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Notebook: Tree CSP Solver #433

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May 28, 2017
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Notebook: Tree CSP Solver #433

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89 changes: 88 additions & 1 deletion csp.ipynb
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Expand Up @@ -15,7 +15,7 @@
},
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Expand Down Expand Up @@ -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
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"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": {
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