CoursMagistral_8.ipynb

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    "# Table of Contents\n",
    " <p><div class=\"lev1 toc-item\"><a href=\"#ALGO1-:-Introduction-à-l'algorithmique\" data-toc-modified-id=\"ALGO1-:-Introduction-à-l'algorithmique-1\"><span class=\"toc-item-num\">1&nbsp;&nbsp;</span><a href=\"https://perso.crans.org/besson/teach/info1_algo1_2019/\" target=\"_blank\">ALGO1 : Introduction à l'algorithmique</a></a></div><div class=\"lev1 toc-item\"><a href=\"#Cours-Magistral-8\" data-toc-modified-id=\"Cours-Magistral-8-2\"><span class=\"toc-item-num\">2&nbsp;&nbsp;</span>Cours Magistral 8</a></div><div class=\"lev2 toc-item\"><a href=\"#Algorithme-de-Ford-Fulkerson\" data-toc-modified-id=\"Algorithme-de-Ford-Fulkerson-21\"><span class=\"toc-item-num\">2.1&nbsp;&nbsp;</span>Algorithme de Ford-Fulkerson</a></div><div class=\"lev3 toc-item\"><a href=\"#Etape-de-base-:-augmente-un-chemin\" data-toc-modified-id=\"Etape-de-base-:-augmente-un-chemin-211\"><span class=\"toc-item-num\">2.1.1&nbsp;&nbsp;</span>Etape de base : augmente un chemin</a></div><div class=\"lev3 toc-item\"><a href=\"#Etape-de-base-:-ajoute-un-arc-retour\" data-toc-modified-id=\"Etape-de-base-:-ajoute-un-arc-retour-212\"><span class=\"toc-item-num\">2.1.2&nbsp;&nbsp;</span>Etape de base : ajoute un arc retour</a></div><div class=\"lev3 toc-item\"><a href=\"#Calcul-du-flot-maximum\" data-toc-modified-id=\"Calcul-du-flot-maximum-213\"><span class=\"toc-item-num\">2.1.3&nbsp;&nbsp;</span>Calcul du flot maximum</a></div><div class=\"lev3 toc-item\"><a href=\"#Exemple\" data-toc-modified-id=\"Exemple-214\"><span class=\"toc-item-num\">2.1.4&nbsp;&nbsp;</span>Exemple</a></div><div class=\"lev2 toc-item\"><a href=\"#Conclusion\" data-toc-modified-id=\"Conclusion-22\"><span class=\"toc-item-num\">2.2&nbsp;&nbsp;</span>Conclusion</a></div>"
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   "source": [
    "# [ALGO1 : Introduction à l'algorithmique](https://perso.crans.org/besson/teach/info1_algo1_2019/)\n",
    "\n",
    "- [Page du cours](https://perso.crans.org/besson/teach/info1_algo1_2019/) : https://perso.crans.org/besson/teach/info1_algo1_2019/\n",
    "- Magistère d'Informatique de Rennes - ENS Rennes - Année 2019/2020\n",
    "- Intervenants :\n",
    "  + Cours : [Lilian Besson](https://perso.crans.org/besson/)\n",
    "  + Travaux dirigés : [Raphaël Truffet](http://perso.eleves.ens-rennes.fr/people/Raphael.Truffet/)\n",
    "- Références :\n",
    "  + [Open Data Structures](http://opendatastructures.org/ods-python.pdf)"
   ]
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   "source": [
    "# Cours Magistral 8\n",
    "\n",
    "- Ce cours traite du problème du flot maximal."
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   "source": [
    "---\n",
    "## Algorithme de Ford-Fulkerson\n",
    "\n",
    "- On va utiliser l'exemple donné sur la page Wikipédia :\n",
    "  <img width=\"30%\" src=\"figures/CM8_Ford-Fulkerson_exemple.svg\">\n",
    "- Cf. [la page Wikipédia](https://en.wikipedia.org/wiki/Ford-Fulkerson_algorithm)."
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   "source": [
    "### Etape de base : augmente un chemin"
   ]
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    "def augment(graph, capacity, flow, val, u, target, visit):\n",
    "    \"\"\" Find an augmenting path from u to target with value at most val.\"\"\"\n",
    "    visit[u] = True\n",
    "    if u == target:\n",
    "        return val\n",
    "    for v in graph[u]:\n",
    "        cuv = capacity[u][v]\n",
    "        if not visit[v] and cuv > flow[u][v]:  # reachable arc\n",
    "            res = min(val, cuv - flow[u][v])\n",
    "            delta = augment(graph, capacity, flow, res, v, target, visit)\n",
    "            if delta > 0:\n",
    "                flow[u][v] += delta            # augment flow\n",
    "                flow[v][u] -= delta\n",
    "                return delta\n",
    "    return 0"
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   "source": [
    "### Etape de base : ajoute un arc retour"
   ]
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    "def add_reverse_arcs(graph, capac=None):\n",
    "    \"\"\" Utility function for flow algorithms that need for every arc (u,v),\n",
    "    the existence of an (v,u) arc, by default with zero capacity.\n",
    "    graph can be in adjacency list, possibly with capacity matrix capac.\n",
    "    or graph can be in adjacency dictionary, then capac parameter is ignored.\n",
    "\n",
    "    :param capac: arc capacity matrix\n",
    "    :param graph: in listlist representation, or in listdict representation,\n",
    "    in this case capac is ignored\n",
    "    :complexity: linear\n",
    "    :returns: nothing, but graph is modified\n",
    "    \"\"\"\n",
    "    for u, _ in enumerate(graph):\n",
    "        for v in graph[u]:\n",
    "            if u not in graph[v]:\n",
    "                if type(graph[v]) is list:\n",
    "                    graph[v].append(u)\n",
    "                    if capac:\n",
    "                        capac[v][u] = 0\n",
    "                else:\n",
    "                    assert type(graph[v]) is dict\n",
    "                    graph[v][u] = 0"
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   "source": [
    "### Calcul du flot maximum"
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    "def ford_fulkerson(graph, capacity, s, t):\n",
    "    \"\"\"Maximum flow by Ford-Fulkerson\n",
    "\n",
    "    :param graph: directed graph in listlist or listdict format\n",
    "    :param capacity: in matrix format or same listdict graph\n",
    "    :param int s: source vertex\n",
    "    :param int t: target vertex\n",
    "\n",
    "    :returns: flow matrix, flow value\n",
    "    :complexity: `O(|V|*|E|*max_capacity)`\n",
    "    \"\"\"\n",
    "    add_reverse_arcs(graph, capacity)\n",
    "    n = len(graph)\n",
    "    flow = [[0] * n for _ in range(n)]\n",
    "    INF = float('inf')\n",
    "    while augment(graph, capacity, flow, INF, s, t, [False] * n) > 0:\n",
    "        pass                         # work already done in _augment\n",
    "    return (flow, sum(flow[s]))      # flow network, amount of flow"
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   "source": [
    "### Exemple\n",
    "Un exemple de graphe, comme celui utilisé [dans la page Wikipédia](https://en.wikipedia.org/wiki/Ford-Fulkerson_algorithm) :\n",
    "\n",
    "<img width=\"30%\" src=\"figures/CM8_Ford-Fulkerson_exemple.svg\">"
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    "A, B, C, D = 0, 1, 2, 3\n",
    "graph = [\n",
    "    [B, C], # A\n",
    "    [C, D], # B\n",
    "    [D], # C\n",
    "    [], # D\n",
    "]\n",
    "capacity = [\n",
    "    [0, 1000, 1000, 0], # A\n",
    "    [0, 0, 1, 1000], # B\n",
    "    [0, 0, 0, 1000], # C\n",
    "    [0, 0, 0, 0], # D\n",
    "]"
   ]
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       "([[0, 1000, 1000, 0],\n",
       "  [-1000, 0, 0, 1000],\n",
       "  [-1000, 0, 0, 1000],\n",
       "  [0, -1000, -1000, 0]],\n",
       " 2000)"
      ]
     },
     "execution_count": 18,
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    "ford_fulkerson(graph, capacity, A, D)"
   ]
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   "source": [
    "On voit que le résultat est le même que celui donné dans l'exemple sur Wikipédia :\n",
    "<img width=\"30%\" src=\"figures/CM8_Ford-Fulkerson_exemple2.svg\">"
   ]
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   "source": [
    "## Conclusion\n",
    "\n",
    "C'est bon pour aujourd'hui !"
   ]
  }
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