Chapter 7: Graphs

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Chapter 7 Graphs
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Introduction

• In nonlinear data structure, each element may
  have several next elements, which introduces
  the concept of branching structures

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Graph

• A graph is a set of points and a set of lines,
  which each line joining one point to another.
• The points are called nodes
• Lines are called edges

• A set of nodes in the graph G
• VG a,b,c,d
• A set of edges in the graph G
• EG 1,2,3,4,5,6,7,8
• The number of elements in VG is called the order
  of graph G.
• A null graph is a graph with order zero

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Graph

• An edge is determined by the node it connects.
• Edge 4, for example, connects nodes c and d and
  is said to be of form (c,d).
• A graph is completely determined by its set of
  nodes and set of edges.
• The actual positioning of these elements on the
  page is unimportant.
• So the above two graphs are equivalent.

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Graph

• There may be multiple edges connecting two nodes,
  e.g. edge 5, 6 and 7 all of form (b,d).
• Some pairs of nodes may not be connected for
  example, there is no edge to form (a,c) or (a,d).
  
• Some edges may connect one node with itself e.g.
  edge 8 is of form (a,a). These are called
  loops.

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Graph

• Simple graph is a graph G where two of the
  following conditions are true
• It has no loop, that is, there does not exist an
  edge in EG of form (V,V) where V is in VG.
• No more than one edge joins any pair of nodes,
  that is, there does not exist more than one edge
  in EG of form (V1, V2) for any pair of elements
  V1, and V2 in VG

d
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Graph

• Multigraph is a graph that is not a simple graph.
• Edges are sometime referred to as arcs
• Nodes are sometime termed vertices
• A connected graph is a graph that cannot be
  partitioned into two graphs without removing at
  least one edge.
• Example of unconnected graph

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Paths

• A path in a graph is a sequence of one or more
  edges that connects two nodes.
• The length of a path is the number of edges that
  it comprises.
• The followings are path between nodes b and d.
• P(b,d) (b,c) (c,d) length 2
• P(b,d) (b,c) (c,b) (b,c) (c,d) length 4
• P(b,d) (b,d) length 1
• P(b,d) (b,c) (c,b) (c,d) length 3
• In general, we are interested only in paths in
  which a given node is visited no more than once.
  This restricts out interest to only the first and
  third paths above. The second path visits both
  node b and c twice the fourth path visits node b
  twice.
• We will always be interested in not traversing
  the same edge more than once in a path.

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Cycles

• A cycle is a path in which both of the following
  conditions are true
• No edge appears more than once in the sequence of
  edges.
• The initial node of the path is the same as the
  terminal node of the path i.e. P(V,V).
• A cycle returns to where it started.
• Examples
• P(a,a) (a,a)
• P(b,b) (b,c) (c,b)
• P(b,b) (b,c) (c,d) (d,b)
• P(d,d) (d,b) (b,d)
• P(d,d) (d,b) (b,c) (c,d)

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Cycles

• A graph with no cycles is said to be acyclic.

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Directed Graphs

• Another special case of the general graph data
  structure is a directed graph, in which
  directionality is assigned to the graphs edges.
• Each edge of a directed graph includes an arrow.
• The in-degree of a node in a directed graph is
  the number of edges that terminate at that node
• The out-degree of a node is the number of edges
  that emanates from that node.
• The degree of a node is the sum of its in- and
  out-degrees.

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Directed Graph

• In-degree(a) 1 out-degree(a) 2 degree(a)
  3
• In-degree(b) 4 out-degree(b) 2 degree(b)
  6
• In-degree(c) 1 out-degree(c) 2 degree(c)
  3
• In-degree(d) ? out-degree(d) ? degree(d)
  ?

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Graphs in Programs

• As with some of the other data structures, the
  common programming languages do not have a
  built-in data type called graph.
• Instead, the graphs characteristics are
  simulated by housing it in another data
  structure.
• There are three major approaches to representing
  graph
• Matrix representation
• List representation
• Multilist representation

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Adjacency Matrix Representation

• One approach to representing this graph is to use
  an adjacency matrix, which is a N-by-N array A.
• If there is an edge connecting nodes i and j,
  then A(i,j) 1. The adjacency matrix for this
  undirected graph is

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Adjacency Matrix Representation

• An edge of a directed graph has its source in one
  node and terminates in another node. The
  adjacency matrix for this directed graph is

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Adjacency Matrix Representation

• The following is example of graph with weighted
  edges.
• The nodes represent cities and the edges
  represent truck routes between cities. Each edge
  is labeled with the distance between the
  connected pair of cities.

• Weighted edges are used frequently.
• In transportation applications, the weights
  represent distances.
• In flow applications, the weights represent
  capacities. For example, the nodes on a graph
  might represent the gallon/minute capacity of a
  pipe-line between the connected location.
• In other applications, the edge weights represent
  time. For instance, graphs can be used to
  represent network of activities.

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Adjacency Matrix Representation
Weighted-edge matrix representation
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Linked representations

• In contrast to the Matrix Representation
  Technique, which stores information about every
  possible edge. The Linked Representations store
  information about only those edges that exist.
• As edges are added or deleted from the graph, the
  linked representation must be modified
  accordingly.
• These are two fundamental types of linked
  structures to represent graphs
• Node directory representation
• Multi-list representation.

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Node Directory Representation

• The node directory representation includes two
  parts a directory and a set of linked list.
• There is one entry in the directory for each node
  of the graph. The directory entry for node I
  points to a linked list that represents the nodes
  that are connected to node i.
• Each record of the linked list has two fields
• One is a node identifier
• One is a link to the next element on the list.
• The directory represents nodes the linked list
  represents edges.

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Node Directory Representation
A node directory representation of the undirected
graph

• An undirected graph of order N with E edges
  requires N entries in the directory and 2E
  linked list entries, except that each loop
  reduces the number of linked list entries by one.

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Node Directory Representation
A node directory representation of the directed
graph

• An directed graph of order N with E edges
  requires N entries in the directory and E linked
  list entries.

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Node Directory Representation

• In the following directed activity graph, each
  node represents an event and each edge represents
  a task whose completion helps to trigger the next
  event, which is the start of other tasks. Each
  edges weight is its required time.

Directory
Edge Information
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Multi-list Representation

• In the multi-list representation of graph
  structure, there are two parts
• A directory of node information
• A set of linked lists of edge information
• There is one entry in the node directory for each
  node of the graph.
• The directory entry for node i points to a linked
  adjacency list of node i.
• Each record of the linked list area appears on
  two adjacency lists one for the node at each end
  of the represented edge.

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Multi-list Representation
Set of edges (1,2),(2,2),(2,3),(4,3),(3,5),(3,6)
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Multi-list Representation
Set of edges (2,1),(2,2),(2,3),(3,4),(5,3),(3,6)
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Graph Traversal

• In many applications it is necessary to visit all
  the nodes of a graph.
• The two basic graph traversal techniques are
• Breadth-first traversal
• Depth-first traversal
• It is usually necessary to take precautions to
  visit each node and edge only once.

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Breadth-first Traversal

• In breadth-first traversal of a graph, one node
  is selected as the start position. It is visited
  and marked, then all unvisited nodes adjacent to
  that nodes are visited and marked in some
  sequential order. Finally, the unvisited nodes
  immediately adjacent to these nodes are visited
  and marked, and so forth, until the entire graph
  has been traversed.

• Breadth-first traversal results in visiting the
  nodes in the following order 1,2,3,4,5,6,7,8.
• The sequence 1,3,2,6,5,4,7,8 is also a valid
  breadth-first traversal visitation order.

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Breadth-first Traversal
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Depth-first Traversal

• Whereas breadth-first traversal of a graph
  proceeds level-by-level, depth-first traversal
  follows first a path from the starting node to an
  ending node, then another path from the start to
  an end, and so forth until all nodes have been
  visited.
• Depth-first traversal results in visiting the
  nodes in the following order 1,2,4,8,5,7,3,6.
• An equally valid sequence to result from
  depth-first traversal is 1,3,6,7,8,2,5,4.

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Depth-first Traversal
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Critical Paths

• An event is critical if slippage of its scheduled
  completion time causes the completion of the
  entire project to be delayed.
• The shortest possible completion time for the
  project is the longest path through the graph.
• The longest path is called the critical path the
  critical events lie along this path.

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Critical Paths

• The earliest start time of event i, ESTi, is the
  earliest possible trigger time for the event.
• The ESTi is calculated from the longest path to
  node i from the start node.
• ESTi maxESTj T(j,i)

• The latest start time of event i, LSTi, is the
  latest possible time that event i can be
  triggered with the graph completing in critical
  path time.
• The set of LSTi are calculated backward from the
  last event.
• LSTi minLSTj - T(i,j)

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Critical Paths

• The difference between the LSTi and ESTi is the
  allowable slippage of event i.
• The events on critical path are 1,3,6,7,8, which
  are called critical events.
• These are the events that need to be controlled
  in order for the project to complete on time.

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Minimal Spanning Trees

• A minimal spanning tree is a tree that contains
  all the nodes of graph where the distances
  occurred in connecting those nodes are minimum.

Minimum total distance is 5,346