Differences Between Recursive and Iterative Implementations of DFS and BFS

Depth-First Search (DFS)

1. Recursive Implementation:

   • Uses the call stack to go back to previous points.
   • Takes up space: about $O(h)$, where $h$ is how tall the graph or tree is.
   • The code is easier to understand and write.

2. Iterative Implementation:

   • Uses a special stack to keep track of where to go next.
   • Takes up more space: about $O(b^d)$, where $b$ is how many branches there are, and $d$ is the depth or how far down we go.
   • Can work well with deeper graphs without causing errors from too much data.

Breadth-First Search (BFS)

1. Recursive Implementation:

   • Not often used because BFS needs a queue instead.
   • Not very practical and can be less efficient.

2. Iterative Implementation:

   • Uses a queue to look at each level of neighbors one by one.
   • Takes up space: about $O(b^d)$, which is similar to DFS but depends on how the graph is made.
   • Helps find the shortest path in graphs that don't have weights.

In general, recursive methods are neat and simple but can hit limits on stack size. On the other hand, iterative methods can be more complex but are great for handling large and deep graphs.