Computer Science II

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Computer Science II

• Graphs

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What is a Directed Graph?

• An ordered pair G (V,E)
• V is a finite, non-empty set of vertices (nodes)
• E is a finite set of ordered pairs, which are the
  edges (arcs) of G
• E ? V ? V

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GraphTerminology

• Vertex b is adjacent to vertex a
• Edge emanates from vertex a
• A(a) is the set of edges emanating from vertex a
• A(a) is the out-degree of vertex a
• Edge is incident on vertex b
• I(b) is the set of edges incident on vertex b
• I(b) is the in-degree of vertex b

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Paths

• A path in a directed graph G (V,E) is a
  non-empty sequence of verticesP
  v1,v2,...,vkwhere vi ? V for 1 ? i ? ksuch
  that (vi,vi1) ? E for 1 ? i lt k
• The length of the path P is k-1

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PathTerminology

• Vertex b is the successor of Vertex a
• Vertex b is the predecessor of Vertex c
• Simple paths have no repeated vertices
• A cycle is where v1 vk

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More on graphs

• An acyclic graph has no cycles
• Undirected graphs are similar to directed graphs
  except that E is a set of unordered pairs

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Representing graphs

• A graph can be represented by an adjacency matrix
• For each edge (vi,vj) ? E, M(i,j) 1
• All other positions in the matrix are set to 0
• Sparse matrices (many vertices, few edges) can be
  implemented as linked list of linked lists

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Questions?

• Given the following graph, how would you
  represent it as an adjacency matrix?
• How would you determine if there is a path from
  vertex a to vertex d?

a
b
d
c