CS 280 - Data Structures

Introduction to Graphs

• One of the most useful data structures. (A very large topic.)
• Related to trees in that a tree is a special kind of graph (much simpler).
• Graphs are more general and have a wider range of use.
• Represent problems involving interconnected (dependent) objects.
• Graph algorithms are more complex. Need to account for cycles; trees have only one path between nodes.
• Graphs can have several (connected) components, a tree is always a single entity.
• Topological ordering, all nodes "dependent" on X are in list L before X. (Prerequistes in a school.)
• Shortest path, weighted graphs to represent travel destinations and times
• Task networks, time it takes to complete a task before starting the next one.

• A collection of points connected by line segments.
• The points are referred to as nodes or vertices, the segments as edges.
• If the edges have a direction (arrowheads in a diagram), the graph is a directed graph or digraph
• A graph that has values (weights or costs) assigned to the edges is called a weighted graph.

Graph



Directed graph (digraph)



Weighted graph

A graph, G, consists of a set of vertices, V, and edges, E where the edges are constructed from pairs of distinct vertices.

G = (V, E)

e = {v1, v2}

e = (v1, v2)

• Two vertices, x and y are said to be adjacent if there is an edge connecting them.
• We use the notation sGd to mean that s is adjacent to d. With a digraph, sGd implies direction. (xGy is not the same as yGx).
• All nodes adjacent to s is called the adjacency set of s.

• Sequence (contiguous) of edges is a path.
• If there is a path from x to y, y is reachable from x.
• The length of a path is the number of edges on the path. (xGx has length of 1).
• Two vertices are connected if there is a path from one to the other.
• A connected component is a subset, S, of vertices that are all connected.
• Digraphs can be strongly connected or weakly connected. (Definitions vary.)

• A path whose source and destination node are the same.
• A self-loop is a cycle (path length of 1)
• A cycle is simple if all nodes on the path are distinct (with the exception of the first and last).
• If a graph has no cycles, it is acyclic. A directed acyclic graph is called a DAG.

For all nodes x in G, xGx is true:

Reflexive

For all nodes x in G, xGx is false:

Irreflexive

For some nodes x in G (but not all), xGx is true:

Neither reflexive nor irreflexive

For all nodes x and y in G, xGy and yGx is true:

Symmetric

For all nodes x and y in G, xGy and yGx is false:

Antisymmetric

For some nodes x and y in G (but not all), xGy and yGx is true:

Neither symmetric nor antisymmetric

For all nodes x, y, and z in G, if xGy and yGz then xGz:

Transitive

For all nodes x, y, and z in G, if xGy and yGz then not xGz:

Not transitive

Degree

• in-degree - number of incoming edges into a node (node is a destination)
• out-degree - number of outgoing edges from a node (node is a source)
• degree - number of edges (in/out) connecting to a node

1. How do directed and undirected graphs differ?
2. What is a path in a graph? What is a cycle in a graph?
3. What is a simple cycle?
4. What is the in-degree and out-degree of a vertex in a directed graph?

Adjacency Matrix

• A graph G with M nodes represented by a M x M boolean array (matrix).
• For each x and y, G(x,y) = TRUE if xGy, otherwise false.
• Determining if two nodes are adjacent is O(1)
• Space required is O(M²) as well as algorithms that must access all elements

A
 
B

Adjacency List

Traversals

• Depth-first, uses stacks
• Breadth-first, uses queues

1. Describe differences between depth-first search and a bread-first search.
2. How do stacks and queues relate to the search methods?