Graphs in data Structures, Slides of Data Structures and Algorithms

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Graphs

Data Structures

What is a graph?

• A data structure that consists of a set of nodes ( vertices ) and a set of edges that relate the nodes to each other
• The set of edges describes relationships among the vertices

Directed vs. undirected graphs

• When the edges in a graph have no

direction, the graph is called undirected

• When the edges in a graph have a direction,

the graph is called directed (or digraph )

Directed vs. undirected graphs (cont.) E(Graph2) = {(1,3) (3,1) (5,9) (9,11) (5,7) Warning : if the graph is directed, the order of the vertices in each edge is important !!

Graph terminology

• Adjacent nodes: two nodes are adjacent if they are connected by an edge
• Path: a sequence of vertices that connect two nodes in a graph
• Complete graph: a graph in which every vertex is directly connected to every other vertex 5 is adjacent to 7 7 is adjacent from 5

• What is the number of edges in a complete directed graph with N vertices? N * (N-1) Graph terminology (cont.) 2 O N ( )

• Weighted graph: a graph in which each edge carries a value Graph terminology (cont.)

Graph implementation

• Array-based implementation
  • (^) A 1D array is used to represent the vertices
  • (^) A 2D array (adjacency matrix) is used to represent the edges

Graph implementation (cont.)

• Linked-list implementation
  • (^) A 1D array is used to represent the vertices
  • (^) A list is used for each vertex v which contains the vertices which are adjacent from v (adjacency list)

Linked-list implementation

Graph searching

• Problem :^ find a path between two nodes of

the graph (e.g., Austin and Washington)

• Methods :^ Depth-First-Search^ (DFS)^ or

Breadth-First-Search (BFS)

Depth-First-Search (DFS)

• What is the idea behind DFS?
  • (^) Travel as far as you can down a path
  • (^) Back up as little as possible when you reach a "dead end" (i.e., next vertex has been "marked" or there is no next vertex)
• DFS can be implemented efficiently using a

stack

start end (initialization)