# Simulated annealing

```ruby
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)
```

## Output:

```
# 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
```


---

# Agent Instructions: Querying This Documentation

If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question.

Perform an HTTP GET request on the current page URL with the `ask` query parameter:

```
GET https://trizen.gitbook.io/sidef-lang/programming_tasks/s/simulated_annealing.md?ask=<question>
```

The question should be specific, self-contained, and written in natural language.
The response will contain a direct answer to the question and relevant excerpts and sources from the documentation.

Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.