Advanced DFS, BFS, Graph Modeling

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Advanced DFS, BFS, Graph Modeling

• 25/2/2006

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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

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What is a graph?

• A set of vertices and edges

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Trees and related terms
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What is a tree?

• A tree is an undirected simple graph G that
  satisfies any of the following equivalent
  conditions
• G is connected and has no simple cycles.
• G has no simple cycles and, if any edge is added
  to G, then a simple cycle is formed.
• G is connected and, if any edge is removed from
  G, then it is not connected anymore.
• Any two vertices in G can be connected by a
  unique simple path.
• G is connected and has n - 1 edges.
• G has no simple cycles and has n - 1 edges.

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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

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Idea of DFS and BFS

• This is a brief idea of DFS and 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

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DFS (pseudo code)

• 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

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DFS (Demonstration)
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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

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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

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DFS (pseudo code)

• DFS (vertex u)
• mark u as visited
• time ? time1 birthutime
• for each vertex v directly reachable from u
• if v is unvisited
• parentvu
• DFS (v)
• time ? time1 deathutime

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DFS forest (Demonstration)
A B C D E F G H
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
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Classification of edges

• Tree edge
• Forward edge
• Back edge
• Cross edge
• Question which type of edges is always absent in
  an undirected graph?

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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 gt birth u and
• death v lt death u
• How about back edge and cross edge?

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Determination of edge types
Tree edge Forward Edge Back Edge Cross Edge
parent v u not a tree edge birthv gt birthu deathv lt deathu birthv lt birthu deathv gt deathu birthv lt birthu deathv lt deathu
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Applications of DFS Forests

• Topological sorting (Tsort)
• Strongly-connected components (SCC)
• Some more advanced algorithms

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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 iltj
• Tsort Number the vertices in topological order

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6
1
7
2
5
4
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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

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Tsort (Demonstration)
S A B C D E F G
birth
death
1
2
3
4
5
8
12
13
6
7
9
10
11
14
15
16
A
B
C
G
S
D
F
E
G C F B A E D
? D E A B F C G
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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

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SCC (Illustration)
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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

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

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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

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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

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BFS (Demonstration)
Queue
A
B
C
F
D
E
H
G
J
I
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Applications of BFS

• Shortest paths finding
• Flood-fill (can also be handled by DFS)

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

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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

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Simple examples (1)

• Given a grid maze with obstacles, find a shortest
  path between two given points

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Simple examples (2)

• A student has the phone numbers of some other
  students
• Suppose you know all pairs (A, B) such that A has
  Bs number
• Now you want to know Alans number, what is the
  minimum number of calls you need to make?

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Simple examples (2)

• Vertex student
• Edge whether A has Bs number
• Add an edge from A to B if A has Bs number
• Problem find a shortest path from your vertex to
  Alans vertex

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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
• What should a vertex represent?
• your position only?
• your position whether you have used the trick

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Complex examples (1)

• A vertex is in the form (position, used)
• The vertices are divided into two groups
• trick used
• trick not used

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

• How about you can walk through obstacles for k
  times?

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Complex examples (1)
k
start
goal
k-1
k-2
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Complex examples (1)
k
start
goal
k-1
k-4
k-2
k-3
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Complex examples (2)

• The famous 8-puzzle
• Given a state, find the moves that bring it to
  the goal state

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Complex examples (2)

• What does a vertex represent?
• 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?

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Complex examples (2)
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Complex examples (3)

• Theseus and Minotaur
• http//www.logicmazes.com/theseus.html
• 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 cant, he will
  test if he could move vertically and get closer
  to Theseus. If he can, he will move one square
  vertically. If he cant move either horizontally
  or vertically, then he just skips that turn.

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Complex examples (3)

• What does a vertex represent?
• Theseus position
• Minotaurs position
• Both

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Some more examples

• How can the followings be modeled?
• Tilt maze (Single-goal mazes only)
• http//www.clickmazes.com/newtilt/ixtilt2d.htm
• Double title maze
• http//www.clickmazes.com/newtilt/ixtilt.htm
• No-left-turn maze
• http//www.clickmazes.com/noleft/ixnoleft.htm
• Same as complex example 1, but you can use the
  trick for k times

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Competition problems

• HKOI2000 S Wormhole Labyrinth
• HKOI2001 S A Node Too Far
• HKOI2004 S Teachers Problem
• TFT2001 OIMan
• TFT2002 Bomber Man
• NOI2001 cung1 ming4 dik7 daa2 zi6 jyun4
• NOI2001 Equation
• IOI2000 Walls
• IOI2002 Troublesome Frog
• IOI2003 Amazing Robots

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Teachers 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

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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)

PWWPPPPPPWWPOMMV
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• What you have learnt
• Graph Modeling

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Variations of BFS and DFS

• Bidirectional Search (BDS)
• Iterative Deepening Search(IDS)

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Bidirectional search (BDS)

• Searches simultaneously from both the start
  vertex and goal vertex
• Commonly implemented as bidirectional BFS

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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
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Bomber Man (1 Bomb)
S








E
1
2
3
4
11
12
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13
4
10
6
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7
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5
9
1
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Shortest Path length 8
What will happen if we stop once we find a path?
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Example
S 1 2 3 4 5 6 7 8 9
10
21 11
20 12
19 13
18 14
17 17 16 15
11 12 13 14 15 16
10
9 8 7 6 5 4 3 2 1 E
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Iterative deepening search (IDS)

• Iteratively performs DFS with increasing depth
  bound
• Shortest paths are guaranteed

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IDS
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IDS (pseudo code)

• DFS (vertex u, depth d)
• mark u as visited
• if (dgt0)
• for each vertex v directly reachable from u
• if v is unvisited
• DFS (v,d-1)
• i0
• Do
• DFS(start vertex,i)
• Increment i
• While (target is not found)

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IDS

• The complexity of IDS is the same as DFS

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Summary of DFS, BFS

• We learned some variations of DFS and BFS
• Bidirectional search (BDS)
• Iterative deepening search (IDS)

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Equation

• Question Find the number of solution of xi,
  given ki,pi. 1ltnlt6, 1ltxilt150
• Vertex possible values of
• k1x1p1 , k1x1p1 k2x2p2 , k1x1p1 k2x2p2
  k3x3p3 , k4x4p4 , k4x4p4 k5x5p5 , k4x4p4
  k5x5p5 k6x6p6