Graph Theory, DFS

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Graph Theory, DFS BFS

• Kelly Choi
• 08-07-2006

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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 Graphs

• 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
Adjacency Matrix Adjacency Linked List Edge List
Memory Storage O(V2) O(VE) O(VE)
Check whether (u,v) is an edge O(1) O(deg(u)) O(deg(u))
Find all adjacent vertices of a vertex u O(V) O(deg(u)) O(deg(u))
deg(u) the number of edges connecting vertex u deg(u) the number of edges connecting vertex u deg(u) the number of edges connecting vertex u deg(u) the number of edges connecting vertex u
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Graph Searching

• Why do we do graph searching? What do we search
  for?
• What information can we find from graph
  searching?
• How do we search the graph? Do we need to visit
  all vertices? In what 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

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Implementation

• 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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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 Teachers Problem

• Emily wants to distribute candies to N students
  one by one, with a rule that if student A is
  teased by B, A can receive candy before B.
• Given lists of students teased by each students,
  find a possible sequence to give the candies

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Shortest Paths?

• Given vertices s and t, if we try to find a path
  from s to t by DFS, the first path found may not
  be the shortest one.
• How can I find a shortest path?
• Lets first review what we have done

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An Idea of What We Did

• Start by visiting a certain vertex
• Vertices fall into 3 categories
• Unvisited
• Visited
• Visited Dead (All reachable vertices from these
  vertices are visited)

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An Idea of What We Did

• Find unvisited vertices by expanding your path
  from visited (but not dead) vertices.
• In DFS, we choose the most recently visited
  vertex to expand.
• Are there any other strategies?

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Breadth-First Search (BFS)

• Instead of going as far as possible, BFS tries to
  search all paths.
• 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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Simulation of BFS

• Queue

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Implementation

• while queue Q not empty
• dequeue the first vertex u from Q
• for each vertex v directly reachable from u
• if v is unvisited
• enqueue v to Q
• 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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There is more

• Other Graph Searching Algorithms
• Bidirectional search (BDS)
• Iterative deepening search (IDS)

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Advantages

• Guarantee shortest paths for unweighted graphs
• Use queue instead of recursive functions
  Avoiding stack overflow

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Flood Fill

• An algorithm that determines the area connected
  to a given node in a multi-dimensional array
• Start BFS/DFS from the given node, counting the
  total number of nodes visited
• Example Squareland (HKOI 2006)

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Graph Modeling

• Conversion of a problem into a graph problem
• Essential in solving most graph problems

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Basics of graph modeling

• 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
• (cx, 2004)

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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?
• A cut vertex is a type of graph vertex, the
  removal of which causes an increase in the number
  of connected components
• What is a cut edge?

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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