Here’s the complete and cohesive lecture note that includes the initial introduction to search strategies, followed by the in-depth examination of BFS, DFS, Limited Depth Search, Iterative Deepening, Bidirectional, and Backward Searches.

Lecture Notes on Search Strategies in Problem Solving

Topic: Search Strategies in Artificial Intelligence
Example: Romania Map Navigation Problem

1. Problem Formulation

In designing an artificial agent to solve a problem, the issue is often framed as a state space represented by a graph. The goal is to search for a path from an initial state to a goal state within this graph, with each node representing a possible state in the problem and each edge representing an action to transition between states.

2. Navigation Problem Example

Consider a common problem-solving example: navigating the Romania map to move from Arad to Bucharest. Starting from Arad, the agent explores reachable cities, forming a tree-like structure of possible routes. At each city, the agent evaluates the next steps, building a search tree within the state space. This tree represents various paths the agent could take, allowing it to systematically explore routes until it reaches the goal state, Bucharest.

3. Tree Search Algorithm

A tree search algorithm explores paths by expanding nodes (states) based on available actions, aiming to find a path from the start to the goal. Key elements of this algorithm include:

• Input: The problem is defined by the initial state, transition model, and goal states.
• Frontier: A set of discovered but unexamined nodes.
• Process:
  • Initialization: The frontier starts with only the initial state.
  • Loop:
    • If the frontier is empty, return failure (no solution exists).
    • Select and analyze a node from the frontier.
    • If this node is the goal state, return the solution path.
    • If not, expand the node to reveal new reachable states, which are then added to the frontier for further exploration.

4. Node Data Structure

Each node in the search tree contains:

• State: The current position within the problem space.
• Parent node: The previous state in the solution path.
• Action: The action that led to this state.
• Path cost (g(n)): The accumulated cost from the initial state to the current node.

5. Frontier Data Structures

The frontier is the set of nodes awaiting expansion and can be structured in several ways, depending on the search strategy:

• FIFO Queue (First In, First Out): Nodes are processed in the order they are added (used in Breadth-First Search).
• LIFO Queue (Last In, First Out): Newest nodes are processed first, useful in Depth-First Search.
• Priority Queue: Nodes are sorted based on priority, such as path cost or heuristic value, as seen in informed search strategies.

6. Types of Search Strategies

Search strategies fall into two main categories:

1. Uninformed (Blind) Search: Uses only information available from the state space, without any extra knowledge of the goal's proximity.
2. Informed (Heuristic) Search: Uses additional data, or heuristics, that estimate the cost to reach the goal, making the search more efficient in finding optimal solutions.

7. Evaluating Search Strategies

Each search strategy is evaluated based on:

• Completeness: Whether it will find a solution if one exists.
• Optimality: Whether it guarantees finding the least costly solution.
• Time Complexity: The time taken to reach the goal.
• Space Complexity: The memory required to complete the search.

8. Breadth-First Search (BFS)

Breadth-First Search (BFS) explores each level of the search tree one at a time before moving to the next level. BFS is well-suited for finding the shortest path in an unweighted graph, as it processes all nodes at each depth sequentially.

• Process: Starting with the initial state, BFS adds all child nodes of each node to the frontier. Nodes are processed in the order they were added, ensuring a level-by-level search.
• Complexity:
  • Time: (O(b^d)), where (b) is the branching factor and (d) is the depth of the goal node.
  • Space: (O(b^d)), as BFS must store all nodes at each depth level.
• Advantages: BFS is complete and optimal for unweighted graphs.
• Disadvantages: It requires significant memory for storing nodes at each level, making it inefficient for deep searches.

9. Depth-First Search (DFS)

Depth-First Search (DFS) explores as far down a branch as possible before backtracking to explore other branches. DFS uses a stack (LIFO structure) to keep track of the current path.

• Process: DFS progresses down a path until it reaches a terminal node or dead end, then backtracks to explore other branches.
• Complexity:
  • Time: (O(b^m)), where (m) is the maximum depth.
  • Space: (O(b \cdot m)), making it more memory-efficient than BFS.
• Advantages: DFS is memory-efficient and can be implemented easily.
• Disadvantages: DFS is neither complete (it may get trapped in loops) nor optimal.

10. Limited Depth Search (LDS) and Iterative Deepening Search (IDS)

To overcome DFS limitations, depth restrictions can be applied.

Limited Depth Search (LDS)

• Concept: DFS is constrained to a certain depth, preventing infinite paths.
• Completeness: Complete if the solution exists within the depth limit.
• Optimality: Not optimal, as solutions found might not be the best.
• Space Complexity: Similar to BFS at the current depth.

Iterative Deepening Search (IDS)

• Concept: IDS combines DFS with gradual depth increases until the goal is found.
• Completeness and Optimality: It’s complete and optimal like BFS.
• Time Complexity: (O(b^d)), matching BFS while using less memory.

11. Bidirectional and Backward Search

Two advanced strategies for further efficiency include bidirectional and backward searches.

Bidirectional Search

• Concept: Searches from the start and goal states, expecting the two to meet.
• Complexity: Reduces time complexity to (O(b^{d/2})).
• Advantages: Faster for large graphs, significantly reducing the number of explored nodes.

Backward Search

• Concept: Begins the search from the goal state and works backward.
• Application: Useful when the goal state is fixed and fewer backward transitions exist.
• Process: Backward search builds a solution path by reaching backward through valid transitions from the goal.

Summary of Algorithm Comparison

Conclusion

Each search strategy has distinct advantages and trade-offs depending on the problem constraints. BFS is optimal for memory-rich scenarios, while DFS suits memory-limited cases. Iterative Deepening effectively combines both strategies, and bidirectional search offers efficient exploration when start and goal states are well-defined. By understanding these algorithms, AI practitioners can choose the most efficient search strategy for a given problem.