Dynamic Graphs: BFS

Chapter 1: Breadth-First Search (BFS) on Dynamic Graphs

Introduction

Breadth-First Search (BFS) is one of the most fundamental algorithms for traversing or searching tree or graph data structures. The concept of dynamic graphs extends the application and makes it more versatile by allowing vertices and edges to be added or removed over time. In this chapter, we'll explore the intricacies of applying BFS on dynamic graphs.

Standard BFS Algorithm

Before diving into dynamic graphs, let's recall the standard BFS algorithm. Given a graph \( G = (V, E) \) where \( V \) is the set of vertices and \( E \) is the set of edges, BFS starts from a source vertex \( s \) and explores its neighbors before moving on to their neighbors.

  BFS(Graph G, Node s):
    1. Initialize queue Q with s
    2. Mark s as visited
    3. while Q is not empty:
        4. u = Q.dequeue()
        5. for each neighbor v of u:
            6. if v is not visited:
                7. Mark v as visited
                8. Q.enqueue(v)

Dynamic Graphs

A dynamic graph \( G_t = (V_t, E_t) \) at time \( t \) can change over time, i.e., vertices and edges can be added or removed. For BFS, this adds additional challenges:

• When vertices are added or removed, the BFS tree may need to be updated.
• When edges are added or removed, the shortest paths may change, requiring an update to the BFS tree.

BFS on Dynamic Graphs

There are various approaches to maintaining a BFS tree over a dynamic graph:

1. Recomputing from Scratch: One straightforward approach is to recompute the BFS tree from the source vertex whenever the graph changes.
2. Incremental Updates: In this method, the BFS tree is updated incrementally when a change occurs, avoiding a full recomputation.

Algorithm for Incremental Updates

The incremental update algorithm works as follows:

  IncrementalBFS(Graph G_t, Node s, BFS_Tree T):
    1. Detect changes (added/removed vertices and edges)
    2. Update T according to the changes
    3. Correct any inconsistencies in T (e.g., recompute paths if needed)

Conclusion

BFS is a robust algorithm that can also be applied to dynamic graphs. Understanding how to efficiently update the BFS tree as the graph changes is key to its effective application in dynamic scenarios.