Euler.67.ipynb

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    "#Project Euler\n",
    "###Problem 67: Maximum path sum II\n",
    "By starting at the top of the triangle below and moving to adjacent numbers on the row below, the maximum total from top to bottom is 23.\n",
    "\n",
    "       3\n",
    "      7 4\n",
    "     2 4 6\n",
    "    8 5 9 3\n",
    "\n",
    "That is, 3 + 7 + 4 + 9 = 23.\n",
    "\n",
    "Find the maximum total from top to bottom in triangle.txt (right click and 'Save Link/Target As...'), a 15K text file containing a triangle with one-hundred rows.\n",
    "\n",
    "**NOTE**: This is a much more difficult version of Problem 18. It is not possible to try every route to solve this problem, as there are 299 altogether! If you could check one trillion (1012) routes every second it would take over twenty billion years to check them all. There is an efficient algorithm to solve it. ;o)\n",
    "___"
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    "Note that this is just a single source path from top to bottom, with a imaginary node at the bottom that has all bottom numbers as parents. This is a DAG graph, which can be solved efficiently. Note that we want the longest path and not the shortest, so we just make the weights negative.\n",
    "\n",
    "First we need a efficient way to save this in a variable. Let's make a graph with nodes that are described by their row and their place in that row. That is, the top is node[1,1] = 75 and by dropping left, left, right we arrive node[3,2] = 35."
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    "raw = raw.split(\"\\n\")\n",
    "\n",
    "for i in range(len(raw)):\n",
    "    raw[i] = [-int(j) for j in raw[i].split(\" \")]\n",
    "\n",
    "graph = {(-1,0): [(0,0)] ,(len(raw),0): []} # add imaginary first and last nodes\n",
    "weights = {((-1,0),(0,0)): raw[0][0]}\n",
    "sorted_graph = [(-1,0)]\n",
    "\n",
    "for row in range(len(raw)-1):              # first rows\n",
    "    for place in range(len(raw[row])):\n",
    "        graph[(row,place)] = [(row+1,place),(row+1,place+1)]\n",
    "        sorted_graph.append((row,place))\n",
    "for place in range(len(raw)):              # final row\n",
    "    graph[(len(raw)-1,place)] = [(len(raw),0)]\n",
    "    sorted_graph.append((len(raw)-1,place))\n",
    "    \n",
    "for u in graph:\n",
    "    for v in graph[u]:\n",
    "        try:\n",
    "            weights[(u,v)] = raw[v[0]][v[1]]\n",
    "        except:\n",
    "            weights[(u,v)] = 0\n",
    "\n",
    "sorted_graph.append((len(raw),0))"
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    "Note that by linearly ordering by row, then place in an array results in the topological sort of this special type of graph. Now we continue with the shortest path algorithm:\n",
    "\n",
    "1) Initialize dist[] = {INF, INF, ….} and dist[s] = 0 where s is the source vertex.\n",
    "\n",
    "2) Do following for every vertex u in topological order.\n",
    "\n",
    "………..Do following for every adjacent vertex v of u\n",
    "\n",
    "………………if (dist[v] > dist[u] + weight(u, v))\n",
    "\n",
    "………………………dist[v] = dist[u] + weight(u, v)"
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    "# step 1\n",
    "distance = {(-1,0): 0}\n",
    "for vertex in sorted_graph[1:]:\n",
    "    distance[vertex] = 10**20\n",
    "\n",
    "# step 2\n",
    "for u in sorted_graph[:-1]:\n",
    "    for v in graph[u]:\n",
    "        if distance[v] > distance[u] + weights[(u,v)]:\n",
    "            distance[v] = distance[u] + weights[(u,v)]"
   ]
  },
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   "execution_count": 3,
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   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The answer is 7273\n"
     ]
    }
   ],
   "source": [
    "print 'The answer is', -distance[sorted_graph[-1]]"
   ]
  },
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