Breadth-first search (BFS) is a fundamental algorithm in computer science and artificial intelligence used for traversing or searching tree or graph data structures. Its time complexity is commonly expressed as $O(V + E)$, where $V$ is the number of vertices and $E$ is the number of edges in the graph. Alternatively, in the context of tree-based searches, BFS time complexity is denoted as $O(b^d)$, where $b$ is the branching factor and $d$ is the depth of the shallowest solution.

This article will explore how these two expressions of time complexity are aligned and explain the reasoning behind them.

Understanding the Complexities: $O(V + E)$ and $O(b^d)$

BFS in Graph Searches: $O(V + E)$

In graph searches, BFS explores each node and all of its neighbors level by level. The overall time complexity of BFS in this context is $O(V + E)$. Here's why:

• Vertices (V): BFS visits every vertex in the graph once. Thus, if there are $V$ vertices, BFS will take $O(V)$ time visiting them all.
• Edges (E): BFS also examines each edge to discover new nodes. In the worst case, the algorithm will explore all edges, leading to an additional $O(E)$ time complexity.

Thus, the time complexity of BFS for graphs is the sum of the time to process vertices and edges, resulting in $O(V + E)$.

BFS in Trees/Search Spaces: $O(b^d)$

In tree-based searches or game theory, BFS is often expressed as $O(b^d)$:

• Branching Factor (b): This denotes the average number of children per node. In other words, at every level, each node generates $b$ children.
• Depth (d): This represents the minimum depth at which BFS finds the goal or solution.

The total number of nodes traversed by BFS in a balanced tree structure can be represented by the sum of a geometric series:

$$N = 1 + b + b^2 + \cdots + b^d$$

In big O notation, this simplifies to $O(b^d)$ because the last term in the series $b^d$ will dominate as $d$ becomes large. This is especially useful in AI applications for determining space complexities of searches in state spaces.

Bridging the Gap Between $O(V + E)$ and $O(b^d)$

Although BFS has different complexity expressions based on context, the underlying algorithm remains the same. While $O(V + E)$ is suitable for sparse graphs, $O(b^d)$ becomes relevant in scenarios where the graph behaves like a perfect tree. Here’s how both concepts can converge:

• Sparse Graphs: If the graph is sparse (i.e., $E \leq b \cdot V$), the time complexity $O(V + E)$ aligns more closely with $O(b^d)$ as the number of vertices and edges directly affects how far and wide the search extends.
• Balanced Trees: In cases where search trees are balanced and well-structured (each node has approximately $b$ children), $O(b^d)$ provides a better representation because it captures how the tree expands into levels.

This dual expression of time complexity in BFS highlights its flexibility across different data structures and problem requirements.

Comparative Table

The following table provides a comparative view of BFS complexity in graph and tree structures:

Conclusion

Breadth-first search remains a versatile and essential algorithm for both graph traversals and tree-based searches. Understanding its complexity through both $O(V + E)$ and $O(b^d)$ perspectives helps practitioners apply BFS efficiently across various domains. Each complexity expression addresses different aspects of how BFS processes nodes and explores search spaces, making it adaptable to a wide range of problem domains, including networking, AI, and beyond.