Graph Theory

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Graph Theory

• Kayman Lui
• 20-09-2007

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Overview

• Graph
• Notation and Implementation
• Tree
• Depth First Search (DFS)
• DFS Forests
• Topology Sort (T-Sort)
• Strongly Connected Component (SCC)
• Breadth First Search (BFS)
• Graph Modeling
• Variations of BFS and DFS
• Bidirectional Search (BDS)
• Iterative Deepening Search(IDS)

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What is a graph?

• A set of vertices and edges
• Directed/Undirected
• Weighted/Unweighted
• Cyclic/Acyclic

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Representation of Graph

• Adjacency Matrix
• A V x V array, with matrixij storing whether
  there is an edge between the ith vertex and the
  jth vertex
• Adjacency Linked List
• One linked list per vertex, each storing directly
  reachable vertices
• Edge List

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Representation of Graphs
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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 Searching

• 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 searching
  algorithms
• Some shortest path algorithms and spanning tree
  algorithms have specific visit order

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Depth-First Search (DFS)

• Strategy Go as far as you can (if you have not
  visit there), otherwise, go back and try another
  way
• Example a person want to visit a place, but do
  not know the path

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DFS (Demonstration)
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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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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 forest (Demonstration)
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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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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
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Applications of DFS Forests

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

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Topological Sort

• 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
• Topological Sort Finding the topological order
  of a directed acyclic graph

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Example

• Assembly Line
• In a factory, there is several process. Some need
  to be done before others. Can you order those
  processes so that they can be done smoothly?
• Studying Order
• Louis is now studying ACM materials. There are
  many topics. He needs to master some basic topics
  before understanding those advanced one. Can you
  help him to plan a smooth study plan?

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T-sort 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)
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? D E A B F C G
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Strongly-connected components (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)
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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)

• Instead of going as far as possible, BFS goes
  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
• BFS makes use of a queue to store visited (but
  not dead) vertices, expanding the path from the
  earliest visited vertices.

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BFS (Demonstration)
Queue
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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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Applications of BFS

• Shortest paths finding
• Flood-fill

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Flood Fill

• An algorithm that determines the area connected
  to a given node in a multi-dimensional array
• Start BFS from the given node, counting the total
  number of nodes visited
• It can also be handled by DFS

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Comparisons of DFS and BFS
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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 in many ACM problems

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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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Examples(1)

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

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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 Alpha number, what is the
  minimum number of calls you need to make?

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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
  Alphas vertex

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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 T-sort

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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.

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Bomber Man (1 Bomb)
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Shortest Path length 8
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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 Complexity (the details can be skipped)

• ( )bm
• ( ) bm

- b is branching factor- td is the number of
vertices visited for depth d
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Conclusion

• The complexity of IDS is the same as DFS

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Other Topics in Graph Theory

• Cut Vertices Cut Edges
• Euler Path/Circuit Hamilton Path/Circuit
• Planarity

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Cut Vertices Cut Edges

• What is a cut vertex?
• The removal of a set of vertices causes a
  connected graph disconnected
• What is a cut edge?
• The removal of a set of edges causes a connected
  graph disconnected

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Euler Path Hamilton Path

• An Euler path is a path in a graph which visits
  each edge exactly once
• A Hamilton path is a path in an undirected graph
  which visits each vertex exactly once.

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Planarity

• A planar graph is a graph that can be drawn so
  that no edges intersect
• K5 and K3,3 are non-planar graphs

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Last QuestionEquation

• 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

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Graph problems in Uva

• 280
• 336
• 532
• 572
• 10592