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Data structures & Algorithms
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What is a Graph? how highly connected is an entity within a network? What is an entity's overall importance in a network ?

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ANSWER ALL QUESTIONS

What is a Graph?

  • A graph is a data structure representing connections between items
    • We're storing values with the potential for connections between any or all elements
  • Examples of graphs in everyday life:
    • PERT charts
    • Flow charts
  • Examples in computer science
    • Networks
    • Social Network diagrams
    • Executing a makefile

 

Social Networks

Using Social Network Analysis, you can answer:

  • How highly connected

is an entity within a network?

  • What is an entity's

overall importance in a network?

  • How central is an entity within a network?
  • How does information

flow within a network?

Makefiles     

How make Works

  • Construct the dependency graph from the target and dependency entries in

the makefile      

  • Do a topological sort to determine an order in which to construct
  • For each target visited, invoke the commands if the target file does not exist or if any dependency file is newer
    • Relies on file modification

dates

Set Theory

  • A set is any collection of objects, e.g. a set of vertices
  • The objects in a set are called the elements of the set

 

  • Repetition and order are not important
    • {2, 3, 5} = {5, 2, 3} = {5, 2, 3, 2, 2, 3}

 

  • Sets can be written in predicate form:
    • {1, 2, 3, 4} = {x : x is a positive integer less than 5}
  • The empty set is {} = 0, all empty sets are equal

 

Elements and Subsets

  • x ∈ A means “x is an element of A”
    • x ∈ A means “x is not an element of A”

 

  • A ⊆ B means “A is a subset of B”
    • x ⊆ A means “x is not a subset of A”

 

  • If A ⊆ B and A ≠ B…

A is a proper subset of B and we write A ⊂                                                                               B

  • A ⊆ A for every set A
    • Every set is a subset of itself

Subsets

  • If A ⊆ B and B ⊆  C then A ⊆  C
  • If A ⊆ B and B ⊆ A then A = B

 

  • The empty set is a subset of every set:

⊆ A for any A

 

  • The subsets of A ={1, 2, 3} are:

 

,{1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}

(Sometimes called the powerset of A, P(A) )

Operations on Sets

  • Union
    • A ∪ B = { x : x ∈ A or x ∈ B }

 

  • Intersection
    • A ∩ B = { x : x ∈ A and x ∈ B }

 

  • Universal Set
    • All sets under consideration will be subsets of a background set, called the Universal Set, U
  • Complement
    • A' = { x : x ∈ U and x ∈ A }

Example

  • Let:
    • U = {a, b, c, d, e, f}
    • A = {a, c}
    • B = {b, c, f}
    • C = {b, d, e, f}

 

  • Then:
    • B ∪ C = {b, c, d, e, f}
    • A ∩ (B ∪ C) = {c}
    • A' = {b, d, e, f}

= C

  • A' ∩ (B ∪ C) = C ∩ (B ∪  C) = {b, d, e, f} = C
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