Question

Suppose we represent a graph G having n vertices and m edges with the edge list structure. Why, in this case, does the insertVertex function run in O(1) time while the eraseVertex function runs in O(m) time?

Short answer

InsertVertex runs in O(1) because it only adds a vertex. EraseVertex runs in O(m) because it must scan and remove all edges.

Step-by-step solution

Understand the Edge List Structure

In an edge list structure, a graph is represented as an array or list of edges. Each edge records the pair of vertices it connects. If there are n vertices and m edges, the edge list has m entries.

Inserting a Vertex

When inserting a new vertex, we only need to expand the list of vertices by one. This operation doesn't depend on the number of edges (m) or vertices (n) and can be done in constant time, O(1).

Erasing a Vertex

When erasing a vertex, we must also remove all edges associated with that vertex. Since we represent the graph with an edge list, this operation involves scanning through all m edges to find and remove those connected to the vertex. Therefore, this operation runs in O(m) time.

Key concepts

Edge List Structure

An edge list structure is a simple way to represent a graph. In this representation, a graph is defined by two main components: vertices and edges. Each edge is an entry that describes a connection between a pair of vertices. When you have n vertices and m edges, you maintain a list that contains all m edges. This structure is versatile and straightforward.

Key characteristics include:

• Ease of understanding and implementation
• Memory efficiency for sparse graphs (with relatively few edges compared to the number of vertices)
• Suitability for algorithms that need to iterate over edges

In this structure, each edge explicitly records its two endpoints. This helps in quickly navigating connections, especially when determining adjacency or traversing the graph.

Vertex Insertion

Inserting a new vertex into a graph using the edge list structure is a quick and straightforward process. The reason for this simplicity lies in the nature of how vertices are stored and managed.

When you add a new vertex:

• You simply extend the list or array of vertices to include the new one
• No need to adjust or interact with the list of edges
• This operation is independent of the number of edges (m) and existing vertices (n)

Because of these points, the insertion of a vertex runs in constant time, denoted as O(1). This means no matter how large your graph is, adding a new vertex will always take the same amount of time - a huge benefit for maintaining efficiency.

Vertex Deletion

Erasing a vertex from a graph in an edge list structure is more involved compared to insertion. This complexity arises because it not only involves removing the vertex but also requires adjusting the edges.

Steps involved in vertex deletion:

• Locate the vertex to be removed
• Scan through the entire list of edges (m in total) to find and delete edges that are connected to that vertex
• Remove the vertex from the list of vertices

The critical part of this process is scanning through all the edges, which makes the time complexity O(m). This dependence on the number of edges is what makes vertex deletion a more time-consuming operation compared to vertex insertion.

Because every edge must be checked to see if it connects to the vertex being erased, this step ensures that the graph remains accurate and connected properly after the removal.