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Intro to

Graphs

Concept

Graphs are a set of vertices connected by edges

 

 

 

 

 

 

Each item in the graph contains:

  1. Stored data -- aka node value
  2. Stored references -- aka relationships aka edges -- to zero or more other nodes

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Vocabulary

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Edges

  represent the connection b/t 2 vertices

  can be directed or undirected

Vertices

  nodes in the graph

Path

  a sequence of connected vertices

  a simple path has no repeated vertices

Cycles

  a path that is cyclical

  an acyclic graph has no cycles

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Adding an edge

Deleting an edge

Detecting an edge

Finding the neighbors of a vertex

Finding a path between two vertices

Common Operations

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Representing

Graphs

Text

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Adjacency Matrix

Undirect Graph

Text

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Adjacency Matrix

Directed Graph

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Adjacency Matrix

Weighted Directed Graph

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Pseudocode

Constructor

addNode()

addEdge()

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Adjacency List

Directed Graph

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Pseudocode

Constructor

addNode()

addEdge()

Consider:

How could we represent (un)directed edges?

How could we represent weighted edges?

Exercise Time!

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Text

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Intro to

Depth-First Search

DFS

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Why? Find paths, cycles, connectivity and more!

Concepts:

Explored (black), Visited (gray), Undiscovered (white)

Graph Traversing

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Pseudocode

//Code here

Exercise Time!

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  1. Mark v as discovered (grey).

  2. For all unvisited (white) neighbors w of v:

    1. Visit vertex w.

  3. Mark v as explored (black).

Procedure

O(n)
O(n)O(n)

Complexity

Depth-First Search (DFS)

Time

Text

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Intro to

Breadth-First Search

BFS

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Concept: BFS

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Diagram: BFS

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Pseudocode

//Code here

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  1. Create a queue Q.

  2. Mark v as discovered (grey) and enqueue v into Q.

  3. While Q is not empty, perform the following steps:

    1. Dequeue u from Q.

    2. Mark u as discovered (grey).

    3. Enqueue all unvisited (white) neighbors w of u.

    4. Mark u as explored (black).

Procedure

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1. BFS Function

   - queue

   - recursion

 

2. Tree vs Graph Data Structure

Interface: BFS

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O(n)

Time Complexity

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Shortest path finding

Web crawlers

Use Cases: BFS

Exercise Time!

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Recap

  • DFS and BFS can be applied to graphs and trees
  • information about how data is organized in the graph or tree can aid in determining whether to use DFS or BFS
  • BFS can be implemented using helper data structures
  • DFS can be implemented recursively and is the simpler of the two methods to implement

Recap

Concepts

Exercise Time!

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Graphs & Paths

By Bianca Gandolfo

Graphs & Paths

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