# Problem Solving Algorithms in AI: Best Answers 007ET

Problem Solving Algorithms and Problem Solving in AI. An algorithm is a defined set of step-by-step procedures that provides the correct answer to a particular problem. An algorithm is often expressed in the form of a graph, where a square represents each step. Arrows then branch off from each step to point to possible directions that you may take to solve the problem.

- Search Algorithms in Artificial Intelligence
- Search Algorithm Terminologies
- Properties of Search Algorithms:
- Types of search algorithms
- Uninformed Search Algorithms
- Informed Search Algorithms
- Pure Heuristic Search Algorithm
- 1.) Best-first Search Algorithm (Greedy Search):
- Advantages of the Best first search algorithm
- Disadvantages of a Best first search algorithm
- Example of the Best first search algorithm
- 2.) A* Search Algorithm
- Advantages of A* Search Algorithm
- Disadvantages of A* Search Algorithm :
- Example of A* Search Algorithm

- Pure Heuristic Search Algorithm
- Hill Climbing Algorithm in Artificial Intelligence
- Features of Hill Climbing Algorithm in AI
- State-space Diagram for Hill Climbing Algorithm in AI
- Different regions in the state space landscape:
- Types of Hill Climbing Algorithm in AI:
- Problems in Hill Climbing Algorithm in AI
- Means-Ends Analysis in Artificial Intelligence
- How means-ends analysis Works
- Operator Subgoaling in AI
- Algorithm for Means-Ends Analysis
- Problem Solving Algorithms in AI FAQ

## Search Algorithms in Artificial Intelligence

Search algorithms are one of the most important areas of Artificial Intelligence. This topic will explain all about the search algorithms in AI.

**Problem Solving Intelligent agents in AI**

In Artificial Intelligence, Search techniques are universal problem-solving methods. **Rational agents** or **Problem-solving agents** in AI mostly used these search strategies or algorithms to solve a specific problem and provide the best result. Problem-solving agents are goal-based agents and use atomic representation. In this topic, we will learn various problem-solving search algorithms.

## Search Algorithm Terminologies

**Search:**Searchingis a step by step procedure to solve a search-problem in a given search space. A search problem can have three main factors:**Search Space:**Search space represents a set of possible solutions, which a system may have.**Start State:**It is a state from where agent begins**the search**.**Goal test:**It is a function which observe the current state and returns whether the goal state is achieved or not.

**Search tree:**A tree representation of search problem is called Search tree. The root of the search tree is the root node which is corresponding to the initial state.**Actions:**It gives the description of all the available actions to the agent.**Transition model:**A description of what each action do, can be represented as a transition model.**Path Cost:**It is a function which assigns a numeric cost to each path.**Solution:**It is an action sequence which leads from the start node to the goal node.**Optimal Solution:**If a solution has the lowest cost among all solutions.

## Properties of Search Algorithms:

Following are the four essential properties of search algorithms to compare the efficiency of these algorithms:

**Completeness:**

A search algorithm is said to be complete if it guarantees to return a solution if at least any solution exists for any random input.McDonald’s to Customize Drive-Through Windows With Artificial Intelligence

**Optimality:**

If a solution found for an algorithm is guaranteed to be the best solution (lowest path cost) among all other solutions, then such a solution is said to be an optimal solution.

**Time Complexity:**

Time complexity is a measure of time for an algorithm to complete its task.

**Space Complexity:**

It is the maximum storage space required at any point during the search, as the complexity of the problem.

## Types of search algorithms

**We can classify the search algorithms into uninformed (Blind search) search and informed search (Heuristic search) algorithms based on the search problems.**

### Uninformed/Blind Search

The uninformed search does not contain any domain knowledge such as closeness, the location of the goal. It operates in a brute-force way as it only includes information about how to traverse the tree and identify leaf and goal nodes. Uninformed search applies how a search tree is searched without any information about the search space like initial state operators and tests for the goal, so it is also called blind search. It examines each node of the tree until it achieves the goal node.

**It can be divided into five main types:**

- Breadth-first search
- Uniform cost search
- Depth-first search
- Iterative deepening depth-first search
- Bidirectional Search

### Informed Search

Informed search algorithms use domain knowledge. In an informed search, problem information is available, which can guide the search. Informed search strategies can find a solution more efficiently than uninformed search strategies. Informed search is also called a Heuristic search.

A heuristic is a way that might not always be guaranteed for best solutions but guaranteed to find a good solution in a reasonable time.

Informed search can solve many complex problems which could not be solved in another way.

An example of informed search algorithms is a traveling salesman problem.

- Greedy Search
- A* Search

## Uninformed Search Algorithms

Uninformed search is a class of general-purpose search algorithms that operates in brute force-way. Uninformed search algorithms do not have additional information about state or search space other than traversing the tree, also called blind search.

**Following are the various types of uninformed search algorithms:**

**Breadth-first Search****Depth-first Search****Depth-limited Search****Iterative deepening depth-first search****Uniform cost search****Bidirectional Search**

### 1. Breadth-first Search Algorithm

- Breadth-first search is the most common search strategy for traversing a tree or graph. This algorithm searches breadthwise in a tree or graph, so it is called breadth-first search.
- BFS algorithm starts searching from the root node of the tree and expands all successor node at the current level before moving to nodes of next level.
- The breadth-first search algorithm is an example of a general-graph search algorithm.
- Breadth-first search implemented using FIFO queue data structure.

**Advantages of Breadth-first search **

- BFS will provide a solution if any solution exists.
- If there are more than one solutions for a given problem, then BFS will provide the minimal solution which requires the least number of steps.

**Disadvantages of Breadth-first search**

- It requires lots of memory since each level of the tree must be saved into memory to expand the next level.
- BFS needs lots of time if the solution is far away from the root node.

**Example of Breadth-first search**

In the below tree structure, we have shown the traversing of the tree using the BFS algorithm from the root node S to goal node K. BFS search algorithm traverse in layers, so it will follow the path which is shown by the dotted arrow, and the traversed path will be:

- S—> A—>B—->C—>D—->G—>H—>E—->F—->I—->K

**Time Complexity:** Time Complexity of the BFS algorithm can be obtained by the number of nodes traversed in BFS until the shallowest Node. Where the d= depth of shallowest solution and b is a node at every state.

**T (b) = 1+b ^{2}+b^{3}+…….+ b^{d}= O (b^{d})**

**Space Complexity:** The space complexity of the BFS algorithm is given by the Memory size of the frontier, which is O(b^{d}).

**Completeness:** BFS is complete, which means if the shallowest goal node is at some finite depth, then BFS will find a solution.

**Optimality:** BFS is optimal if path cost is a non-decreasing function of the depth of the node.

### 2. Depth-first Search Algorithm

- Depth-first search isa recursive algorithm for traversing a tree or graph data structure.
- It is called the depth-first search because it starts from the root node and follows each path to its greatest depth node before moving to the next path.
- DFS uses a stack data structure for its implementation.
- The process of the DFS algorithm is similar to the BFS algorithm.

**Note: Backtracking is an algorithm technique for finding all possible solutions using recursion.**

**Advantage of Depth-first Search**

- DFS requires very less memory as it only needs to store a stack of the nodes on the path from root node to the current node.
- It takes less time to reach to the goal node than BFS algorithm (if it traverses in the right path).

**Disadvantage of Depth-first Search**

- There is the possibility that many states keep re-occurring, and there is no guarantee of finding the solution.
- DFS algorithm goes for deep down searching and sometime it may go to the infinite loop.

#### Example of Depth-first Search

In the below search tree, we have shown the flow of depth-first search, and it will follow the order as:

Root node—>Left node —-> right node.

It will start searching from root node S, traverse A, then B, then D and E; after traversing E, it will backtrack the tree as E has no other successor and still goal node is not found. After backtracking, it will traverse node C and then G, and here it will terminate as it found the goal node.

**Completeness:** DFS search algorithm is complete within finite state space as it will expand every node within a limited search tree.

**Time Complexity:** The time complexity of DFS will be equivalent to the node traversed by the algorithm. It is given by:

**T(n)= 1+ n ^{2}+ n^{3} +………+ n^{m}=O(n^{m})**

**Where m= maximum depth of any node and this can be much larger than d (Shallowest solution depth)**

**Space Complexity:** DFS algorithm needs to store only a single path from the root node. Hence space complexity of DFS is equivalent to the size of the fringe set, which is **O(bm)**.

**Optimal:** DFS search algorithm is non-optimal, as it may generate a large number of steps or high cost to reach the goal node.

### 3. Depth-Limited Search Algorithm:

A depth-limited search algorithm is similar to a depth-first search with a predetermined limit. Depth-limited search can solve the drawback of the infinite path in the Depth-first search. In this algorithm, the node at the depth limit will treat as it has no successor nodes further.

Depth-limited search can be terminated with two Conditions of failure:

- Standard failure value: It indicates that problem does not have any solution.
- Cutoff failure value: It defines no solution for the problem within a given depth limit.

**Advantages of the depth-limited search algorithm**

Depth-limited search is Memory efficient.

**Disadvantages depth-limited search algorithm**

- Depth-limited search also has a disadvantage of incompleteness.
- It may not be optimal if the problem has more than one solution.

**Example of depth-limited search algorithm**

**Completeness:** DLS search algorithm is complete if the solution is above the depth limit.

**Time Complexity:** The time complexity of the DLS algorithm is **O(b ^{ℓ})**.

**Space Complexity:** The space complexity of the DLS algorithm is O**(b×ℓ)**.

**Optimal:** Depth-limited search can be viewed as a special case of DFS, and it is also not optimal even if ℓ>d.

### 4. Uniform-cost Search Algorithm

Uniform-cost search is a searching algorithm used for traversing a weighted tree or graph. This algorithm comes into play when a different cost is available for each edge. The primary goal of the uniform-cost search is to find a path to the goal node which has the lowest cumulative cost. Uniform-cost search expands nodes according to their path costs from the root node.

It can be used to solve any graph/tree where the optimal cost is in demand. A uniform-cost search algorithm is implemented by the priority queue. It gives maximum priority to the lowest cumulative cost. Uniform cost search is equivalent to BFS algorithm if the path cost of all edges is the same.

**Advantages: of Uniform-cost Search Algorithm **

- Uniform cost search is optimal because at every state the path with the least cost is chosen.

**Disadvantages of Uniform-cost Search Algorithm :**

- It does not care about the number of steps involve in searching and only concerned about path cost. Due to which this algorithm may be stuck in an infinite loop.

**Example: of Uniform-cost Search Algorithm **

**Completeness:**

Uniform-cost search is complete, such as if there is a solution, UCS will find it.

**Time Complexity:**

Let C* **is the Cost of the optimal solution**, and **ε** is each step to get closer to the goal node. Then the number of steps is = C*/ε+1. Here we have taken +1, starting from state 0 and end to C*/ε.

Hence, the worst-case time complexity of Uniform-cost search is**O(b ^{1 + [C*/ε]})/**.

**Space Complexity:**

The same logic is for space complexity so, the worst-case space complexity of Uniform-cost search is **O(b ^{1 + [C*/ε]})**.

**Optimal:**

Uniform-cost search is always optimal as it only selects a path with the lowest path cost.

### 5. Iterative deepeningdepth-first Search Algorithm

The iterative deepening algorithm is a combination of DFS and BFS algorithms. This search algorithm finds out the best depth limit and does it by gradually increasing the limit until a goal is found.

This algorithm performs a depth-first search up to a certain “depth limit,” It keeps increasing the depth limit after each iteration until the goal node is found.

This Search algorithm combines the benefits of Breadth-first search’s fast search and depth-first search’s memory efficiency.

The iterative search algorithm is useful for uninformed search when search space is large, and the depth of the goal node is unknown.

**Advantages of Iterative deepening depth-first Search Algorithm :**

- Itcombines the benefits of BFS and DFS search algorithm in terms of fast search and memory efficiency.

**Disadvantages of Iterative deepening depth-first Search Algorithm :**

- The main drawback of IDDFS is that it repeats all the work of the previous phase.

**Example of Iterative deepening depth-first Search Algorithm **

The following tree structure is showing the iterative deepening depth-first search. IDDFS algorithm performs various iterations until it does not find the goal node. The iteration performed by the algorithm is given as:

1’st Iteration—–> A

2’nd Iteration—-> A, B, C

3’rd Iteration——>A, B, D, E, C, F, G

4’th Iteration——>A, B, D, H, I, E, C, F, K, G

In the fourth iteration, the algorithm will find the goal node.

**Completeness:**

This algorithm is complete if the branching factor is finite.

**Time Complexity:**

Let’s suppose b is the branching factor and depth is d, then the worst-case complexity is **O(b ^{d})**.

**Space Complexity:**

The space complexity of IDDFS will be **O(bd)**.

**Optimal:**

IDDFS algorithm is optimal if path cost is a non-decreasing function of the depth of the node.

### 6. Bidirectional Search Algorithm

Bidirectional search algorithm runs two simultaneous searches, one from the initial state called forward-search and the other from the goal node called backward-search, to find the goal node. Bidirectional search replaces one single search graph with two small subgraphs in which one starts the search from an initial vertex and the other start from the goal vertex. The search stops when these two graphs intersect each other.

**Bidirectional search can use search techniques such as BFS, DFS, DLS, etc.**

**Advantages of Bidirectional Search Algorithm **

- Bidirectional search is fast.
- Bidirectional search requires less memory

**Disadvantages of Bidirectional Search Algorithm **

- Implementation of the bidirectional search tree is difficult.
- In bidirectional search, one should know the goal state in advance.

**Example of Bidirectional Search Algorithm **

In the below search tree, the bidirectional search algorithm is applied. This algorithm divides one graph/tree into two sub-graphs. It starts traversing from node 1 in the forward direction and starts from goal node 16 in the backward direction.

The algorithm terminates at node 9, where two searches meet.

**Completeness:** Bidirectional Search is complete if we use BFS in both searches.

**Time Complexity:** The time complexity of bidirectional search using BFS is **O(b ^{d})**.

**Space Complexity:** The space complexity of the bidirectional search is **O(b ^{d})**.

**Optimal:** Bidirectional search is Optimal.

## Informed Search Algorithms

So far, we have talked about the uninformed search algorithms, which looked through search space for all possible solutions to the problem without additional knowledge about search space. But informed search algorithm contains an array of knowledge such as how far we are from the goal, path cost, how to reach the goal node, etc. This knowledge help agents to explore less to the search space and find more efficiently the goal node.

The informed search algorithm is more useful for large search spaces. An informed search algorithm uses the idea of heuristic, so it is also called Heuristic search.

**Heuristics function:** Heuristic is a function used in Informed Search, and it finds the most promising path. It takes the current state of the agent as its input and produces the estimation of how close the agent is to the goal. The heuristic method, however, might not always give the best solution, but it guaranteed to find a good solution in a reasonable time.

Heuristic function estimates how close a state is to the goal. It is represented by h(n), and it calculates the cost of an optimal path between the pair of states. The value of the heuristic function is always positive.

**Admissibility of the heuristic function is given as **How to find Nth Highest Salary in SQL.

- h(n) <= h*(n)

**Here h(n) is heuristic cost, and h*(n) is the estimated cost. Hence heuristic cost should be less than or equal to the estimated cost.**

### Pure Heuristic Search Algorithm

Pure heuristic search is the simplest form of heuristic search algorithms. It expands nodes based on their heuristic value, h(n). It maintains two lists, OPEN and CLOSED lists. In the CLOSED list, it places those nodes which have already expanded, and in the OPEN list, it places nodes that have not yet been expanded.

On each iteration, each node n with the lowest heuristic value is expanded and generates all its successors, and n is placed to the closed list. The algorithm continues unit a goal state is found.

In the informed search, we will discuss two main algorithms, which are given below:

**Best First Search Algorithm(Greedy search)****A* Search Algorithm**

#### 1.) Best-first Search Algorithm (Greedy Search):

Greedy best-first search algorithm always selects the path which appears best at that moment. It is the combination of depth-first search and breadth-first search algorithms. It uses the heuristic function and search. Best-first search allows us to take advantage of both algorithms.

With the help of the best-first search, at each step, we can choose the most promising node. In the best-first search algorithm, we expand the node which is closest to the goal node, and the closest cost is estimated by heuristic function, i.e.

- f(n)= g(n).

Were, h(n)= estimated cost from node n to the goal.

The greedy best-first algorithm is implemented by the priority queue.

**Best first search algorithm:**

**Step 1:**Place the starting node into the OPEN list.**Step 2:**If the OPEN list is empty, Stop and return failure.**Step 3:**Remove the node n, from the OPEN list which has the lowest value of h(n), and places it in the CLOSED list.**Step 4:**Expand the node n, and generate the successors of node n.**Step 5:**Check each successor of node n, and find whether any node is a goal node or not. If any successor node is goal node, then return success and terminate the search, else proceed to Step 6.**Step 6:**For each successor node, algorithm checks for evaluation function f(n), and then check if the node has been in either OPEN or CLOSED list. If the node has not been in both list, then add it to the OPEN list.**Step 7:**Return to Step 2.

**Advantages of the Best first search algorithm**

- Best first search can switch between BFS and DFS by gaining the advantages of both the algorithms.
- This algorithm is more efficient than BFS and DFS algorithms.

**Disadvantages of a Best first search algorithm **

- It can behave as an unguided depth-first search in the worst case scenario.
- It can get stuck in a loop as DFS.
- This algorithm is not optimal.

**Example of the Best first search algorithm **

Consider the below search problem, and we will traverse it using greedy best-first search. At each iteration, each node is expanded using evaluation function f(n)=h(n), which is given in the below table.

In this search example, we are using two lists which are **OPEN** and **CLOSED** Lists. Following are the iteration for traversing the above example.

**Expand the nodes of S and put them in the CLOSED list**

**Initialization:** Open [A, B], Closed [S]

**Iteration 1:** Open [A], Closed [S, B]

**Iteration 2:** Open [E, F, A], Closed [S, B] : Open [E, A], Closed [S, B, F]

**Iteration 3:** Open [I, G, E, A], Closed [S, B, F] : Open [I, E, A], Closed [S, B, F, G]

Hence the final solution path will be: **S—-> B—–>F—-> G**

**Time Complexity:** The worst-case time complexity of Greedy best-first search is O(b^{m}).

**Space Complexity:** The worst-case space complexity of Greedy best-first search is O(bm). Where m is the maximum depth of the search space.

**Complete:** Greedy best-first search is also incomplete, even if the given state space is finite.

**Optimal:** Greedy best-first search algorithm is not optimal.

#### 2.) A* Search Algorithm

A* search is the most commonly known form of best-first search. It uses the heuristic function h(n) and cost to reach the node n from the start state g(n). It has combined features of UCS and greedy best-first search, by which it solve the problem efficiently. A* search algorithm finds the shortest path through the search space using the heuristic function

. This search algorithm expands fewer search trees and provides optimal results faster. A* algorithm is similar to UCS except that it uses g(n)+h(n) instead of g(n).

In the A* search algorithm, we use the search heuristic and the cost to reach the node. Hence we can combine both costs as following, and this sum is called a **fitness number**.

At each point in the search space, only those nodes are expanded with the lowest value of f(n), and the algorithm terminates when the goal node is found.

**Algorithm of A* search:**

**Step1:** Place the starting node in the OPEN list.

**Step 2:** Check if the OPEN list is empty or not. If the list is empty, then return failure and stops.

**Step 3:** Select the node from the OPEN list which has the smallest value of evaluation function (g+h); if node n is the goal node, then return success and stop, otherwise

**Step 4:** Expand node n and generate all of its successors, and put n into the closed list. For each successor n’, check whether n’ is already in the OPEN or CLOSED list; if not, then compute the evaluation function for n’ and place it into the Open list.

**Step 5:** Else, if node n’ is already in OPEN and CLOSED, then it should be attached to the back pointer, which reflects the lowest g(n’) value.

**Step 6:** Return to **Step 2**.

**Advantages of A* Search Algorithm**

- A* search algorithm is the best algorithm than other search algorithms.
- A* search algorithm is optimal and complete.
- This algorithm can solve very complex problems.

**Disadvantages of A* Search Algorithm :**

- It does not always produce the shortest path as it mostly based on heuristics and approximation.
- A* search algorithm has some complexity issues.
- The main drawback of A* is memory requirement as it keeps all generated nodes in the memory, so it is not practical for various large-scale problems.

**Example of A* Search Algorithm**

In this example, we will traverse the given graph using the A* algorithm. The heuristic value of all states is given in below table so we will calculate the f(n) of each state using the formula f(n)= g(n) + h(n), where g(n) is the cost to reach any node from the start state. Here we will use the OPEN and CLOSED list.

**Solution:**

**Initialization:** {(S, 5)}

**Iteration1:** {(S–> A, 4), (S–>G, 10)}

**Iteration2:** {(S–> A–>C, 4), (S–> A–>B, 7), (S–>G, 10)}

**Iteration3:** {(S–> A–>C—>G, 6), (S–> A–>C—>D, 11), (S–> A–>B, 7), (S–>G, 10)}

**Iteration 4** will give the final result, as **S—>A—>C—>G** it provides the optimal path with cost 6.

**Points to remember:**

- A* algorithm returns the path which occurred first, and it does not search for all remaining paths.
- The efficiency of A* algorithm depends on the quality of heuristic.
- A* algorithm expands all nodes which satisfy the condition f(n)

**Complete:** A* algorithm is complete as long as:

- Branching factor is finite.
- Cost at every action is fixed.

**Optimal:** A* search algorithm is optimal if it follows below two conditions:

**Admissible:**the first condition requires for optimality is that h(n) should be an admissible heuristic for A* tree search. An admissible heuristic is optimistic in nature.**Consistency:**Second required condition is consistency for only A* graph-search.

If the heuristic function is admissible, then A* tree search will always find the least cost path.

**Time Complexity:** The time complexity of the A* search algorithm depends on the heuristic function, and the number of nodes expanded is exponential to the depth of solution d. So the time complexity is O(b^d), where b is the branching factor.

**Space Complexity:** The space complexity of the A* search algorithm is **O(b^d)**

## Hill Climbing Algorithm in Artificial Intelligence

- Hill climbing algorithm is a local search algorithm which continuously moves in the direction of increasing elevation/value to find the peak of the mountain or best solution to the problem. It terminates when it reaches a peak value where no neighbor has a higher value.
- Hill climbing algorithm is a technique which is used for optimizing the mathematical problems. One of the widely discussed examples of Hill climbing algorithm is Traveling-salesman Problem in which we need to minimize the distance traveled by the salesman.
- It is also called greedy local search as it only looks to its good immediate neighbor state and not beyond that.
- A node of hill climbing algorithm has two components which are state and value.
- Hill Climbing is mostly used when a good heuristic is available.
- In this algorithm, we don’t need to maintain and handle the search tree or graph as it only keeps a single current state.

## Features of Hill Climbing Algorithm in AI

Following are some main features of the Hill Climbing Algorithm:

**Generate and Test variant:**Hill Climbing is the variant of Generate and Test method. The Generate and Test method produce feedback which helps to decide which direction to move in the search space.**Greedy approach:**Hill-climbing algorithm search moves in the direction which optimizes the cost.**No backtracking:**It does not backtrack the search space, as it does not remember the previous states.

## State-space Diagram for Hill Climbing Algorithm in AI

The state-space landscape is a graphical representation of the hill-climbing algorithm showing a graph between various states of algorithm and Objective function/Cost.

On Y-axis, we have taken the function, which can be an objective function or cost function, and state-space on the x-axis. If the function on Y-axis is cost then, the goal of the search is to find the global minimum and local minimum. If the function of the Y-axis is the Objective function, then the goal of the search is to find the global maximum and local maximum.

## Different regions in the state space landscape:

**Local Maximum:** Local maximum is a state which is better than its neighbor states, but there is also another state which is higher than it.

**Global Maximum:** Global maximum is the best possible state of the state-space landscape. It has the highest value of the objective function.

**Current state:** It is a state in a landscape diagram where an agent is currently present.

**Flat local maximum:** It is a flat space in the landscape where all the neighbor states of current states have the same value.

**Shoulder:** It is a plateau region that has an uphill edge.

## Types of Hill Climbing Algorithm in AI:

- Simple hill Climbing:
- Steepest-Ascent hill-climbing:
- Stochastic hill Climbing:

### 1. Simple Hill Climbing

Simple hill climbing is the simplest way to implement a hill-climbing algorithm. It only evaluates the neighbor node state and selects the first one, which optimizes the current cost and sets it as a current state. It only checks its one successor state, and if it finds better than the current state, then move else be in the same state. This algorithm has the following features:

- Less time consuming
- Less optimal solution and the solution is not guaranteed

### Algorithm for Simple Hill Climbing

**Step 1:**Evaluate the initial state, if it is goal state then return success and Stop.**Step 2:**Loop Until a solution is found or there is no new operator left to apply.**Step 3:**Select and apply an operator to the current state.**Step 4:**Check new state:- If it is goal state, then return success and quit.
- Else if it is better than the current state then assign new state as a current state.
- Else if not better than the current state, then return to step2.

**Step 5:**Exit.

### 2. Steepest-Ascent hill climbing

The steepest-Ascent algorithm is a variation of a simple hill-climbing algorithm. This algorithm examines all the neighboring nodes of the current state and selects one neighbor node closest to the goal state. This algorithm consumes more time as it searches for multiple neighbors.

### Algorithm for Steepest-Ascent hill climbing

**Step 1:**Evaluate the initial state, if it is goal state then return success and stop, else make current state as initial state.**Step 2:**Loop until a solution is found or the current state does not change.- Let SUCC be a state such that any successor of the current state will be better than it.
- For each operator that applies to the current state:
- Apply the new operator and generate a new state.
- Evaluate the new state.
- If it is goal state, then return it and quit, else compare it to the SUCC.
- If it is better than SUCC, then set new state as SUCC.
- If the SUCC is better than the current state, then set current state to SUCC.

**Step 5:**Exit.

### 3. Stochastic hill climbing:

Stochastic hill climbing does not examine all its neighbors before moving. Rather, this search algorithm selects one neighbor node at random and decides whether to choose it as a current state or examine another state.

## Problems in Hill Climbing Algorithm in AI

**1. Local Maximum:** A local maximum is a peak state in the landscape that is better than each of its neighboring states, but there is another state also present which is higher than the local maximum.https://1dda26b3a7617abe11f2baab2820b78a.safeframe.googlesyndication.com/safeframe/1-0-38/html/container.html

**Solution:** The backtracking technique can be a solution to the local maximum in the state-space landscape. Create a list of the promising path so that the algorithm can backtrack the search space and explore other paths.

**2. Plateau:** A plateau is the flat area of the search space in which all the neighbor states of the current state contain the same value; this algorithm does not find any best direction to move. A hill-climbing search might be lost in the plateau area.

**Solution:** The solution for the plateau is to take big steps or very few steps while searching to solve the problem. Randomly select a state far away from the current state, so it is possible that the algorithm could find a non-plateau region.

**3. Ridges:** A ridge is a special form of the local maximum. It has an area higher than its surrounding areas, but itself has a slope and cannot be reached in a single move.

**Solution:** With bidirectional search, or by moving in different directions, we can improve this problem.

### Simulated Annealing

A hill-climbing algorithm that never makes a move towards a lower value is guaranteed to be incomplete because it can get stuck on a local maximum. And if the algorithm applies a random walk by moving a successor, then it may be complete but not efficient. Simulated Annealing is an algorithm that yields both efficiency and completeness.

In mechanical terms, Annealing is a process of hardening a metal or glass to a high temperature then cooling gradually, allowing the metal to reach a low-energy crystalline state. The same process is used in simulated annealing, in which the algorithm picks a random move instead of picking the best move. If the random move improves the state, then it follows the same path. Otherwise, the algorithm follows the path with a probability of less than 1, or it moves downhill and chooses another path.

## Means-Ends Analysis in Artificial Intelligence

- We have studied the strategies which can reason either in forward or backward, but a mixture of the two directions is appropriate for solving a complex and large problem. Such a mixed strategy, make it possible that first to solve the major part of a problem and then go back and solve the small problems arise during combining the big parts of the problem. Such a technique is called
**Means-Ends Analysis**. - Means-Ends Analysis is problem-solving techniques used in Artificial intelligence for limiting search in AI programs.
- It is a mixture of Backward and forward search technique.
- The MEA technique was first introduced in 1961 by Allen Newell, and Herbert A. Simon in their problem-solving computer program, which was named as General Problem Solver (GPS).
- The MEA analysis process centered on the evaluation of the difference between the current state and goal state.

## How means-ends analysis Works

The means-ends analysis process can be applied recursively for a problem. It is a strategy to control search in problem-solving. Following are the main steps that describe the working of the MEA technique for solving a problem.

- First, evaluate the difference between Initial State and final State.
- Select the various operators which can be applied for each difference.
- Apply the operator at each difference, which reduces the difference between the current state and goal state.

## Operator Subgoaling in AI

In the MEA process, we detect the differences between the current state and the goal state. Once these differences occur, then we can apply an operator to reduce the differences. But sometimes, it is possible that an operator cannot be applied to the current state. So we create the subproblem of the current state, in which the operator can be applied, such as a type of backward chaining in which operators are selected. Then sub-goals are set up to establish the preconditions of the operator is called **Operator Subgoaling**.

## Algorithm for Means-Ends Analysis

Let’s take the Current state as CURRENT and Goal State as GOAL; then following are the steps for the MEA algorithm.

**Step 1:**Compare CURRENT to GOAL, if there are no differences between both then return Success and Exit.**Step 2:**Else, select the most significant difference and reduce it by doing the following steps until the success or failure occurs.- Select a new operator O which is applicable for the current difference, and if there is no such operator, then signal failure.
- Attempt to apply operator O to CURRENT. Make a description of two states.

i) O-Start, a state in which O?s preconditions are satisfied.

ii) O-Result, the state that would result if O were applied In O-start. - If
**(First-Part <—— MEA (CURRENT, O-START)**

And**(LAST-Part <—– MEA (O-Result, GOAL)**, are successful, then signal Success and return the result of combining FIRST-PART, O, and LAST-PART.

The above-discussed algorithm is more suitable for a simple problem and not adequate for solving complex problems.https://imasdk.googleapis.com/js/core/bridge3.472.0_en.html#goog_1615571450HTML Tutorial

**Example of Mean-Ends Analysis**

Let’s take an example where we know the initial state and goal state as given below. In this problem, we need to get the goal state by finding differences between the initial state and goal state and applying operators.

### Solution:

To solve the above problem, we will first find the differences between initial states and goal states, and for each difference, we will generate a new state and apply the operators. The operators we have for this problem are:

**Move****Delete****Expand**

**1. Evaluating the initial state:** In the first step, we will evaluate the initial state and will compare the initial and Goal state to find the differences between both states.

**2. Applying Delete operator:** As we can check, the first difference is that in the goal state, no dot symbol is present in the initial state, so, first, we will apply the **Delete operator** to remove this dot.

**3. Applying Move Operator:** After applying the Delete operator, the new state will again compare with the goal state. After comparing these states, another difference is the square is outside the circle, so that we will apply the **Move Operator**.

**4. Applying Expand Operator:** Now, a new state is generated in the third step, and we will compare this state with the goal state. After comparing the states, there is still one difference which is the size of the square, so that we will apply **Expand operator**, and finally, it will generate the goal state.

## Problem Solving Algorithms in AI FAQ

### What kind of problem-solving strategy is an algorithm?

**A mathematical formula**is a good example of a problem-solving algorithm. While an algorithm guarantees an accurate answer, it is not always the best approach to problem-solving.

### What are the 4 types of algorithm?

**Algorithm types we will consider include:**

Simple recursive algorithms.

Backtracking algorithms.

Divide and conquer algorithms.

Dynamic programming algorithms.

Greedy algorithms.

Branch and bound algorithms.

Brute force algorithms.

Randomized algorithms.

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