More work on tsp

notebooks/tsp.ipynb

@@ -2,7 +2,7 @@
  "cells": [
   {
    "cell_type": "markdown",
-   "id": "533363df",
+   "id": "e3666bcd",
    "metadata": {},
    "source": [
     "<img src=\"https://upload.wikimedia.org/wikipedia/commons/thumb/3/39/VU_logo.png/800px-VU_logo.png?20161029201021\" width=\"350\">\n",
@@ -14,7 +14,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "98ebaf0c",
+   "id": "1e74dfd9",
    "metadata": {},
    "source": [
     "## Contents\n",
@@ -27,7 +27,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "14864a1b",
+   "id": "bf3dd0e6",
    "metadata": {},
    "source": [
     "## The traveling sales person (TSP) problem\n",
@@ -48,33 +48,484 @@
   },
   {
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     }
    },
    "cell_type": "markdown",
-   "id": "14ae7875",
+   "id": "dad0547d",
    "metadata": {},
    "source": [
     "<div>\n",
-    "<img src=\"attachment:g18228.png\"  align=\"left\" width=\"450\"/>\n",
+    "<img src=\"attachment:g22650.png\"  align=\"left\" width=\"450\"/>\n",
     "</div>"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "aca8b2c4",
+   "id": "c303dddf",
    "metadata": {},
    "source": [
     "### Branch and bound algorithm\n",
     "\n",
+    "We consider a [branch and bound strategy](https://en.wikipedia.org/wiki/Branch_and_bound) to solve the ASP problem. The first ingredient of this strategy is to enumerate all potential solutions of the problem using a search tree. The tree is defined as follows:\n",
     "\n",
-    "\n"
+    "- The root is the initial city\n",
+    "- The children of a node are the neighbor cities not visited so-far\n",
+    "- **We sort the children using the nearest city first heuristic**\n",
+    "- A node will be a leaf of the tree when all neighbor cities are already visited\n",
+    "\n",
+    "\n",
+    "The nearest city first heuristic is a way to speed up the search as we increase the possibility of finding the solution in the first paths enumerated in the tree."
+   ]
+  },
+  {
+   "attachments": {
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truncated
+    }
+   },
+   "cell_type": "markdown",
+   "id": "9797bb92",
+   "metadata": {},
+   "source": [
+    "<div>\n",
+    "<img src=\"attachment:g25091.png\"  align=\"left\" width=\"525\"/>\n",
+    "</div>"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "ecf89636",
+   "metadata": {},
+   "source": [
+    "### Pruning the search tree\n",
+    "\n",
+    "We look for a solution by going over the branches of the search tree via first-depth search. Each time we hit a lead in the tree we have a route between all cities and this a possible solution of the TSP problem (called a partial solution). In this process, we keep track of which is the shortest partial solution found so far. Note that we don't need to completely traverse all branches. In some cases, the distance at an intermediate point in a route will be already equal or greater than the best distance found so-far. In this case, we don't need to continue moving within this branch and we can jump to the next one. This is the \"bound\" criterion in our branch and bound strategy, which is also referred to as \"pruning\" the search tree.\n",
+    "\n",
+    "In the example below, the first route has distance 6. At an intermediate point of the 2nd route we have already covered a distance equal to 6 which guarantees that the solution will not be in this part of the tree.\n"
+   ]
+  },
+  {
+   "attachments": {
+    "g23181.png": {
+     "image/png": 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truncated
+    }
+   },
+   "cell_type": "markdown",
+   "id": "f18d02d6",
+   "metadata": {},
+   "source": [
+    "<div>\n",
+    "<img src=\"attachment:g23181.png\"  align=\"left\" width=\"300\"/>\n",
+    "</div>"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "239b7362",
+   "metadata": {},
+   "source": [
+    "Thus, there are only 2 more complete routes that we need to visit. Using this strategy we have significantly reduced the routes visited: from 6 possible routes to 3 visited routes."
+   ]
+  },
+  {
+   "attachments": {
+    "g23549.png": {
+     "image/png": 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truncated
+    }
+   },
+   "cell_type": "markdown",
+   "id": "0fbe1ee0",
+   "metadata": {},
+   "source": [
+    "<div>\n",
+    "<img src=\"attachment:g23549.png\"  align=\"left\" width=\"300\"/>\n",
+    "</div>"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "4fe017a4",
+   "metadata": {},
+   "source": [
+    "## Serial implementation"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "40027c85",
+   "metadata": {},
+   "source": [
+    "### Input distance"
    ]
   },
   {
    "cell_type": "code",
    "execution_count": 1,
+   "id": "2eeecdd6",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "4×4 Matrix{Int64}:\n",
+       " 0  2  3  2\n",
+       " 2  0  4  1\n",
+       " 3  4  0  3\n",
+       " 2  1  3  0"
+      ]
+     },
+     "execution_count": 1,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "C = [\n",
+    "    0 2 3 2\n",
+    "    2 0 4 1\n",
+    "    3 4 0 3\n",
+    "    2 1 3 0    \n",
+    "]"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 24,
+   "id": "03cac0a1",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "4-element Vector{Vector{Tuple{Int64, Int64}}}:\n",
+       " [(1, 0), (2, 2), (3, 3), (4, 4)]\n",
+       " [(2, 0), (4, 1), (1, 2), (3, 3)]\n",
+       " [(3, 0), (1, 3), (2, 3), (4, 10)]\n",
+       " [(4, 0), (2, 1), (1, 4), (3, 10)]"
+      ]
+     },
+     "execution_count": 24,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "C = [\n",
+    "    [(1,0), (2,2), (3,3), (4,4)],\n",
+    "    [(2,0), (4,1), (1,2), (3,3)],\n",
+    "    [(3,0), (1,3), (2,3), (4,10)], \n",
+    "    [(4,0), (2,1), (1,4), (3,10)]\n",
+    "]"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "f2b70f85",
+   "metadata": {},
+   "source": [
+    "### Nearest-city first heuristic"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "id": "a50706bc",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "sort_neighbors (generic function with 1 method)"
+      ]
+     },
+     "execution_count": 2,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "function sort_neighbors(C)\n",
+    "    n = size(C,1)\n",
+    "    map(1:n) do i\n",
+    "        Ci = C[i,:]\n",
+    "        cities = sortperm(Ci)\n",
+    "        distances = Ci[cities]\n",
+    "        collect(zip(cities,distances))[2:end]\n",
+    "    end\n",
+    "end"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "6dd0288e",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "connections = sort_neighbors(C)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "4f8cf018",
+   "metadata": {},
+   "source": [
+    "### Implementation"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 25,
+   "id": "d00beaba",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "(path, distance, min_distance) = ([1, 2, 4, 3], 13, 13)\n",
+      "(path, distance, min_distance) = ([1, 3, 2, 4], 7, 7)\n"
+     ]
+    },
+    {
+     "data": {
+      "text/plain": [
+       "7"
+      ]
+     },
+     "execution_count": 25,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "function tsp_serial(C,city)\n",
+    "    #connections = sort_neighbors(C)\n",
+    "    num_cities = length(connections)\n",
+    "    path=zeros(Int,num_cities)\n",
+    "    hops = 1\n",
+    "    path[hops] = city\n",
+    "    distance = 0\n",
+    "    min_distance = typemax(Int)\n",
+    "    tsp_serial_recursive!(connections,hops,path,distance,min_distance)\n",
+    "end\n",
+    "function tsp_serial_recursive!(connections,hops,path,distance,min_distance)\n",
+    "    num_cities = length(connections)\n",
+    "    if hops == num_cities\n",
+    "        min_distance = min(distance,min_distance)\n",
+    "        @show path, distance, min_distance\n",
+    "        return min_distance\n",
+    "    else\n",
+    "        next_hops = hops + 1\n",
+    "        current_city = path[hops]\n",
+    "        for (next_city,distance_increment) in connections[current_city]\n",
+    "            if !visited(next_city,hops,path)\n",
+    "                next_distance = distance + distance_increment\n",
+    "                path[next_hops] = next_city\n",
+    "                if next_distance < min_distance\n",
+    "                    min_distance = tsp_serial_recursive!(connections,next_hops,path,next_distance,min_distance)\n",
+    "                end\n",
+    "            end\n",
+    "        end        \n",
+    "    end\n",
+    "    return min_distance\n",
+    "end\n",
+    "function visited(city,hops,path)\n",
+    "    for i = 1:hops\n",
+    "        if path[i] == city\n",
+    "            return true\n",
+    "        end\n",
+    "    end\n",
+    "    return false\n",
+    "end\n",
+    "\n",
+    "city = 1\n",
+    "tsp_serial(C,city)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "505e00f7",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "function tsp_serial(C,city)\n",
+    "    connections = sort_neighbors(C)\n",
+    "    num_cities = length(connections)\n",
+    "    current_route = (;path=zeros(Int,num_cities),distance=Ref(0))\n",
+    "    min_route = (;path=zeros(Int,num_cities),distance=Ref(typemax(Int)))\n",
+    "    hops = 1\n",
+    "    current_route.path[hops] = city\n",
+    "    tsp_serial_recursive!(connections,hops,current_route,min_route)\n",
+    "    min_route\n",
+    "end\n",
+    "function tsp_serial_recursive(connections,hops,current_route,min_route)\n",
+    "    num_cities = length(connections)\n",
+    "    if hops == num_cities\n",
+    "        if current_route.distance < min_route.distance\n",
+    "            min_route.path .= current_route.path\n",
+    "            min_route.distance[] = current_route.distance \n",
+    "            return\n",
+    "        end\n",
+    "    else\n",
+    "        next_hops = hops + 1\n",
+    "        current_city = current_route.path[hops]\n",
+    "        for (next_city,distance_increment) in connections[current_city]\n",
+    "            if !visited(next_city,hops,current_route.path)\n",
+    "                \n",
+    "                next_distance = current_distance + distance_increment\n",
+    "                \n",
+    "                if next_distance < min_distance\n",
+    "                    return tsp_serial_recursive(connections,hops+1,current_route,min_route)\n",
+    "                end\n",
+    "            end\n",
+    "        end        \n",
+    "    end\n",
+    "    min_distance\n",
+    "end\n",
+    "\n",
+    "function visited(city,hops,path)\n",
+    "    for i = 1:hops\n",
+    "        if path[i] == city\n",
+    "            return true\n",
+    "        end\n",
+    "    end\n",
+    "    return false\n",
+    "end\n",
+    "\n",
+    "city = 1\n",
+    "tsp_serial(C,city)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "94d8193f",
+   "metadata": {},
+   "source": [
+    "### Generating some test data\n",
+    "\n",
+    "The following function generates a random symmetric distance matrix, we have used a large. Note that the TSP problem at least requires that the underlying graph is connected (otherwise there is no route that visits all nodes). The next function does not guarantee this. In any case, we can still used a non-connected graph as input. The outcome of the search will be that there is no possible route."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "88fc4499",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "function rand_symmetric_distance_table(n)\n",
+    "  threshold = 0.2\n",
+    "  mincost = 3\n",
+    "  maxcost = 10\n",
+    "  infinity = 10000*maxcost\n",
+    "  C = fill(infinity,n,n)\n",
+    "  for j in 1:n\n",
+    "    for i in 1:j\n",
+    "      if rand() > threshold\n",
+    "        C[i,j] = rand(mincost:maxcost)\n",
+    "        C[j,i] = C[i,j]\n",
+    "      end\n",
+    "    end\n",
+    "    C[j,j] = 0\n",
+    "  end\n",
+    "  C,infinity\n",
+    "end"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "d649cf80",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "n = 10000\n",
+    "C, inf = rand_symmetric_distance_table(n)\n",
+    "city = 1\n",
+    "(;path,distance) = tsp_serial(C,city)\n",
+    "if  distance >= inf\n",
+    "    println(\"The input graph is nonconnected\")\n",
+    "else\n",
+    "    (;path,distance)\n",
+    "end"
+   ]
+  },
+  {
+   "attachments": {
+    "g23360.png": {
+     "image/png": 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truncated
+    }
+   },
+   "cell_type": "markdown",
+   "id": "b0ea4b99",
+   "metadata": {},
+   "source": [
+    "<div>\n",
+    "<img src=\"attachment:g23360.png\"  align=\"left\" width=\"300\"/>\n",
+    "</div>"
+   ]
+  },
+  {
+   "attachments": {
+    "g22999.png": {
+     "image/png": 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truncated
+    }
+   },
+   "cell_type": "markdown",
+   "id": "e734b6dc",
+   "metadata": {},
+   "source": [
+    "<div>\n",
+    "<img src=\"attachment:g22999.png\"  align=\"left\" width=\"300\"/>\n",
+    "</div>"
+   ]
+  },
+  {
+   "attachments": {
+    "g22822.png": {
+     "image/png": "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 truncated
+    }
+   },
+   "cell_type": "markdown",
+   "id": "c8f723c4",
+   "metadata": {},
+   "source": [
+    "<div>\n",
+    "<img src=\"attachment:g22822.png\"  align=\"left\" width=\"300\"/>\n",
+    "</div>"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "6f8100ad",
+   "metadata": {},
+   "source": []
+  },
+  {
+   "cell_type": "markdown",
+   "id": "bef63110",
+   "metadata": {},
+   "source": []
+  },
+  {
+   "cell_type": "markdown",
+   "id": "9f66ca90",
+   "metadata": {},
+   "source": []
+  },
+  {
+   "cell_type": "markdown",
+   "id": "88b90b22",
+   "metadata": {},
+   "source": []
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
    "id": "8850d90e",
    "metadata": {},
    "outputs": [],
@@ -84,25 +535,10 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 2,
+   "execution_count": null,
    "id": "5d4935ee",
    "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "text/plain": [
-       "4-element Vector{Int64}:\n",
-       " 2\n",
-       " 3\n",
-       " 4\n",
-       " 5"
-      ]
-     },
-     "execution_count": 2,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
+   "outputs": [],
    "source": [
     "if procs() == workers()\n",
     "    addprocs(4)\n",
@@ -111,7 +547,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 3,
+   "execution_count": null,
    "id": "4a2756ae",
    "metadata": {},
    "outputs": [],
@@ -128,92 +564,10 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 4,
-   "id": "39e9e667",
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "text/plain": [
-       "tsp_serial_impl (generic function with 1 method)"
-      ]
-     },
-     "execution_count": 4,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "function tsp_serial_impl(connections,hops,path,current_distance,min_distance)\n",
-    "    num_cities = length(connections)\n",
-    "    if hops == num_cities\n",
-    "        if current_distance < min_distance\n",
-    "            return current_distance\n",
-    "        end\n",
-    "    else\n",
-    "        current_city = path[hops]\n",
-    "        next_hops = hops + 1\n",
-    "        for (next_city,distance_increment) in connections[current_city]\n",
-    "            if !visited(next_city,hops,path)\n",
-    "                path[next_hops] = next_city\n",
-    "                next_distance = current_distance + distance_increment\n",
-    "                if next_distance < min_distance\n",
-    "                    return tsp_serial_impl(connections,next_hops,path,next_distance,min_distance)\n",
-    "                end\n",
-    "            end\n",
-    "        end        \n",
-    "    end\n",
-    "    min_distance\n",
-    "end"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 5,
-   "id": "83b58881",
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "text/plain": [
-       "tsp_serial (generic function with 1 method)"
-      ]
-     },
-     "execution_count": 5,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "function tsp_serial(connections,city)\n",
-    "    num_cities = length(connections)\n",
-    "    path=zeros(Int,num_cities)\n",
-    "    hops = 1\n",
-    "    path[hops] = city\n",
-    "    current_distance = 0\n",
-    "    min_distance = typemax(Int)\n",
-    "    min_distance = tsp_serial_impl(connections,hops,path,current_distance,min_distance)\n",
-    "    (;path=path,distance=min_distance)\n",
-    "end"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 6,
+   "execution_count": null,
    "id": "78095098",
    "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "text/plain": [
-       "(path = [1, 4, 5, 2, 3, 6], distance = 222)"
-      ]
-     },
-     "execution_count": 6,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
+   "outputs": [],
    "source": [
     "connections = [\n",
     "    [(1,0),(4,39),(5,76), (6,78),(3,94),(2,97)],\n",
@@ -229,21 +583,10 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 10,
+   "execution_count": null,
    "id": "03f0dd8e",
    "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "text/plain": [
-       "(path = [1, 4, 5, 2, 3, 6], distance = 222)"
-      ]
-     },
-     "execution_count": 10,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
+   "outputs": [],
    "source": [
     "@everywhere function tsp_dist_impl(connections,hops,path,current_distance,min_distance,max_hops,jobs_chnl,ftr_result)\n",
     "    num_cities = length(connections)\n",