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},
{
"cell_type": "code",
"execution_count": 43,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"8.53 ms ± 601 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)\n"
]
}
],
"source": [
"%%timeit\n",
"astar_search(puzzle_1, max_heuristic)\n",
"astar_search(puzzle_2, max_heuristic)\n",
"astar_search(puzzle_3, max_heuristic)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We can infer that the `manhattan` heuristic function works the fastest.\n",
"<br>\n",
"`sqrt_manhattan` has an extra `sqrt` operation which makes it quite a lot slower than the others.\n",
"<br>\n",
"`max_heuristic` should have been a bit slower as it calls two functions, but in this case, those values were already calculated which saved some time.\n",
"Feel free to play around with these functions."
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{
"cell_type": "markdown",
"metadata": {},
"source": [
"## HILL CLIMBING\n",
"\n",
"Hill Climbing is a heuristic search used for optimization problems.\n",
"Given a large set of inputs and a good heuristic function, it tries to find a sufficiently good solution to the problem. \n",
"This solution may or may not be the global optimum.\n",
"The algorithm is a variant of generate and test algorithm. \n",
"<br>\n",
"As a whole, the algorithm works as follows:\n",
"- Evaluate the initial state.\n",
"- If it is equal to the goal state, return.\n",
"- Find a neighboring state (one which is heuristically similar to the current state)\n",
"- Evaluate this state. If it is closer to the goal state than before, replace the initial state with this state and repeat these steps.\n",
"<br>"
]
},
{
"cell_type": "code",
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"execution_count": 44,
"metadata": {},
"outputs": [
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"\n",
"<div class=\"highlight\"><pre><span></span><span class=\"k\">def</span> <span class=\"nf\">hill_climbing</span><span class=\"p\">(</span><span class=\"n\">problem</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""From the initial node, keep choosing the neighbor with highest value,</span>\n",
"<span class=\"sd\"> stopping when no neighbor is better. [Figure 4.2]"""</span>\n",
" <span class=\"n\">current</span> <span class=\"o\">=</span> <span class=\"n\">Node</span><span class=\"p\">(</span><span class=\"n\">problem</span><span class=\"o\">.</span><span class=\"n\">initial</span><span class=\"p\">)</span>\n",
" <span class=\"k\">while</span> <span class=\"bp\">True</span><span class=\"p\">:</span>\n",
" <span class=\"n\">neighbors</span> <span class=\"o\">=</span> <span class=\"n\">current</span><span class=\"o\">.</span><span class=\"n\">expand</span><span class=\"p\">(</span><span class=\"n\">problem</span><span class=\"p\">)</span>\n",
" <span class=\"k\">if</span> <span class=\"ow\">not</span> <span class=\"n\">neighbors</span><span class=\"p\">:</span>\n",
" <span class=\"k\">break</span>\n",
" <span class=\"n\">neighbor</span> <span class=\"o\">=</span> <span class=\"n\">argmax_random_tie</span><span class=\"p\">(</span><span class=\"n\">neighbors</span><span class=\"p\">,</span>\n",
" <span class=\"n\">key</span><span class=\"o\">=</span><span class=\"k\">lambda</span> <span class=\"n\">node</span><span class=\"p\">:</span> <span class=\"n\">problem</span><span class=\"o\">.</span><span class=\"n\">value</span><span class=\"p\">(</span><span class=\"n\">node</span><span class=\"o\">.</span><span class=\"n\">state</span><span class=\"p\">))</span>\n",
" <span class=\"k\">if</span> <span class=\"n\">problem</span><span class=\"o\">.</span><span class=\"n\">value</span><span class=\"p\">(</span><span class=\"n\">neighbor</span><span class=\"o\">.</span><span class=\"n\">state</span><span class=\"p\">)</span> <span class=\"o\"><=</span> <span class=\"n\">problem</span><span class=\"o\">.</span><span class=\"n\">value</span><span class=\"p\">(</span><span class=\"n\">current</span><span class=\"o\">.</span><span class=\"n\">state</span><span class=\"p\">):</span>\n",
" <span class=\"k\">break</span>\n",
" <span class=\"n\">current</span> <span class=\"o\">=</span> <span class=\"n\">neighbor</span>\n",
" <span class=\"k\">return</span> <span class=\"n\">current</span><span class=\"o\">.</span><span class=\"n\">state</span>\n",
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"source": [
"psource(hill_climbing)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We will find an approximate solution to the traveling salespersons problem using this algorithm.\n",
"<br>\n",
"We need to define a class for this problem.\n",
"<br>\n",
"`Problem` will be used as a base class."
]
},
{
"cell_type": "code",
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"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"class TSP_problem(Problem):\n",
"\n",
" \"\"\" subclass of Problem to define various functions \"\"\"\n",
"\n",
" def two_opt(self, state):\n",
" \"\"\" Neighbour generating function for Traveling Salesman Problem \"\"\"\n",
" neighbour_state = state[:]\n",
" left = random.randint(0, len(neighbour_state) - 1)\n",
" right = random.randint(0, len(neighbour_state) - 1)\n",
" if left > right:\n",
" left, right = right, left\n",
" neighbour_state[left: right + 1] = reversed(neighbour_state[left: right + 1])\n",
" return neighbour_state\n",
"\n",
" def actions(self, state):\n",
" \"\"\" action that can be excuted in given state \"\"\"\n",
" return [self.two_opt]\n",
"\n",
" def result(self, state, action):\n",
" \"\"\" result after applying the given action on the given state \"\"\"\n",
" return action(state)\n",
"\n",
" def path_cost(self, c, state1, action, state2):\n",
" \"\"\" total distance for the Traveling Salesman to be covered if in state2 \"\"\"\n",
" cost = 0\n",
" for i in range(len(state2) - 1):\n",
" cost += distances[state2[i]][state2[i + 1]]\n",
" cost += distances[state2[0]][state2[-1]]\n",
" return cost\n",
"\n",
" def value(self, state):\n",
" \"\"\" value of path cost given negative for the given state \"\"\"\n",
" return -1 * self.path_cost(None, None, None, state)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We will use cities from the Romania map as our cities for this problem.\n",
"<br>\n",
"A list of all cities and a dictionary storing distances between them will be populated."
]
},
{
"cell_type": "code",
"execution_count": 46,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"['Arad', 'Bucharest', 'Craiova', 'Drobeta', 'Eforie', 'Fagaras', 'Giurgiu', 'Hirsova', 'Iasi', 'Lugoj', 'Mehadia', 'Neamt', 'Oradea', 'Pitesti', 'Rimnicu', 'Sibiu', 'Timisoara', 'Urziceni', 'Vaslui', 'Zerind']\n"
]
}
],
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"source": [
"distances = {}\n",
"all_cities = []\n",
"\n",
"for city in romania_map.locations.keys():\n",
" distances[city] = {}\n",
" all_cities.append(city)\n",
" \n",
"all_cities.sort()\n",
"print(all_cities)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Next, we need to populate the individual lists inside the dictionary with the manhattan distance between the cities."
]
},
{
"cell_type": "code",
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"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"import numpy as np\n",
"for name_1, coordinates_1 in romania_map.locations.items():\n",
" for name_2, coordinates_2 in romania_map.locations.items():\n",
" distances[name_1][name_2] = np.linalg.norm(\n",
" [coordinates_1[0] - coordinates_2[0], coordinates_1[1] - coordinates_2[1]])\n",
" distances[name_2][name_1] = np.linalg.norm(\n",
" [coordinates_1[0] - coordinates_2[0], coordinates_1[1] - coordinates_2[1]])"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The way neighbours are chosen currently isn't suitable for the travelling salespersons problem.\n",
"We need a neighboring state that is similar in total path distance to the current state.\n",
"<br>\n",
"We need to change the function that finds neighbors."
]
},
{
"cell_type": "code",
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"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"def hill_climbing(problem):\n",
" \n",
" \"\"\"From the initial node, keep choosing the neighbor with highest value,\n",
" stopping when no neighbor is better. [Figure 4.2]\"\"\"\n",
" \n",
" def find_neighbors(state, number_of_neighbors=100):\n",
" \"\"\" finds neighbors using two_opt method \"\"\"\n",
" \n",
" neighbors = []\n",
" \n",
" for i in range(number_of_neighbors):\n",
" new_state = problem.two_opt(state)\n",
" neighbors.append(Node(new_state))\n",
" state = new_state\n",
" \n",
" return neighbors\n",
"\n",
" # as this is a stochastic algorithm, we will set a cap on the number of iterations\n",
" iterations = 10000\n",
" \n",
" current = Node(problem.initial)\n",
" while iterations:\n",
" neighbors = find_neighbors(current.state)\n",
" if not neighbors:\n",
" break\n",
" neighbor = argmax_random_tie(neighbors,\n",
" key=lambda node: problem.value(node.state))\n",
" if problem.value(neighbor.state) <= problem.value(current.state):\n",
" current.state = neighbor.state\n",
" iterations -= 1\n",
" \n",
" return current.state"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"An instance of the TSP_problem class will be created."
]
},
{
"cell_type": "code",
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"tsp = TSP_problem(all_cities)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We can now generate an approximate solution to the problem by calling `hill_climbing`.\n",
"The results will vary a bit each time you run it."
]
},
{
"cell_type": "code",
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"execution_count": 50,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"['Arad',\n",
" 'Timisoara',\n",
" 'Lugoj',\n",
" 'Mehadia',\n",
" 'Drobeta',\n",
" 'Craiova',\n",
" 'Pitesti',\n",
" 'Giurgiu',\n",
" 'Bucharest',\n",
" 'Urziceni',\n",
" 'Eforie',\n",
" 'Hirsova',\n",
" 'Vaslui',\n",
" 'Iasi',\n",
" 'Neamt',\n",
" 'Fagaras',\n",
" 'Rimnicu',\n",
" 'Sibiu',\n",
" 'Oradea',\n",
" 'Zerind']"
]
},
"execution_count": 50,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"hill_climbing(tsp)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The solution looks like this.\n",
"It is not difficult to see why this might be a good solution.\n",
"<br>\n",
""
]
},
{
"cell_type": "markdown",
"\n",
"Genetic algorithms (or GA) are inspired by natural evolution and are particularly useful in optimization and search problems with large state spaces.\n",
"\n",
"Given a problem, algorithms in the domain make use of a *population* of solutions (also called *states*), where each solution/state represents a feasible solution. At each iteration (often called *generation*), the population gets updated using methods inspired by biology and evolution, like *crossover*, *mutation* and *natural selection*."
]
},
{
"cell_type": "markdown",
"source": [
"### Overview\n",
"\n",
"A genetic algorithm works in the following way:\n",
"\n",
"1) Initialize random population.\n",
"\n",
"2) Calculate population fitness.\n",
"\n",
"3) Select individuals for mating.\n",
"\n",
"4) Mate selected individuals to produce new population.\n",
"\n",
" * Random chance to mutate individuals.\n",
"\n",
"5) Repeat from step 2) until an individual is fit enough or the maximum number of iterations was reached."
]
},
{
"cell_type": "markdown",
"### Glossary\n",
"\n",
"Before we continue, we will lay the basic terminology of the algorithm.\n",
"\n",
"* Individual/State: A list of elements (called *genes*) that represent possible solutions.\n",
"* Population: The list of all the individuals/states.\n",
"\n",
"* Gene pool: The alphabet of possible values for an individual's genes.\n",
"\n",
"* Generation/Iteration: The number of times the population will be updated.\n",
"\n",
"* Fitness: An individual's score, calculated by a function specific to the problem."
]
},
{
"cell_type": "markdown",
"### Crossover\n",
"\n",
"Two individuals/states can \"mate\" and produce one child. This offspring bears characteristics from both of its parents. There are many ways we can implement this crossover. Here we will take a look at the most common ones. Most other methods are variations of those below.\n",
"\n",
"* Point Crossover: The crossover occurs around one (or more) point. The parents get \"split\" at the chosen point or points and then get merged. In the example below we see two parents get split and merged at the 3rd digit, producing the following offspring after the crossover.\n",
"\n",
"\n",
"\n",
"* Uniform Crossover: This type of crossover chooses randomly the genes to get merged. Here the genes 1, 2 and 5 were chosen from the first parent, so the genes 3, 4 were added by the second parent.\n",
"\n",
""
]
},
{
"cell_type": "markdown",
"### Mutation\n",
"\n",
"When an offspring is produced, there is a chance it will mutate, having one (or more, depending on the implementation) of its genes altered.\n",
"\n",
"For example, let's say the new individual to undergo mutation is \"abcde\". Randomly we pick to change its third gene to 'z'. The individual now becomes \"abzde\" and is added to the population."
]
},
{
"cell_type": "markdown",
"At each iteration, the fittest individuals are picked randomly to mate and produce offsprings. We measure an individual's fitness with a *fitness function*. That function depends on the given problem and it is used to score an individual. Usually the higher the better.\n",
"The selection process is this:\n",
"1) Individuals are scored by the fitness function.\n",
"\n",
"2) Individuals are picked randomly, according to their score (higher score means higher chance to get picked). Usually the formula to calculate the chance to pick an individual is the following (for population *P* and individual *i*):\n",
"\n",
"$$ chance(i) = \\dfrac{fitness(i)}{\\sum_{k \\, in \\, P}{fitness(k)}} $$"
]
},
{
"cell_type": "markdown",
"### Implementation\n",
"\n",
"Below we look over the implementation of the algorithm in the `search` module.\n",
"\n",
"First the implementation of the main core of the algorithm:"
]
},
{
"cell_type": "code",
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"<div class=\"highlight\"><pre><span></span><span class=\"k\">def</span> <span class=\"nf\">genetic_algorithm</span><span class=\"p\">(</span><span class=\"n\">population</span><span class=\"p\">,</span> <span class=\"n\">fitness_fn</span><span class=\"p\">,</span> <span class=\"n\">gene_pool</span><span class=\"o\">=</span><span class=\"p\">[</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"mi\">1</span><span class=\"p\">],</span> <span class=\"n\">f_thres</span><span class=\"o\">=</span><span class=\"bp\">None</span><span class=\"p\">,</span> <span class=\"n\">ngen</span><span class=\"o\">=</span><span class=\"mi\">1000</span><span class=\"p\">,</span> <span class=\"n\">pmut</span><span class=\"o\">=</span><span class=\"mf\">0.1</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""[Figure 4.8]"""</span>\n",
" <span class=\"k\">for</span> <span class=\"n\">i</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"n\">ngen</span><span class=\"p\">):</span>\n",
" <span class=\"n\">population</span> <span class=\"o\">=</span> <span class=\"p\">[</span><span class=\"n\">mutate</span><span class=\"p\">(</span><span class=\"n\">recombine</span><span class=\"p\">(</span><span class=\"o\">*</span><span class=\"n\">select</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"p\">,</span> <span class=\"n\">population</span><span class=\"p\">,</span> <span class=\"n\">fitness_fn</span><span class=\"p\">)),</span> <span class=\"n\">gene_pool</span><span class=\"p\">,</span> <span class=\"n\">pmut</span><span class=\"p\">)</span>\n",
" <span class=\"k\">for</span> <span class=\"n\">i</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"nb\">len</span><span class=\"p\">(</span><span class=\"n\">population</span><span class=\"p\">))]</span>\n",
"\n",
" <span class=\"n\">fittest_individual</span> <span class=\"o\">=</span> <span class=\"n\">fitness_threshold</span><span class=\"p\">(</span><span class=\"n\">fitness_fn</span><span class=\"p\">,</span> <span class=\"n\">f_thres</span><span class=\"p\">,</span> <span class=\"n\">population</span><span class=\"p\">)</span>\n",
" <span class=\"k\">if</span> <span class=\"n\">fittest_individual</span><span class=\"p\">:</span>\n",
" <span class=\"k\">return</span> <span class=\"n\">fittest_individual</span>\n",
"\n",
"\n",
" <span class=\"k\">return</span> <span class=\"n\">argmax</span><span class=\"p\">(</span><span class=\"n\">population</span><span class=\"p\">,</span> <span class=\"n\">key</span><span class=\"o\">=</span><span class=\"n\">fitness_fn</span><span class=\"p\">)</span>\n",
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Aman Deep Singh
a validé
"psource(genetic_algorithm)"
]
},
{
"cell_type": "markdown",
"source": [
"The algorithm takes the following input:\n",
"\n",
"* `population`: The initial population.\n",
"\n",
"* `fitness_fn`: The problem's fitness function.\n",
"\n",
"* `gene_pool`: The gene pool of the states/individuals. By default 0 and 1.\n",
"* `f_thres`: The fitness threshold. If an individual reaches that score, iteration stops. By default 'None', which means the algorithm will not halt until the generations are ran.\n",
"\n",
"* `ngen`: The number of iterations/generations.\n",
"\n",
"* `pmut`: The probability of mutation.\n",
"\n",
"The algorithm gives as output the state with the largest score."
]
},
{
"cell_type": "markdown",
"For each generation, the algorithm updates the population. First it calculates the fitnesses of the individuals, then it selects the most fit ones and finally crosses them over to produce offsprings. There is a chance that the offspring will be mutated, given by `pmut`. If at the end of the generation an individual meets the fitness threshold, the algorithm halts and returns that individual.\n",
"\n",
Aman Deep Singh
a validé
"The function of mating is accomplished by the method `recombine`:"
]
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{
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"\n",
"<div class=\"highlight\"><pre><span></span><span class=\"k\">def</span> <span class=\"nf\">recombine</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">y</span><span class=\"p\">):</span>\n",
" <span class=\"n\">n</span> <span class=\"o\">=</span> <span class=\"nb\">len</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n",
" <span class=\"n\">c</span> <span class=\"o\">=</span> <span class=\"n\">random</span><span class=\"o\">.</span><span class=\"n\">randrange</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">n</span><span class=\"p\">)</span>\n",
" <span class=\"k\">return</span> <span class=\"n\">x</span><span class=\"p\">[:</span><span class=\"n\">c</span><span class=\"p\">]</span> <span class=\"o\">+</span> <span class=\"n\">y</span><span class=\"p\">[</span><span class=\"n\">c</span><span class=\"p\">:]</span>\n",
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Aman Deep Singh
a validé
"source": [
"psource(recombine)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The method picks at random a point and merges the parents (`x` and `y`) around it.\n",
"\n",
"The mutation is done in the method `mutate`:"
]
},
{
"cell_type": "code",
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"<div class=\"highlight\"><pre><span></span><span class=\"k\">def</span> <span class=\"nf\">mutate</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">gene_pool</span><span class=\"p\">,</span> <span class=\"n\">pmut</span><span class=\"p\">):</span>\n",
" <span class=\"k\">if</span> <span class=\"n\">random</span><span class=\"o\">.</span><span class=\"n\">uniform</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"mi\">1</span><span class=\"p\">)</span> <span class=\"o\">>=</span> <span class=\"n\">pmut</span><span class=\"p\">:</span>\n",
" <span class=\"k\">return</span> <span class=\"n\">x</span>\n",
"\n",
" <span class=\"n\">n</span> <span class=\"o\">=</span> <span class=\"nb\">len</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span>\n",
" <span class=\"n\">g</span> <span class=\"o\">=</span> <span class=\"nb\">len</span><span class=\"p\">(</span><span class=\"n\">gene_pool</span><span class=\"p\">)</span>\n",
" <span class=\"n\">c</span> <span class=\"o\">=</span> <span class=\"n\">random</span><span class=\"o\">.</span><span class=\"n\">randrange</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">n</span><span class=\"p\">)</span>\n",
" <span class=\"n\">r</span> <span class=\"o\">=</span> <span class=\"n\">random</span><span class=\"o\">.</span><span class=\"n\">randrange</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"n\">g</span><span class=\"p\">)</span>\n",
"\n",
" <span class=\"n\">new_gene</span> <span class=\"o\">=</span> <span class=\"n\">gene_pool</span><span class=\"p\">[</span><span class=\"n\">r</span><span class=\"p\">]</span>\n",
" <span class=\"k\">return</span> <span class=\"n\">x</span><span class=\"p\">[:</span><span class=\"n\">c</span><span class=\"p\">]</span> <span class=\"o\">+</span> <span class=\"p\">[</span><span class=\"n\">new_gene</span><span class=\"p\">]</span> <span class=\"o\">+</span> <span class=\"n\">x</span><span class=\"p\">[</span><span class=\"n\">c</span><span class=\"o\">+</span><span class=\"mi\">1</span><span class=\"p\">:]</span>\n",
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Aman Deep Singh
a validé
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"psource(mutate)"
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"We pick a gene in `x` to mutate and a gene from the gene pool to replace it with.\n",
"\n",
"To help initializing the population we have the helper function `init_population`\":"
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