What makes this problem so famous and so studied is the fact that it has no "quick" solution as the complexity of calculating the best path increases adding more points. And the complexity increases so fast that, even with modern hardware, it can be impossible to compute an exact solution in a reasonable time. In more rigorous terms, it is an NP-hard problem. Many heuristics are known to solve this problem and in this post we will see a solution based on Self-organizing Maps (SOM). A SOM is a Neural Network that is capable of mapping an input point into a bi-dimnsional space placing points that are close to each other into the same area. Hence, the idea to solve our problem is to train the SOM in order to map the points to visit in single dimension map and visit the points from the one mapped to the first cell (the one on the left) to the last cell (on the right). Points that are mapped to the same cell are visited consecutively.

Let's generate a set of points to test this idea:

import numpy as np import matplotlib.pyplot as plt np.random.RandomState(10) N_points = 20 N_neurons = N_points*2 t = np.linspace(0, np.pi*2, N_points) x = np.cos(t)+(np.random.rand(N_points)-.5)*.3 y = np.sin(t)*.8+(np.random.rand(N_points)-.5)*.2 points = np.array([x,y]).T plt.scatter(x, y)

We can now import MiniSom, our favorite implementation of the Self_Organizing Maps, and see what path it's able to produce:

from minisom import MiniSom som = MiniSom(1, N_neurons*2, 2, sigma=10, neighborhood_function='gaussian', random_seed=50) max_iter = 2000 som.pca_weights_init(points) paths_x = [] paths_y = [] for i in np.arange(max_iter): i_point = i % len(points) som.update(points[i_point], som.winner(points[i_point]), i, max_iter) visit_order = np.argsort([som.winner(p)[1] for p in points]) visit_order = np.concatenate((visit_order, [visit_order[0]])) paths_x.append(points[visit_order][:,0]) paths_y.append(points[visit_order][:,1]) plt.scatter(x, y, label='point to visit') plt.plot(paths_x[-1], paths_y[-1], 'C3', linewidth=2, label='path')

In the snippet above we initialized the SOM and run 2000 training iterations (check this out to discover how that works). At each iteration we have saved the path found and visualized the last solution. As we can see, the line covers all the points and it's easy to see that it's the best possible path with just a glance. However, it's interesting to see how the solution evolves at each iteration:

from matplotlib.animation import FuncAnimation from IPython.display import HTML fig, ax = plt.subplots() plt.scatter(x, y, label='point to visit') ln, = plt.plot([], [], 'C3', linewidth=2, label='path') plt.legend() def update(frame): ln.set_data(paths_x[frame], paths_y[frame]) plt.title('iteration = %d' % frame) return ln, ani = FuncAnimation(fig, update, frames=np.arange(max_iter), interval=10, repeat=False, blit=False) HTML(ani.to_html5_video())

Here we note that the initial path is very messy and presents various loops and that the more the network is trained the more optimal the solution becomes. Notice that the snippet above uses the object

*HTML*from the IPython library and it will automatically display the video if a Jupyter notebook is used. The video can be saved in a specific location using

*ani.save(filename.mp4)*.

## No comments:

## Post a Comment