question: A teacher wants to distribute sweets to students in an order such that, if student u tease student v, u should not get the sweet before v
Vertex: student
Edge: directed, (v,u) is a directed edge if student v tease u
Algorithm: Tsort

59. OI Man
Question: OIMan (O) has 2 kinds of form: H- and S-form. He can transform to S-form for m minutes by battery. He can only kill monsters (M) and virus (V) in S-form. Given the number of battery and m, find the minimum time needed to kill the virus.
Vertex: position, form, time left for S-form, number of batteries left
Edge: directed
Move to P in H-form (position)
Move to P/M/V in S-form (position, S-form time)
Use battery and move (position, form, S-form time, number of batteries)
PWWP PPPP PWWP OMMV

60. Equation
Question: Find the number of solution of xi, given ki,pi. 1<=n<=6, 1<=xi<=150
Vertex: possible values of
k1x1p1 , k1x1p1 + k2x2p2 , k1x1p1 + k2x2p2 + k3x3p3 , k4x4p4 , k4x4p4 + k5x5p5 , k4x4p4 + k5x5p5 + k6x6p6