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|
############################################################################
#
# File: travels.icn
#
# Subject: Program to animate the traveling salesman problem
#
# Author: Gregg M. Townsend
#
# Date: September 17, 1997
#
############################################################################
#
# This file is in the public domain.
#
############################################################################
#
# Usage: travels [window options] [-q] [npoints]
#
# -q (quiet) suppresses commentary normally written to stdout
#
# npoints seeds the field with that many initial cities
# and sets the count for the "reseed" button
#
#
# travels illustrates several heuristic algorithms for obtaining
# approximate solutions to the traveling salesman problem. Cities may
# be seeded randomly or entered with the mouse. Speed may be controlled
# using a slider. The CPU time, number of cities, and path length are
# displayed on a status line and written to standard output after every
# major action.
#
############################################################################
#
# Several types of controls are provided. New cities may be added
# at any time, invalidating any current path. At least two cities must
# be seeded before a path can be constructed. A path must be constructed
# before any of the optimization algorithms can be applied.
#
# For a description on of the algorithms used, see:
# David S. Johnson
# Local Optimization and the Traveling Salesman Problem
# Proc. 17th Colloquium on Automata, Languages, & Programming
# Springer-Verlag (1990), pp. 446-461
#
#
# Mouse Actions:
#
# Clicking the left mouse button adds a new point.
#
#
# Keyboard Actions:
#
# The digit 0 clears all points.
# The digits 1 through 9 seed 1 to 81 (n ^ 2) new points.
#
# Each of the pushbuttons below also has a keyboard equivalent
# which is indicated on the pushbutton.
#
#
# Pushbuttons:
#
# Removing and adding points:
# Clear Remove all points
# Reseed Add n random points (a command option, default 20)
#
# Path construction:
# Initial Connect points in order of initialization
# Random Random path
# Strip Strip-wise construction
# NearNbr Nearest-neighbor algorithm
# NearIns Nearest-insertion algorithm
# FarIns Farthest-insertion algorithm
# Greedy Greedy algorithm
#
# Optimizations:
# 2-Adj Swap pairs of adjacent points
# Uncross Swap pairs of intersecting segments
# 2-Opt Swap all segment pairs that shorten the path
#
# Control:
# List List coordinates of points on standard output
# Refresh Redraw the screen
# Quit Exit the program
#
#
# Delay Slider:
#
# The delay slider can be used to slow down the action. It specifies a
# number of milliseconds to pause before visiting a new point or drawing
# a new path segment. Its response is nonlinear in order to allow finer
# control of short delays. Delays are inexact due to system granularity
# and other problems.
#
# Unfortunately, the delay slider can only be changed between actions,
# not during construction or optimization.
#
############################################################################
#
# Requires: Version 9 graphics
#
############################################################################
#
# Links: options, optwindw, button, slider, evmux, random, graphics
#
############################################################################
link options
link optwindw
link button
link slider
link evmux
link random
link graphics
$define EColor "dark blue" # emphasis color
global ptlist # list of point records (permanent id order, not route)
record point(
id, # permanent id
x, y, # location
nxt, prv, # forward and backward links for route
t1, t2) # scratch cells for traversal algorithms
global distlist # list of distance recs (linearized triangular matrix)
global distsrt # sorted distance list (created when needed)
record dstrec(
d, # distance between two points (x1000, stored as int)
p, q) # the two points
global newpts # non-null if points are new since last report
global havepath # non-null if we have a valid path
# (start from any point and follow links)
global lastclk # value of &time before last computation
global delaytime # delay time between steps, in msec
global opts # command line options
global nseed # number of points to seed
global win # main window
global fgwin # binding for drawing in foreground color
global emwin # binding for drawing in emphasis color
global bgwin # binding for erasing in background color
global m, w, h, bw, bh, fh # screen layout parameters
global ax, ay, aw, ah # corners and size of arena
######################### main program #########################
procedure main(args)
local base, pt, sg, hl
# get options and open a window
opts := options(args, "qE:" || winoptions()) # get options
/opts["W"] := 700 # default width
/opts["H"] := 500 # default height
/opts["E"] := EColor # default emphasis
/opts["T"] := "sans,bold,12" # default font
/opts["M"] := -1 # use standard margin
win := optwindow(opts, "linewidth=2") # open window
m := opts["M"] # save specified margin
h := opts["H"] # save usable height
w := opts["W"] # save usable width
bw := 100 # button width
bh := 18 # button height
fh := 20 # footer height
ax := m + bw + m # arena bounds and size
ay := m
aw := w - bw - m
ah := h - fh - m
fgwin := Clone(win)
emwin := Clone(win, "fg=" || (opts["E"] | EColor | "black"), "linewidth=4")
bgwin := Clone(win, "fg=" || Bg(win), "linewidth=4")
# set up sensor for adding points
sensor(win, &lrelease, addpt, &null, ax, ay, aw, ah)
# set up buttons
buttonrow(win, m, m, bw, bh, 0, bh + (2 > m | 2),
"seeding", &null, &null,
"Clear 0", argless, clrpts,
"Reseed D", argless, reseed,
&null, &null, &null, # spacing
"construction", &null, &null,
"Initial I", argless, initpath,
"Random R", argless, randpath,
"Strip S", argless, strippath,
"NearNbr B", argless, nearnbr,
"NearIns N", argless, nearins,
"FarIns F", argless, farins,
"Greedy G", argless, greedypath,
&null, &null, &null,
"optimization", &null, &null,
"2-Adj A", argless, twoadj,
"Uncross U", argless, uncross,
"2-Opt T", argless, twoopt,
&null, &null, &null,
"control", &null, &null,
"Refresh H", argless, refresh,
"List L", argless, listpath,
&null, &null, &null,
"Quit Q", argless, exit,
)
# set up corresponding keyboard handlers
quitsensor(win) # q and Q
sensor(win, 'Ii', argless, initpath)
sensor(win, 'Rr', argless, randpath)
sensor(win, 'Ss', argless, strippath)
sensor(win, 'Bb', argless, nearnbr)
sensor(win, 'Nn', argless, nearins)
sensor(win, 'Ff', argless, farins)
sensor(win, 'Gg', argless, greedypath)
sensor(win, 'Aa', argless, twoadj)
sensor(win, 'Uu', argless, uncross)
sensor(win, 'Tt', argless, twoopt)
sensor(win, 'Ll', argless, listpath)
sensor(win, 'Dd', argless, reseed)
sensor(win, 'Hh', argless, refresh)
sensor(win, '0', argless, clrpts)
sensor(win, '123456789', reseed)
# set up speed slider
slider(win, setdly, 0, m, m + h - bh, bw, bh, 0, 0, 1)
setdly(win, 0, 0)
# initialize
randomize()
clrpts()
lastclk := &time
if nseed := integer(args[1]) then
reseed()
else
nseed := 20
# process events
evmux(win)
end
# setdly(win, arg, value) -- set delay time
procedure setdly(win, arg, value)
local s, l
value := integer(10001 ^ value + 0.5) - 1
delaytime := value
s := " delay " || value || " "
l := TextWidth(win, s)
GotoXY(win, m + (bw - l) / 2, m + h - bh - m / 2)
writes(win, s)
return
end
# pause() -- delay according to the current setting
procedure pause()
if delaytime > 0 then
WDelay(win, delaytime)
return
end
######################### path constructions #########################
# initpath() -- connect in initial placement order
procedure initpath()
local i
bgnpath(0, "placement order...") | fail
ptlist[1].nxt := &null
every i := 2 to *ptlist do {
follow(ptlist[i-1], ptlist[i])
pause()
}
ptlist[-1].nxt := ptlist[1]
ptlist[1].prv := ptlist[-1]
drawpath(fgwin, ptlist[-1], ptlist[1])
havepath := 1
report("initial path")
return
end
# randpath() -- make random connections
procedure randpath()
local l, i, p, q
bgnpath(0, "connecting randomly...") | fail
l := copy(ptlist) # get copy of point list
every i := 1 to *l do # shuffle it
l[i] :=: l[?i]
p := l[1]
q := l[-1]
p.nxt := &null
every i := 2 to *l do {
follow(l[i-1], l[i])
pause()
}
p.prv := q
q.nxt := p
drawpath(fgwin, q, p)
havepath := 1
report("random path")
return
end
# strippath() -- construct using strips
procedure strippath()
local i, l, n, p, q, r
if *ptlist < 3 then
return
bgnpath(0, "stripwise algorithm")
n := integer(sqrt(*ptlist) + .5)
l := list(n)
every !l := list()
every p := !ptlist do {
i := integer(1 + n * (p.x - ax) / real(aw + 1))
put(l[i], p)
}
every i := 1 to n do
l[i] := sortf(l[i], 3)
every i := 2 to n by 2 do {
r := []
every push(r, !l[i])
l[i] := r
}
q := !!l # get first point from first non-empty bin
every p := !!l do {
q.nxt := p
p.prv := q
drawpath(fgwin, q, p)
q := p
pause()
}
q := !!l
p.nxt := q
q.prv := p
drawpath(fgwin, p, q)
havepath := 1
report("stripwise algorithm")
return
end
# nearnbr() -- nearest neighbor
procedure nearnbr()
local f, p, q, s, d
bgnpath(1, "nearest neighbor...") | fail
f := p := ?ptlist
p.nxt := p.prv := &null
s := set([p])
while *s < *ptlist do {
every d := !distsrt do {
if d.p === p then
q := d.q
else if d.q === p then
q := d.p
else
next
if member(s, q) then
next
insert(s, q)
p := follow(p, q)
p.nxt := &null
pause()
break
}
}
p.nxt := f
f.prv := p
drawpath(fgwin, p, f)
havepath := 1
report("nearest neighbor")
return
end
# nearins() -- make path using nearest-insertion algorithm
procedure nearins()
local d, p, q, t, todo, mind
bgnpath(0, "nearest insertion...") | fail
# init path with the two closest points
mind := 1000000000
every d := !distlist do
if mind >:= d.d then {
p := d.p
q := d.q
}
p.nxt := p.prv := q
q.nxt := q.prv := p
drawpath(fgwin, p, q)
pause()
todo := set(ptlist) # set of points not yet on path
every delete(todo, p | q)
every t := !todo do
t.t1 := dist(t, q) # point.t1 = distance to nearest point on path
while *todo > 0 do { # repeat for each new point added to path
mind := 1000000000 # mind = minimum distance this pass
every t := !todo do {
t.t1 >:= dist(t, p) # update pt's dist to path if latest pt closer
if mind >:= t.t1 then # check for better (smaller) min d this pass
q := t # if nearest so far
}
# point q is the remaining point nearest from any point on the path
joinpath(p, q)
delete(todo, q)
pause()
p := q
}
havepath := 1
redraw()
report("nearest insertion")
return
end
# farins() -- make path using farthest-insertion algorithm
procedure farins()
local d, p, q, t, todo, maxd
bgnpath(0, "farthest insertion...") | fail
# init path with the two most distant points
maxd := -1
every d := !distlist do
if maxd <:= d.d then {
p := d.p
q := d.q
}
p.nxt := p.prv := q
q.nxt := q.prv := p
drawpath(fgwin, p, q)
pause()
todo := set(ptlist) # set of points not yet on path
every delete(todo, p | q)
every t := !todo do
t.t1 := dist(t, q) # point.t1 = distance to nearest point on path
while *todo > 0 do { # repeat for each new point added to path
maxd := -1 # maxd = furthest distance this pass
every t := !todo do {
t.t1 >:= dist(t, p) # update pt's dist to path if latest pt closer
if maxd <:= t.t1 then # check for better (larger) maxd this pass
q := t # if farthest so far
}
# point q is the remaining point farthest from any point on the path
joinpath(p, q)
delete(todo, q)
pause()
p := q
}
havepath := 1
redraw()
report("farthest insertion")
return
end
# joinpath(p, q) -- add q at best place in path beginning at p
procedure joinpath(p, q)
local start, best, d
d := dist(p, q) + dist(q, p.nxt) - dist(p, p.nxt)
start := best := p
while (p := p.nxt) ~=== start do
if d >:= dist(p, q) + dist(q, p.nxt) - dist(p, p.nxt) then
best := p
follow(best, q)
return
end
# greedypath() -- make path using greedy algorithm
procedure greedypath()
local p, q, d, g, need
bgnpath(1, "greedy algorithm...") | fail
every p := !ptlist do {
p.nxt := p.prv := &null
p.t1 := p.id # point.t1 = group membership
p.t2 := 0 # point.t2 = degree of node
}
need := *ptlist # number of edges we still need
every d := |!distsrt do { # |! is to handle 2-pt case
p := d.p
q := d.q
if p.t2 > 1 | q.t2 > 1 then # if either is fully connected
next
if p.t1 = q.t1 & need > 1 then # if would be cycle & not done
next
# now we are committed to adding the point
pause()
DrawLine(fgwin, p.x, p.y, q.x, q.y) # draw new edge
p.t2 +:= 1 # increase degree counts
q.t2 +:= 1
if /p.nxt <- q & /q.prv := p then { # if q can follow p easily
g := q.t1 ~=:= p.t1 | break # break if the final connection
while q := \q.nxt do
q.t1 := g
}
else if /q.nxt <- p & /p.prv := q then { # if p can follow q easily
g := p.t1 ~=:= q.t1 | break # break if the final connection
while p := \p.nxt do
p.t1 := g
}
else if /p.nxt := q then { # implies /q.nxt -- both are chain tails
g := p.t1
repeat {
q.t1 := g
q.nxt := q.prv
q.prv := p
p := q
q := \q.nxt | break
}
}
else { # /p.prv & /q.prv -- both are chain heads
p.prv := q
g := p.t1
repeat {
q.t1 := g
q.prv := q.nxt
q.nxt := p
p := q
q := \q.prv | break
}
}
if (need -:= 1) = 0 then # quit when have all edges
break
}
havepath := 1
report("greedy algorithm")
return
end
# bgnpath(i, msg) -- common setup for path construction
#
# i > 0 if *sorted* distance table will be needed
# msg is status message
procedure bgnpath(i, msg)
if *ptlist < 2 then
fail
prepdist(i)
status(msg)
if \havepath then
erasepath()
havepath := &null
lastclk := &time
return
end
######################### optimizations #########################
# twoadj() -- swap pairs of adjacent points
procedure twoadj()
local lastchg, nflips, p, q
if /havepath then
return
status("2-adj...")
lastclk := &time
nflips := 0
lastchg := p := ?ptlist # pick random starting point
repeat {
q := p.nxt.nxt
repeat {
DrawLine(emwin, p.x, p.y, p.nxt.x, p.nxt.y) # mark current spot
if not pairtest(p, q) then # if swap doesn't help
break
flip(p, q) # do the swap
nflips +:= 1 # count it
lastchg := p # update point of last change
}
pause()
p := p.nxt
if p === lastchg then
break # have made complete circuit without changes
}
report("2-adj (" || nflips || " flips)")
refresh()
return
end
procedure adjtest(p, q)
return ((p.nxt.nxt === q) | (q.nxt.nxt === p)) & pairtest(p, q)
end
# twoopt() -- swap segments if total path shortens
procedure twoopt()
pairdriver("2-opt", pairtest)
return
end
# pairtest(p, q) -- succeed if swapping out-segments from p and q shortens path
procedure pairtest(p, q)
return (dist(p,q) + dist(p.nxt,q.nxt)) < (dist(p,p.nxt) + dist(q,q.nxt)) &
(not (p === (q.prv | q | q.nxt)))
end
# uncross() -- swap intersecting segments
procedure uncross()
pairdriver("uncross", intersect)
return
end
# intersect(p, q) -- succeed if outward segments from p and q intersect
#
# from comp.graphics.algorithms FAQ, by O'Rourke
procedure intersect(p, q)
local a, b, c, d
local xac, xdc, xba, yac, ydc, yba
local n1, n2, d12, r, s
a := p
b := p.nxt
c := q
d := q.nxt
xac := a.x - c.x
xdc := d.x - c.x
xba := b.x - a.x
yac := a.y - c.y
ydc := d.y - c.y
yba := b.y - a.y
n1 := yac * xdc - xac * ydc
n2 := yac * xba - xac * yba
d12 := real(xba * ydc - yba * xdc)
if d12 = 0.0 then
fail # lines are parallel or coincident
r := n1 / d12
s := n2 / d12
# intersection point is: (a.x + r * xba, a.y + r * yba)
if 0.0 < r < 1.0 & 0.0 < s < 1.0 then
return # segments AB and CD do intersect
else
fail # segments do not intersect (though extensions do)
end
# pairdriver(label, tproc) -- driver for "uncross" and "2-opt"
procedure pairdriver(label, tproc)
local slist, todo, nflips, a, p, q
if /havepath then
return
status(label || "...")
lastclk := &time
nflips := 0
slist := list() # initial list of segments
every put(slist, path())
todo := set() # segments to reconsider
while p := get(slist) | ?todo do { # pick candidate segment
delete(todo, p)
pause()
# restart search every time p's outgoing edge changes
repeat {
DrawLine(emwin, p.x, p.y, p.nxt.x, p.nxt.y) # mark segment in progress
# check for swap with every other edge
every q := !ptlist do {
if tproc(p, q) then { # if test procedure succeeds,
# a swap is worthwhile
# the path from p.nxt through q will reverse direction;
# this will change segment labelings; so fix up "todo" set
a := q.prv
while a ~=== p do {
if member(todo, a) then { # if segment is on list
delete(todo, a) # remove under old name
insert(todo, a.nxt) # add under new name
}
a := a.prv
}
# new segment from p will be done when we loop again
# other new segment to list
insert(todo, p.nxt) # add to list
# now flip the edges
flip(p, q) # flip the edges
nflips +:= 1 # count the flip
break next # restart search loop using new edge
}
}
break # if no improvement for one full loop
}
}
report(label || " (" || nflips || " flips)")
refresh()
return
end
######################### point maintenance #########################
# clrpts() -- remove all points
procedure clrpts()
ptlist := []
distlist := []
distsrt := []
havepath := &null
refresh()
fillrect(bgwin)
status("0 points")
return
end
# reseed() -- add random points to the list
procedure reseed(win, dummy, x, y, event)
local p, v, n
n := integer(\event)^2 | nseed
every 1 to n do
addpt(win, &null, ax + ?aw, ay + ?ah)
return
end
# addpt(win, dummy, x, y) -- add one point to the list
procedure addpt(win, dummy, x, y)
local n, p, q
if \havepath then {
erasepath()
havepath := &null
}
n := *ptlist
p := point(n + 1, x, y)
every q := !ptlist do
put(distlist, dstrec(integer(1000 * sqrt((q.x-x)^2 + (q.y-y)^2)), p, q))
put(ptlist, p)
drawpt(p)
status(*ptlist || " points")
newpts := 1
return p
end
# prepdist(i) -- prepare distance data for path construction
#
# copy the distance list, if not already done, so it can be indexed quickly.
# also create the sorted list if i > 0.
procedure prepdist(i)
static c, n
if c ~=== distlist | n ~= *distlist then {
c := distlist := copy(distlist)
n := *distlist
}
if \i > 0 & *distsrt < *distlist then {
status("sorting distances... ")
lastclk := &time
WFlush(win)
distsrt := sortf(distlist, 1)
report("distance sort")
}
return
end
# dist(p, q) -- return distance between p and q assuming p ~=== q
procedure dist(p, q)
local m, n
m := p.id
n := q.id
if m < n then
m :=: n
return distlist[((m - 1) * (m - 2)) / 2 + n].d
end
# path() -- generate current path, even if it changes during generation
procedure path()
local l, p, q
p := q := ptlist[1] | fail
l := [p]
while (p := p.nxt) ~=== q do
put(l, p)
suspend !l
end
# follow(p, q) -- insert q to follow p (erases old path from p, draws new)
procedure follow(p, q)
DrawLine(bgwin, p.x, p.y, (p.prv~===\p.nxt).x, p.nxt.y)
every drawpt(p | \p.nxt)
q.nxt := p.nxt
q.prv := p
(\p.nxt).prv := q
p.nxt := q
DrawLine(fgwin, p.x, p.y, q.x, q.y)
DrawLine(fgwin, q.x, q.y, (\q.nxt).x, q.nxt.y)
return q
end
# flip(p, q) -- link p to q, and their successors to each other
procedure flip(p, q)
local a, b
DrawLine(bgwin, p.x, p.y, p.nxt.x, p.nxt.y)
DrawLine(bgwin, q.x, q.y, q.nxt.x, q.nxt.y)
# relink half of the chain backwards
a := q
while a ~=== p do {
a.prv :=: a.nxt
a := a.nxt
}
a := p.nxt
b := q.prv
p.nxt := q
q.prv := p
a.nxt := b
b.prv := a
DrawLine(fgwin, p.x, p.y, q.x, q.y)
DrawLine(fgwin, a.x, a.y, b.x, b.y)
every drawpt(p | q | a | b)
return
end
# linkpath(p, q, ...) -- link points p, q, ... in order
procedure linkpath(l[])
local i, p, q, v
i := p := get(l)
v := [fgwin, p.x, p.y]
every q := !l do {
p.nxt := q
q.prv := p
p := q
put(v, p.x, p.y)
}
DrawLine ! v
every drawpt(i | !l)
return
end
######################### drawing #########################
# refresh() -- redraw screen to repair segments and points
procedure refresh()
fillrect(bgwin) # erase segs
redraw()
return
end
# redraw() -- redraw path without erasing
procedure redraw()
local p
every drawpt(!ptlist)
every p := !ptlist do
DrawLine(fgwin, p.x, p.y, (\p.nxt).x, p.nxt.y)
return
end
# erasepath() -- erase path, redraw points if necessary
procedure erasepath()
local l, p, v
v := [bgwin]
every p := ptlist[1].prv | path() do
put(v, p.x, p.y)
DrawLine ! v
every drawpt(!ptlist)
return
end
# drawpath(win, p, q) -- draw the path from p to q
#
# (of course, depending on the foreground color, this can hide a path, too.)
procedure drawpath(win, p, q)
local v
v := [win, p.x, p.y]
while p ~=== q do {
p := p.nxt
put(v, p.x)
put(v, p.y)
}
DrawLine ! v
return
end
# drawpt(p) -- draw the single point p
procedure drawpt(p)
FillRectangle(fgwin, p.x - 2, p.y - 2, 5, 5)
return
end
# fillrect(win) -- fill the working area
procedure fillrect(win)
FillRectangle(win, ax - m + 1, ay - m + 1, aw + 2 * m - 1, ah + 2 * m - 1)
return
end
######################### reporting #########################
# listpath() -- list the coordinates of each point on standard output
procedure listpath()
local p
if \havepath then {
write("\point list in order of traversal:")
every listpt(path())
}
else {
write("\point list (no path established):")
every listpt(!ptlist)
}
return
end
# listpt(p) - list one point
procedure listpt(p)
write(right(p.id, 3), ".", right(p.x, 5), right(p.y, 5),
right((\p.prv).id | "", 6), right((\p.nxt).id | "", 6))
return
end
# report(text) -- display statistics on screen and stdout
#
# The statistics include the delta time since lastclk was last set.
#
# Output to stdout is suppressed if the "-q" option was given.
# Output to stdout is double spaced if the set of points has changed.
procedure report(text)
local p, n, d, s, dt
dt := ((((&time - lastclk) / 1000.0) || "000") ? (tab(upto(".")) || move(3)))
s := right(*ptlist, 4) || " pts "
if \havepath then {
d := 0
every p := !ptlist do
d +:= dist(p, p.nxt)
d := (d + 500) / 1000
s ||:= right("d = " || d, 10)
}
else
s ||:= " "
s ||:= right(dt , 8) || " sec " || text
status(s)
if /opts["q"] then {
if \newpts then
write()
write(s)
}
newpts := &null
return
end
# status(s) -- write s as a status message
procedure status(s)
EraseArea(win, m + bw + m, m + h - fh)
GotoXY(win, m + bw + m, m + h - (fh / 4))
writes(win, s)
return
end
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