From 79359461b71553663f3340a0ee30c7e87bfcf97d Mon Sep 17 00:00:00 2001
From: Antonis Maronikolakis <antmarakis@programmers.gr>
Date: Wed, 29 Mar 2017 01:16:59 +0300
Subject: [PATCH] Update csp.ipynb

---
 csp.ipynb | 89 ++++++++++++++++++++++++++++++++++++++++++++++++++++++-
 1 file changed, 88 insertions(+), 1 deletion(-)

diff --git a/csp.ipynb b/csp.ipynb
index 66c7eac6d..5255ff1d8 100644
--- a/csp.ipynb
+++ b/csp.ipynb
@@ -15,7 +15,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 3,
+   "execution_count": 1,
    "metadata": {
     "collapsed": true,
     "deletable": true,
@@ -627,6 +627,93 @@
     "solve_parameters.nassigns"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Tree CSP Solver\n",
+    "\n",
+    "The `tree_csp_solver` function (**Figure 6.11** in the book) can be used to solve problems whose constraint graph is a tree. Given a CSP, with `neighbors` forming a tree, it returns an assignement that satisfies the given constraints. The algorithm works as follows:\n",
+    "\n",
+    "First it finds the *topological sort* of the tree. This is an ordering of the tree where each variable/node comes after its parent in the tree. The function that accomplishes this is `topological_sort`, which builds the topological sort using the recursive function `build_topological`. That function is an augmented DFS, where each newly visited node of the tree is pushed on a stack. The stack in the end holds the variables topologically sorted.\n",
+    "\n",
+    "Then the algorithm makes arcs between each parent and child consistent. *Arc-consistency* between two variables, *a* and *b*, occurs when for every possible value of *a* there is an assignment in *b* that satisfies the problem's constraints. If such an assignment cannot be found, then the problematic value is removed from *a*'s possible values. This is done with the use of the function `make_arc_consistent` which takes as arguments a variable `Xj` and its parent, and makes the arc between them consistent by removing any values from the parent which do not allow for a consistent assignment in `Xj`.\n",
+    "\n",
+    "If an arc cannot be made consistent, the solver fails. If every arc is made consistent, we move to assigning values.\n",
+    "\n",
+    "First we assign a random value to the root from its domain and then we start assigning values to the rest of the variables. Since the graph is now arc-consistent, we can simply move from variable to variable picking any remaining consistent values. At the end we are left with a valid assignment. If at any point though we find a variable where no consistent value is left in its domain, the solver fails.\n",
+    "\n",
+    "The implementation of the algorithm:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "metadata": {
+    "collapsed": true
+   },
+   "outputs": [],
+   "source": [
+    "%psource tree_csp_solver"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "We will now use the above function to solve a problem. More specifically, we will solve the problem of coloring the map of Australia. At our disposal we have two colors: Red and Blue. As a reminder, this is the graph of Australia:\n",
+    "\n",
+    "`\"SA: WA NT Q NSW V; NT: WA Q; NSW: Q V; T: \"`\n",
+    "\n",
+    "Unfortunately as you can see the above is not a tree. If, though, we remove `SA`, which has arcs to `WA`, `NT`, `Q`, `NSW` and `V`, we are left with a tree (we also remove `T`, since it has no in-or-out arcs). We can now solve this using our algorithm. Let's define the map coloring problem at hand:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "metadata": {
+    "collapsed": true
+   },
+   "outputs": [],
+   "source": [
+    "australia_small = MapColoringCSP(list('RB'),\n",
+    "                           'NT: WA Q; NSW: Q V')"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "We will input `australia_small` to the `tree_csp_solver` and we will print the given assignment."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 5,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "{'WA': 'B', 'NT': 'R', 'Q': 'B', 'V': 'B', 'NSW': 'R'}\n"
+     ]
+    }
+   ],
+   "source": [
+    "assignment = tree_csp_solver(australia_small)\n",
+    "print(assignment)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "`WA`, `Q` and `V` got painted Blue, while `NT` and `NSW` got painted Red."
+   ]
+  },
   {
    "cell_type": "markdown",
    "metadata": {