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Data Structures - Graph

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.

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Data Structures - Graph

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  1. 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.

  2. Data Structures - Graph

  3. Data Structures - Graph THESE ARE ALL GRAPHS.

  4. 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.

  5. 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.

  6. wgc| m mwgc| mwg|c mwc| g mgc| w mc| wg mg| wc mw|gc | mwgc wg|mc wc| mg gc| mw c| mwg g| mwc w|mgc m| wgc Example Graph Problem - Puzzle

  7. wgc| m mwgc| mwg|c mwc| g mgc| w mc| wg mg| wc mw|gc | mwgc wg|mc wc| mg gc| mw c| mwg g| mwc w|mgc m| wgc Example Graph Problem - Puzzle Solved

  8. 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.

  9. 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?

  10. 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?

  11. 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?

  12. Cost of Graph Problems

  13. Data Structures - Tree

  14. Data Structures - Tree ROOT - has no parent, only one in the tree. LEAVES - have no children.

  15. Data Structures - Heap

  16. Data Structures - Graph THESE ARE TREES (ABOVE). ARE THEY HEAPS? THESE ARE NOT TREES (ABOVE). ARE THEY HEAPS?

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