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{{draft task}} Quoted from the [https://en.wikipedia.org/wiki/Simulated_annealing Wikipedia page] : '''Simulated annealing (SA)''' is a probabilistic technique for approximating the global optimum of a given function. Simulated annealing interprets slow cooling as a slow decrease in the probability of temporarily accepting worse solutions as it explores the solution space.
'''Pseudo code''' from [https://en.wikipedia.org/wiki/Simulated_annealing Wikipedia]
Notations :
T : temperature. Decreases to 0.
s : a system state
E(s) : Energy at s. The function we want to minimize
∆E : variation of E, from state s to state s_next
P(∆E , T) : Probability to move from s to s_next.
if ( ∆E < 0 ) P = 1
else P = exp ( - ∆E / T) . Decreases as T → 0
Pseudo-code:
Let s = s0 -- initial state
For k = 0 through kmax (exclusive):
T ← temperature(k , kmax)
Pick a random neighbour state , s_next ← neighbour(s)
∆E ← E(s) - E(s_next)
If P(∆E , T) ≥ random(0, 1), move to the new state:
s ← s_next
Output: the final state s
'''Problem statement'''
We want to apply SA to the travelling salesman problem. There are 100 cities, numbered 0 to 99, located on a plane, at integer coordinates i,j : 0 <= i,j < 10 . The city at (i,j) has number 10*i + j. The cities are '''all''' connected : the graph is complete : you can go from one city to any other city in one step.
The salesman wants to start from city 0, visit all cities, each one time, and go back to city 0. The travel cost between two cities is the euclidian distance between there cities. The total travel cost is the total path length.
A path '''s''' is a sequence (0 a b ...z 0) where (a b ..z) is a permutation of the numbers (1 2 .. 99). The path length = E(s) is the sum d(0,a) + d(a,b) + ... + d(z,0) , where d(u,v) is the distance between two cities. Naturally, we want to minimize E(s).
Definition : The neighbours of a city are the closest cities at distance 1 horizontally/vertically, or √2 diagonally. A corner city (0,9,90,99) has 3 neighbours. A center city has 8 neighbours.
Distances between cities
d ( 0, 7) → 7
d ( 0, 99) → 12.7279
d ( 23, 78) → 7.0711
d ( 33, 44) → 1.4142 // sqrt(2)
'''Task'''
Apply SA to the travelling salesman problem, using the following set of parameters/functions :
- kT = 1 (Multiplication by kT is a placeholder, representing computing temperature as a function of 1-k/kmax):
- temperature (k, kmax) = kT * (1 - k/kmax)
- neighbour (s) : Pick a random city u > 0 . Pick a random neighbour city v > 0 of u , among u's 8 (max) neighbours on the grid. Swap u and v in s . This gives the new state ''s_next''.
- kmax = 1000_000
- s0 = a random permutation
For k = 0 to kmax by step kmax/10 , display k, T, E(s). Display the final state s_final, and E(s_final).
You will see that the Energy may grow to a local optimum, before decreasing to a global optimum.
'''Illustrated example''' [http://www.echolalie.org/echolisp/annealing.html Temperature charts]
'''Numerical example'''
kT = 1
E(s0) = 529.9158
k: 0 T: 1 Es: 529.9158
k: 100000 T: 0.9 Es: 201.1726
k: 200000 T: 0.8 Es: 178.1723
k: 300000 T: 0.7 Es: 154.7069
k: 400000 T: 0.6 Es: 158.1412 <== local optimum
k: 500000 T: 0.5 Es: 133.856
k: 600000 T: 0.4 Es: 129.5684
k: 700000 T: 0.3 Es: 112.6919
k: 800000 T: 0.2 Es: 105.799
k: 900000 T: 0.1 Es: 102.8284
k: 1000000 T: 0 Es: 102.2426
E(s_final) = 102.2426
Path s_final = ( 0 10 11 21 31 20 30 40 50 60 70 80 90 91 81 71 73 83 84 74 64 54 55 65 75 76 66
67 77 78 68 58 48 47 57 56 46 36 37 27 26 16 15 5 6 7 17 18 8 9 19 29 28 38 39 49 59 69
79 89 99 98 88 87 97 96 86 85 95 94 93 92 82 72 62 61 51 41 42 52 63 53 43 32 22 12 13
23 33 34 44 45 35 25 24 14 4 3 2 1 0)
'''Extra credit'''
Tune the parameters kT, kmax, or use different temperature() and/or neighbour() functions to demonstrate a quicker convergence, or a better optimum.
EchoLisp
(lib 'math)
;; distances
(define (d ci cj)
(distance (% ci 10) (quotient ci 10) (% cj 10) (quotient cj 10)))
(define _dists
(build-vector 10000 (lambda (ij) (d (quotient ij 100) (% ij 100)))))
(define-syntax-rule (dist ci cj)
[_dists (+ ci (* 100 cj))])
;; E(s) = length(path)
(define (Es path)
(define lpath (vector->list path))
(for/sum ((ci lpath) (cj (rest lpath))) (dist ci cj)))
;; temperature() function
(define (T k kmax kT)
(* kT (- 1 (// k kmax))))
#|
;; alternative temperature()
;; must be decreasing with k increasing and → 0
(define (T k kmax kT)
(* kT (- 1 (sin (* PI/2 (// k kmax))))))
|#
;; ∆E = Es_new - Es_old > 0
;; probability to move if ∆E > 0, → 0 when T → 0 (frozen state)
(define (P ∆E k kmax kT)
(exp (// (- ∆E ) (T k kmax kT))))
;; ∆E from path ( .. a u b .. c v d ..) to (.. a v b ... c u d ..)
;; ∆E before swapping (u,v)
;; Quicker than Es(s_next) - Es(s)
(define (dE s u v)
;;old
(define a (dist [s (1- u)] [s u]))
(define b (dist [s (1+ u)] [s u]))
(define c (dist [s (1- v)] [s v]))
(define d (dist [s (1+ v)] [s v]))
;; new
(define na (dist [s (1- u)] [s v]))
(define nb (dist [s (1+ u)] [s v]))
(define nc (dist [s (1- v)] [s u]))
(define nd (dist [s (1+ v)] [s u]))
(cond
((= v (1+ u)) (- (+ na nd) (+ a d)))
((= u (1+ v)) (- (+ nc nb) (+ c b)))
(else (- (+ na nb nc nd) (+ a b c d)))))
;; all 8 neighbours
(define dirs #(1 -1 10 -10 9 11 -11 -9))
(define (sa kmax (kT 10))
(define s (list->vector (cons 0 (append (shuffle (range 1 100)) 0))))
(printf "E(s0) %d" (Es s)) ;; random starter
(define Emin (Es s)) ;; E0
(for ((k kmax))
(when (zero? (% k (/ kmax 10)))
(printf "k: %10d T: %8.4d Es: %8.4d" k (T k kmax kT) (Es s))
)
(define u (1+ (random 99))) ;; city index 1 99
(define cv (+ [s u] [dirs (random 8)])) ;; city number
#:continue (or (> cv 99) (<= cv 0))
#:continue (> (dist [s u] cv) 5) ;; check true neighbour (eg 0 9)
(define v (vector-index cv s 1)) ;; city index
(define ∆e (dE s u v))
(when (or
(< ∆e 0) ;; always move if negative
(>= (P ∆e k kmax kT) (random)))
(vector-swap! s u v)
(+= Emin ∆e))
;; (assert (= (round Emin) (round (Es s))))
) ;; for
(printf "k: %10d T: %8.4d Es: %8.4d" kmax (T (1- kmax) kmax kT) (Es s))
(s-plot s 0)
(printf "E(s_final) %d" Emin)
(writeln 'Path s))
{{out}}
(sa 1000000 1)
E(s0) 501.0909
k: 0 T: 1 Es: 501.0909
k: 100000 T: 0.9 Es: 167.3632
k: 200000 T: 0.8 Es: 160.7791
k: 300000 T: 0.7 Es: 166.8746
k: 400000 T: 0.6 Es: 142.579
k: 500000 T: 0.5 Es: 131.0657
k: 600000 T: 0.4 Es: 116.9214
k: 700000 T: 0.3 Es: 110.8569
k: 800000 T: 0.2 Es: 103.3137
k: 900000 T: 0.1 Es: 102.4853
k: 1000000 T: 0 Es: 102.4853
E(s_final) 102.4853
Path #( 0 10 20 30 40 50 60 70 71 61 62 53 63 64 54 44 45 55 65
74 84 83 73 72 82 81 80 90 91 92 93 94 95 85 75 76 86 96 97 98 99
88 89 79 69 59 49 48 47 57 58 68 78 87 77 67 66 56 46 36 35 25 24
34 33 32 43 42 52 51 41 31 21 11 12 22 23 13 14 15 16 17 26 27 37 38
39 29 28 18 19 9 8 7 6 5 4 3 2 1 0)
Go
{{trans|zkl}}
package main import ( "fmt" "math" "math/rand" "time" ) var ( dists = calcDists() dirs = [8]int{1, -1, 10, -10, 9, 11, -11, -9} // all 8 neighbors ) // distances func calcDists() []float64 { dists := make([]float64, 10000) for i := 0; i < 10000; i++ { ab, cd := math.Floor(float64(i)/100), float64(i%100) a, b := math.Floor(ab/10), float64(int(ab)%10) c, d := math.Floor(cd/10), float64(int(cd)%10) dists[i] = math.Hypot(a-c, b-d) } return dists } // index into lookup table of float64s func dist(ci, cj int) float64 { return dists[cj*100+ci] } // energy at s, to be minimized func Es(path []int) float64 { d := 0.0 for i := 0; i < len(path)-1; i++ { d += dist(path[i], path[i+1]) } return d } // temperature function, decreases to 0 func T(k, kmax, kT int) float64 { return (1 - float64(k)/float64(kmax)) * float64(kT) } // variation of E, from state s to state s_next func dE(s []int, u, v int) float64 { su, sv := s[u], s[v] // old a, b, c, d := dist(s[u-1], su), dist(s[u+1], su), dist(s[v-1], sv), dist(s[v+1], sv) // new na, nb, nc, nd := dist(s[u-1], sv), dist(s[u+1], sv), dist(s[v-1], su), dist(s[v+1], su) if v == u+1 { return (na + nd) - (a + d) } else if u == v+1 { return (nc + nb) - (c + b) } else { return (na + nb + nc + nd) - (a + b + c + d) } } // probability to move from s to s_next func P(deltaE float64, k, kmax, kT int) float64 { return math.Exp(-deltaE / T(k, kmax, kT)) } func sa(kmax, kT int) { rand.Seed(time.Now().UnixNano()) temp := make([]int, 99) for i := 0; i < 99; i++ { temp[i] = i + 1 } rand.Shuffle(len(temp), func(i, j int) { temp[i], temp[j] = temp[j], temp[i] }) s := make([]int, 101) // all 0 by default copy(s[1:], temp) // random path from 0 to 0 fmt.Println("kT =", kT) fmt.Printf("E(s0) %f\n\n", Es(s)) // random starter Emin := Es(s) // E0 for k := 0; k <= kmax; k++ { if k%(kmax/10) == 0 { fmt.Printf("k:%10d T: %8.4f Es: %8.4f\n", k, T(k, kmax, kT), Es(s)) } u := 1 + rand.Intn(99) // city index 1 to 99 cv := s[u] + dirs[rand.Intn(8)] // city number if cv <= 0 || cv >= 100 { // bogus city continue } if dist(s[u], cv) > 5 { // check true neighbor (eg 0 9) continue } v := s[cv] // city index deltae := dE(s, u, v) if deltae < 0 || // always move if negative P(deltae, k, kmax, kT) >= rand.Float64() { s[u], s[v] = s[v], s[u] Emin += deltae } } fmt.Printf("\nE(s_final) %f\n", Emin) fmt.Println("Path:") // output final state for i := 0; i < len(s); i++ { if i > 0 && i%10 == 0 { fmt.Println() } fmt.Printf("%4d", s[i]) } fmt.Println() } func main() { sa(1e6, 1) }
{{out}} Sample run:
kT = 1
E(s0) 520.932463
k: 0 T: 1.0000 Es: 520.9325
k: 100000 T: 0.9000 Es: 185.1279
k: 200000 T: 0.8000 Es: 167.7657
k: 300000 T: 0.7000 Es: 158.6923
k: 400000 T: 0.6000 Es: 151.6564
k: 500000 T: 0.5000 Es: 139.9185
k: 600000 T: 0.4000 Es: 132.9964
k: 700000 T: 0.3000 Es: 121.8962
k: 800000 T: 0.2000 Es: 120.0445
k: 900000 T: 0.1000 Es: 116.8476
k: 1000000 T: 0.0000 Es: 116.5565
E(s_final) 116.556509
Path:
0 11 21 31 41 51 52 61 62 72
82 73 74 64 44 45 55 54 63 53
42 32 43 33 35 34 24 23 22 13
12 2 3 4 14 25 26 7 6 16
15 5 17 27 36 46 56 66 65 75
77 78 68 69 59 49 39 38 37 28
29 19 9 8 18 47 48 58 57 67
76 86 85 95 96 97 87 88 79 89
99 98 84 94 83 93 92 91 90 80
81 71 70 60 50 40 30 20 10 1
0
J
Implementation:
dist=: +/&.:*:@:-"1/~10 10#:i.100
satsp=:4 :0
kT=. 1
pathcost=. [: +/ 2 {&y@<\ 0 , ] , 0:
neighbors=. 0 (0}"1) y e. 1 2{/:~~.,y
s=. (?~#y)-.0
d=. pathcost s
step=. x%10
for_k. i.x+1 do.
T=. kT*1-k%x
u=. ({~ ?@#)s
v=. ({~ ?@#)I.u{neighbors
sk=. (<s i.u,v) C. s
dk=. pathcost sk
dE=. dk-d
if. (^-dE%T) >?0 do.
s=.sk
d=.dk
end.
if. 0=step|k do.
echo k,T,d
end.
end.
0,s,0
)
Notes:
E(s_final) gets displayed on the kmax progress line.
We do not do anything special for negative deltaE because the exponential will be greater than 1 for that case and that will always be greater than our random number from the range 0..1.
Also, while we leave connection distances (and, thus, number of cities) as a parameter, some other aspects of this problem made more sense when included in the implementation:
We leave city 0 out of our data structure, since it can't appear in the middle of our path. But we bring it back in when computing path distance.
Neighbors are any city which have one of the two closest non-zero distances from the current city (and specifically excluding city 0, since that is anchored as our start and end city).
Sample run:
1e6 satsp dist
0 1 538.409
100000 0.9 174.525
200000 0.8 165.541
300000 0.7 173.348
400000 0.6 168.188
500000 0.5 134.983
600000 0.4 121.585
700000 0.3 111.443
800000 0.2 101.657
900000 0.1 101.657
1e6 0 101.657
0 1 2 3 4 13 23 24 34 44 43 33 32 31 41 42 52 51 61 62 53 54 64 65 55 45 35 25 15 14 5 6 7 17 16 26 27 37 36 46 47 48 38 28 18 8 9 19 29 39 49 59 69 79 78 68 58 57 56 66 67 77 76 75 85 86 87 88 89 99 98 97 96 95 94 84 74 73 63 72 82 83 93 92 91 90 80 81 71 70 60 50 40 30 20 21 22 12 11 10 0
Julia
{{trans|EchoLisp}}
'''Module''':
module TravelingSalesman using Random, Printf # Eₛ: length(path) Eₛ(distances, path) = sum(distances[ci, cj] for (ci, cj) in zip(path, Iterators.drop(path, 1))) # T: temperature T(k, kmax, kT) = kT * (1 - k / kmax) # Alternative temperature: #T(k, kmax, kT) = kT * (1 - sin(π / 2 * k / kmax)) # ΔE = Eₛ_new - Eₛ_old > 0 # Prob. to move if ΔE > 0, → 0 when T → 0 (fronzen state) P(ΔE, k, kmax, kT) = exp(-ΔE / T(k, kmax, kT)) # ∆E from path ( .. a u b .. c v d ..) to (.. a v b ... c u d ..) # ∆E before swapping (u,v) # Quicker than Eₛ(s_next) - Eₛ(path) function dE(distances, path, u, v) a = distances[path[u - 1], path[u]] b = distances[path[u + 1], path[u]] c = distances[path[v - 1], path[v]] d = distances[path[v + 1], path[v]] na = distances[path[u - 1], path[v]] nb = distances[path[u + 1], path[v]] nc = distances[path[v - 1], path[u]] nd = distances[path[v + 1], path[u]] if v == u + 1 return (na + nd) - (a + d) elseif u == v + 1 return (nc + nb) - (c + b) else return (na + nb + nc + nd) - (a + b + c + d) end end const dirs = [1, -1, 10, -10, 9, 11, -11, -9] function _prettypath(path) r = IOBuffer() for g in Iterators.partition(path, 10) println(r, join(lpad.(g, 3), ", ")) end return String(take!(r)) end function findpath(distances, kmax, kT) n = size(distances, 1) path = vcat(1, shuffle(2:n), 1) Emin = Eₛ(distances, path) @printf("\n# Entropy(s₀) = %10.2f\n", Emin) println("# Random path: \n", _prettypath(path)) for k in Base.OneTo(kmax) if iszero(k % (kmax ÷ 10)) @printf("k: %10d | T: %8.4f | Eₛ: %8.4f\n", k, T(k, kmax, kT), Eₛ(distances, path)) end u = rand(2:n) v = path[u] + rand(dirs) v ∈ 2:n || continue δE = dE(distances, path, u, v) if δE < 0 || P(δE, k, kmax, kT) ≥ rand() path[u], path[v] = path[v], path[u] Emin += δE end end @printf("k: %10d | T: %8.4f | Eₛ: %8.4f\n", kmax, T(kmax, kmax, kT), Eₛ(distances, path)) println("\n# Found path:\n", _prettypath(path)) return path end end # module TravelingSalesman
'''Main''':
distance(a, b) = sqrt(sum((a .- b) .^ 2)) const _citydist = collect(distance((ci % 10, ci ÷ 10), (cj % 10, cj ÷ 10)) for ci in 1:100, cj in 1:100) TravelingSalesman.findpath(_citydist, 1_000_000, 1)
{{out}}
# Entropy(s₀) = 521.86
# Random path:
1, 2, 11, 80, 78, 73, 68, 19, 43, 69
86, 79, 66, 67, 77, 96, 26, 62, 60, 98
71, 3, 59, 37, 18, 40, 34, 92, 97, 6
84, 94, 29, 63, 36, 50, 87, 45, 83, 90
76, 28, 15, 38, 91, 58, 47, 44, 85, 17
25, 33, 31, 99, 27, 74, 53, 95, 16, 13
42, 88, 8, 4, 7, 64, 54, 9, 14, 41
5, 81, 65, 23, 75, 100, 89, 51, 20, 48
82, 12, 21, 55, 24, 70, 49, 10, 35, 72
52, 22, 61, 32, 46, 57, 30, 93, 39, 56
1
k: 100000 | T: 0.9000 | Eₛ: 184.4448
k: 200000 | T: 0.8000 | Eₛ: 175.3662
k: 300000 | T: 0.7000 | Eₛ: 169.0505
k: 400000 | T: 0.6000 | Eₛ: 160.8328
k: 500000 | T: 0.5000 | Eₛ: 147.1973
k: 600000 | T: 0.4000 | Eₛ: 132.9186
k: 700000 | T: 0.3000 | Eₛ: 126.9931
k: 800000 | T: 0.2000 | Eₛ: 122.0656
k: 900000 | T: 0.1000 | Eₛ: 119.7924
k: 1000000 | T: 0.0000 | Eₛ: 119.7924
k: 1000000 | T: 0.0000 | Eₛ: 119.7924
# Found path:
1, 2, 12, 13, 3, 4, 6, 7, 8, 9
19, 18, 17, 5, 14, 15, 16, 27, 28, 29
39, 38, 26, 25, 24, 23, 22, 10, 21, 20
30, 31, 32, 33, 34, 35, 36, 37, 49, 48
47, 46, 45, 44, 43, 42, 41, 40, 50, 51
52, 53, 54, 55, 56, 57, 58, 59, 69, 68
67, 65, 64, 63, 62, 61, 71, 60, 70, 80
81, 82, 72, 73, 74, 66, 78, 79, 89, 99
98, 97, 96, 95, 94, 85, 86, 87, 88, 77
76, 75, 84, 83, 93, 92, 91, 100, 90, 11
1
Nim
import math, random, sugar, strformat from times import cpuTime const kT = 1 kMax = 1_000_000 proc randomNeighbor(x: int): int = case x of 0: rand([1, 10, 11]) of 9: rand([8, 18, 19]) of 90: rand([80, 81, 91]) of 99: rand([88, 89, 98]) elif x > 0 and x < 9: # top ceiling rand [x-1, x+1, x+9, x+10, x+11] elif x > 90 and x < 99: # bottom floor rand [x-11, x-10, x-9, x-1, x+1] elif x mod 10 == 0: # left wall rand([x-10, x-9, x+1, x+10, x+11]) elif (x+1) mod 10 == 0: # right wall rand([x-11, x-10, x-1, x+9, x+10]) else: # center rand([x-11, x-10, x-9, x-1, x+1, x+9, x+10, x+11]) proc neighbor(s: seq[int]): seq[int] = result = s var city = rand s var cityNeighbor = city.randomNeighbor while cityNeighbor == 0 or city == 0: city = rand s cityNeighbor = city.randomNeighbor result[s.find city].swap result[s.find cityNeighbor] func distNeighbor(a, b: int): float = template divmod(a: int): (int, int) = (a div 10, a mod 10) let (diva, moda) = a.divmod (divb, modb) = b.divmod hypot((diva-divb).float, (moda-modb).float) func temperature(k, kmax: float): float = kT * (1 - (k / kmax)) func pdelta(eDelta, temp: float): float = if eDelta < 0: 1.0 else: exp(-eDelta / temp) func energy(path: seq[int]): float = var sum = 0.distNeighbor path[0] for i in 1 ..< path.len: sum += path[i-1].distNeighbor(path[i]) sum + path[^1].distNeighbor 0 proc main = randomize() var s = block: var x = lc[x | (x <- 0 .. 99), int] template shuffler: int = rand(1 .. x.len-1) for i in 1 .. x.len-1: x[i].swap x[shuffler()] x let startTime = cpuTime() echo fmt"E(s0): {energy s:6.4f}" for k in 0 .. kMax: var temp = temperature(float k, float kMax) lastenergy = energy s newneighbor = s.neighbor newenergy = newneighbor.energy if k mod (kMax div 10) == 0: echo fmt"k: {k:7} T: {temp:6.2f} Es: {lastenergy:6.4f}" var deltaEnergy = newenergy - lastenergy if pDelta(deltaEnergy, temp) >= rand(1.0): s = newneighbor s.add 0 echo fmt"E(sFinal): {energy s:6.4f}" echo fmt"path: {s}" #echo fmt"ended after: {cpuTime() - startTime}" main()
Compile and run:
nim c -r -d:release --opt:speed travel_sa.nim
{{out}} Sample run:
E(s0): 505.1591
k: 0 T: 1.00 Es: 505.1591
k: 100000 T: 0.90 Es: 196.5216
k: 200000 T: 0.80 Es: 165.6735
k: 300000 T: 0.70 Es: 159.3411
k: 400000 T: 0.60 Es: 144.8330
k: 500000 T: 0.50 Es: 131.7888
k: 600000 T: 0.40 Es: 127.6914
k: 700000 T: 0.30 Es: 113.9280
k: 800000 T: 0.20 Es: 104.7279
k: 900000 T: 0.10 Es: 103.3137
k: 1000000 T: 0.00 Es: 103.3137
E(sFinal): 103.3137
path: @[0, 10, 11, 22, 21, 20, 30, 31, 41, 40, 50, 51, 61, 60, 70, 71, 81, 80, 90, 91, 92, 93, 82, 83, 73, 72, 62, 63, 53, 52, 42, 32, 33, 23, 13, 14, 24, 34, 35, 25, 15, 16, 26, 36, 47, 48, 38, 39, 49, 59, 58, 57, 68, 69, 79, 89, 99, 98, 97, 96, 95, 94, 84, 74, 75, 85, 86, 87, 88, 78, 77, 67, 76, 66, 65, 64, 54, 43, 44, 45, 55, 56, 46, 37, 27, 28, 29, 19, 9, 8, 18, 17, 7, 6, 5, 4, 3, 2, 12, 1, 0]
Phix
{{trans|zkl}} Note that the standard builtin exp() suffered occasional overflows, so this uses b_a_exp() from bigatom.e, but it does make it much slower.
function hypot(atom a,b) return sqrt(a*a+b*b) end function
function calc_dists()
sequence dists = repeat(0,10000)
for abcd=1 to 10000 do
integer {ab,cd} = {floor(abcd/100),mod(abcd,100)},
{a,b,c,d} = {floor(ab/10),mod(ab,10),
floor(cd/10),mod(cd,10)}
dists[abcd] = hypot(a-c,b-d)
end for
return dists
end function
constant dists = calc_dists()
function dist(integer ci,cj) return dists[cj*100+ci] end function
function Es(sequence path)
atom d = 0
for i=1 to length(path)-1 do
d += dist(path[i],path[i+1])
end for
return d
end function
-- temperature() function
function T(integer k, kmax, kT) return (1-k/kmax)*kT end function
include bigatom.e -- (just for b_a_exp())
-- deltaE = Es_new - Es_old > 0
-- probability to move if deltaE > 0, -->0 when T --> 0 (frozen state)
function P(atom deltaE, integer k, kmax, kT) return b_a_exp(-deltaE/T(k,kmax,kT)) end function
-- deltaE from path ( .. a u b .. c v d ..) to (.. a v b ... c u d ..)
function dE(sequence s, integer u,v)
-- (note that u,v are 0-based, but 1..99 here)
-- integer sum1 = s[u-1], su = s[u], sup1 = s[u+1],
-- svm1 = s[v-1], sv = s[v], svp1 = s[v+1]
integer sum1 = s[u], su = s[u+1], sup1 = s[u+2],
svm1 = s[v], sv = s[v+1], svp1 = s[v+2]
-- old
atom {a,b,c,d}:={dist(sum1,su), dist(su,sup1), dist(svm1,sv), dist(sv,svp1)},
-- new
{na,nb,nc,nd}:={dist(sum1,sv), dist(sv,sup1), dist(svm1,su), dist(su,svp1)}
return iff(v==u+1?(na+nd)-(a+d):
iff(u==v+1?(nc+nb)-(c+b):
(na+nb+nc+nd)-(a+b+c+d)))
end function
-- all 8 neighbours
constant dirs = {1, -1, 10, -10, 9, 11, -11, -9}
procedure sa(integer kmax, kT=10)
sequence s = 0&shuffle(tagset(99))&0
atom Emin:=Es(s) -- E0
printf(1,"E(s0) %f\n",Emin) -- random starter
for k=0 to kmax do
if mod(k,kmax/10)=0 then
printf(1,"k:%,10d T: %8.4f Es: %8.4f\n",{k,T(k,kmax,kT),Es(s)})
end if
integer u = rand(99), -- city index 1 99
cv = s[u+1]+dirs[rand(8)] -- city number
if cv>0 and cv<100 -- not bogus city
and dist(s[u+1],cv)<5 then -- and true neighbour
integer v = s[cv+1] -- city index
atom deltae := dE(s,u,v);
if deltae<0 -- always move if negative
or P(deltae,k,kmax,kT)>=rnd() then
{s[u+1],s[v+1]} = {s[v+1],s[u+1]}
Emin += deltae
end if
end if
end for
printf(1,"E(s_final) %f\n",Emin)
printf(1,"Path:\n")
pp(s,{pp_IntFmt,"%2d",pp_StrFmt,-2})
end procedure
sa(1_000_000,1)
{{out}}
E(s0) 515.164811
k: 0 T: 1.0000 Es: 515.1648
k: 100,000 T: 0.9000 Es: 189.3123
k: 200,000 T: 0.8000 Es: 198.7498
k: 300,000 T: 0.7000 Es: 158.2189
k: 400,000 T: 0.6000 Es: 165.4813
k: 500,000 T: 0.5000 Es: 156.3467
k: 600,000 T: 0.4000 Es: 142.7928
k: 700,000 T: 0.3000 Es: 128.0352
k: 800,000 T: 0.2000 Es: 121.7794
k: 900,000 T: 0.1000 Es: 121.2328
k: 1,000,000 T: 0.0000 Es: 121.1291
E(s_final) 121.129115
Path:
{ 0,10,62,63,64,65,76,75,84,85,95,86,96,97,87,77,67,66,56,46,47,48,49,59,69,
79,89,99,98,88,78,68,58,57,37,38,27,26,36,35,45,55,54,53,52,43,33,23,22,32,
42,41,51,61,60,50,40,30,31,21,20,11,12, 2, 3, 4, 5, 6,17,18,28,39,29,19, 9,
8, 7,16,15,24,44,74,83,93,94,92,91,71,70,90,80,81,82,72,73,34,25,14,13, 1,
0}
Sidef
{{trans|Julia}}
module TravelingSalesman { # Eₛ: length(path) func Eₛ(distances, path) { var total = 0 [path, path.slice(1)].zip {|ci,cj| total += distances[ci-1][cj-1] } total } # T: temperature func T(k, kmax, kT) { kT * (1 - k/kmax) } # ΔE = Eₛ_new - Eₛ_old > 0 # Prob. to move if ΔE > 0, → 0 when T → 0 (fronzen state) func P(ΔE, k, kmax, kT) { exp(-ΔE / T(k, kmax, kT)) } # ∆E from path ( .. a u b .. c v d ..) to (.. a v b ... c u d ..) # ∆E before swapping (u,v) # Quicker than Eₛ(s_next) - Eₛ(path) func dE(distances, path, u, v) { var a = distances[path[u-1]-1][path[u]-1] var b = distances[path[u+1]-1][path[u]-1] var c = distances[path[v-1]-1][path[v]-1] var d = distances[path[v+1]-1][path[v]-1] var na = distances[path[u-1]-1][path[v]-1] var nb = distances[path[u+1]-1][path[v]-1] var nc = distances[path[v-1]-1][path[u]-1] var nd = distances[path[v+1]-1][path[u]-1] if (v == u+1) { return ((na+nd) - (a+d)) } if (u == v+1) { return ((nc+nb) - (c+b)) } return ((na+nb+nc+nd) - (a+b+c+d)) } const dirs = [1, -1, 10, -10, 9, 11, -11, -9] func _prettypath(path) { path.slices(10).map { .map{ "%3s" % _ }.join(', ') }.join("\n") } func findpath(distances, kmax, kT) { const n = distances.len const R = 2..n var path = [1, R.shuffle..., 1] var Emin = Eₛ(distances, path) printf("# Entropy(s₀) = s%10.2f\n", Emin) printf("# Random path:\n%s\n\n", _prettypath(path)) for k in (1 .. kmax) { if (k % (kmax//10) == 0) { printf("k: %10d | T: %8.4f | Eₛ: %8.4f\n", k, T(k, kmax, kT), Eₛ(distances, path)) } var u = R.rand var v = (path[u-1] + dirs.rand) v ~~ R || next var δE = dE(distances, path, u-1, v-1) if ((δE < 0) || (P(δE, k, kmax, kT) >= 1.rand)) { path.swap(u-1, v-1) Emin += δE } } printf("k: %10d | T: %8.4f | Eₛ: %8.4f\n", kmax, T(kmax, kmax, kT), Eₛ(distances, path)) say ("\n# Found path:\n", _prettypath(path)) return path } } var citydist = {|ci| { |cj| var v1 = Vec(ci%10, ci//10) var v2 = Vec(cj%10, cj//10) v1.dist(v2) }.map(1..100) }.map(1..100) TravelingSalesman::findpath(citydist, 1e6, 1)
{{out}}
# Entropy(s₀) = 520.29
# Random path:
1, 10, 79, 52, 24, 9, 58, 11, 42, 4
15, 87, 62, 88, 21, 91, 99, 84, 61, 14
5, 17, 33, 95, 74, 31, 40, 13, 37, 69
6, 22, 97, 45, 56, 63, 75, 83, 53, 41
3, 47, 89, 80, 78, 98, 46, 18, 25, 51
93, 16, 50, 30, 48, 8, 66, 68, 59, 73
49, 96, 36, 32, 100, 27, 76, 44, 64, 39
90, 82, 20, 12, 54, 86, 29, 81, 26, 72
60, 94, 35, 92, 43, 7, 85, 55, 28, 57
23, 34, 65, 71, 38, 2, 77, 70, 19, 67
1
k: 100000 | T: 0.9000 | Eₛ: 185.1809
k: 200000 | T: 0.8000 | Eₛ: 168.6262
k: 300000 | T: 0.7000 | Eₛ: 146.5948
k: 400000 | T: 0.6000 | Eₛ: 140.1441
k: 500000 | T: 0.5000 | Eₛ: 129.5132
k: 600000 | T: 0.4000 | Eₛ: 132.8942
k: 700000 | T: 0.3000 | Eₛ: 124.2865
k: 800000 | T: 0.2000 | Eₛ: 120.0859
k: 900000 | T: 0.1000 | Eₛ: 115.0771
k: 1000000 | T: 0.0000 | Eₛ: 114.9728
k: 1000000 | T: 0.0000 | Eₛ: 114.9728
# Found path:
1, 2, 13, 3, 4, 5, 6, 7, 8, 9
19, 29, 18, 28, 27, 17, 16, 26, 25, 15
14, 24, 23, 12, 11, 10, 20, 21, 30, 40
41, 31, 32, 44, 45, 46, 47, 48, 49, 39
38, 37, 36, 35, 34, 42, 51, 50, 60, 61
52, 53, 54, 55, 56, 57, 58, 59, 69, 68
77, 67, 66, 65, 64, 62, 72, 71, 70, 80
81, 82, 74, 75, 76, 87, 88, 78, 79, 89
99, 98, 97, 96, 86, 85, 83, 91, 90, 100
92, 93, 94, 95, 84, 73, 63, 43, 33, 22
1
zkl
{{trans|EchoLisp}}
var [const] _dists=(0d10_000).pump(List,fcn(abcd){ // two points (a,b) & (c,d), calc distance
ab,cd,a,b,c,d:=abcd/100, abcd%100, ab/10,ab%10, cd/10,cd%10;
(a-c).toFloat().hypot(b-d)
});
fcn dist(ci,cj){ _dists[cj*100 + ci] } // index into lookup table of floats
fcn Es(path) // E(s) = length(path): E(a,b,c)--> dist(a,b) + dist(b,c)
{ d:=Ref(0.0); path.reduce('wrap(a,b){ d.apply('+,dist(a,b)); b }); d.value }
// temperature() function
fcn T(k,kmax,kT){ (1.0 - k.toFloat()/kmax)*kT }
// deltaE = Es_new - Es_old > 0
// probability to move if deltaE > 0, -->0 when T --> 0 (frozen state)
fcn P(deltaE,k,kmax,kT){ (-deltaE/T(k,kmax,kT)).exp() } //-->Float
// deltaE from path ( .. a u b .. c v d ..) to (.. a v b ... c u d ..)
// deltaE before swapping (u,v)
fcn dE(s,u,v){ su,sv:=s[u],s[v]; //-->Float
// old
a,b,c,d:=dist(s[u-1],su), dist(s[u+1],su), dist(s[v-1],sv), dist(s[v+1],sv);
// new
na,nb,nc,nd:=dist(s[u-1],sv), dist(s[u+1],sv), dist(s[v-1],su), dist(s[v+1],su);
if (v==u+1) (na+nd) - (a+d);
else if(u==v+1) (nc+nb) - (c+b);
else (na+nb+nc+nd) - (a+b+c+d);
}
// all 8 neighbours
var [const] dirs=ROList(1, -1, 10, -10, 9, 11, -11, -9),
fmt="k:%10,d T: %8.4f Es: %8.4f".fmt; // since we use it twice
fcn sa(kmax,kT=10){
s:=List(0, [1..99].walk().shuffle().xplode(), 0); // random path from 0 to 0
println("E(s0) %f".fmt(Es(s))); // random starter
Emin:=Es(s); // E0
foreach k in (kmax){
if(0==k%(kmax/10)) println(fmt(k,T(k,kmax,kT),Es(s)));
u:=(1).random(100); // city index 1 99
cv:=s[u] + dirs[(0).random(8)]; // city number
if(not (0<cv<100)) continue; // bogus city
if(dist(s[u],cv)>5) continue; // check true neighbour (eg 0 9)
v:=s.index(cv,1); // city index
deltae:=dE(s,u,v);
if(deltae<0 or // always move if negative
P(deltae,k,kmax,kT)>=(0.0).random(1)){
s.swap(u,v);
Emin+=deltae;
}
// (assert (= (round Emin) (round (Es s))))
}//foreach
println(fmt(kmax,T(kmax-1,kmax,kT),Es(s)));
println("E(s_final) %f".fmt(Emin));
println("Path: ",s.toString(*));
}
sa(0d1_000_000,1);
{{out}}
E(s0) 540.897080
k: 0 T: 1.0000 Es: 540.8971
k: 100,000 T: 0.9000 Es: 181.5102
k: 200,000 T: 0.8000 Es: 167.1944
k: 300,000 T: 0.7000 Es: 159.0975
k: 400,000 T: 0.6000 Es: 170.2344
k: 500,000 T: 0.5000 Es: 130.9919
k: 600,000 T: 0.4000 Es: 115.3422
k: 700,000 T: 0.3000 Es: 113.9280
k: 800,000 T: 0.2000 Es: 106.7924
k: 900,000 T: 0.1000 Es: 103.7213
k: 1,000,000 T: 0.0000 Es: 103.7213
E(s_final) 103.721349
Path: L(0,10,11,21,20,30,40,50,60,70,80,81,71,72,73,63,52,62,61,51,41,31,32,22,12,13,14,15,25,16,17,18,28,27,26,36,35,45,34,24,23,33,42,43,44,54,53,64,74,84,83,82,90,91,92,93,94,95,85,86,96,97,87,88,98,99,89,79,69,68,78,77,67,66,76,75,65,55,56,46,37,38,48,47,57,58,59,49,39,29,19,9,8,7,6,5,4,3,2,1,0)