I am going to implement a hill climbing search algorithm on the traveling salesman problem in this tutorial. Hill-climbing is a local search algorithm that starts with an initial solution, it then tries to improve that solution until no more improvement can be made. This algorithm works for large real-world problems in which the path to the goal is irrelevant.
Hill-climbing is a simple algorithm that can be used to find a satisfactory solution fast, without any need to use a lot of memory. Hill-climbing can be used on real-world problems with a lot of permutations or combinations. The algorithm is often referred to as greedy local search because it iteratively searchs for a better solution.
Hill climbing uses randomly generated solutions that can be more or less guided by what the person implementing it thinks is the best solution. Hill-climbing can be implemented in many variants: stochastic hill climbing, first-choice hill climbing, random-restart hill climbing and more custom variants. The algorithm can be used to find a satisfactory solution to a problem of finding a configuration when it is impossible to test all permutations or combinations.
Traveling Salesman Problem (TSP)
The traveling salesman problem is famous because it is difficult to give an optimal solution in an reasonable time as the number of cities in the problem increases. I have found distance data for 13 cities (Traveling Salesman Problem). The problem is to find the shortest route from a starting location and back to the starting location after visiting all the other cities. This problem has 479001600 ((13-1)!) permutations and if we added one more city it would have 6227020800 ((14-1)!) permutations.
It would take to long to test all permutations, we use hill-climbing to find a satisfactory solution. The initial solution can be random, random with distance weights or a guessed best solution based on the shortest distance between cities.
# 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()Output
I choosed to use the best solution by distance as an initial solution, the best solution is mutated in each iteration and a mutated solution will be the new best solution if the total distance is less than the distance for the current best solution. I am using extra iterations to give the algorithm more time to find a better solution. The best solution is 7293 miles.
-- 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