This notebook serves as supporting material for topics covered in Chapter 3 - Solving Problems by Searching and Chapter 4 - Beyond Classical Search from the book Artificial Intelligence: A Modern Approach. This notebook uses implementations from search.py module. Let's start by importing everything from search module.
In [1]:
from search import *
# Needed to hide warnings in the matplotlib sections
import warnings
warnings.filterwarnings("ignore")
Here, we learn about problem solving. Building goal-based agents that can plan ahead to solve problems, in particular, navigation problem/route finding problem. First, we will start the problem solving by precisely defining problems and their solutions. We will look at several general-purpose search algorithms. Broadly, search algorithms are classified into two types:
Don't miss the visualisations of these algorithms solving the route-finding problem defined on Romania map at the end of this notebook.
In [2]:
%psource Problem
The Problem class has six methods.
__init__(self, initial, goal) : This is what is called a constructor and is the first method called when you create an instance of the class. initial specifies the initial state of our search problem. It represents the start state from where our agent begins its task of exploration to find the goal state(s) which is given in the goal parameter.actions(self, state) : This method returns all the possible actions agent can execute in the given state state.result(self, state, action) : This returns the resulting state if action action is taken in the state state. This Problem class only deals with deterministic outcomes. So we know for sure what every action in a state would result to.goal_test(self, state) : Given a graph state, it checks if it is a terminal state. If the state is indeed a goal state, value of True is returned. Else, of course, False is returned.path_cost(self, c, state1, action, state2) : Return the cost of the path that arrives at state2 as a result of taking action from state1, assuming total cost of c to get up to state1.value(self, state) : This acts as a bit of extra information in problems where we try to optimise a value when we cannot do a goal test.We will use the abstract class Problem to define our real problem named GraphProblem. You can see how we define GraphProblem by running the next cell.
In [3]:
%psource GraphProblem
Now it's time to define our problem. We will define it by passing initial, goal, graph to GraphProblem. So, our problem is to find the goal state starting from the given initial state on the provided graph. Have a look at our romania_map, which is an Undirected Graph containing a dict of nodes as keys and neighbours as values.
In [4]:
romania_map = UndirectedGraph(dict(
Arad=dict(Zerind=75, Sibiu=140, Timisoara=118),
Bucharest=dict(Urziceni=85, Pitesti=101, Giurgiu=90, Fagaras=211),
Craiova=dict(Drobeta=120, Rimnicu=146, Pitesti=138),
Drobeta=dict(Mehadia=75),
Eforie=dict(Hirsova=86),
Fagaras=dict(Sibiu=99),
Hirsova=dict(Urziceni=98),
Iasi=dict(Vaslui=92, Neamt=87),
Lugoj=dict(Timisoara=111, Mehadia=70),
Oradea=dict(Zerind=71, Sibiu=151),
Pitesti=dict(Rimnicu=97),
Rimnicu=dict(Sibiu=80),
Urziceni=dict(Vaslui=142)))
romania_map.locations = dict(
Arad=(91, 492), Bucharest=(400, 327), Craiova=(253, 288),
Drobeta=(165, 299), Eforie=(562, 293), Fagaras=(305, 449),
Giurgiu=(375, 270), Hirsova=(534, 350), Iasi=(473, 506),
Lugoj=(165, 379), Mehadia=(168, 339), Neamt=(406, 537),
Oradea=(131, 571), Pitesti=(320, 368), Rimnicu=(233, 410),
Sibiu=(207, 457), Timisoara=(94, 410), Urziceni=(456, 350),
Vaslui=(509, 444), Zerind=(108, 531))
It is pretty straightforward to understand this romania_map. The first node Arad has three neighbours named Zerind, Sibiu, Timisoara. Each of these nodes are 75, 140, 118 units apart from Arad respectively. And the same goes with other nodes.
And romania_map.locations contains the positions of each of the nodes. We will use the straight line distance (which is different from the one provided in romania_map) between two cities in algorithms like A*-search and Recursive Best First Search.
Define a problem: Hmm... say we want to start exploring from Arad and try to find Bucharest in our romania_map. So, this is how we do it.
In [5]:
romania_problem = GraphProblem('Arad', 'Bucharest', romania_map)
Have a look at romania_locations. It is a dictionary defined in search module. We will use these location values to draw the romania graph using networkx.
In [6]:
romania_locations = romania_map.locations
print(romania_locations)
Let's start the visualisations by importing necessary modules. We use networkx and matplotlib to show the map in the notebook and we use ipywidgets to interact with the map to see how the searching algorithm works.
In [7]:
%matplotlib inline
import networkx as nx
import matplotlib.pyplot as plt
from matplotlib import lines
from ipywidgets import interact
import ipywidgets as widgets
from IPython.display import display
import time
Let's get started by initializing an empty graph. We will add nodes, place the nodes in their location as shown in the book, add edges to the graph.
In [8]:
# initialise a graph
G = nx.Graph()
# use this while labeling nodes in the map
node_labels = dict()
# use this to modify colors of nodes while exploring the graph.
# This is the only dict we send to `show_map(node_colors)` while drawing the map
node_colors = dict()
for n, p in romania_locations.items():
# add nodes from romania_locations
G.add_node(n)
# add nodes to node_labels
node_labels[n] = n
# node_colors to color nodes while exploring romania map
node_colors[n] = "white"
# we'll save the initial node colors to a dict to use later
initial_node_colors = dict(node_colors)
# positions for node labels
node_label_pos = { k:[v[0],v[1]-10] for k,v in romania_locations.items() }
# use this while labeling edges
edge_labels = dict()
# add edges between cities in romania map - UndirectedGraph defined in search.py
for node in romania_map.nodes():
connections = romania_map.get(node)
for connection in connections.keys():
distance = connections[connection]
# add edges to the graph
G.add_edge(node, connection)
# add distances to edge_labels
edge_labels[(node, connection)] = distance
In [9]:
# initialise a graph
G = nx.Graph()
# use this while labeling nodes in the map
node_labels = dict()
# use this to modify colors of nodes while exploring the graph.
# This is the only dict we send to `show_map(node_colors)` while drawing the map
node_colors = dict()
for n, p in romania_locations.items():
# add nodes from romania_locations
G.add_node(n)
# add nodes to node_labels
node_labels[n] = n
# node_colors to color nodes while exploring romania map
node_colors[n] = "white"
# we'll save the initial node colors to a dict to use later
initial_node_colors = dict(node_colors)
# positions for node labels
node_label_pos = { k:[v[0],v[1]-10] for k,v in romania_locations.items() }
# use this while labeling edges
edge_labels = dict()
# add edges between cities in romania map - UndirectedGraph defined in search.py
for node in romania_map.nodes():
connections = romania_map.get(node)
for connection in connections.keys():
distance = connections[connection]
# add edges to the graph
G.add_edge(node, connection)
# add distances to edge_labels
edge_labels[(node, connection)] = distance
We have completed building our graph based on romania_map and its locations. It's time to display it here in the notebook. This function show_map(node_colors) helps us do that. We will be calling this function later on to display the map at each and every interval step while searching, using variety of algorithms from the book.
In [9]:
def show_map(node_colors):
# set the size of the plot
plt.figure(figsize=(18,13))
# draw the graph (both nodes and edges) with locations from romania_locations
nx.draw(G, pos = romania_locations, node_color = [node_colors[node] for node in G.nodes()])
# draw labels for nodes
node_label_handles = nx.draw_networkx_labels(G, pos = node_label_pos, labels = node_labels, font_size = 14)
# add a white bounding box behind the node labels
[label.set_bbox(dict(facecolor='white', edgecolor='none')) for label in node_label_handles.values()]
# add edge lables to the graph
nx.draw_networkx_edge_labels(G, pos = romania_locations, edge_labels=edge_labels, font_size = 14)
# add a legend
white_circle = lines.Line2D([], [], color="white", marker='o', markersize=15, markerfacecolor="white")
orange_circle = lines.Line2D([], [], color="orange", marker='o', markersize=15, markerfacecolor="orange")
red_circle = lines.Line2D([], [], color="red", marker='o', markersize=15, markerfacecolor="red")
gray_circle = lines.Line2D([], [], color="gray", marker='o', markersize=15, markerfacecolor="gray")
green_circle = lines.Line2D([], [], color="green", marker='o', markersize=15, markerfacecolor="green")
plt.legend((white_circle, orange_circle, red_circle, gray_circle, green_circle),
('Un-explored', 'Frontier', 'Currently Exploring', 'Explored', 'Final Solution'),
numpoints=1,prop={'size':16}, loc=(.8,.75))
# show the plot. No need to use in notebooks. nx.draw will show the graph itself.
plt.show()
We can simply call the function with node_colors dictionary object to display it.
In [10]:
show_map(node_colors)
Voila! You see, the romania map as shown in the Figure[3.2] in the book. Now, see how different searching algorithms perform with our problem statements.
In this section, we have visualizations of the following searching algorithms:
We add the colors to the nodes to have a nice visualisation when displaying. So, these are the different colors we are using in these visuals:
Now, we will define some helper methods to display interactive buttons and sliders when visualising search algorithms.
In [11]:
def final_path_colors(problem, solution):
"returns a node_colors dict of the final path provided the problem and solution"
# get initial node colors
final_colors = dict(initial_node_colors)
# color all the nodes in solution and starting node to green
final_colors[problem.initial] = "green"
for node in solution:
final_colors[node] = "green"
return final_colors
def display_visual(user_input, algorithm=None, problem=None):
if user_input == False:
def slider_callback(iteration):
# don't show graph for the first time running the cell calling this function
try:
show_map(all_node_colors[iteration])
except:
pass
def visualize_callback(Visualize):
if Visualize is True:
button.value = False
global all_node_colors
iterations, all_node_colors, node = algorithm(problem)
solution = node.solution()
all_node_colors.append(final_path_colors(problem, solution))
slider.max = len(all_node_colors) - 1
for i in range(slider.max + 1):
slider.value = i
#time.sleep(.5)
slider = widgets.IntSlider(min=0, max=1, step=1, value=0)
slider_visual = widgets.interactive(slider_callback, iteration = slider)
display(slider_visual)
button = widgets.ToggleButton(value = False)
button_visual = widgets.interactive(visualize_callback, Visualize = button)
display(button_visual)
if user_input == True:
node_colors = dict(initial_node_colors)
if algorithm == None:
algorithms = {"Breadth First Tree Search": breadth_first_tree_search,
"Breadth First Search": breadth_first_search,
"Uniform Cost Search": uniform_cost_search,
"A-star Search": astar_search}
algo_dropdown = widgets.Dropdown(description = "Search algorithm: ",
options = sorted(list(algorithms.keys())),
value = "Breadth First Tree Search")
display(algo_dropdown)
def slider_callback(iteration):
# don't show graph for the first time running the cell calling this function
try:
show_map(all_node_colors[iteration])
except:
pass
def visualize_callback(Visualize):
if Visualize is True:
button.value = False
problem = GraphProblem(start_dropdown.value, end_dropdown.value, romania_map)
global all_node_colors
if algorithm == None:
user_algorithm = algorithms[algo_dropdown.value]
# print(user_algorithm)
# print(problem)
iterations, all_node_colors, node = user_algorithm(problem)
solution = node.solution()
all_node_colors.append(final_path_colors(problem, solution))
slider.max = len(all_node_colors) - 1
for i in range(slider.max + 1):
slider.value = i
# time.sleep(.5)
start_dropdown = widgets.Dropdown(description = "Start city: ",
options = sorted(list(node_colors.keys())), value = "Arad")
display(start_dropdown)
end_dropdown = widgets.Dropdown(description = "Goal city: ",
options = sorted(list(node_colors.keys())), value = "Fagaras")
display(end_dropdown)
button = widgets.ToggleButton(value = False)
button_visual = widgets.interactive(visualize_callback, Visualize = button)
display(button_visual)
slider = widgets.IntSlider(min=0, max=1, step=1, value=0)
slider_visual = widgets.interactive(slider_callback, iteration = slider)
display(slider_visual)
In [12]:
def tree_search(problem, frontier):
"""Search through the successors of a problem to find a goal.
The argument frontier should be an empty queue.
Don't worry about repeated paths to a state. [Figure 3.7]"""
# we use these two variables at the time of visualisations
iterations = 0
all_node_colors = []
node_colors = dict(initial_node_colors)
#Adding first node to the queue
frontier.append(Node(problem.initial))
node_colors[Node(problem.initial).state] = "orange"
iterations += 1
all_node_colors.append(dict(node_colors))
while frontier:
#Popping first node of queue
node = frontier.pop()
# modify the currently searching node to red
node_colors[node.state] = "red"
iterations += 1
all_node_colors.append(dict(node_colors))
if problem.goal_test(node.state):
# modify goal node to green after reaching the goal
node_colors[node.state] = "green"
iterations += 1
all_node_colors.append(dict(node_colors))
return(iterations, all_node_colors, node)
frontier.extend(node.expand(problem))
for n in node.expand(problem):
node_colors[n.state] = "orange"
iterations += 1
all_node_colors.append(dict(node_colors))
# modify the color of explored nodes to gray
node_colors[node.state] = "gray"
iterations += 1
all_node_colors.append(dict(node_colors))
return None
def breadth_first_tree_search(problem):
"Search the shallowest nodes in the search tree first."
iterations, all_node_colors, node = tree_search(problem, FIFOQueue())
return(iterations, all_node_colors, node)
Now, we use ipywidgets to display a slider, a button and our romania map. By sliding the slider we can have a look at all the intermediate steps of a particular search algorithm. By pressing the button Visualize, you can see all the steps without interacting with the slider. These two helper functions are the callback functions which are called when we interact with the slider and the button.
In [ ]:
all_node_colors = []
romania_problem = GraphProblem('Arad', 'Fagaras', romania_map)
display_visual(user_input = False, algorithm = breadth_first_tree_search, problem = romania_problem)
In [13]:
def breadth_first_search(problem):
"[Figure 3.11]"
# we use these two variables at the time of visualisations
iterations = 0
all_node_colors = []
node_colors = dict(initial_node_colors)
node = Node(problem.initial)
node_colors[node.state] = "red"
iterations += 1
all_node_colors.append(dict(node_colors))
if problem.goal_test(node.state):
node_colors[node.state] = "green"
iterations += 1
all_node_colors.append(dict(node_colors))
return(iterations, all_node_colors, node)
frontier = FIFOQueue()
frontier.append(node)
# modify the color of frontier nodes to blue
node_colors[node.state] = "orange"
iterations += 1
all_node_colors.append(dict(node_colors))
explored = set()
while frontier:
node = frontier.pop()
node_colors[node.state] = "red"
iterations += 1
all_node_colors.append(dict(node_colors))
explored.add(node.state)
for child in node.expand(problem):
if child.state not in explored and child not in frontier:
if problem.goal_test(child.state):
node_colors[child.state] = "green"
iterations += 1
all_node_colors.append(dict(node_colors))
return(iterations, all_node_colors, child)
frontier.append(child)
node_colors[child.state] = "orange"
iterations += 1
all_node_colors.append(dict(node_colors))
node_colors[node.state] = "gray"
iterations += 1
all_node_colors.append(dict(node_colors))
return None
In [ ]:
all_node_colors = []
romania_problem = GraphProblem('Arad', 'Bucharest', romania_map)
display_visual(user_input = False, algorithm = breadth_first_search, problem = romania_problem)
In [14]:
def best_first_graph_search(problem, f):
"""Search the nodes with the lowest f scores first.
You specify the function f(node) that you want to minimize; for example,
if f is a heuristic estimate to the goal, then we have greedy best
first search; if f is node.depth then we have breadth-first search.
There is a subtlety: the line "f = memoize(f, 'f')" means that the f
values will be cached on the nodes as they are computed. So after doing
a best first search you can examine the f values of the path returned."""
# we use these two variables at the time of visualisations
iterations = 0
all_node_colors = []
node_colors = dict(initial_node_colors)
f = memoize(f, 'f')
node = Node(problem.initial)
node_colors[node.state] = "red"
iterations += 1
all_node_colors.append(dict(node_colors))
if problem.goal_test(node.state):
node_colors[node.state] = "green"
iterations += 1
all_node_colors.append(dict(node_colors))
return(iterations, all_node_colors, node)
frontier = PriorityQueue(min, f)
frontier.append(node)
node_colors[node.state] = "orange"
iterations += 1
all_node_colors.append(dict(node_colors))
explored = set()
while frontier:
node = frontier.pop()
node_colors[node.state] = "red"
iterations += 1
all_node_colors.append(dict(node_colors))
if problem.goal_test(node.state):
node_colors[node.state] = "green"
iterations += 1
all_node_colors.append(dict(node_colors))
return(iterations, all_node_colors, node)
explored.add(node.state)
for child in node.expand(problem):
if child.state not in explored and child not in frontier:
frontier.append(child)
node_colors[child.state] = "orange"
iterations += 1
all_node_colors.append(dict(node_colors))
elif child in frontier:
incumbent = frontier[child]
if f(child) < f(incumbent):
del frontier[incumbent]
frontier.append(child)
node_colors[child.state] = "orange"
iterations += 1
all_node_colors.append(dict(node_colors))
node_colors[node.state] = "gray"
iterations += 1
all_node_colors.append(dict(node_colors))
return None
def uniform_cost_search(problem):
"[Figure 3.14]"
iterations, all_node_colors, node = best_first_graph_search(problem, lambda node: node.path_cost)
return(iterations, all_node_colors, node)
In [ ]:
all_node_colors = []
romania_problem = GraphProblem('Arad', 'Bucharest', romania_map)
display_visual(user_input = False, algorithm = uniform_cost_search, problem = romania_problem)
In [15]:
def best_first_graph_search(problem, f):
"""Search the nodes with the lowest f scores first.
You specify the function f(node) that you want to minimize; for example,
if f is a heuristic estimate to the goal, then we have greedy best
first search; if f is node.depth then we have breadth-first search.
There is a subtlety: the line "f = memoize(f, 'f')" means that the f
values will be cached on the nodes as they are computed. So after doing
a best first search you can examine the f values of the path returned."""
# we use these two variables at the time of visualisations
iterations = 0
all_node_colors = []
node_colors = dict(initial_node_colors)
f = memoize(f, 'f')
node = Node(problem.initial)
node_colors[node.state] = "red"
iterations += 1
all_node_colors.append(dict(node_colors))
if problem.goal_test(node.state):
node_colors[node.state] = "green"
iterations += 1
all_node_colors.append(dict(node_colors))
return(iterations, all_node_colors, node)
frontier = PriorityQueue(min, f)
frontier.append(node)
node_colors[node.state] = "orange"
iterations += 1
all_node_colors.append(dict(node_colors))
explored = set()
while frontier:
node = frontier.pop()
node_colors[node.state] = "red"
iterations += 1
all_node_colors.append(dict(node_colors))
if problem.goal_test(node.state):
node_colors[node.state] = "green"
iterations += 1
all_node_colors.append(dict(node_colors))
return(iterations, all_node_colors, node)
explored.add(node.state)
for child in node.expand(problem):
if child.state not in explored and child not in frontier:
frontier.append(child)
node_colors[child.state] = "orange"
iterations += 1
all_node_colors.append(dict(node_colors))
elif child in frontier:
incumbent = frontier[child]
if f(child) < f(incumbent):
del frontier[incumbent]
frontier.append(child)
node_colors[child.state] = "orange"
iterations += 1
all_node_colors.append(dict(node_colors))
node_colors[node.state] = "gray"
iterations += 1
all_node_colors.append(dict(node_colors))
return None
def astar_search(problem, h=None):
"""A* search is best-first graph search with f(n) = g(n)+h(n).
You need to specify the h function when you call astar_search, or
else in your Problem subclass."""
h = memoize(h or problem.h, 'h')
iterations, all_node_colors, node = best_first_graph_search(problem, lambda n: n.path_cost + h(n))
return(iterations, all_node_colors, node)
In [ ]:
all_node_colors = []
romania_problem = GraphProblem('Arad', 'Bucharest', romania_map)
display_visual(user_input = False, algorithm = astar_search, problem = romania_problem)
In [ ]:
all_node_colors = []
# display_visual(user_input = True, algorithm = breadth_first_tree_search)
display_visual(user_input = True)
Genetic algorithms (or GA) are inspired by natural evolution and are particularly useful in optimization and search problems with large state spaces.
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.
A genetic algorithm works in the following way:
1) Initialize random population.
2) Calculate population fitness.
3) Select individuals for mating.
4) Mate selected individuals to produce new population.
* Random chance to mutate individuals.
5) Repeat from step 2) until an individual is fit enough or the maximum number of iterations was reached.
Before we continue, we will lay the basic terminology of the algorithm.
Individual/State: A list of elements (called genes) that represent possible solutions.
Population: The list of all the individuals/states.
Gene pool: The alphabet of possible values for an individual's genes.
Generation/Iteration: The number of times the population will be updated.
Fitness: An individual's score, calculated by a function specific to the problem.
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.
When an offspring is produced, there is a chance it will mutate, having one (or more, depending on the implementation) of its genes altered.
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.
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.
The selection process is this:
1) Individuals are scored by the fitness function.
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):
$$ chance(i) = \dfrac{fitness(i)}{\sum_{k \, in \, P}{fitness(k)}} $$
In [2]:
%psource genetic_algorithm
The algorithm takes the following input:
population: The initial population.
fitness_fn: The problem's fitness function.
gene_pool: The gene pool of the states/individuals. By default 0 and 1.
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.
ngen: The number of iterations/generations.
pmut: The probability of mutation.
The algorithm gives as output the state with the largest score.
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.
The function of mating is accomplished by the method reproduce:
In [3]:
%psource reproduce
The method picks at random a point and merges the parents (x and y) around it.
The mutation is done in the method mutate:
In [4]:
%psource mutate
We pick a gene in x to mutate and a gene from the gene pool to replace it with.
To help initializing the population we have the helper function init_population":
In [5]:
%psource init_population
The function takes as input the number of individuals in the population, the gene pool and the length of each individual/state. It creates individuals with random genes and returns the population when done.
Below we give two example usages for the genetic algorithm, for a graph coloring problem and the 8 queens problem.
First we will take on the simpler problem of coloring a small graph with two colors. Before we do anything, let's imagine how a solution might look. First, we have to represent our colors. Say, 'R' for red and 'G' for green. These make up our gene pool. What of the individual solutions though? For that, we will look at our problem. We stated we have a graph. A graph has nodes and edges, and we want to color the nodes. Naturally, we want to store each node's color. If we have four nodes, we can store their colors in a list of genes, one for each node. A possible solution will then look like this: ['R', 'R', 'G', 'R']. In the general case, we will represent each solution with a list of chars ('R' and 'G'), with length the number of nodes.
Next we need to come up with a fitness function that appropriately scores individuals. Again, we will look at the problem definition at hand. We want to color a graph. For a solution to be optimal, no edge should connect two nodes of the same color. How can we use this information to score a solution? A naive (and ineffective) approach would be to count the different colors in the string. So ['R', 'R', 'R', 'R'] has a score of 1 and ['R', 'R', 'G', 'G'] has a score of 2. Why that fitness function is not ideal though? Why, we forgot the information about the edges! The edges are pivotal to the problem and the above function only deals with node colors. We didn't use all the information at hand and ended up with an ineffective answer. How, then, can we use that information to our advantage?
We said that the optimal solution will have all the edges connecting nodes of different color. So, to score a solution we can count how many edges are valid (aka connecting nodes of different color). That is a great fitness function!
Let's jump into solving this problem using the genetic_algorithm function.
First we need to represent the graph. Since we mostly need information about edges, we will just store the edges. We will denote edges with capital letters and nodes with integers:
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edges = {
'A': [0, 1],
'B': [0, 3],
'C': [1, 2],
'D': [2, 3]
}
Edge 'A' connects nodes 0 and 1, edge 'B' connects nodes 0 and 3 etc.
We already said our gene pool is 'R' and 'G', so we can jump right into initializing our population. Since we have only four nodes, state_length should be 4. For the number of individuals, we will try 8. We can increase this number if we need higher accuracy, but be careful! Larger populations need more computating power and take longer. You need to strike that sweet balance between accuracy and cost (the ultimate dilemma of the programmer!).
In [7]:
population = init_population(8, ['R', 'G'], 4)
print(population)
We created and printed the population. You can see that the genes in the individuals are random and there are 8 individuals each with 4 genes.
Next we need to write our fitness function. We previously said we want the function to count how many edges are valid. So, given a coloring/individual c, we will do just that:
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def fitness(c):
return sum(c[n1] != c[n2] for (n1, n2) in edges.values())
Great! Now we will run the genetic algorithm and see what solution it gives.
In [9]:
solution = genetic_algorithm(population, fitness, gene_pool=['R', 'G'])
print(solution)
The algorithm converged to a solution. Let's check its score:
In [10]:
print(fitness(solution))
The solution has a score of 4. Which means it is optimal, since we have exactly 4 edges in our graph, meaning all are valid!
NOTE: Because the algorithm is non-deterministic, there is a chance a different solution is given. It might even be wrong, if we are very unlucky!
Let's take a look at a more complicated problem.
In the Eight Queens problem, we are tasked with placing eight queens on an 8x8 chessboard without any queen threatening the others (aka queens should not be in the same row, column or diagonal). In its general form the problem is defined as placing N queens in an NxN chessboard without any conflicts.
First we need to think about the representation of each solution. We can go the naive route of representing the whole chessboard with the queens' placements on it. That is definitely one way to go about it, but for the purpose of this tutorial we will do something different. We have eight queens, so we will have a gene for each of them. The gene pool will be numbers from 0 to 7, for the different columns. The position of the gene in the state will denote the row the particular queen is placed in.
For example, we can have the state "03304577". Here the first gene with a value of 0 means "the queen at row 0 is placed at column 0", for the second gene "the queen at row 1 is placed at column 3" and so forth.
We now need to think about the fitness function. On the graph coloring problem we counted the valid edges. The same thought process can be applied here. Instead of edges though, we have positioning between queens. If two queens are not threatening each other, we say they are at a "non-attacking" positioning. We can, therefore, count how many such positionings are there.
Let's dive right in and initialize our population:
In [11]:
population = init_population(100, range(8), 8)
print(population[:5])
We have a population of 100 and each individual has 8 genes. The gene pool is the integers from 0 to 7, in string form. Above you can see the first five individuals.
Next we need to write our fitness function. Remember, queens threaten each other if they are at the same row, column or diagonal.
Since positionings are mutual, we must take care not to count them twice. Therefore for each queen, we will only check for conflicts for the queens after her.
A gene's value in an individual q denotes the queen's column, and the position of the gene denotes its row. We can check if the aforementioned values between two genes are the same. We also need to check for diagonals. A queen a is in the diagonal of another queen, b, if the difference of the rows between them is equal to either their difference in columns (for the diagonal on the right of a) or equal to the negative difference of their columns (for the left diagonal of a). Below is given the fitness function.
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def fitness(q):
non_attacking = 0
for row1 in range(len(q)):
for row2 in range(row1+1, len(q)):
col1 = int(q[row1])
col2 = int(q[row2])
row_diff = row1 - row2
col_diff = col1 - col2
if col1 != col2 and row_diff != col_diff and row_diff != -col_diff:
non_attacking += 1
return non_attacking
Note that the best score achievable is 28. That is because for each queen we only check for the queens after her. For the first queen we check 7 other queens, for the second queen 6 others and so on. In short, the number of checks we make is the sum 7+6+5+...+1. Which is equal to 7*(7+1)/2 = 28.
Because it is very hard and will take long to find a perfect solution, we will set the fitness threshold at 25. If we find an individual with a score greater or equal to that, we will halt. Let's see how the genetic algorithm will fare.
In [16]:
solution = genetic_algorithm(population, fitness, f_thres=25, gene_pool=range(8))
print(solution)
print(fitness(solution))
Above you can see the solution and its fitness score, which should be no less than 25.
With that this tutorial on the genetic algorithm comes to an end. Hope you found this guide helpful!