Bergsklättringsalgoritm för sökning i Python

Jag kommer att implementera en bergsklättringsalgoritm för sökning i denna handledning, algoritmen kommer att användas på handelsresandeproblemet. Bergsklättringsalgoritmen är en lokal sökalgoritm som startar med en initial lösning, algoritmen försöker sedan att förbättra lösningen tills det att ingen förbättring kan ske. Denna algoritm fungerar på stora verkliga problem där sökvägen till målet är irrelevant.

Bergsklättringsalgoritmen är en enkel algoritm som kan användas för att hitta en tillfredsställande lösning snabbt, detta utan att behöva använda mycket minne. Bergsklättringssökning kan användas på verkliga problem med många permutationer eller kombinationer. Algoritmen benämns ofta som girig lokal sökning eftersom den iterativt söker efter en bättre lösning.

Bergsklättringsalgoritmen använder slumpmässigt genererade lösningar som kan vara mer eller mindre styrda av den bästa lösningen enligt den person som implementerar algoritmen. Bergsklättringssökning kan implementeras i många varianter: stokastisk bergsklättring, förstavalsbergsklättring, slumpmässig omstart av bergsklättring och andra anpassade varianter. Algoritmen kan användas för att hitta en tillfredsställande lösning på ett problem avseende en konfiguration när det är omöjligt att testa alla permutationer eller kombinationer.

Handelsresandeproblemet

Handelsresandeproblemet är känt eftersom det är svårt att generera en optimal lösning inom rimlig tid när antalet städer i problemet ökar i antal. Jag har hittat distansdata för 13 städer (Traveling Salesman Problem). Målet är att hitta den kortaste vägen från en startplats och tillbaka till startplatsen efter att ha besökt alla andra städer. Det här problemet har 479001600 ((13-1)!) permutationer och om vi lägger till ytterligare en stad skulle det ha 6227020800 ((14-1)!) permutationer.

Det tar för lång tid att testa alla permutationer, vi använder bergsklättringssökning för att hitta en tillfredsställande lösning inom rimlig tid. Den initiala lösningen kan vara slumpmässig, slumpmässig med avståndsvikter eller en gissad bästa lösning baserat på det kortaste avståndet mellan städer.

# Import libraries
import random
import copy

# This class represent a state
class State:

    # Create a new state
    def __init__(self, route:[], distance:int=0):
        self.route = route
        self.distance = distance

    # Compare states
    def __eq__(self, other):
        for i in range(len(self.route)):
            if(self.route[i] != other.route[i]):
                return False
        return True

    # Sort states
    def __lt__(self, other):
         return self.distance < other.distance

    # Print a state
    def __repr__(self):
        return ('({0},{1})\n'.format(self.route, self.distance))

    # Create a shallow copy
    def copy(self):
        return State(self.route, self.distance)

    # Create a deep copy
    def deepcopy(self):
        return State(copy.deepcopy(self.route), copy.deepcopy(self.distance))

    # Update distance
    def update_distance(self, matrix, home):
        
        # Reset distance
        self.distance = 0

        # Keep track of departing city
        from_index = home

        # Loop all cities in the current route
        for i in range(len(self.route)):
            self.distance += matrix[from_index][self.route[i]]
            from_index = self.route[i]

        # Add the distance back to home
        self.distance += matrix[from_index][home]

# This class represent a city (used when we need to delete cities)
class City:

    # Create a new city
    def __init__(self, index:int, distance:int):
        self.index = index
        self.distance = distance

    # Sort cities
    def __lt__(self, other):
         return self.distance < other.distance

# Get the best random solution from a population
def get_random_solution(matrix:[], home:int, city_indexes:[], size:int, use_weights=False):

    # Create a list with city indexes
    cities = city_indexes.copy()

    # Remove the home city
    cities.pop(home)

    # Create a population
    population = []
    for i in range(size):

        if(use_weights == True):
            state = get_random_solution_with_weights(matrix, home)
        else:
            # Shuffle cities at random
            random.shuffle(cities)

            # Create a state
            state = State(cities[:])
            state.update_distance(matrix, home)

        # Add an individual to the population
        population.append(state)

    # Sort population
    population.sort()

    # Return the best solution
    return population[0]

# Get best solution by distance
def get_best_solution_by_distance(matrix:[], home:int):
    
    # Variables
    route = []
    from_index = home
    length = len(matrix) - 1

    # Loop until route is complete
    while len(route) < length:

         # Get a matrix row
        row = matrix[from_index]

        # Create a list with cities
        cities = {}
        for i in range(len(row)):
            cities[i] = City(i, row[i])

        # Remove cities that already is assigned to the route
        del cities[home]
        for i in route:
            del cities[i]

        # Sort cities
        sorted = list(cities.values())
        sorted.sort()

        # Add the city with the shortest distance
        from_index = sorted[0].index
        route.append(from_index)

    # Create a new state and update the distance
    state = State(route)
    state.update_distance(matrix, home)

    # Return a state
    return state

# Get a random solution by using weights
def get_random_solution_with_weights(matrix:[], home:int):
    
    # Variables
    route = []
    from_index = home
    length = len(matrix) - 1

    # Loop until route is complete
    while len(route) < length:

         # Get a matrix row
        row = matrix[from_index]

        # Create a list with cities
        cities = {}
        for i in range(len(row)):
            cities[i] = City(i, row[i])

        # Remove cities that already is assigned to the route
        del cities[home]
        for i in route:
            del cities[i]

        # Get the total weight
        total_weight = 0
        for key, city in cities.items():
            total_weight += city.distance

        # Add weights
        weights = []
        for key, city in cities.items():
            weights.append(total_weight / city.distance)

        # Add a city at random
        from_index = random.choices(list(cities.keys()), weights=weights)[0]
        route.append(from_index)

    # Create a new state and update the distance
    state = State(route)
    state.update_distance(matrix, home)

    # Return a state
    return state

# Mutate a solution
def mutate(matrix:[], home:int, state:State, mutation_rate:float=0.01):
    
    # Create a copy of the state
    mutated_state = state.deepcopy()

    # Loop all the states in a route
    for i in range(len(mutated_state.route)):

        # Check if we should do a mutation
        if(random.random() < mutation_rate):

            # Swap two cities
            j = int(random.random() * len(state.route))
            city_1 = mutated_state.route[i]
            city_2 = mutated_state.route[j]
            mutated_state.route[i] = city_2
            mutated_state.route[j] = city_1

    # Update the distance
    mutated_state.update_distance(matrix, home)

    # Return a mutated state
    return mutated_state

# Hill climbing algorithm
def hill_climbing(matrix:[], home:int, initial_state:State, max_iterations:int, mutation_rate:float=0.01):

    # Keep track of the best state
    best_state = initial_state

    # An iterator can be used to give the algorithm more time to find a solution
    iterator = 0

    # Create an infinite loop
    while True:

        # Mutate the best state
        neighbor = mutate(matrix, home, best_state, mutation_rate)

        # Check if the distance is less than in the best state
        if(neighbor.distance >= best_state.distance):
            iterator += 1
            if (iterator > max_iterations):
                break
        if(neighbor.distance < best_state.distance):
            best_state = neighbor

    # Return the best state
    return best_state

# The main entry point for this module
def main():

    # Cities to travel
    cities = ['New York', 'Los Angeles', 'Chicago', 'Minneapolis', 'Denver', 'Dallas', 'Seattle', 'Boston', 'San Francisco', 'St. Louis', 'Houston', 'Phoenix', 'Salt Lake City']
    city_indexes = [0,1,2,3,4,5,6,7,8,9,10,11,12]

    # Index of start location
    home = 2 # Chicago

    # Max iterations
    max_iterations = 1000

    # Distances in miles between cities, same indexes (i, j) as in the cities array
    matrix = [[0, 2451, 713, 1018, 1631, 1374, 2408, 213, 2571, 875, 1420, 2145, 1972],
            [2451, 0, 1745, 1524, 831, 1240, 959, 2596, 403, 1589, 1374, 357, 579],
            [713, 1745, 0, 355, 920, 803, 1737, 851, 1858, 262, 940, 1453, 1260],
            [1018, 1524, 355, 0, 700, 862, 1395, 1123, 1584, 466, 1056, 1280, 987],
            [1631, 831, 920, 700, 0, 663, 1021, 1769, 949, 796, 879, 586, 371],
            [1374, 1240, 803, 862, 663, 0, 1681, 1551, 1765, 547, 225, 887, 999],
            [2408, 959, 1737, 1395, 1021, 1681, 0, 2493, 678, 1724, 1891, 1114, 701],
            [213, 2596, 851, 1123, 1769, 1551, 2493, 0, 2699, 1038, 1605, 2300, 2099],
            [2571, 403, 1858, 1584, 949, 1765, 678, 2699, 0, 1744, 1645, 653, 600],
            [875, 1589, 262, 466, 796, 547, 1724, 1038, 1744, 0, 679, 1272, 1162],
            [1420, 1374, 940, 1056, 879, 225, 1891, 1605, 1645, 679, 0, 1017, 1200],
            [2145, 357, 1453, 1280, 586, 887, 1114, 2300, 653, 1272, 1017, 0, 504],
            [1972, 579, 1260, 987, 371, 999, 701, 2099, 600, 1162, 1200, 504, 0]]


    # Get the best route by distance
    state = get_best_solution_by_distance(matrix, home)
    print('-- Best solution by distance --')
    print(cities[home], end='')
    for i in range(0, len(state.route)):
       print(' -> ' + cities[state.route[i]], end='')
    print(' -> ' + cities[home], end='')
    print('\n\nTotal distance: {0} miles'.format(state.distance))
    print()

    # Get the best random route
    state = get_random_solution(matrix, home, city_indexes, 100)
    print('-- Best random solution --')
    print(cities[home], end='')
    for i in range(0, len(state.route)):
       print(' -> ' + cities[state.route[i]], end='')
    print(' -> ' + cities[home], end='')
    print('\n\nTotal distance: {0} miles'.format(state.distance))
    print()

    # Get a random solution with weights
    state = get_random_solution(matrix, home, city_indexes, 100, use_weights=True)
    print('-- Best random solution with weights --')
    print(cities[home], end='')
    for i in range(0, len(state.route)):
       print(' -> ' + cities[state.route[i]], end='')
    print(' -> ' + cities[home], end='')
    print('\n\nTotal distance: {0} miles'.format(state.distance))
    print()

    # Run hill climbing to find a better solution
    state = get_best_solution_by_distance(matrix, home)
    state = hill_climbing(matrix, home, state, 1000, 0.1)
    print('-- Hill climbing solution --')
    print(cities[home], end='')
    for i in range(0, len(state.route)):
       print(' -> ' + cities[state.route[i]], end='')
    print(' -> ' + cities[home], end='')
    print('\n\nTotal distance: {0} miles'.format(state.distance))
    print()

# Tell python to run main method
if __name__ == "__main__": main()

Resultat

Jag valde att använda en initial lösning baserad på det kortaste avståndet mellan städer, den bästa lösningen muteras i varje iteration och en muterad lösning blir den nya bästa lösningen om det totala avståndet är mindre än avståndet för den nuvarande bästa lösningen. Jag använder extra iterationer för att ge algoritmen mer tid till att hitta en bättre lösning. Den bästa lösningen är 7293 miles enligt det ursprunliga exemplet i länken.

-- Best solution by distance --
Chicago -> St. Louis -> Minneapolis -> Denver -> Salt Lake City -> Phoenix -> Los Angeles -> San Francisco -> Seattle -> Dallas -> Houston -> New York -> Boston -> Chicago

Total distance: 8131 miles

-- Best random solution --
Chicago -> Boston -> Salt Lake City -> Los Angeles -> San Francisco -> Seattle -> Denver -> Houston -> Dallas -> Phoenix -> St. Louis -> Minneapolis -> New York -> Chicago

Total distance: 11091 miles

-- Best random solution with weights --
Chicago -> Boston -> New York -> St. Louis -> Dallas -> Houston -> Phoenix -> Seattle -> Denver -> Salt Lake City -> Los Angeles -> San Francisco -> Minneapolis -> Chicago

Total distance: 9155 miles

-- Hill climbing solution --
Chicago -> St. Louis -> Minneapolis -> Denver -> Salt Lake City -> Seattle -> San Francisco -> Los Angeles -> Phoenix -> Dallas -> Houston -> New York -> Boston -> Chicago

Total distance: 7534 miles
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