# 3: Solving Problems by Searching

### **3.1 Problem-Solving Agents**

<figure><img src="https://402344287-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2FmN9WztUWeaD35CTdgQpe%2Fuploads%2FtyqPRz2Gck19Du2Slbuo%2Fivanpicazo_AI_Educative_by_NEIROSHI._Neiroshi_is_the_develope_dbe45904-82b7-4840-bfe8-66e2c5867d17_2%20(1).png?alt=media&#x26;token=c9f88609-33a3-47e7-8f66-b2a019968bde" alt=""><figcaption></figcaption></figure>

Problem-solving agents are goal-driven systems that aim to achieve specific objectives by searching for solutions. These agents operate in a well-defined problem space and use systematic methods to explore possible actions.

#### **3.1.1 Components of Problem-Solving**

A problem-solving agent requires:

1. **Initial State**: The starting point in the problem space.\
   Example: In a maze, the initial state is the agent’s starting position.
2. **Actions**: Possible moves the agent can take.\
   Example: Moving up, down, left, or right in a maze.
3. **Transition Model**: Describes the result of each action.\
   Example: Moving "right" from a cell leads to the adjacent cell.
4. **Goal Test**: Determines whether the agent has achieved its objective.\
   Example: Reaching the maze’s exit.
5. **Path Cost**: The cumulative cost of the path taken.\
   Example: The number of steps or energy consumed in the maze.

***

### **3.2 Problem Types**

Understanding the nature of a problem helps in selecting the most suitable algorithm.

#### **3.2.1 Single-State Problems**

The agent knows exactly where it is at all times. The environment is fully observable and deterministic.

Example: Solving a jigsaw puzzle.

***

#### **3.2.2 Multi-State Problems**

The agent operates with partial knowledge and must account for uncertainty.

Example: Navigating an unfamiliar city without a map.

***

#### **3.2.3 Adversarial Problems**

In these problems, the agent competes against another agent, such as in games.

Example: Chess, where each player’s move influences the other’s strategy.

***

### **3.3 Search Strategies**

Search algorithms explore the problem space to find solutions. Strategies are broadly categorized into uninformed and informed search.

#### **3.3.1 Uninformed Search**

These algorithms have no additional knowledge about the problem beyond the definition of actions and goal tests.

**Breadth-First Search (BFS)**

* Explores all nodes at a given depth before moving to the next level.
* Guaranteed to find the shortest path in an unweighted graph.

**Depth-First Search (DFS)**

* Explores as deeply as possible along one branch before backtracking.
* Memory-efficient but may not find the optimal solution.

***

#### **3.3.2 Informed Search**

These algorithms use heuristics—additional information about the problem—to guide the search.

**Greedy Best-First Search**

* Prioritizes nodes that appear closest to the goal based on a heuristic.
* Fast but not always optimal.

**A\* Search**

* Combines path cost and heuristic to guarantee the optimal solution.
* Widely used in applications like navigation systems.

***

### **3.4 Problem-Solving in Action**

Consider a robot tasked with finding the shortest path to a goal in a grid-like environment.

1. **Initial State**: Robot’s starting position.
2. **Actions**: Move up, down, left, or right.
3. **Goal Test**: Reaching the target position.
4. **Path Cost**: Minimize the total number of moves.

#### **Uninformed Approach**

Breadth-first search ensures the robot finds the shortest path but explores many unnecessary nodes.

#### **Informed Approach**

A\* search uses heuristics like the Manhattan distance to efficiently guide the robot toward the goal.

***

### **3.5 Uninformed Search Strategies**

Uninformed search strategies, also called blind searches, do not use additional information about the goal beyond the problem definition. They systematically explore the problem space.

#### **3.5.1 Breadth-First Search (BFS)**

* Explores all nodes at the current depth level before moving deeper.
* Uses a queue to track frontier nodes.

**Properties:**

* **Completeness**: Guaranteed to find a solution if one exists.
* **Optimality**: Finds the shortest path in an unweighted graph.
* **Time Complexity**: O(b^d), where b is the branching factor and d is the depth of the shallowest solution.
* **Space Complexity**: O(b^d).

**Example**: BFS is useful in social networks to find the shortest connection between two users.

***

#### **3.5.2 Depth-First Search (DFS)**

* Explores as deeply as possible along a branch before backtracking.
* Uses a stack to manage frontier nodes.

**Properties:**

* **Completeness**: Not guaranteed in infinite-depth spaces.
* **Optimality**: Not guaranteed—it may find a suboptimal solution.
* **Time Complexity**: O(b^m), where b is the branching factor and m is the maximum depth of the search tree.
* **Space Complexity**: O(b \* m).

**Example**: DFS is useful in scenarios like solving mazes or puzzles, where depth exploration is prioritized.

***

#### **3.5.3 Uniform-Cost Search (UCS)**

* Expands the node with the lowest path cost first.
* Uses a priority queue to organize nodes based on cost.

**Properties:**

* **Completeness**: Guaranteed if path costs are positive.
* **Optimality**: Finds the least-cost solution.
* **Time Complexity**: O(b^(1 + (C\*/e))), where C\* is the cost of the optimal solution and e is the minimum cost of any action.
* **Space Complexity**: Same as time complexity.

**Example**: UCS is ideal for pathfinding in weighted graphs, such as GPS navigation systems.

***

### **3.6 Informed Search Strategies**

Informed strategies, or heuristic searches, use domain-specific knowledge to guide the search efficiently.

#### **3.6.1 Greedy Best-First Search**

* Expands the node that appears closest to the goal, based on a heuristic function h(n).
* Heuristic h(n): Estimates the cost to reach the goal from node n.

**Properties:**

* **Completeness**: Not guaranteed if the search gets stuck in loops.
* **Optimality**: Not guaranteed—it may follow suboptimal paths.
* **Time Complexity**: O(b^m), where b is the branching factor and m is the maximum depth of the search tree.
* **Space Complexity**: O(b^m).

**Example**: Used in routing problems where speed is prioritized over guaranteed optimality.

***

#### *3.6.2 A-Star (A) Search*\*

* Combines path cost and heuristic, using the formula f(n) = g(n) + h(n):
  * g(n): Path cost from the start to node n.
  * h(n): Heuristic estimate from node n to the goal.

**Properties:**

* **Completeness**: Guaranteed if the branching factor is finite.
* **Optimality**: Guaranteed if h(n) is admissible (never overestimates the true cost).
* **Time Complexity**: O(b^d), where b is the branching factor and d is the depth of the solution.
* **Space Complexity**: Same as time complexity.

**Example**: Widely used in pathfinding algorithms like Google Maps, where both distance and travel time are considered.

***

#### **3.6.3 Heuristics**

Heuristics are domain-specific functions that estimate the cost of reaching the goal. They guide the search process effectively.

**Types of Heuristics:**

1. **Admissible Heuristic**: Never overestimates the true cost to the goal.\
   Example: Straight-line distance in navigation problems.
2. **Consistent Heuristic**: Satisfies the condition h(n) <= c(n, n') + h(n'), where c(n, n') is the cost of moving from node n to n'.\
   Example: Manhattan distance in grid-based problems.

***

### **3.7 Comparing Search Strategies**

| **Search Strategy**        | **Complete** | **Optimal**         | **Time Complexity** | **Space Complexity**    |
| -------------------------- | ------------ | ------------------- | ------------------- | ----------------------- |
| Breadth-First Search (BFS) | Yes          | Yes (if unweighted) | O(b^d)              | O(b^d)                  |
| Depth-First Search (DFS)   | No           | No                  | O(b^m)              | O(b \* m)               |
| Uniform-Cost Search (UCS)  | Yes          | Yes                 | O(b^(1 + (C\*/e)))  | Same as time complexity |
| Greedy Best-First Search   | No           | No                  | O(b^m)              | O(b^m)                  |
| A-Star (A\*) Search        | Yes          | Yes                 | O(b^d)              | Same as time complexity |

***

***

### **3.8 Optimizing Search Algorithms**

To improve efficiency, search algorithms often incorporate optimization techniques that reduce computation and memory usage.

#### **3.8.1 Graph Search**

Graph search is an enhancement over tree search that avoids redundant exploration by keeping track of previously visited states. This eliminates cycles and revisits.

**Implementation**:

* Maintain a **closed list** (set of explored states) alongside the **frontier** (set of unexplored states).
* Skip any state that is already in the closed list.

**Example**:\
In a maze, if the agent revisits previously explored paths, graph search ensures they are skipped, preventing wasted computation.

***

#### **3.8.2 Bidirectional Search**

Bidirectional search works by running two simultaneous searches:

1. One forward from the initial state.
2. One backward from the goal state.

When the two searches meet, a solution is found.

**Advantages**:

* Significantly reduces search depth compared to unidirectional search.
* Time complexity is approximately O(b^(d/2)) instead of O(b^d), where b is the branching factor and d is the depth of the solution.

**Example**:\
Finding the shortest path in a social network, where one search starts from a person and the other from their target connection.

***

#### **3.8.3 Iterative Deepening Search**

This combines the space efficiency of depth-first search (DFS) with the completeness of breadth-first search (BFS). The algorithm performs DFS with increasing depth limits until a solution is found.

**Advantages**:

* Minimal memory usage, like DFS.
* Guaranteed to find the optimal solution, like BFS.

**Use Case**:\
Suitable for problems with large state spaces but where memory is limited, such as puzzle-solving.

***

#### **3.8.4 Memory-Bounded Search**

To address the high memory requirements of algorithms like A\*, memory-bounded variants like **Iterative Deepening A\*** (IDA\*) and **Simplified Memory-Bounded A\*** (SMA\*) are used.

**IDA\***:

* Uses iterative deepening with A\*-style evaluation.
* Expands nodes within a threshold cost, increasing the threshold iteratively.

**SMA\***:

* Maintains only the most promising nodes within a fixed memory limit.
* Discards less promising nodes and revisits them only if needed.

***

### **3.9 Practical Considerations for Search Algorithms**

When implementing search algorithms in real-world applications, several factors must be considered:

#### **3.9.1 State Representation**

Efficient representation of states is crucial for reducing memory usage and speeding up processing. States can be encoded as:

* **Arrays or Matrices**: For grid-based problems.
* **Graphs**: For network-related problems.
* **Binary Strings**: For optimization problems like the traveling salesman.

***

#### **3.9.2 Heuristic Design**

Heuristics significantly influence the performance of informed search algorithms. Designing effective heuristics involves:

* **Accuracy**: The heuristic should closely approximate the actual cost to the goal.
* **Efficiency**: Computing the heuristic should not be more expensive than the search itself.
* **Admissibility and Consistency**: Ensure the heuristic meets these properties to guarantee optimality with A\*.

***

#### **3.9.3 Scalability**

Large problem spaces require scalable solutions. Techniques include:

* Using **distributed computing** to parallelize searches.
* Applying **approximation algorithms** when exact solutions are computationally infeasible.

***

### **3.10 Use Cases of Search Algorithms**

Search algorithms form the backbone of many real-world applications. Here are some key examples:

#### **3.10.1 Navigation Systems**

Search algorithms like A\* and Dijkstra's are used in GPS systems to compute the shortest routes between two locations, considering factors like traffic and road closures.

***

#### **3.10.2 Game Playing**

Games like chess and Go rely on adversarial search strategies, including:

* **Minimax Algorithm**: Evaluates moves to minimize the opponent’s maximum advantage.
* **Alpha-Beta Pruning**: Optimizes Minimax by eliminating branches that cannot influence the final decision.

***

#### **3.10.3 Robotics**

Robots use search algorithms to navigate dynamic environments and avoid obstacles. Pathfinding in grid-based maps is often achieved using algorithms like A\*.

***

#### **3.10.4 Network Optimization**

Search is applied to optimize communication networks, routing packets efficiently across nodes while minimizing delays and avoiding congestion.

***

### **3.11 Summary**

In this chapter, we explored:

1. The foundations of problem-solving agents and their interaction with environments.
2. Various search strategies, including uninformed and informed approaches.
3. Optimization techniques like graph search, bidirectional search, and memory-bounded methods.
4. Practical considerations for implementing search algorithms in real-world applications.
5. Real-world use cases, including navigation, gaming, robotics, and networking.

**Next Steps**: In the next chapter, we’ll delve into **Adversarial Search**, exploring competitive environments where multiple agents interact, such as games. Let me know when you're ready to continue!
