"""
tsp.py: Construction and local optimization for the TSP.
The Traveling Salesman Problem (TSP) is a combinatorial optimization
problem, where given a map (a set of cities and their positions), one
wants to find an order for visiting all the cities in such a way that
the travel distance is minimal.
This file contains a set of functions to illustrate:
- construction heuristics for the TSP
- improvement heuristics for a previously constructed solution
- local search, and random-start local search.
Copyright (c) by Joao Pedro PEDROSO and Mikio KUBO, 2007
"""
import math
import random
def distL2((x1,y1), (x2,y2)):
"""Compute the L2-norm (Euclidean) distance between two points.
The distance is rounded to the closest integer, for compatibility
with the TSPLIB convention.
The two points are located on coordinates (x1,y1) and (x2,y2),
sent as parameters"""
xdiff = x2 - x1
ydiff = y2 - y1
return int(math.sqrt(xdiff*xdiff + ydiff*ydiff) + .5)
def distL1((x1,y1), (x2,y2)):
"""Compute the L1-norm (Manhattan) distance between two points.
The distance is rounded to the closest integer, for compatibility
with the TSPLIB convention.
The two points are located on coordinates (x1,y1) and (x2,y2),
sent as parameters"""
return int(abs(x2-x1) + abs(y2-y1)+.5)
def mk_matrix(coord, dist):
"""Compute a distance matrix for a set of points.
Uses function 'dist' to calculate distance between
any two points. Parameters:
-coord -- list of tuples with coordinates of all points, [(x1,y1),...,(xn,yn)]
-dist -- distance function
"""
n = len(coord)
D = {} # dictionary to hold n times n matrix
for i in range(n-1):
for j in range(i+1,n):
(x1,y1) = coord[i]
(x2,y2) = coord[j]
D[i,j] = dist((x1,y1), (x2,y2))
D[j,i] = D[i,j]
return n,D
def read_tsplib(filename):
"basic function for reading a TSP problem on the TSPLIB format"
"NOTE: only works for 2D euclidean or manhattan distances"
f = open(filename, 'r');
line = f.readline()
while line.find("EDGE_WEIGHT_TYPE") == -1:
line = f.readline()
if line.find("EUC_2D") != -1:
dist = distL2
elif line.find("MAN_2D") != -1:
dist = distL1
else:
print "cannot deal with non-euclidean or non-manhattan distances"
raise Exception
while line.find("NODE_COORD_SECTION") == -1:
line = f.readline()
xy_positions = []
while 1:
line = f.readline()
if line.find("EOF") != -1: break
(i,x,y) = line.split()
x = float(x)
y = float(y)
xy_positions.append((x,y))
n,D = mk_matrix(xy_positions, dist)
return n, xy_positions, D
def mk_closest(D, n):
"""Compute a sorted list of the distances for each of the nodes.
For each node, the entry is in the form [(d1,i1), (d2,i2), ...]
where each tuple is a pair (distance,node).
"""
C = []
for i in range(n):
dlist = [(D[i,j], j) for j in range(n) if j != i]
dlist.sort()
C.append(dlist)
return C
def length(tour, D):
"""Calculate the length of a tour according to distance matrix 'D'."""
z = D[tour[-1], tour[0]] # edge from last to first city of the tour
for i in range(1,len(tour)):
z += D[tour[i], tour[i-1]] # add length of edge from city i-1 to i
return z
def randtour(n):
"""Construct a random tour of size 'n'."""
sol = range(n) # set solution equal to [0,1,...,n-1]
random.shuffle(sol) # place it in a random order
return sol
def nearest(last, unvisited, D):
"""Return the index of the node which is closest to 'last'."""
near = unvisited[0]
min_dist = D[last, near]
for i in unvisited[1:]:
if D[last,i] < min_dist:
near = i
min_dist = D[last, near]
return near
def nearest_neighbor(n, i, D):
"""Return tour starting from city 'i', using the Nearest Neighbor.
Uses the Nearest Neighbor heuristic to construct a solution:
- start visiting city i
- while there are unvisited cities, follow to the closest one
- return to city i
"""
unvisited = range(n)
unvisited.remove(i)
last = i
tour = [i]
while unvisited != []:
next = nearest(last, unvisited, D)
tour.append(next)
unvisited.remove(next)
last = next
return tour
def exchange_cost(tour, i, j, D):
"""Calculate the cost of exchanging two arcs in a tour.
Determine the variation in the tour length if
arcs (i,i+1) and (j,j+1) are removed,
and replaced by (i,j) and (i+1,j+1)
(note the exception for the last arc).
Parameters:
-t -- a tour
-i -- position of the first arc
-j>i -- position of the second arc
"""
n = len(tour)
a,b = tour[i],tour[(i+1)%n]
c,d = tour[j],tour[(j+1)%n]
return (D[a,c] + D[b,d]) - (D[a,b] + D[c,d])
def exchange(tour, tinv, i, j):
"""Exchange arcs (i,i+1) and (j,j+1) with (i,j) and (i+1,j+1).
For the given tour 't', remove the arcs (i,i+1) and (j,j+1) and
insert (i,j) and (i+1,j+1).
This is done by inverting the sublist of cities between i and j.
"""
n = len(tour)
if i>j:
i,j = j,i
assert i>=0 and i<j-1 and j<n
path = tour[i+1:j+1]
path.reverse()
tour[i+1:j+1] = path
for k in range(i+1,j+1):
tinv[tour[k]] = k
def improve(tour, z, D, C):
"""Try to improve tour 't' by exchanging arcs; return improved tour length.
If possible, make a series of local improvements on the solution 'tour',
using a breadth first strategy, until reaching a local optimum.
"""
n = len(tour)
tinv = [0 for i in tour]
for k in range(n):
tinv[tour[k]] = k # position of each city in 't'
for i in range(n):
a,b = tour[i],tour[(i+1)%n]
dist_ab = D[a,b]
improved = False
for dist_ac,c in C[a]:
if dist_ac >= dist_ab:
break
j = tinv[c]
d = tour[(j+1)%n]
dist_cd = D[c,d]
dist_bd = D[b,d]
delta = (dist_ac + dist_bd) - (dist_ab + dist_cd)
if delta < 0: # exchange decreases length
exchange(tour, tinv, i, j);
z += delta
improved = True
break
if improved:
continue
for dist_bd,d in C[b]:
if dist_bd >= dist_ab:
break
j = tinv[d]-1
if j==-1:
j=n-1
c = tour[j]
dist_cd = D[c,d]
dist_ac = D[a,c]
delta = (dist_ac + dist_bd) - (dist_ab + dist_cd)
if delta < 0: # exchange decreases length
exchange(tour, tinv, i, j);
z += delta
break
return z
def localsearch(tour, z, D, C=None):
"""Obtain a local optimum starting from solution t; return solution length.
Parameters:
tour -- initial tour
z -- length of the initial tour
D -- distance matrix
"""
n = len(tour)
if C == None:
C = mk_closest(D, n) # create a sorted list of distances to each node
while 1:
newz = improve(tour, z, D, C)
if newz < z:
z = newz
else:
break
return z
def multistart_localsearch(k, n, D, report=None):
"""Do k iterations of local search, starting from random solutions.
Parameters:
-k -- number of iterations
-D -- distance matrix
-report -- if not None, call it to print verbose output
Returns best solution and its cost.
"""
C = mk_closest(D, n) # create a sorted list of distances to each node
bestt=None
bestz=None
for i in range(0,k):
tour = randtour(n)
z = length(tour, D)
z = localsearch(tour, z, D, C)
if z < bestz or bestz == None:
bestz = z
bestt = list(tour)
if report:
report(z, tour)
return bestt, bestz
if __name__ == "__main__":
"""Local search for the Travelling Saleman Problem: sample usage."""
#
# test the functions:
#
# random.seed(1) # uncomment for having always the same behavior
import sys
if len(sys.argv) == 1:
# create a graph with several cities' coordinates
coord = [(4,0),(5,6),(8,3),(4,4),(4,1),(4,10),(4,7),(6,8),(8,1)]
n, D = mk_matrix(coord, distL2) # create the distance matrix
instance = "toy problem"
else:
instance = sys.argv[1]
n, coord, D = read_tsplib(instance) # create the distance matrix
# n, coord, D = read_tsplib('INSTANCES/TSP/eil51.tsp') # create the distance matrix
# function for printing best found solution when it is found
from time import clock
init = clock()
def report_sol(obj, s=""):
print "cpu:%g\tobj:%g\ttour:%s" % \
(clock(), obj, s)
print "*** travelling salesman problem ***"
print
# random construction
print "random construction + local search:"
tour = randtour(n) # create a random tour
z = length(tour, D) # calculate its length
print "random:", tour, z, ' --> ',
z = localsearch(tour, z, D) # local search starting from the random tour
print tour, z
print
# greedy construction
print "greedy construction with nearest neighbor + local search:"
for i in range(n):
tour = nearest_neighbor(n, i, D) # create a greedy tour, visiting city 'i' first
z = length(tour, D)
print "nneigh:", tour, z, ' --> ',
z = localsearch(tour, z, D)
print tour, z
print
# multi-start local search
print "random start local search:"
niter = 100
tour,z = multistart_localsearch(niter, n, D, report_sol)
assert z == length(tour, D)
print "best found solution (%d iterations): z = %g" % (niter, z)
print tour
print