Data Structures - Graph

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Data Structures - Graph

• Graphs are simply a collection of nodes (points)
  and edges connecting the nodes.
• Typical notation lookslike this G N, E .
• G the graph.
• N node set
• E edge set
• Nodes represent items under consideration - can
  be anything.
• Edges represent relationships between the items -
  can be anything.
• Edges can be undirected, meaning the relationship
  holds both ways between nodes.
• Edges can be directed, meaning the relationship
  is only one way.

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Data Structures - Graph
Name Graph
Definition A set of nodes (data elements) connected by edges in an arbitrary manner.
Std Functions None
Non-Std Functions None
Notes The most versatile data structure (linked lists, trees and heaps are special instances of graphs). Standard Problems Graph Coloring Coloring the nodes of a graph such that adjacent nodes do not have the same color. Traveling Salesman Visiting each node in the graph exactly once for the least cost (start and stop at the same node). Maximum Flow Determine the amount of flow through a network.
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Data Structures - Graph
THESE ARE ALL GRAPHS.
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Graph Problems

• There are literally thousands of graph problems,
  but we will focus on three that are occur very
  commonly and show the diversity of the graph
  structure
• The Traveling Salesman Problem.
• Graph Coloring Problem.
• Maximum Flow Problem.
• At least one of these problems is solved by you
  every day without you realizing it (until now).
• The fact that the nodes and edges can represent
  anything means that the graph structure is very
  versatile and virtually any problem can be mapped
  to a graph problem.

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Example Graph Problem - Puzzle

• This is an old puzzle and has many variants A
  man is returning home from market with a wolf,
  bag of corn, and goose. He has to cross a lttle
  river on the way and the only way is to use a
  little boat. The boats holds him and one other
  item.
• He cannot leave the wolf and goose alone, as the
  wolf eats the goose.
• He cannot leave the corn and goose alone as the
  goose eats the corn.
• He can leave the corn and wolf alone.
• How does he get everything across the river and
  bring everything home (uneaten)?
• Build a graph
• N the set of possible arrangements of man,
  wolf, goose, corn, and river.
• E Possible transitions due to one boat trip.

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Example Graph Problem - Puzzle
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Example Graph Problem - Puzzle Solved
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Graph Problems

• There are literally thousands of graph problems,
  but we will focus on three that are occur very
  commonly and show the diversity of the graph
  structure
• The Traveling Salesman Problem.
• Graph Coloring Problem.
• Maximum Flow Problem.
• Each problem has a decision form and an
  optimization form. The decision form asks "Can
  we do it?" and the optimization form asks "How
  well can we do it?"
• At least one of these problems is solved by you
  every day without you realizing it (until now).
• The fact that the nodes and edges can represent
  anything means that the graph structure is very
  versatile and virtually any problem can be mapped
  to a graph problem.

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Graph Problems - Traveling Salesman

• Description Given a graph, G N, E, where
• N a set of cities.
• E travel routes between the cities, each having
  a cost associated with it.
• One special node, s.
• You must begin at city sand travel to each of the
  other cities exactly once and then return to city
  s. Thus you make a complete cycle of all cities
  in the graph.
• Decision form of the problem Can a route be
  found where the total cost of the trip is less
  than X? (Answer is yes or no).
• Optimization form of the problem What is the
  absolute lowest cost?

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Graph Problems - Graph Coloring

• Description Given a graph, G N, E, where
• N a set of nodes.
• E edges between the nodes.
• The object is to color the graph such that no
  nodes connecte by an edge have the same color.
• Decision form of the problem Can the graph be
  colored with X or less colors? (Answer is yes or
  no).
• Optimization form of the problem What is the
  fewest number of colors required to color the
  graph?

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Graph Problems - Maximum Flow

• Description Given a graph, G N, E, where
• N a set of nodes.
• E edges representing pipes, each assigned a
  given capacity.
• Two special nodes. Node s is a source node that
  can potentially spit out an infinite amount of
  material. Node f is a sink node that can
  potentially absorb an infinite amount of
  material.
• The object is to determine the maximum amount of
  material that can flow through the network for
  the source to the sink.
• Decision form of the problem Can X amount of
  material be pushed through the network from the
  source to the sink? (Answer is yes or no).
• Optimization form of the problem What is the
  maximum amount of material that can flow through
  the material from the source to the sink?

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Cost of Graph Problems
Name Description Cost Comments
Traveling Salesman Decision Does a route exist with cost less than X? O(n!) Hard to solve, easy to verify.
Traveling Salesman Optimization What is the least cost route. O(n!) Hard to solve, hard to verify.
Graph Coloring Decision Can a graph be colored properly using X colors? O(n!) Hard to solve, easy to verify.
Graph Coloring Optimization What is the least number of colrs required to properly color a graph? O(n!) Hard to solve, hard to verify.
Maximum Flow What is the most material that can be pushed through a network? O(n3) Polynomial solution time means easy to solve, easy to verify.
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Data Structures - Tree
Name Tree
Definition A graph with directed edges connecting parent to child such that there is exactly one node with no parents and all other nodes have exactly one parent.
Std Functions None
Non-Std Functions None
Notes The first element in the tree is the root node, which has no parents, and from which all others can be reached. Nodes with no children are "leaf" nodes. If nodes a and b are connected by an edge, then a is a child of b if b is closer to the root than a. a is a parent of b if a is closer to the root than b Useful in making decisions and categorizing data.
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Data Structures - Tree
ROOT - has no parent, only one in the tree.
LEAVES - have no children.
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Data Structures - Heap
Name Heap
Definition A tree in which a parent node has a value larger than all its children.
Std Functions Heap(a, H) Add new node a to heap H. Unheap(H) Remove the root element from heap H and re-establish the heap.
Non-Std Functions None
Notes Flexible data structure useful for sorting elements as they arrive. This allows sorting on lists whoses size change constantly. Used in priority queues or other situations where maintaining and accessing a maximum element is important.
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Data Structures - Graph
THESE ARE TREES (ABOVE). ARE THEY HEAPS?
THESE ARE NOT TREES (ABOVE). ARE THEY HEAPS?