├── requirements.txt
├── report
    ├── report.pdf
    ├── slides.pdf
    ├── img
    │   ├── italy.png
    │   ├── uruguay.png
    │   └── finnish_som.png
    ├── slides.org
    └── report.org
├── diagrams
    └── uruguay.gif
├── src
    ├── distance.py
    ├── neuron.py
    ├── plot.py
    ├── io_helper.py
    └── main.py
├── README.org
├── LICENSE
├── .gitignore
└── assets
    ├── qa194.tsp
    └── uy734.tsp


/requirements.txt:
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1 | matplotlib==2.1.0
2 | pandas==0.20.3
3 | numpy==1.14.0
4 | 


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/report/report.pdf:
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https://raw.githubusercontent.com/OCEChain/SOM/HEAD/report/report.pdf


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/report/slides.pdf:
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https://raw.githubusercontent.com/OCEChain/SOM/HEAD/report/slides.pdf


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/diagrams/uruguay.gif:
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https://raw.githubusercontent.com/OCEChain/SOM/HEAD/diagrams/uruguay.gif


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/report/img/italy.png:
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https://raw.githubusercontent.com/OCEChain/SOM/HEAD/report/img/italy.png


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/report/img/uruguay.png:
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https://raw.githubusercontent.com/OCEChain/SOM/HEAD/report/img/uruguay.png


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/report/img/finnish_som.png:
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https://raw.githubusercontent.com/OCEChain/SOM/HEAD/report/img/finnish_som.png


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/src/distance.py:
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 1 | import numpy as np
 2 | 
 3 | def select_closest(candidates, origin):
 4 |     """Return the index of the closest candidate to a given point."""
 5 |     return euclidean_distance(candidates, origin).argmin()
 6 | 
 7 | def euclidean_distance(a, b):
 8 |     """Return the array of distances of two numpy arrays of points."""
 9 |     return np.linalg.norm(a - b, axis=1)
10 | 
11 | def route_distance(cities):
12 |     """Return the cost of traversing a route of cities in a certain order."""
13 |     points = cities[['x', 'y']]
14 |     distances = euclidean_distance(points, np.roll(points, 1, axis=0))
15 |     return np.sum(distances)
16 | 


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/README.org:
--------------------------------------------------------------------------------
 1 | #+TITLE:  Solving the Traveling Salesman Problem using Self-Organizing Maps
 2 | #+AUTHOR: Diego Vicente Martín
 3 | #+EMAIL:  mail@diego.codes
 4 | 
 5 | This repository contains an implementation of a Self Organizing Map that can be
 6 | used to find sub-optimal solutions for the Traveling Salesman Problem. The
 7 | 
 8 | 
 9 | #+BEGIN_SRC sh
10 | pip install requirements.txt
11 | #+END_SRC
12 | 
13 | To run the code, simply execute:
14 | 
15 | #+BEGIN_SRC sh
16 | cd som-tsp
17 | python src/main.py assets/<instance>.tsp
18 | #+END_SRC
19 | 
20 | The images generated will be stored in the =diagrams= folder. Using a tool like
21 | =convert=, you can easily generate an animation like the one in this file by
22 | running:
23 | 
24 | #+BEGIN_SRC sh
25 | convert -delay 10 -loop 0 *.png animation.gif
26 | #+END_SRC
27 | 
28 | This code is licensed under MIT License, so feel free to modify and/or use it
29 | in your projects. If you have any doubts, feel free to contact me or contribute
30 | to this repository by creating an issue.
31 | 
32 | -----
33 | 
34 | 
35 | 


--------------------------------------------------------------------------------
/LICENSE:
--------------------------------------------------------------------------------
 1 | MIT License
 2 | 
 3 | Copyright (c) 2018 Diego Vicente
 4 | 
 5 | Permission is hereby granted, free of charge, to any person obtaining a copy
 6 | of this software and associated documentation files (the "Software"), to deal
 7 | in the Software without restriction, including without limitation the rights
 8 | to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
 9 | copies of the Software, and to permit persons to whom the Software is
10 | furnished to do so, subject to the following conditions:
11 | 
12 | The above copyright notice and this permission notice shall be included in all
13 | copies or substantial portions of the Software.
14 | 
15 | THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
16 | IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
17 | FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
18 | AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
19 | LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
20 | OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
21 | SOFTWARE.


--------------------------------------------------------------------------------
/src/neuron.py:
--------------------------------------------------------------------------------
 1 | import numpy as np
 2 | 
 3 | from distance import select_closest
 4 | 
 5 | def generate_network(size):
 6 |     """
 7 |     Generate a neuron network of a given size.
 8 | 
 9 |     Return a vector of two dimensional points in the interval [0,1].
10 |     """
11 |     return np.random.rand(size, 2)
12 | 
13 | def get_neighborhood(center, radix, domain):
14 |     """Get the range gaussian of given radix around a center index."""
15 | 
16 |     # Impose an upper bound on the radix to prevent NaN and blocks
17 |     if radix < 1:
18 |         radix = 1
19 | 
20 |     # Compute the circular network distance to the center
21 |     deltas = np.absolute(center - np.arange(domain))
22 |     distances = np.minimum(deltas, domain - deltas)
23 | 
24 |     # Compute Gaussian distribution around the given center
25 |     return np.exp(-(distances*distances) / (2*(radix*radix)))
26 | 
27 | def get_route(cities, network):
28 |     """Return the route computed by a network."""
29 |     cities['winner'] = cities[['x', 'y']].apply(
30 |         lambda c: select_closest(network, c),
31 |         axis=1, raw=True)
32 | 
33 |     return cities.sort_values('winner').index
34 | 


--------------------------------------------------------------------------------
/src/plot.py:
--------------------------------------------------------------------------------
 1 | import matplotlib.pyplot as plt
 2 | import matplotlib as mpl
 3 | 
 4 | def plot_network(cities, neurons, name='diagram.png', ax=None):
 5 |     """Plot a graphical representation of the problem"""
 6 |     mpl.rcParams['agg.path.chunksize'] = 10000
 7 | 
 8 |     if not ax:
 9 |         fig = plt.figure(figsize=(5, 5), frameon = False)
10 |         axis = fig.add_axes([0,0,1,1])
11 | 
12 |         axis.set_aspect('equal', adjustable='datalim')
13 |         plt.axis('off')
14 | 
15 |         axis.scatter(cities['x'], cities['y'], color='red', s=4)
16 |         axis.plot(neurons[:,0], neurons[:,1], 'r.', ls='-', color='#0063ba', markersize=2)
17 | 
18 |         plt.savefig(name, bbox_inches='tight', pad_inches=0, dpi=200)
19 |         plt.close()
20 | 
21 |     else:
22 |         ax.scatter(cities['x'], cities['y'], color='red', s=4)
23 |         ax.plot(neurons[:,0], neurons[:,1], 'r.', ls='-', color='#0063ba', markersize=2)
24 |         return ax
25 | 
26 | def plot_route(cities, route, name='diagram.png', ax=None):
27 |     """Plot a graphical representation of the route obtained"""
28 |     mpl.rcParams['agg.path.chunksize'] = 10000
29 | 
30 |     if not ax:
31 |         fig = plt.figure(figsize=(5, 5), frameon = False)
32 |         axis = fig.add_axes([0,0,1,1])
33 | 
34 |         axis.set_aspect('equal', adjustable='datalim')
35 |         plt.axis('off')
36 | 
37 |         axis.scatter(cities['x'], cities['y'], color='red', s=4)
38 |         route = cities.reindex(route)
39 |         route.loc[route.shape[0]] = route.iloc[0]
40 |         axis.plot(route['x'], route['y'], color='purple', linewidth=1)
41 | 
42 |         plt.savefig(name, bbox_inches='tight', pad_inches=0, dpi=200)
43 |         plt.close()
44 | 
45 |     else:
46 |         ax.scatter(cities['x'], cities['y'], color='red', s=4)
47 |         route = cities.reindex(route)
48 |         route.loc[route.shape[0]] = route.iloc[0]
49 |         ax.plot(route['x'], route['y'], color='purple', linewidth=1)
50 |         return ax
51 | 


--------------------------------------------------------------------------------
/src/io_helper.py:
--------------------------------------------------------------------------------
 1 | import pandas as pd
 2 | import numpy as np
 3 | 
 4 | def read_tsp(filename):
 5 |     """
 6 |     Read a file in .tsp format into a pandas DataFrame
 7 | 
 8 |     The .tsp files can be found in the TSPLIB project. Currently, the library
 9 |     only considers the possibility of a 2D map.
10 |     """
11 |     with open(filename) as f:
12 |         node_coord_start = None
13 |         dimension = None
14 |         lines = f.readlines()
15 | 
16 |         # Obtain the information about the .tsp
17 |         i = 0
18 |         while not dimension or not node_coord_start:
19 |             line = lines[i]
20 |             if line.startswith('DIMENSION :'):
21 |                 dimension = int(line.split()[-1])
22 |             if line.startswith('NODE_COORD_SECTION'):
23 |                 node_coord_start = i
24 |             i = i+1
25 | 
26 |         print('Problem with {} cities read.'.format(dimension))
27 | 
28 |         f.seek(0)
29 | 
30 |         # Read a data frame out of the file descriptor
31 |         cities = pd.read_csv(
32 |             f,
33 |             skiprows=node_coord_start + 1,
34 |             sep=' ',
35 |             names=['city', 'y', 'x'],
36 |             dtype={'city': str, 'x': np.float64, 'y': np.float64},
37 |             header=None,
38 |             nrows=dimension
39 |         )
40 | 
41 |         # cities.set_index('city', inplace=True)
42 | 
43 |         return cities
44 | 
45 | def normalize(points):
46 |     """
47 |     Return the normalized version of a given vector of points.
48 | 
49 |     For a given array of n-dimensions, normalize each dimension by removing the
50 |     initial offset and normalizing the points in a proportional interval: [0,1]
51 |     on y, maintining the original ratio on x.
52 |     """
53 |     ratio = (points.x.max() - points.x.min()) / (points.y.max() - points.y.min()), 1
54 |     ratio = np.array(ratio) / max(ratio)
55 |     norm = points.apply(lambda c: (c - c.min()) / (c.max() - c.min()))
56 |     return norm.apply(lambda p: ratio * p, axis=1)
57 | 


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/src/main.py:
--------------------------------------------------------------------------------
 1 | from sys import argv
 2 | 
 3 | import numpy as np
 4 | 
 5 | from io_helper import read_tsp, normalize
 6 | from neuron import generate_network, get_neighborhood, get_route
 7 | from distance import select_closest, euclidean_distance, route_distance
 8 | from plot import plot_network, plot_route
 9 | 
10 | def main():
11 |     if len(argv) != 2:
12 |         print("Correct use: python src/main.py <filename>.tsp")
13 |         return -1
14 | 
15 |     problem = read_tsp(argv[1])
16 | 
17 |     route = som(problem, 100000)
18 | 
19 |     problem = problem.reindex(route)
20 | 
21 |     distance = route_distance(problem)
22 | 
23 |     print('Route found of length {}'.format(distance))
24 | 
25 | 
26 | def som(problem, iterations, learning_rate=0.8):
27 |     """Solve the TSP using a Self-Organizing Map."""
28 | 
29 |     # Obtain the normalized set of cities (w/ coord in [0,1])
30 |     cities = problem.copy()
31 | 
32 |     cities[['x', 'y']] = normalize(cities[['x', 'y']])
33 | 
34 |     # The population size is 8 times the number of cities
35 |     n = cities.shape[0] * 8
36 | 
37 |     # Generate an adequate network of neurons:
38 |     network = generate_network(n)
39 |     print('Network of {} neurons created. Starting the iterations:'.format(n))
40 | 
41 |     for i in range(iterations):
42 |         if not i % 100:
43 |             print('\t> Iteration {}/{}'.format(i, iterations), end="\r")
44 |         # Choose a random city
45 |         city = cities.sample(1)[['x', 'y']].values
46 |         winner_idx = select_closest(network, city)
47 |         # Generate a filter that applies changes to the winner's gaussian
48 |         gaussian = get_neighborhood(winner_idx, n//10, network.shape[0])
49 |         # Update the network's weights (closer to the city)
50 |         network += gaussian[:,np.newaxis] * learning_rate * (city - network)
51 |         # Decay the variables
52 |         learning_rate = learning_rate * 0.99997
53 |         n = n * 0.9997
54 | 
55 |         # Check for plotting interval
56 |         if not i % 1000:
57 |             plot_network(cities, network, name='diagrams/{:05d}.png'.format(i))
58 | 
59 |         # Check if any parameter has completely decayed.
60 |         if n < 1:
61 |             print('Radius has completely decayed, finishing execution',
62 |             'at {} iterations'.format(i))
63 |             break
64 |         if learning_rate < 0.001:
65 |             print('Learning rate has completely decayed, finishing execution',
66 |             'at {} iterations'.format(i))
67 |             break
68 |     else:
69 |         print('Completed {} iterations.'.format(iterations))
70 | 
71 |     plot_network(cities, network, name='diagrams/final.png')
72 | 
73 |     route = get_route(cities, network)
74 |     plot_route(cities, route, 'diagrams/route.png')
75 |     return route
76 | 
77 | if __name__ == '__main__':
78 |     main()
79 | 


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/.gitignore:
--------------------------------------------------------------------------------
  1 | 
  2 | # Created by https://www.gitignore.io/api/macos,emacs,python
  3 | 
  4 | ### Emacs ###
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200 | 


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/report/slides.org:
--------------------------------------------------------------------------------
  1 | #+TITLE:  Self-Organizing Maps para resolver Traveling Salesman Problem
  2 | #+AUTHOR: Diego Vicente Martín (=100317150@alumnos.uc3m.es=)
  3 | #+EMAIL:  100317150@alumnos.uc3m.es
  4 | #+DATE:   15 de enero, 2018
  5 | #+STARTUP: beamer
  6 | #+OPTIONS: H:2
  7 | #+LaTeX_CLASS: beamer
  8 | #+BEAMER_FRAME_LEVEL: 2
  9 | #+LANGUAGE: es
 10 | #+LaTeX_HEADER: \usepackage[export]{adjustbox}[2011/08/13]
 11 | #+LaTeX_HEADER: \usepackage[spanish]{babel}
 12 | 
 13 | * Resumen del artículo original
 14 | 
 15 | ** Introducción
 16 | 
 17 | - *Self-organizing maps*: estudiados por cite:kohonen-1998-maps
 18 |   - Técnica de organización y visualización inspirada en ANNs.
 19 |   - Simula un modelo dado a través de una regresión en una red.
 20 |   - Se /auto-organiza/ para poner cerca entre sí nodos que representan partes
 21 |     similares.
 22 | 
 23 | #+ATTR_LATEX: :width 0.8\textwidth
 24 | [[./img/finnish_som.png]]
 25 | 
 26 | ** Regresión al modelo
 27 | 
 28 | - Las neuronas del SOM se organizan espacialmente para juntar nodos similares.
 29 | - La regresión se realiza elemento por elemento del modelo:
 30 | 
 31 | \[
 32 | n_{t+1} = n_{t} + h(w_{e}) \cdot \Delta(e, n_{t})
 33 | \]
 34 | 
 35 |   - \(n_{i}\) :: neurona en el momento \(i\).
 36 |   - \(w_{e}\) :: neurona ganadora del elemento.
 37 |   - \(h(n)\) :: factor de vecindario de una neurona \(n\).
 38 |   - \(\Delta(x, y)\) :: vector de distancia entre \(x\) e \(y\)
 39 | 
 40 | ** Definiendo conceptos
 41 | 
 42 | - La elección del *ganador* se define por similitud:
 43 |   - El ejemplo más cercano usando la distancia euclídea.
 44 |   - Las dos distribuciones con mayor correlación.
 45 |   - Elemento de mayor heurística.
 46 | 
 47 | - El vecindario actúa como un filtro de convolución.
 48 |   - Un filtro de suavizado (normalmente Gaussiano).
 49 |   - Se encarga de actualizar zonas concretas del mapa.
 50 | 
 51 | ** Otras cosas a tener en cuenta
 52 | 
 53 | - La conectividad de la red puede cambiar
 54 |   - No solo una red rectangular, sino hexagonal, octogonal...
 55 | 
 56 | - El modelo no siempre converge si los parámetros no son correctos.
 57 | 
 58 | - Si se usa como un LVQ, se pueden aplicar regiones de Voronoi.
 59 | 
 60 | * Modificando SOM para resolver TSP
 61 | 
 62 | ** Usando SOM para resolver TSP
 63 | 
 64 | - Si usamos una conectividad 2 (1D) en el mapa, la red es un anillo.
 65 |   - Las neuronas intentarán reducir la distancia con su sucesor y predecesor.
 66 |   - Aún así, se aplicará una regresión al modelo.
 67 | 
 68 | - Usando un mapa para resolver un TSP en 2D:
 69 |   - Similitud: distancia euclídea.
 70 |   - Conectividad de una dimensión (circular).
 71 |   - Vecindario: filtro Gaussiano de 1D.
 72 | 
 73 | ** Otras modificaciones
 74 | 
 75 | - El mapa no siempre converge, hace falta una forma de equilibrar la
 76 |   exploración y la explotación del modelo.
 77 | - Solución: introducción factor de aprendizaje (\alpha) y descuentos en el factor de
 78 |   aprendizaje y el tamaño del vecindario.
 79 |   - Reducir el factor de aprendizaje permite forzar la convergencia.
 80 |   - Reducir el vecindario fuerza la exploración primero para luego explotar
 81 |     zonas más locales.
 82 | 
 83 | \[
 84 | n_{t+1} = n_{t} + \alpha_{t} \cdot g(w_{e}, h_{t}) \cdot \Delta(e, n_{t})
 85 | \]
 86 | 
 87 | 
 88 | \[
 89 | \alpha_{t+1} = \gamma_{\alpha} \cdot \alpha_{t} , \ \ h_{t+1} = \gamma_{h} \cdot h_{t}
 90 | \]
 91 | 
 92 | * Implementación y evaluación
 93 | 
 94 | ** Implementación
 95 | 
 96 | - Python 3 y =numpy= para la vectorización de operaciones en el mapa.
 97 | - Parámetros configurables, valores elegidos por defecto basados en
 98 |   cite:brocki-2010-somtsp:
 99 |   - Tamaño de la red: 8 veces el número de ciudades.
100 |   - \alpha = 0.8, \gamma_{\alpha} = 0.99997
101 |   - h = número de ciudades, \gamma_{h}= 0.9997
102 | 
103 | ** Evaluación
104 | 
105 | - Métricas:
106 |   - Calidad de la solución en función de la óptima.
107 |   - Tiempo consumido hasta devolver una solución.
108 | 
109 | ** Resultados
110 | 
111 | - Pruebas del mapa en 3 países:
112 |   - Qatar, con 194 ciudades.
113 |   - Uruguay, con 734 ciudades.
114 |   - Filandia, con 10639 ciudades.
115 |   - Italia, con 16862 ciudades.
116 | 
117 | | Instancia | Iteraciones | Tiempo (s) |  Longitud | Calidad |
118 | |-----------+-------------+------------+-----------+---------|
119 | | Qatar     |       14690 |       14.3 |  10233.89 |    9.4% |
120 | | Uruguay   |       17351 |       23.4 |  85072.35 |    7.5% |
121 | | Finlandia |       37833 |      284.0 | 636580.27 |   22.3% |
122 | | Italia    |       39368 |      401.1 | 723212.87 |   29.7% |
123 | 
124 | ** Visualización: Uruguay
125 | 
126 | #+ATTR_LATEX: :width 1.2\textwidth,center
127 | [[./img/uruguay.png]]
128 | 
129 | 
130 | ** Visualización: Italia
131 | 
132 | #+ATTR_LATEX: :width 1.2\textwidth,center
133 | [[./img/italy.png]]
134 | 
135 | * Conclusiones
136 | 
137 | ** Conclusiones
138 | 
139 | - SOM es una técnica muy interesante que ofrece buenos resultados
140 | - Potente herramiento para la visualización
141 | - Aplicar SOM al TSP resulta en una técnica muy sensible a los parámetros.
142 | - Es posible encontrar una ruta subóptima en menos de 25 segundos para más de
143 |   700 ciudades.
144 | 
145 | ** Referencias
146 | 
147 | bibliography:~/Dropbox/org/bibliography/main.bib
148 | bibliographystyle:apalike
149 | 


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/assets/qa194.tsp:
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  1 | NAME : qa194
  2 | COMMENT : 194 locations in Qatar
  3 | COMMENT : Derived from National Imagery and Mapping Agency data
  4 | TYPE : TSP
  5 | DIMENSION : 194
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199 | 192 26133.3333 51216.6667
200 | 193 26133.3333 51300.0000
201 | 194 26150.2778 51108.0556
202 | EOF
203 | 


--------------------------------------------------------------------------------
/report/report.org:
--------------------------------------------------------------------------------
  1 | #+TITLE:  Self-Organizing Maps for solving the Traveling Salesman Problem
  2 | #+AUTHOR: Diego Vicente Martín (=100317150@alumnos.uc3m.es=)
  3 | #+EMAIL:  100317150@alumnos.uc3m.es
  4 | #+LaTeX_CLASS: article
  5 | #+LaTeX_CLASS_OPTIONS: [10pt]
  6 | #+LaTeX_HEADER: \usepackage[export]{adjustbox}[2011/08/13]
  7 | #+LATEX_HEADER: \setlength{\parskip}{\baselineskip}
  8 | #+LATEX_HEADER: \usepackage{enumitem}
  9 | #+LATEX_HEADER: \setlist[itemize]{noitemsep,
 10 | #+LATEX_HEADER:     topsep=0pt, before={\vspace*{-0.7\baselineskip}}}
 11 | #+LATEX_HEADER: \usepackage{amsmath}
 12 | 
 13 | #+OPTIONS: toc:nil date:nil H:2
 14 | 
 15 | #+BEGIN_abstract
 16 | Self-organizing maps (also known as Kohonen maps after its creator, Teuvo
 17 | Kohonen) are an representation technique based on abstracting the data in a
 18 | grid representation in order to obtain a low-dimensional version of the
 19 | geometric relations in between. Using this technique, we are able to apply some
 20 | slight modifications to the grid in order to solve the Traveling Salesman
 21 | Problem and find sub-optimal solutions for it. In this report, there is an
 22 | introduction to the concepts of the problem and the techniques used, as well as
 23 | an evaluation of the Python representation.
 24 | #+END_abstract
 25 | 
 26 | * Summary of the original paper
 27 | 
 28 | The original paper released by Teuvo Kohonen in 1998 cite:kohonen-1998-maps
 29 | consists on a brief, masterful description of the technique. In there, it is
 30 | explained that a /self-organizing map/ is described as an (usually
 31 | two-dimensional) grid of nodes, inspired in a neural network. Closely related
 32 | to the map, is the idea of the /model/, that is, the real world observation the
 33 | map is trying to represent. The purpose of the technique is to represent the
 34 | model with a lower number of dimensions, while maintaining the relations of
 35 | similarity of the nodes contained in it.
 36 | 
 37 | To capture this similarity, the nodes in the map are spatially organized to be
 38 | closer the more similar they are with each other. For that reason, SOM are a
 39 | great way for pattern visualization and organization of data. To obtain this
 40 | structure, the map is applied a regression operation to modify the nodes
 41 | position in order update the nodes, one element from the model (\(e\)) at a
 42 | time. The expression used for the regression is:
 43 | 
 44 | \[
 45 | n_{t+1} = n_{t} + h(w_{e}) \cdot \Delta(e, n_{t})
 46 | \]
 47 | 
 48 | This implies that the position of the node \(n\) is updated adding the distance
 49 | from it to the given element, multiplied by the neighborhood factor of the
 50 | winner neuron, \(w_{e}\). The /winner/ of an element is the more similar node in
 51 | the map to it, usually measured by the closer node using the Euclidean distance
 52 | (although it is possible to use a different similarity measure if appropriate).
 53 | 
 54 | On the other side, the /neighborhood/ is defined as a convolution-like kernel
 55 | for the map around the winner. Doing this, we are able to update the winner and
 56 | the neurons nearby closer to the element, obtaining a soft and proportional
 57 | result. The function is usually defined as a Gaussian distribution, but other
 58 | implementations are as well. One worth mentioning is a bubble neighborhood,
 59 | that updates the neurons that are within a radius of the winner (based on a
 60 | discrete Kronecker delta function).
 61 | 
 62 | After this presentation to the technique, the paper suggest several nuances for
 63 | the implementation that can be useful: using eigenvectors as initialization
 64 | positions for the model instead of random, using Voronoi regions to speed up
 65 | the computation of neighborhoods, and how to apply learning vector quantization
 66 | for discrete classes in the sample vector. Finally, the last remark is the
 67 | suggestion of using a hexagonal grid when using SOM to create a similarity
 68 | graph or visualization.
 69 | 
 70 | * Using SOM to solve the Traveling Salesman Problem
 71 | 
 72 | The Traveling Salesman Problem (TSP) is an NP-Complete problem consisting of
 73 | finding the shortest (or, in case of having defined costs, the cheapest) route
 74 | that traverses all the cities given in a problem exactly once, with or without
 75 | coming back to the initial point cite:hoffman-2013-tsp.
 76 | 
 77 | In this work, the solution proposed uses some slight modifications on the
 78 | concept of self-organizing maps to use them as a tool to solve the problem.
 79 | Most of the modifications are inspired by cite:angeniol-1988-somtsp and some of
 80 | the parametrization techniques are studied in cite:brocki-2010-somtsp.
 81 | 
 82 | ** Modifying the algorithm
 83 | 
 84 | To use the network to solve the TSP, there main concept to understand is how to
 85 | modify the neighborhood function. If instead of a grid we declare a /circular
 86 | array of neurons/, each node will only be conscious of the neurons in front of
 87 | it and behind. That is, the similarity will work just in one dimension. Making
 88 | this slight modification, the self-organizing map will behave as an elastic
 89 | ring, getting closer to the cities but trying to minimize the perimeter of it
 90 | thanks to the neighborhood function.
 91 | 
 92 | Although this modification is the main idea behind the technique, it will not
 93 | work as is: the algorithm will hardly converge any of the times. To ensure the
 94 | convergence of it, we can include a learning rate, \alpha, to control the
 95 | exploration and exploitation of the algorithm. To obtain high exploration
 96 | first, and high exploitation after that in the execution, we must include
 97 | /decay/ in both the neighborhood function and the learning rate. Decaying the
 98 | learning rate will ensure less aggressive displacement of the neurons around
 99 | the model, and decaying the neighborhood will result in a more moderate
100 | exploitation of the local minima of each part of the model. Then, our
101 | regression can be expressed as:
102 | 
103 | \[
104 | n_{t+1} = n_{t} + \alpha_{t} \cdot g(w_{e}, h_{t}) \cdot \Delta(e, n_{t})
105 | \]
106 | 
107 | Where \alpha is the learning rate at a given time, and \(g\) is the Gaussian
108 | function centered in a winner and with a neighborhood dispersion of \(h\). The
109 | decay function consists on simply multiplying the two given discounts, \gamma, for
110 | the learning rate and the neighborhood distance.
111 | 
112 | \[
113 | \alpha_{t+1} = \gamma_{\alpha} \cdot \alpha_{t} , \ \ h_{t+1} = \gamma_{h} \cdot h_{t}
114 | \]
115 | 
116 | This expression is indeed quite similar to that of Q-Learning, and the
117 | convergence is search in a similar fashion to this technique. Decaying the
118 | parameters can be useful in unsupervised learning tasks like the aforementioned
119 | ones.
120 | 
121 | Finally, to obtain the route from the SOM, it is only necessary to associate a
122 | city with its winner neuron, traverse the ring starting from any point and sort
123 | the cities by order of appearance of their winner neuron in the ring. As it is
124 | stated in cite:brocki-2010-somtsp, if several cities map to the same neuron, it
125 | is because the order of traversing such cities have not been contemplated by
126 | the SOM (due to lack of relevance for the final distance or because of not
127 | enough precision). In that case, any possible ordered can be considered for
128 | such cities.
129 | 
130 | ** Implementation
131 | 
132 | For the task, an implementation of the previously explained technique is
133 | provided in Python 3. It is able to parse and load any 2D instance problem
134 | modelled as a =TSPLIB= file and run the regression to obtain the shortest
135 | route. This format is chosen because for the testing and evaluation of the
136 | solution the problems in the National Traveling Salesman Problem instances
137 | offered by the University of Waterloo.
138 | 
139 | All the functionalities of the implementation are collected in several files
140 | included in the =src= directory:
141 | - =distance.py=: contains the 2-Dimensional (Euclidean) distance functions,
142 |   as well as other related functions for computing and evaluating distances.
143 | - =io_helper.py=: functions to open =.tsp= files and load them into valid
144 |   runtime objects.
145 | - =main.py=: includes the general execution control of the self-organizing map.
146 | - =neuron.py=: includes all the network generation, neighborhood computation
147 |   and route computation functions.
148 | - =plot.py=: some functions to generate graphical representations of different
149 |   snapshots during and after the execution.
150 | 
151 | On a lower level, the =numpy= package was used for the computations, which
152 | enables vectorization of the computations and higher performance in the
153 | execution, as well as more expressive and concise code. =pandas= is used for
154 | loading the =.tsp= files to memory easily, and =matplotlib= is used to plot the
155 | graphical representation. These dependencies are all included in the Anaconda
156 | distribution of Python, or can be easily installed using =pip=.
157 | 
158 | * Evaluation
159 | 
160 | To evaluate the implementation, we will use some instances provided by the
161 | National Traveling Salesman Problem library. These instances are inspired in
162 | real countries and also include the optimal route for most of them, which is a
163 | key part of our evaluation. The evaluation strategy consists in running several
164 | instances of the problem and study some metrics:
165 | - *Execution time* invested by the technique to find a solution.
166 | - *Quality* of the solution, measured in function of the optimal route: a route
167 |   that we say is "10% longer that the optimal route" is exactly 1.1 times the
168 |   length of the optimal one.
169 | 
170 | The parameters used in the evaluation are the ones found by parametrization of
171 | the technique, by using the ones provided in cite:brocki-2010-somtsp as a
172 | starting point. These parameters are:
173 | - A population size of 8 times the cities in the problem.
174 | - An initial learning rate of 0.8, with a discount rate of 0.99997.
175 | - An initial neighbourhood of the number of cities, decayed by 0.9997.
176 | 
177 | These parameters were applied to the following instances:
178 | - *Qatar*, containing 194 cities with an optimal tour of 9352.
179 | - *Uruguay*, containing 734 cities with an optimal tour of 79114.
180 | - *Finland*, containing 10639 cities with an optimal tour of 520527.
181 | - *Italy*, containing 16862 cities with an optimal tour of 557315.
182 | 
183 | The implementation also stops the execution if some of the variables decays
184 | under the useful threshold. An uniform way of running the algorithm is tested,
185 | although a finer grained parameters can be found for each instance. The
186 | following table gathers the evaluation results, with the average result of 5
187 | executions in each of the instances.
188 | 
189 | | Instance | Iterations | Time (s) |    Length | Quality |
190 | |----------+------------+----------+-----------+---------|
191 | | Qatar    |      14690 |     14.3 |  10233.89 |    9.4% |
192 | | Uruguay  |      17351 |     23.4 |  85072.35 |    7.5% |
193 | | Finland  |      37833 |    284.0 | 636580.27 |   22.3% |
194 | | Italy    |      39368 |    401.1 | 723212.87 |   29.7% |
195 | 
196 | We can see how the time consumption is directly related to the number of
197 | cities, since the complexity of finding the closest neuron of a city depends on
198 | the population to traverse to find it. It is possible to obtain better results
199 | in the large set of cities, but for time constraints it is usually cropped
200 | shorter. The results are never 30% longer than the optimal. In the other
201 | shorter cases, it is interesting to see how a better quality is obtained in
202 | Uruguay even the problem is harder, due to the topography of the country.
203 | 
204 | #+CAPTION: Three different steps of execution of the Uruguay instance.
205 | #+ATTR_LATEX: :width 1.3\textwidth,center
206 | [[./img/uruguay.png]]
207 | 
208 | It is possible to see in the graphical representation of the executions how the
209 | technique finds its way: first starting as an elastic, fast changing ring
210 | trying to adapt to its shorter form possible, and then starting to fix its
211 | position to find better local ways inside of it. For that purpose, a great
212 | number of neurons are added to the network and the parameters are decayed, to
213 | ensure that first exploration is used and then local exploitation is performed
214 | to refine the solution.
215 | 
216 | #+CAPTION: Three different steps of execution of the Italy instance.
217 | #+ATTR_LATEX: :width 1.3\textwidth,center
218 | [[./img/italy.png]]
219 | 
220 | The results are overall acceptable, except in extremely demanding settings for
221 | the TSP. We can see how in less than half a minute, a suboptimal route crossing
222 | more than 700 cities is found.
223 | 
224 | * Conclusions
225 | 
226 | The experiment show a really interesting way to use an organization technique
227 | like the self-organizing maps to find sub-optimal solutions for an NP-Complete
228 | problem like the traveling salesman problem. By modifying the technique used to
229 | find similarity, we are able to create a network that organizes the cities in
230 | the one of the shortest routes possible. Even though the technique is sensible
231 | to parametrization and may need to run for a great number of iterations in the
232 | biggest instances provided, the results are satisfactory and inspiring to find
233 | new uses in well-known, established techniques.
234 | 
235 | #+LATEX: \newpage
236 | 
237 | bibliography:~/Dropbox/org/bibliography/main.bib
238 | bibliographystyle:ieeetr
239 | 
240 | #  LocalWords:  Teuvo Kohonen SOM eigenvectors Voronoi parametrization minima
241 | 


--------------------------------------------------------------------------------
/assets/uy734.tsp:
--------------------------------------------------------------------------------
  1 | NAME : uy734
  2 | COMMENT : 734 locations in Uruguay
  3 | COMMENT : Derived from National Imagery and Mapping Agency data
  4 | TYPE : TSP
  5 | DIMENSION : 734
  6 | EDGE_WEIGHT_TYPE : EUC_2D
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610 | 603 -34370.0000 -55225.0000
611 | 604 -34382.7778 -56541.6667
612 | 605 -34383.3333 -56283.3333
613 | 606 -34383.3333 -56383.3333
614 | 607 -34383.3333 -56766.6667
615 | 608 -34400.0000 -53783.3333
616 | 609 -34400.0000 -56333.3333
617 | 610 -34400.0000 -56533.3333
618 | 611 -34400.0000 -57166.6667
619 | 612 -34400.0000 -57333.3333
620 | 613 -34400.0000 -57666.6667
621 | 614 -34400.0000 -57883.3333
622 | 615 -34411.6667 -56402.2222
623 | 616 -34416.6667 -56466.6667
624 | 617 -34416.6667 -57283.3333
625 | 618 -34416.6667 -57733.3333
626 | 619 -34419.7222 -56422.5000
627 | 620 -34433.3333 -55983.3333
628 | 621 -34433.3333 -57416.6667
629 | 622 -34433.3333 -57716.6667
630 | 623 -34433.3333 -57883.3333
631 | 624 -34442.2222 -56443.6111
632 | 625 -34449.7222 -56079.1667
633 | 626 -34450.0000 -55783.3333
634 | 627 -34450.0000 -56166.6667
635 | 628 -34450.0000 -56250.0000
636 | 629 -34450.0000 -56283.3333
637 | 630 -34450.0000 -57600.0000
638 | 631 -34453.3333 -56390.5556
639 | 632 -34466.6667 -57716.6667
640 | 633 -34466.6667 -57850.0000
641 | 634 -34483.3333 -54333.3333
642 | 635 -34483.3333 -55633.3333
643 | 636 -34483.3333 -55650.0000
644 | 637 -34483.3333 -56850.0000
645 | 638 -34497.5000 -56037.2222
646 | 639 -34500.0000 -55583.3333
647 | 640 -34500.0000 -56383.3333
648 | 641 -34516.6667 -55250.0000
649 | 642 -34519.7222 -56798.8889
650 | 643 -34521.9444 -56393.6111
651 | 644 -34522.7778 -56277.7778
652 | 645 -34533.3333 -56300.0000
653 | 646 -34533.3333 -56316.6667
654 | 647 -34550.0000 -55883.3333
655 | 648 -34566.6667 -55700.0000
656 | 649 -34583.3333 -55833.3333
657 | 650 -34583.8889 -56699.4444
658 | 651 -34589.4444 -56252.7778
659 | 652 -34600.0000 -54550.0000
660 | 653 -34600.0000 -55483.3333
661 | 654 -34600.0000 -56133.3333
662 | 655 -34605.0000 -56356.3889
663 | 656 -34633.3333 -55616.6667
664 | 657 -34633.3333 -55966.6667
665 | 658 -34633.3333 -56333.3333
666 | 659 -34633.3333 -56619.1667
667 | 660 -34646.9444 -56062.7778
668 | 661 -34650.0000 -55233.3333
669 | 662 -34650.0000 -55866.6667
670 | 663 -34650.0000 -56600.0000
671 | 664 -34650.0000 -56716.6667
672 | 665 -34650.0000 -56750.0000
673 | 666 -34665.0000 -56219.4444
674 | 667 -34666.6667 -54166.6667
675 | 668 -34666.6667 -55266.6667
676 | 669 -34666.6667 -56000.0000
677 | 670 -34683.3333 -55683.3333
678 | 671 -34683.3333 -56733.3333
679 | 672 -34698.3333 -56217.5000
680 | 673 -34700.0000 -55900.0000
681 | 674 -34715.2778 -56073.3333
682 | 675 -34716.6667 -55416.6667
683 | 676 -34716.6667 -55533.3333
684 | 677 -34716.6667 -55950.0000
685 | 678 -34726.3889 -56220.0000
686 | 679 -34733.3333 -55266.6667
687 | 680 -34733.3333 -55716.6667
688 | 681 -34733.6111 -56036.6667
689 | 682 -34742.2222 -56098.3333
690 | 683 -34750.0000 -55333.3333
691 | 684 -34750.0000 -55683.3333
692 | 685 -34761.6667 -56223.6111
693 | 686 -34762.5000 -56020.5556
694 | 687 -34763.3333 -56385.2778
695 | 688 -34764.1667 -56213.8889
696 | 689 -34766.6667 -55750.0000
697 | 690 -34781.1111 -56053.3333
698 | 691 -34790.2778 -56350.0000
699 | 692 -34800.0000 -54916.6667
700 | 693 -34800.0000 -55233.3333
701 | 694 -34800.0000 -55366.6667
702 | 695 -34800.0000 -56216.6667
703 | 696 -34800.0000 -56233.3333
704 | 697 -34801.6667 -56334.1667
705 | 698 -34803.0556 -56085.8333
706 | 699 -34813.0556 -56044.7222
707 | 700 -34816.6667 -56100.0000
708 | 701 -34816.6667 -56183.3333
709 | 702 -34828.6111 -56139.4444
710 | 703 -34833.3333 -56116.6667
711 | 704 -34833.3333 -56216.6667
712 | 705 -34833.3333 -56233.3333
713 | 706 -34850.0000 -56083.3333
714 | 707 -34850.0000 -56116.6667
715 | 708 -34850.0000 -56216.6667
716 | 709 -34850.0000 -56233.3333
717 | 710 -34858.0556 -56170.8333
718 | 711 -34860.2778 -56052.2222
719 | 712 -34866.6667 -54866.6667
720 | 713 -34866.6667 -56166.6667
721 | 714 -34870.5556 -56133.6111
722 | 715 -34875.0000 -56258.8889
723 | 716 -34877.2222 -56029.7222
724 | 717 -34883.3333 -56150.0000
725 | 718 -34883.3333 -56183.3333
726 | 719 -34883.3333 -56200.0000
727 | 720 -34883.3333 -56216.6667
728 | 721 -34883.3333 -56233.3333
729 | 722 -34885.2778 -56060.5556
730 | 723 -34900.0000 -54950.0000
731 | 724 -34900.0000 -55283.3333
732 | 725 -34900.0000 -56100.0000
733 | 726 -34900.0000 -56133.3333
734 | 727 -34900.0000 -56150.0000
735 | 728 -34900.0000 -56183.3333
736 | 729 -34905.0000 -56168.3333
737 | 730 -34905.2778 -56194.4444
738 | 731 -34908.3333 -56208.3333
739 | 732 -34909.7222 -56152.2222
740 | 733 -34922.7778 -56159.7222
741 | 734 -34966.6667 -54950.0000
742 | EOF
743 | 


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