Download Graph representation

Graphs
Representations
© Gunnar Gotshalks
GraphRepresentation–1
Adjacency Matrix
• Row and column indices are vertex number
• Matrix entries
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< p , q > ∈ G → Matrix [ p , q ] = True
< p , q > ∉ G → Matrix [ p , q ] = False
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© Gunnar Gotshalks
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Space
Big O?
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GraphRepresentation–2
Adjacency Matrix – 2
• In general the matrix entries can be any data representing
the edge
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True / False are the minimum – an edge exists / ~exists
Integer – 0 / 1 for False / True
Number of parallel edges
Airline flight number
Time to do task (vertices represent checkpoints)
Maximum rate of oil flow in a pipeline
© Gunnar Gotshalks
GraphRepresentation–3
Adjacency List
• The adjacency matrix is very useful but is wasteful of
space due to many ‘zero’ entries
• Instead of a matrix use sequence representations based
on lists – not arrays
• Each vertex is represented as a sequence of adjacent
vertices
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Each vertex in an adjacency list points to its entry in the
list of all vertices
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Each vertex in an adjacency list can also point to edge
data
© Gunnar Gotshalks
GraphRepresentation–4
Adjacency List – Abstract Diagram
© Gunnar Gotshalks
GraphRepresentation–5
Adjacency List – Example
Space
Big O?
Adjacency list is itself a graph
with vertices of mixed types.
© Gunnar Gotshalks
GraphRepresentation–6
Graph Descriptors Vertex Adjacency List
© Gunnar Gotshalks
GraphRepresentation–7
Incidence Matrix – Undirected Graph
• Row indices are edge numbers
Column indices are vertex numbers
• Matrix entries
e = < p , q > ↔ Matrix [ e , p ] = Matrix [ e , q ] = True
e
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© Gunnar Gotshalks
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Space
Big O?
GraphRepresentation–8
Graph Descriptors Edge Incident Lists
© Gunnar Gotshalks
GraphRepresentation–9
Edge List
Space
Big O?
© Gunnar Gotshalks
GraphRepresentation–10
Graph Representation Summary
• Need to maintain three types of sequences
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All vertices sequence
All edge sequence
Adjacency list (slide 11) or incidence list (slide 12)
> One for each vertex
© Gunnar Gotshalks
GraphRepresentation–11
Graph Representation Summary – 2
• Each of the sequence types can be in arrays or linked lists
depending upon
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How dynamic the structures are
> Many insertions and deletions during use, linked
lists are better
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The variance in sequence sizes for adjacency or
incidence lists
> Low variance – all lists are roughly the same length
– can use array, as there would be few unused
positions
> High variance – many short and long lists
– use linked lists
– Linked lists have overhead space for pointers and
extra time to dereference a link.
© Gunnar Gotshalks
GraphRepresentation–12