Given below are two statements:
Statement I: Breadth-First Search is optimal when all the step costs are equal, whereas uniform-cost search is optimal with any step cost.
Statement II: When all the step costs are the same, uniform-cost search expends more nodes at depth d than the Breadth-First Search.
In light of the above statements, choose the correct answer from the options given below:

Correct option is A

Statement I correctly identifies the conditions for the optimality of Breadth-First Search (BFS) and Uniform-Cost Search (UCS). Statement II accurately describes the behavior of Uniform-Cost Search when step costs are uniform, where it expands more nodes at depth d compared to BFS.
Information Booster
1. Statement I:
· Breadth-First Search (BFS): BFS is optimal when all step costs are equal. It explores all nodes at a given depth before moving to the next depth, ensuring the shortest path in terms of the number of edges.
· Uniform-Cost Search (UCS): UCS is optimal regardless of the step costs because it expands the least-cost node first, guaranteeing the shortest path in terms of total cost.
2. Statement II: When step costs are uniform, UCS behaves similarly to BFS in terms of traversing layers. However, it evaluates nodes based on their cumulative cost (depth times uniform cost), which can result in additional nodes being expanded at depth d.
3. Key Differences:
· BFS uses a queue and explores level-by-level.
· UCS uses a priority queue based on path cost, expanding nodes in increasing cost order.
4. Example: Consider a graph where all step costs are equal to 1. BFS and UCS will both explore the shallowest nodes first but UCS may perform extra checks (expanding more nodes) based on cumulative cost calculations.
Additional Knowledge
· Breadth-First Search:
· Time Complexity: O(V + E), where V is vertices and E is edges.
· Space Complexity: O(V).
· Uniform-Cost Search:
· Time Complexity: Depends on the branching factor and cost distribution but typically O(bd), where b is the branching factor and d is the depth of the solution.
· Space Complexity: O(bd).