import math, random, time, numpy as np

def distance(city1, city2):
    return math.sqrt((city1[0] - city2[0]) ** 2 + (city1[1] - city2[1]) ** 2) + 1e-10

def total_distance(cities):
    return sum([distance(cities[i - 1], cities[i]) for i in range(len(cities))])

class AntColony:
    def __init__(self, cities, n_ants, alpha=1, beta=2, evaporation=0.5, intensification=2, max_time=0.1):
        self.cities = cities
        self.n = len(cities)
        self.n_ants = n_ants
        self.alpha = alpha
        self.beta = beta
        self.evaporation = evaporation
        self.intensification = intensification
        self.max_time = max_time
        self.pheromones = [[1 / self.n for _ in range(self.n)] for __ in range(self.n)]

    def choose_next_city(self, ant):
        unvisited_cities = [i for i in range(self.n) if i not in ant]
        probabilities = [self.pheromones[ant[-1]][i] ** self.alpha * ((1 / self.distance(self.cities[ant[-1]], self.cities[i])) ** self.beta) for i in unvisited_cities]
        total = sum(probabilities)
        if total == 0:
            probabilities = [1 / len(unvisited_cities) for _ in unvisited_cities]
        else:
            probabilities = [p / total for p in probabilities]
        return np.random.choice(unvisited_cities, p=probabilities)

    def update_pheromones(self, ant):
        pheromones_delta = self.intensification / self.total_distance([self.cities[i] for i in ant])
        for i in range(len(ant) - 1):
            self.pheromones[ant[i]][ant[i+1]] += pheromones_delta

    def run(self):
        best_ant = []
        best_distance = float('inf')
        start_time = time.time()
        while time.time() - start_time < self.max_time:
            ants = [[random.randint(0, self.n - 1)] for _ in range(self.n_ants)]
            for ant in ants:
                for _ in range(self.n - 1):
                    ant.append(self.choose_next_city(ant))
                ant_distance = self.total_distance([self.cities[i] for i in ant])
                if ant_distance < best_distance:
                    best_distance = ant_distance
                    best_ant = ant.copy()
                self.update_pheromones(ant)
            self.pheromones = [[(1 - self.evaporation) * p for p in row] for row in self.pheromones]
        return [self.cities[i] for i in best_ant]