# Python hill climbing example

Hill climbing search algorithm is one of the simplest algorithms which falls under local search and optimization techniques. Evaluation function at step 3 calculates the distance of the current state from the final state.

So in case of 3x3 Slide Puzzle we have:. Evaluation Function dF calculates the sum of the moves required for each tile to reach its final state. However, tile 8 is 1 move away from its final position.

Hill climbing evaluates the possible next moves and picks the one which has the least distance. It also checks if the new state after the move was already observed. If true, then it skips the move and picks the next best move. Another drawback which is highly documented is local optima.

The algorithm decides the next move state based on immediate distance costassuming that the small improvement now is the best way to reach the final state. However, the path chosen may lead to higher cost more steps later. Analogues to entering a valley after climbing a small hill.

In order to get around the local optima, I propose the usage of depth-first approach. I observed that the depth-first approach improves the overall efficiency of reaching the final state. However, its memory intensive, proportional to the depth value used. This is because the system has to keep track of future states as per the depth used.

A live evaluation can be performed here. Try out various depths and complexities and see the evaluation graphs. Sometimes, the puzzle remains unresolved due to lockdown no new state. Or, if you are just in the mood of solving the puzzle, try yourself against the bot powered by Hill Climbing Algorithm. Hit the like button on this article every time you lose against the bot Sign in.

Rahul Aware Follow. Towards Data Science A Medium publication sharing concepts, ideas, and codes. Towards Data Science Follow. A Medium publication sharing concepts, ideas, and codes. Write the first response. More From Medium. More from Towards Data Science. Rhea Moutafis in Towards Data Science. Taylor Brownlow in Towards Data Science. Discover Medium. Make Medium yours. Become a member. About Help Legal.Hill climbing search is a local search problem.

It is based on the heuristic search technique where the person who is climbing up on the hill estimates the direction which will lead him to the highest peak. The topographical regions shown in the figure can be defined as:. Simple hill climbing is the simplest technique to climb a hill.

The task is to reach the highest peak of the mountain.

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If he finds his next step better than the previous one, he continues to move else remain in the same state. This search focus only on his previous and next step. Steepest-ascent hill climbing is different from simple hill climbing search. Unlike simple hill climbing search, It considers all the successive nodes, compares them, and choose the node which is closest to the solution.

Steepest hill climbing search is similar to best-first search because it focuses on each node instead of one. Note: Both simple, as well as steepest-ascent hill climbing search, fails when there is no closer node.

Stochastic hill climbing does not focus on all the nodes. It selects one node at random and decides whether it should be expanded or search for a better one. Random-restart algorithm is based on try and try strategy. It iteratively searches the node and selects the best one at each step until the goal is not found.

The success depends most commonly on the shape of the hill. If there are few plateaus, local maxima, and ridges, it becomes easy to reach the destination. Hill climbing algorithm is a fast and furious approach. It finds the solution state rapidly because it is quite easy to improve a bad state. But, there are following limitations of this search:. Simulated annealing is similar to the hill climbing algorithm.

It works on the current situation. It picks a random move instead of picking the best move. If the move leads to the improvement of the current situation, it is always accepted as a step towards the solution state, else it accepts the move having a probability less than 1.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service.

Stack Overflow for Teams is a private, secure spot for you and your coworkers to find and share information. I am a little confused with Hill Climbing algorithm. I want to "run" the algorithm until i found the first solution in that tree "a" is initial and h and k are final states and it says that the numbers near the states are the heuristic values. Here's the tree:. Is this right? If i can go back then how? A common way to avoid getting stuck in local maxima with Hill Climbing is to use random restarts.

In your example if G is a local maxima, the algorithm would stop there and then pick another random node to restart from. Note that Local Search like Hill Climbing isn't complete and can't guarantee to find the global maxima. The benefit, of course, is that it requires a fraction of the resources. In practice and applied to the right problems, it's a very effective solution. You could try to use a technique called simulated annealing to prevent your search to get stuck on to local minimums.

Essentially, in simulated annealing, there is a parameter T that controls your likelihood to make a move to sub-optimal neighborhood states. If T is high, you are more likely to make a sub-optimal move to a neighboring state and thereby might escape a local minimum when stuck there, which you wouldn't if you used normal hill climbing.

However, only the purest form of hill climbing doesn't allow you to either backtrack. A simple riff on hill climbing that will avoid the local minima issue at the expense of more time and memory is a tabu search, where you remember previous bad results and purposefully avoid them. If once again you get stuck at some local minima you have to restart again with some other random node.

Generally there is a limit on the no. After you reach this limit, you select the least amongst all the local minimas you reached during the process. Though it is still not complete but this one has better chances of finding the global optimal solution. Actually in Hill Climbing you don't generally backtrack, because you're not keeping track of state it's local search and you would be moving away from a maxima.

Neither backtracking or Tabu Search help answer the question either: the former just moves you away from a local maxima and the latter keeps you from revisiting the same local maxima. Neither would help you hit a global maxima.

NOTE: The proposed variation of hill-climbing to pick a point randomly, but picking the least cost other than the already visited nodes is better than picking randomly. ANOTHER NOTE: is that when node E didn't visit I because I has higher value than E, the algorithm already inserted it in the data structure, thus picking the least cost other than the already visited would create a new path from I because I was never visited and thus it has lower value than J, this is the only path that i've skipped.

How are we doing? Please help us improve Stack Overflow. Take our short survey. Learn more. Hill climbing algorithm simple example Ask Question. Asked 8 years, 2 months ago.

Active 1 year, 6 months ago. Viewed 64k times. Iakob Fokas Iakob Fokas 1 1 gold badge 2 2 silver badges 6 6 bronze badges. Hill climbing is local search. You need to define some kind of neighbour relation between states. Usually this relation is symmetric. You have a directed tree there, which reminds me of a search tree.Hill Climbing is a heuristic search used for mathematical optimization problems in the field of Artificial Intelligence.

Given a large set of inputs and a good heuristic function, it tries to find a sufficiently good solution to the problem. This solution may not be the global optimal maximum. Generate possible solutions.

Test to see if this is the expected solution. If the solution has been found quit else go to step 1. Hence we call Hill climbing as a variant of generate and test algorithm as it takes the feedback from the test procedure. Then this feedback is utilized by the generator in deciding the next move in search space. Uses the Greedy approach : At any point in state space, the search moves in that direction only which optimizes the cost of function with the hope of finding the optimal solution at the end.

Types of Hill Climbing. Step 1 : Evaluate the initial state. If it is a goal state then stop and return success. Otherwise, make initial state as current state. Step 2 : Loop until the solution state is found or there are no new operators present which can be applied to the current state. If the current state is a goal state, then stop and return success. If it is better than the current state, then make it current state and proceed further.

If it is not better than the current state, then continue in the loop until a solution is found. Step 3 : Exit. If it is goal state then exit else make the current state as initial state Step 2 : Repeat these steps until a solution is found or current state does not change. State space diagram is a graphical representation of the set of states our search algorithm can reach vs the value of our objective function the function which we wish to maximize.

X-axis : denotes the state space ie states or configuration our algorithm may reach. Y-axis : denotes the values of objective function corresponding to a particular state. The best solution will be that state space where objective function has maximum value global maximum. To overcome plateaus : Make a big jump. Randomly select a state far away from the current state.

Chances are that we will land at a non-plateau region Ridge : Any point on a ridge can look like peak because movement in all possible directions is downward. Hence the algorithm stops when it reaches this state. To overcome Ridge : In this kind of obstacle, use two or more rules before testing. It implies moving in several directions at once.

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Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below. Writing code in comment? Please use ide. How to be a Good Programmer in College? What Should I Do? How to Think Like a Programmer? In the above definition, mathematical optimization problems implies that hill-climbing solves the problems where we need to maximize or minimize a given real function by choosing values from the given inputs.

Example- Travelling salesman problem where we need to minimize the distance traveled by the salesman.This is a template method for the hill climbing algorithm.

It doesn't guarantee that it will return the optimal solution. Because you have found a configuration that is better than the initial configuration A, so you want to go on using hill-climbing instead of returning the initial configuration.

### Hill Climbing Algorithm

Also, you mixed up the variable names a bit in the end I think, but that just as a side note. You are right. Should it be random? Privacy Policy Contact Us Support. All rights reserved. All other marks are property of their respective owners. Languages Tags Authors Sets.

Python, 89 lines Download. Copy to clipboard. In particular, the son is obtained changing the row index of one of the queen. The data structure representation for the chessboard will be a list of tuples each of them representing the coordinates of a the position of a queen. It has to be passed as argument of the function sons A ans value A that determines, respectively, the next configuration starting from A where Hill Climbing Algorithm needs to restart to evaluate and the heuristic function h.

### Hill Climbing Algorithm in Artificial Intelligence

This function represents a template method. Hi, thanks for the code! Helped me a lot. Thanks Robo! You are right, I did some mistakes there. Changed the recipe. I am happy that this recipe helped you. Hi, You are right. Required Modules none specified. Accounts Create Account Free! Sign In. Filippo Squillace's recipes 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 search is to find the global minimum and local minimum. If the function of Y-axis is Objective function, then the goal of the search is to find the global maximum and local maximum. 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 state space landscape. It has the highest value of 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.

Simple hill climbing is the simplest way to implement a hill climbing algorithm. It only evaluates the neighbor node state at a time and selects the first one which optimizes current cost and set it as a current state.

## What is Heuristic Search – Techniques & Hill Climbing in AI

It only checks it's 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:. The steepest-Ascent algorithm is a variation of simple hill climbing algorithm. This algorithm examines all the neighboring nodes of the current state and selects one neighbor node which is closest to the goal state.

This algorithm consumes more time as it searches for multiple neighbors. Stochastic hill climbing does not examine for all its neighbor 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.

Local Maximum: A local maximum is a peak state in the landscape which is better than each of its neighboring states, but there is another state also present which is higher than the local maximum.

Solution: Backtracking technique can be a solution of the local maximum in state space landscape. Create a list of the promising path so that the algorithm can backtrack the search space and explore other paths as well.

Plateau: A plateau is the flat area of the search space in which all the neighbor states of the current state contains the same value, because of this algorithm does not find any best direction to move.In this tutorial, we will discuss what is meant by an optimization problem and step through an example of how mlrose can be used to solve them.

This is the first in a series of three tutorials. Part 2 can be found here and Part 3 can be found here. Some examples of states are:. What is important, for our purposes, is that the state can be represented numerically, ideally as a one-dimensional array or vector of values. The output fitness values allow us to compare the inputted state to other states we might be considering. In this context, the elements of the state array can be thought of as the variables or parameters of the function.

For the One-Max example given above, even if the solution was not immediately obvious, it would be possible to calculate the fitness value for all possible state vectors, xand then select the best of those vectors. However, for more complicated problems, this cannot always be done within a reasonable period of time.

Randomized optimization overcomes this issue. There is no guarantee a randomized optimization algorithm will find the optimal solution to a given optimization problem for example, it is possible that the algorithm may find a local maximum of the fitness function, instead of the global maximum. There is a trade-off between the time spent searching for the optimal solution to an optimization problem and the quality of the solution ultimately found.

Solving an optimization problem using mlrose involves three simple steps:. To illustrate each of these steps, we will work through the example of the 8-Queens optimization problem, described below:. In chess, the queen is the most powerful piece on the board. It can attack any piece in the same row, column or diagonal. In the 8-Queens problem, you are given an 8 x 8 chessboard with eight queens and no other pieces and the aim is to place the queens on the board so that none of them can attack each other.

Clearly, in an optimal solution to this problem, there will be exactly one queen in each column of the chessboard. This is not an optimal solution to the 8-Queens problem, since the three queens in columns 5, 6 and 7 are attacking each other diagonally, as are the queens in columns 2 and 6. Before starting with this example, you will need to import the mlrose and Numpy Python packages. The first step in solving any optimization problem is to define the fitness function.

This is the function we would ultimately like to maximize or minimize, and which can be used to evaluate the fitness of a given state vector, x. In the context of the 8-Queens problem, our goal is to find a state vector for which no pairs of attacking queens exist. Therefore, we could define our fitness function as evaluating the number of pairs of attacking queens for a given state and try to minimize this function.

The pre-defined Queens class includes an implementation of the 8- Queens fitness function described above. We can initialize a fitness function object for this class, as follows:.

Alternatively, we could look at the 8-Queens problem as one where the aim is to find a state vector for which all pairs of queens do not attack each other. In this context, we could define our fitness function as evaluating the number of pairs of non-attacking queens for a given state and try to maximize this function.

Once we have created a fitness function object, we can use it as an input into an optimization problem object. In mlroseoptimization problem objects are used to contain all of the important information about the optimization problem we are trying to solve. The 8-Queens problem is an example of a discrete-state optimization problem, since each of the elements of the state vector must take on an integer value in the range 0 to 7. To initialize a discrete-state optimization problem object, it is necessary to specify the problem length i.

For this example, we will use the first of the two fitness function objects defined above, so we want to solve a minimization problem. Now that we have defined an optimization problem object, we are ready to solve our optimization problem.

For discrete-state and travelling salesperson optimization problems, we can choose any of these algorithms. For our example, suppose we wish to use simulated annealing. To specify the schedule object, mlrose includes pre-defined decay schedule classes for geometric, arithmetic and expontential decay, as well as a class for defining your own decay schedule in a manner similar to the way in which we created a customized fitness function object.

This can be done using the following code.

The algorithm returns the best state it can find, given the parameter values it has been provided, as well as the fitness value for that state. Running this code gives us a good solution to the 8-Queens problem, but not the optimal solution.