"vscode:/vscode.git/clone" n'existait pas sur "3778544dc9afa87e420817f138ac9a3d45872808"
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"# CONSTRAINT SATISFACTION PROBLEMS\n",
"This IPy notebook acts as supporting material for topics covered in **Chapter 6 Constraint Satisfaction Problems** of the book* Artificial Intelligence: A Modern Approach*. We make use of the implementations in **csp.py** module. Even though this notebook includes a brief summary of the main topics, familiarity with the material present in the book is expected. We will look at some visualizations and solve some of the CSP problems described in the book. Let us import everything from the csp module to get started."
{
"cell_type": "code",
"from notebook import psource, pseudocode, plot_NQueens\n",
"%matplotlib inline\n",
"# Hide warnings in the matplotlib sections\n",
"import warnings\n",
"warnings.filterwarnings(\"ignore\")"
"* Overview\n",
"* Graph Coloring\n",
"* N-Queens\n",
"* Backtracking Search\n",
"* Tree CSP Solver\n",
"* Graph Coloring Visualization\n",
"* N-Queens Visualization"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## OVERVIEW\n",
"\n",
"CSPs are a special kind of search problems. Here we don't treat the space as a black box but the state has a particular form and we use that to our advantage to tweak our algorithms to be more suited to the problems. A CSP State is defined by a set of variables which can take values from corresponding domains. These variables can take only certain values in their domains to satisfy the constraints. A set of assignments which satisfies all constraints passes the goal test. Let us start by exploring the CSP class which we will use to model our CSPs. You can keep the popup open and read the main page to get a better idea of the code."
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"\n",
"<div class=\"highlight\"><pre><span></span><span class=\"k\">class</span> <span class=\"nc\">CSP</span><span class=\"p\">(</span><span class=\"n\">search</span><span class=\"o\">.</span><span class=\"n\">Problem</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""This class describes finite-domain Constraint Satisfaction Problems.</span>\n",
"<span class=\"sd\"> A CSP is specified by the following inputs:</span>\n",
"<span class=\"sd\"> variables A list of variables; each is atomic (e.g. int or string).</span>\n",
"<span class=\"sd\"> domains A dict of {var:[possible_value, ...]} entries.</span>\n",
"<span class=\"sd\"> neighbors A dict of {var:[var,...]} that for each variable lists</span>\n",
"<span class=\"sd\"> the other variables that participate in constraints.</span>\n",
"<span class=\"sd\"> constraints A function f(A, a, B, b) that returns true if neighbors</span>\n",
"<span class=\"sd\"> A, B satisfy the constraint when they have values A=a, B=b</span>\n",
"\n",
"<span class=\"sd\"> In the textbook and in most mathematical definitions, the</span>\n",
"<span class=\"sd\"> constraints are specified as explicit pairs of allowable values,</span>\n",
"<span class=\"sd\"> but the formulation here is easier to express and more compact for</span>\n",
"<span class=\"sd\"> most cases. (For example, the n-Queens problem can be represented</span>\n",
"<span class=\"sd\"> in O(n) space using this notation, instead of O(N^4) for the</span>\n",
"<span class=\"sd\"> explicit representation.) In terms of describing the CSP as a</span>\n",
"<span class=\"sd\"> problem, that's all there is.</span>\n",
"\n",
"<span class=\"sd\"> However, the class also supports data structures and methods that help you</span>\n",
"<span class=\"sd\"> solve CSPs by calling a search function on the CSP. Methods and slots are</span>\n",
"<span class=\"sd\"> as follows, where the argument 'a' represents an assignment, which is a</span>\n",
"<span class=\"sd\"> dict of {var:val} entries:</span>\n",
"<span class=\"sd\"> assign(var, val, a) Assign a[var] = val; do other bookkeeping</span>\n",
"<span class=\"sd\"> unassign(var, a) Do del a[var], plus other bookkeeping</span>\n",
"<span class=\"sd\"> nconflicts(var, val, a) Return the number of other variables that</span>\n",
"<span class=\"sd\"> conflict with var=val</span>\n",
"<span class=\"sd\"> curr_domains[var] Slot: remaining consistent values for var</span>\n",
"<span class=\"sd\"> Used by constraint propagation routines.</span>\n",
"<span class=\"sd\"> The following methods are used only by graph_search and tree_search:</span>\n",
"<span class=\"sd\"> actions(state) Return a list of actions</span>\n",
"<span class=\"sd\"> result(state, action) Return a successor of state</span>\n",
"<span class=\"sd\"> goal_test(state) Return true if all constraints satisfied</span>\n",
"<span class=\"sd\"> The following are just for debugging purposes:</span>\n",
"<span class=\"sd\"> nassigns Slot: tracks the number of assignments made</span>\n",
"<span class=\"sd\"> display(a) Print a human-readable representation</span>\n",
"<span class=\"sd\"> """</span>\n",
"\n",
" <span class=\"k\">def</span> <span class=\"fm\">__init__</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"p\">,</span> <span class=\"n\">variables</span><span class=\"p\">,</span> <span class=\"n\">domains</span><span class=\"p\">,</span> <span class=\"n\">neighbors</span><span class=\"p\">,</span> <span class=\"n\">constraints</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""Construct a CSP problem. If variables is empty, it becomes domains.keys()."""</span>\n",
" <span class=\"n\">variables</span> <span class=\"o\">=</span> <span class=\"n\">variables</span> <span class=\"ow\">or</span> <span class=\"nb\">list</span><span class=\"p\">(</span><span class=\"n\">domains</span><span class=\"o\">.</span><span class=\"n\">keys</span><span class=\"p\">())</span>\n",
"\n",
" <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">variables</span> <span class=\"o\">=</span> <span class=\"n\">variables</span>\n",
" <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">domains</span> <span class=\"o\">=</span> <span class=\"n\">domains</span>\n",
" <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">neighbors</span> <span class=\"o\">=</span> <span class=\"n\">neighbors</span>\n",
" <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">constraints</span> <span class=\"o\">=</span> <span class=\"n\">constraints</span>\n",
" <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">initial</span> <span class=\"o\">=</span> <span class=\"p\">()</span>\n",
" <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">curr_domains</span> <span class=\"o\">=</span> <span class=\"bp\">None</span>\n",
" <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">nassigns</span> <span class=\"o\">=</span> <span class=\"mi\">0</span>\n",
"\n",
" <span class=\"k\">def</span> <span class=\"nf\">assign</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"p\">,</span> <span class=\"n\">var</span><span class=\"p\">,</span> <span class=\"n\">val</span><span class=\"p\">,</span> <span class=\"n\">assignment</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""Add {var: val} to assignment; Discard the old value if any."""</span>\n",
" <span class=\"n\">assignment</span><span class=\"p\">[</span><span class=\"n\">var</span><span class=\"p\">]</span> <span class=\"o\">=</span> <span class=\"n\">val</span>\n",
" <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">nassigns</span> <span class=\"o\">+=</span> <span class=\"mi\">1</span>\n",
"\n",
" <span class=\"k\">def</span> <span class=\"nf\">unassign</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"p\">,</span> <span class=\"n\">var</span><span class=\"p\">,</span> <span class=\"n\">assignment</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""Remove {var: val} from assignment.</span>\n",
"<span class=\"sd\"> DO NOT call this if you are changing a variable to a new value;</span>\n",
"<span class=\"sd\"> just call assign for that."""</span>\n",
" <span class=\"k\">if</span> <span class=\"n\">var</span> <span class=\"ow\">in</span> <span class=\"n\">assignment</span><span class=\"p\">:</span>\n",
" <span class=\"k\">del</span> <span class=\"n\">assignment</span><span class=\"p\">[</span><span class=\"n\">var</span><span class=\"p\">]</span>\n",
"\n",
" <span class=\"k\">def</span> <span class=\"nf\">nconflicts</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"p\">,</span> <span class=\"n\">var</span><span class=\"p\">,</span> <span class=\"n\">val</span><span class=\"p\">,</span> <span class=\"n\">assignment</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""Return the number of conflicts var=val has with other variables."""</span>\n",
" <span class=\"c1\"># Subclasses may implement this more efficiently</span>\n",
" <span class=\"k\">def</span> <span class=\"nf\">conflict</span><span class=\"p\">(</span><span class=\"n\">var2</span><span class=\"p\">):</span>\n",
" <span class=\"k\">return</span> <span class=\"p\">(</span><span class=\"n\">var2</span> <span class=\"ow\">in</span> <span class=\"n\">assignment</span> <span class=\"ow\">and</span>\n",
" <span class=\"ow\">not</span> <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">constraints</span><span class=\"p\">(</span><span class=\"n\">var</span><span class=\"p\">,</span> <span class=\"n\">val</span><span class=\"p\">,</span> <span class=\"n\">var2</span><span class=\"p\">,</span> <span class=\"n\">assignment</span><span class=\"p\">[</span><span class=\"n\">var2</span><span class=\"p\">]))</span>\n",
" <span class=\"k\">return</span> <span class=\"n\">count</span><span class=\"p\">(</span><span class=\"n\">conflict</span><span class=\"p\">(</span><span class=\"n\">v</span><span class=\"p\">)</span> <span class=\"k\">for</span> <span class=\"n\">v</span> <span class=\"ow\">in</span> <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">neighbors</span><span class=\"p\">[</span><span class=\"n\">var</span><span class=\"p\">])</span>\n",
"\n",
" <span class=\"k\">def</span> <span class=\"nf\">display</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"p\">,</span> <span class=\"n\">assignment</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""Show a human-readable representation of the CSP."""</span>\n",
" <span class=\"c1\"># Subclasses can print in a prettier way, or display with a GUI</span>\n",
" <span class=\"k\">print</span><span class=\"p\">(</span><span class=\"s1\">'CSP:'</span><span class=\"p\">,</span> <span class=\"bp\">self</span><span class=\"p\">,</span> <span class=\"s1\">'with assignment:'</span><span class=\"p\">,</span> <span class=\"n\">assignment</span><span class=\"p\">)</span>\n",
"\n",
" <span class=\"c1\"># These methods are for the tree and graph-search interface:</span>\n",
"\n",
" <span class=\"k\">def</span> <span class=\"nf\">actions</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"p\">,</span> <span class=\"n\">state</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""Return a list of applicable actions: nonconflicting</span>\n",
"<span class=\"sd\"> assignments to an unassigned variable."""</span>\n",
" <span class=\"k\">if</span> <span class=\"nb\">len</span><span class=\"p\">(</span><span class=\"n\">state</span><span class=\"p\">)</span> <span class=\"o\">==</span> <span class=\"nb\">len</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">variables</span><span class=\"p\">):</span>\n",
" <span class=\"k\">return</span> <span class=\"p\">[]</span>\n",
" <span class=\"k\">else</span><span class=\"p\">:</span>\n",
" <span class=\"n\">assignment</span> <span class=\"o\">=</span> <span class=\"nb\">dict</span><span class=\"p\">(</span><span class=\"n\">state</span><span class=\"p\">)</span>\n",
" <span class=\"n\">var</span> <span class=\"o\">=</span> <span class=\"n\">first</span><span class=\"p\">([</span><span class=\"n\">v</span> <span class=\"k\">for</span> <span class=\"n\">v</span> <span class=\"ow\">in</span> <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">variables</span> <span class=\"k\">if</span> <span class=\"n\">v</span> <span class=\"ow\">not</span> <span class=\"ow\">in</span> <span class=\"n\">assignment</span><span class=\"p\">])</span>\n",
" <span class=\"k\">return</span> <span class=\"p\">[(</span><span class=\"n\">var</span><span class=\"p\">,</span> <span class=\"n\">val</span><span class=\"p\">)</span> <span class=\"k\">for</span> <span class=\"n\">val</span> <span class=\"ow\">in</span> <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">domains</span><span class=\"p\">[</span><span class=\"n\">var</span><span class=\"p\">]</span>\n",
" <span class=\"k\">if</span> <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">nconflicts</span><span class=\"p\">(</span><span class=\"n\">var</span><span class=\"p\">,</span> <span class=\"n\">val</span><span class=\"p\">,</span> <span class=\"n\">assignment</span><span class=\"p\">)</span> <span class=\"o\">==</span> <span class=\"mi\">0</span><span class=\"p\">]</span>\n",
"\n",
" <span class=\"k\">def</span> <span class=\"nf\">result</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"p\">,</span> <span class=\"n\">state</span><span class=\"p\">,</span> <span class=\"n\">action</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""Perform an action and return the new state."""</span>\n",
" <span class=\"p\">(</span><span class=\"n\">var</span><span class=\"p\">,</span> <span class=\"n\">val</span><span class=\"p\">)</span> <span class=\"o\">=</span> <span class=\"n\">action</span>\n",
" <span class=\"k\">return</span> <span class=\"n\">state</span> <span class=\"o\">+</span> <span class=\"p\">((</span><span class=\"n\">var</span><span class=\"p\">,</span> <span class=\"n\">val</span><span class=\"p\">),)</span>\n",
"\n",
" <span class=\"k\">def</span> <span class=\"nf\">goal_test</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"p\">,</span> <span class=\"n\">state</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""The goal is to assign all variables, with all constraints satisfied."""</span>\n",
" <span class=\"n\">assignment</span> <span class=\"o\">=</span> <span class=\"nb\">dict</span><span class=\"p\">(</span><span class=\"n\">state</span><span class=\"p\">)</span>\n",
" <span class=\"k\">return</span> <span class=\"p\">(</span><span class=\"nb\">len</span><span class=\"p\">(</span><span class=\"n\">assignment</span><span class=\"p\">)</span> <span class=\"o\">==</span> <span class=\"nb\">len</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">variables</span><span class=\"p\">)</span>\n",
" <span class=\"ow\">and</span> <span class=\"nb\">all</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">nconflicts</span><span class=\"p\">(</span><span class=\"n\">variables</span><span class=\"p\">,</span> <span class=\"n\">assignment</span><span class=\"p\">[</span><span class=\"n\">variables</span><span class=\"p\">],</span> <span class=\"n\">assignment</span><span class=\"p\">)</span> <span class=\"o\">==</span> <span class=\"mi\">0</span>\n",
" <span class=\"k\">for</span> <span class=\"n\">variables</span> <span class=\"ow\">in</span> <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">variables</span><span class=\"p\">))</span>\n",
"\n",
" <span class=\"c1\"># These are for constraint propagation</span>\n",
"\n",
" <span class=\"k\">def</span> <span class=\"nf\">support_pruning</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""Make sure we can prune values from domains. (We want to pay</span>\n",
"<span class=\"sd\"> for this only if we use it.)"""</span>\n",
" <span class=\"k\">if</span> <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">curr_domains</span> <span class=\"ow\">is</span> <span class=\"bp\">None</span><span class=\"p\">:</span>\n",
" <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">curr_domains</span> <span class=\"o\">=</span> <span class=\"p\">{</span><span class=\"n\">v</span><span class=\"p\">:</span> <span class=\"nb\">list</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">domains</span><span class=\"p\">[</span><span class=\"n\">v</span><span class=\"p\">])</span> <span class=\"k\">for</span> <span class=\"n\">v</span> <span class=\"ow\">in</span> <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">variables</span><span class=\"p\">}</span>\n",
"\n",
" <span class=\"k\">def</span> <span class=\"nf\">suppose</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"p\">,</span> <span class=\"n\">var</span><span class=\"p\">,</span> <span class=\"n\">value</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""Start accumulating inferences from assuming var=value."""</span>\n",
" <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">support_pruning</span><span class=\"p\">()</span>\n",
" <span class=\"n\">removals</span> <span class=\"o\">=</span> <span class=\"p\">[(</span><span class=\"n\">var</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">)</span> <span class=\"k\">for</span> <span class=\"n\">a</span> <span class=\"ow\">in</span> <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">curr_domains</span><span class=\"p\">[</span><span class=\"n\">var</span><span class=\"p\">]</span> <span class=\"k\">if</span> <span class=\"n\">a</span> <span class=\"o\">!=</span> <span class=\"n\">value</span><span class=\"p\">]</span>\n",
" <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">curr_domains</span><span class=\"p\">[</span><span class=\"n\">var</span><span class=\"p\">]</span> <span class=\"o\">=</span> <span class=\"p\">[</span><span class=\"n\">value</span><span class=\"p\">]</span>\n",
" <span class=\"k\">return</span> <span class=\"n\">removals</span>\n",
"\n",
" <span class=\"k\">def</span> <span class=\"nf\">prune</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"p\">,</span> <span class=\"n\">var</span><span class=\"p\">,</span> <span class=\"n\">value</span><span class=\"p\">,</span> <span class=\"n\">removals</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""Rule out var=value."""</span>\n",
" <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">curr_domains</span><span class=\"p\">[</span><span class=\"n\">var</span><span class=\"p\">]</span><span class=\"o\">.</span><span class=\"n\">remove</span><span class=\"p\">(</span><span class=\"n\">value</span><span class=\"p\">)</span>\n",
" <span class=\"k\">if</span> <span class=\"n\">removals</span> <span class=\"ow\">is</span> <span class=\"ow\">not</span> <span class=\"bp\">None</span><span class=\"p\">:</span>\n",
" <span class=\"n\">removals</span><span class=\"o\">.</span><span class=\"n\">append</span><span class=\"p\">((</span><span class=\"n\">var</span><span class=\"p\">,</span> <span class=\"n\">value</span><span class=\"p\">))</span>\n",
"\n",
" <span class=\"k\">def</span> <span class=\"nf\">choices</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"p\">,</span> <span class=\"n\">var</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""Return all values for var that aren't currently ruled out."""</span>\n",
" <span class=\"k\">return</span> <span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">curr_domains</span> <span class=\"ow\">or</span> <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">domains</span><span class=\"p\">)[</span><span class=\"n\">var</span><span class=\"p\">]</span>\n",
"\n",
" <span class=\"k\">def</span> <span class=\"nf\">infer_assignment</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""Return the partial assignment implied by the current inferences."""</span>\n",
" <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">support_pruning</span><span class=\"p\">()</span>\n",
" <span class=\"k\">return</span> <span class=\"p\">{</span><span class=\"n\">v</span><span class=\"p\">:</span> <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">curr_domains</span><span class=\"p\">[</span><span class=\"n\">v</span><span class=\"p\">][</span><span class=\"mi\">0</span><span class=\"p\">]</span>\n",
" <span class=\"k\">for</span> <span class=\"n\">v</span> <span class=\"ow\">in</span> <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">variables</span> <span class=\"k\">if</span> <span class=\"mi\">1</span> <span class=\"o\">==</span> <span class=\"nb\">len</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">curr_domains</span><span class=\"p\">[</span><span class=\"n\">v</span><span class=\"p\">])}</span>\n",
"\n",
" <span class=\"k\">def</span> <span class=\"nf\">restore</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"p\">,</span> <span class=\"n\">removals</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""Undo a supposition and all inferences from it."""</span>\n",
" <span class=\"k\">for</span> <span class=\"n\">B</span><span class=\"p\">,</span> <span class=\"n\">b</span> <span class=\"ow\">in</span> <span class=\"n\">removals</span><span class=\"p\">:</span>\n",
" <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">curr_domains</span><span class=\"p\">[</span><span class=\"n\">B</span><span class=\"p\">]</span><span class=\"o\">.</span><span class=\"n\">append</span><span class=\"p\">(</span><span class=\"n\">b</span><span class=\"p\">)</span>\n",
"\n",
" <span class=\"c1\"># This is for min_conflicts search</span>\n",
"\n",
" <span class=\"k\">def</span> <span class=\"nf\">conflicted_vars</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"p\">,</span> <span class=\"n\">current</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""Return a list of variables in current assignment that are in conflict"""</span>\n",
" <span class=\"k\">return</span> <span class=\"p\">[</span><span class=\"n\">var</span> <span class=\"k\">for</span> <span class=\"n\">var</span> <span class=\"ow\">in</span> <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">variables</span>\n",
" <span class=\"k\">if</span> <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">nconflicts</span><span class=\"p\">(</span><span class=\"n\">var</span><span class=\"p\">,</span> <span class=\"n\">current</span><span class=\"p\">[</span><span class=\"n\">var</span><span class=\"p\">],</span> <span class=\"n\">current</span><span class=\"p\">)</span> <span class=\"o\">></span> <span class=\"mi\">0</span><span class=\"p\">]</span>\n",
"</pre></div>\n",
"</body>\n",
"</html>\n"
],
]
},
{
"cell_type": "markdown",
"The __ _ _init_ _ __ method parameters specify the CSP. Variables can be passed as a list of strings or integers. Domains are passed as dict (dictionary datatpye) where \"key\" specifies the variables and \"value\" specifies the domains. The variables are passed as an empty list. Variables are extracted from the keys of the domain dictionary. Neighbor is a dict of variables that essentially describes the constraint graph. Here each variable key has a list of its values which are the variables that are constraint along with it. The constraint parameter should be a function **f(A, a, B, b**) that **returns true** if neighbors A, B **satisfy the constraint** when they have values **A=a, B=b**. We have additional parameters like nassings which is incremented each time an assignment is made when calling the assign method. You can read more about the methods and parameters in the class doc string. We will talk more about them as we encounter their use. Let us jump to an example."
{
"cell_type": "markdown",
"We use the graph coloring problem as our running example for demonstrating the different algorithms in the **csp module**. The idea of map coloring problem is that the adjacent nodes (those connected by edges) should not have the same color throughout the graph. The graph can be colored using a fixed number of colors. Here each node is a variable and the values are the colors that can be assigned to them. Given that the domain will be the same for all our nodes we use a custom dict defined by the **UniversalDict** class. The **UniversalDict** Class takes in a parameter and returns it as a value for all the keys of the dict. It is very similar to **defaultdict** in Python except that it does not support item assignment."
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
]
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"execution_count": 3,
"metadata": {},
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],
"s = UniversalDict(['R','G','B'])\n",
"s[5]"
"For our CSP we also need to define a constraint function **f(A, a, B, b)**. In this, we need to ensure that the neighbors don't have the same color. This is defined in the function **different_values_constraint** of the module."
]
},
{
"cell_type": "code",
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"outputs": [
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"<div class=\"highlight\"><pre><span></span><span class=\"k\">def</span> <span class=\"nf\">different_values_constraint</span><span class=\"p\">(</span><span class=\"n\">A</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">B</span><span class=\"p\">,</span> <span class=\"n\">b</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""A constraint saying two neighboring variables must differ in value."""</span>\n",
" <span class=\"k\">return</span> <span class=\"n\">a</span> <span class=\"o\">!=</span> <span class=\"n\">b</span>\n",
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"The CSP class takes neighbors in the form of a Dict. The module specifies a simple helper function named **parse_neighbors** which allows us to take input in the form of strings and return a Dict of a form that is compatible with the **CSP Class**."
]
},
{
"cell_type": "code",
"outputs": [],
"source": [
"%pdoc parse_neighbors"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The **MapColoringCSP** function creates and returns a CSP with the above constraint function and states. The variables are the keys of the neighbors dict and the constraint is the one specified by the **different_values_constratint** function. **Australia**, **USA** and **France** are three CSPs that have been created using **MapColoringCSP**. **Australia** corresponds to ** Figure 6.1 ** in the book."
]
},
{
"cell_type": "code",
"execution_count": 6,
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"outputs": [
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"<div class=\"highlight\"><pre><span></span><span class=\"k\">def</span> <span class=\"nf\">MapColoringCSP</span><span class=\"p\">(</span><span class=\"n\">colors</span><span class=\"p\">,</span> <span class=\"n\">neighbors</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""Make a CSP for the problem of coloring a map with different colors</span>\n",
"<span class=\"sd\"> for any two adjacent regions. Arguments are a list of colors, and a</span>\n",
"<span class=\"sd\"> dict of {region: [neighbor,...]} entries. This dict may also be</span>\n",
"<span class=\"sd\"> specified as a string of the form defined by parse_neighbors."""</span>\n",
" <span class=\"k\">if</span> <span class=\"nb\">isinstance</span><span class=\"p\">(</span><span class=\"n\">neighbors</span><span class=\"p\">,</span> <span class=\"nb\">str</span><span class=\"p\">):</span>\n",
" <span class=\"n\">neighbors</span> <span class=\"o\">=</span> <span class=\"n\">parse_neighbors</span><span class=\"p\">(</span><span class=\"n\">neighbors</span><span class=\"p\">)</span>\n",
" <span class=\"k\">return</span> <span class=\"n\">CSP</span><span class=\"p\">(</span><span class=\"nb\">list</span><span class=\"p\">(</span><span class=\"n\">neighbors</span><span class=\"o\">.</span><span class=\"n\">keys</span><span class=\"p\">()),</span> <span class=\"n\">UniversalDict</span><span class=\"p\">(</span><span class=\"n\">colors</span><span class=\"p\">),</span> <span class=\"n\">neighbors</span><span class=\"p\">,</span>\n",
" <span class=\"n\">different_values_constraint</span><span class=\"p\">)</span>\n",
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"The N-queens puzzle is the problem of placing N chess queens on an N×N chessboard in a way such that no two queens threaten each other. Here N is a natural number. Like the graph coloring problem, NQueens is also implemented in the csp module. The **NQueensCSP** class inherits from the **CSP** class. It makes some modifications in the methods to suit this particular problem. The queens are assumed to be placed one per column, from left to right. That means position (x, y) represents (var, val) in the CSP. The constraint that needs to be passed to the CSP is defined in the **queen_constraint** function. The constraint is satisfied (true) if A, B are really the same variable, or if they are not in the same row, down diagonal, or up diagonal. "
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"<div class=\"highlight\"><pre><span></span><span class=\"k\">def</span> <span class=\"nf\">queen_constraint</span><span class=\"p\">(</span><span class=\"n\">A</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">B</span><span class=\"p\">,</span> <span class=\"n\">b</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""Constraint is satisfied (true) if A, B are really the same variable,</span>\n",
"<span class=\"sd\"> or if they are not in the same row, down diagonal, or up diagonal."""</span>\n",
" <span class=\"k\">return</span> <span class=\"n\">A</span> <span class=\"o\">==</span> <span class=\"n\">B</span> <span class=\"ow\">or</span> <span class=\"p\">(</span><span class=\"n\">a</span> <span class=\"o\">!=</span> <span class=\"n\">b</span> <span class=\"ow\">and</span> <span class=\"n\">A</span> <span class=\"o\">+</span> <span class=\"n\">a</span> <span class=\"o\">!=</span> <span class=\"n\">B</span> <span class=\"o\">+</span> <span class=\"n\">b</span> <span class=\"ow\">and</span> <span class=\"n\">A</span> <span class=\"o\">-</span> <span class=\"n\">a</span> <span class=\"o\">!=</span> <span class=\"n\">B</span> <span class=\"o\">-</span> <span class=\"n\">b</span><span class=\"p\">)</span>\n",
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"The **NQueensCSP** method implements methods that support solving the problem via **min_conflicts** which is one of the many popular techniques for solving CSPs. Because **min_conflicts** hill climbs the number of conflicts to solve, the CSP **assign** and **unassign** are modified to record conflicts. More details about the structures: **rows**, **downs**, **ups** which help in recording conflicts are explained in the docstring."
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"<div class=\"highlight\"><pre><span></span><span class=\"k\">class</span> <span class=\"nc\">NQueensCSP</span><span class=\"p\">(</span><span class=\"n\">CSP</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""Make a CSP for the nQueens problem for search with min_conflicts.</span>\n",
"<span class=\"sd\"> Suitable for large n, it uses only data structures of size O(n).</span>\n",
"<span class=\"sd\"> Think of placing queens one per column, from left to right.</span>\n",
"<span class=\"sd\"> That means position (x, y) represents (var, val) in the CSP.</span>\n",
"<span class=\"sd\"> The main structures are three arrays to count queens that could conflict:</span>\n",
"<span class=\"sd\"> rows[i] Number of queens in the ith row (i.e val == i)</span>\n",
"<span class=\"sd\"> downs[i] Number of queens in the \\ diagonal</span>\n",
"<span class=\"sd\"> such that their (x, y) coordinates sum to i</span>\n",
"<span class=\"sd\"> ups[i] Number of queens in the / diagonal</span>\n",
"<span class=\"sd\"> such that their (x, y) coordinates have x-y+n-1 = i</span>\n",
"<span class=\"sd\"> We increment/decrement these counts each time a queen is placed/moved from</span>\n",
"<span class=\"sd\"> a row/diagonal. So moving is O(1), as is nconflicts. But choosing</span>\n",
"<span class=\"sd\"> a variable, and a best value for the variable, are each O(n).</span>\n",
"<span class=\"sd\"> If you want, you can keep track of conflicted variables, then variable</span>\n",
"<span class=\"sd\"> selection will also be O(1).</span>\n",
"<span class=\"sd\"> >>> len(backtracking_search(NQueensCSP(8)))</span>\n",
"<span class=\"sd\"> 8</span>\n",
"<span class=\"sd\"> """</span>\n",
"\n",
" <span class=\"k\">def</span> <span class=\"fm\">__init__</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"p\">,</span> <span class=\"n\">n</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""Initialize data structures for n Queens."""</span>\n",
" <span class=\"n\">CSP</span><span class=\"o\">.</span><span class=\"fm\">__init__</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"p\">,</span> <span class=\"nb\">list</span><span class=\"p\">(</span><span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">)),</span> <span class=\"n\">UniversalDict</span><span class=\"p\">(</span><span class=\"nb\">list</span><span class=\"p\">(</span><span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">))),</span>\n",
" <span class=\"n\">UniversalDict</span><span class=\"p\">(</span><span class=\"nb\">list</span><span class=\"p\">(</span><span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">))),</span> <span class=\"n\">queen_constraint</span><span class=\"p\">)</span>\n",
"\n",
" <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">rows</span> <span class=\"o\">=</span> <span class=\"p\">[</span><span class=\"mi\">0</span><span class=\"p\">]</span><span class=\"o\">*</span><span class=\"n\">n</span>\n",
" <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">ups</span> <span class=\"o\">=</span> <span class=\"p\">[</span><span class=\"mi\">0</span><span class=\"p\">]</span><span class=\"o\">*</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">n</span> <span class=\"o\">-</span> <span class=\"mi\">1</span><span class=\"p\">)</span>\n",
" <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">downs</span> <span class=\"o\">=</span> <span class=\"p\">[</span><span class=\"mi\">0</span><span class=\"p\">]</span><span class=\"o\">*</span><span class=\"p\">(</span><span class=\"mi\">2</span><span class=\"o\">*</span><span class=\"n\">n</span> <span class=\"o\">-</span> <span class=\"mi\">1</span><span class=\"p\">)</span>\n",
"\n",
" <span class=\"k\">def</span> <span class=\"nf\">nconflicts</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"p\">,</span> <span class=\"n\">var</span><span class=\"p\">,</span> <span class=\"n\">val</span><span class=\"p\">,</span> <span class=\"n\">assignment</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""The number of conflicts, as recorded with each assignment.</span>\n",
"<span class=\"sd\"> Count conflicts in row and in up, down diagonals. If there</span>\n",
"<span class=\"sd\"> is a queen there, it can't conflict with itself, so subtract 3."""</span>\n",
" <span class=\"n\">n</span> <span class=\"o\">=</span> <span class=\"nb\">len</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">variables</span><span class=\"p\">)</span>\n",
" <span class=\"n\">c</span> <span class=\"o\">=</span> <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">rows</span><span class=\"p\">[</span><span class=\"n\">val</span><span class=\"p\">]</span> <span class=\"o\">+</span> <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">downs</span><span class=\"p\">[</span><span class=\"n\">var</span><span class=\"o\">+</span><span class=\"n\">val</span><span class=\"p\">]</span> <span class=\"o\">+</span> <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">ups</span><span class=\"p\">[</span><span class=\"n\">var</span><span class=\"o\">-</span><span class=\"n\">val</span><span class=\"o\">+</span><span class=\"n\">n</span><span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">]</span>\n",
" <span class=\"k\">if</span> <span class=\"n\">assignment</span><span class=\"o\">.</span><span class=\"n\">get</span><span class=\"p\">(</span><span class=\"n\">var</span><span class=\"p\">,</span> <span class=\"bp\">None</span><span class=\"p\">)</span> <span class=\"o\">==</span> <span class=\"n\">val</span><span class=\"p\">:</span>\n",
" <span class=\"n\">c</span> <span class=\"o\">-=</span> <span class=\"mi\">3</span>\n",
" <span class=\"k\">return</span> <span class=\"n\">c</span>\n",
"\n",
" <span class=\"k\">def</span> <span class=\"nf\">assign</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"p\">,</span> <span class=\"n\">var</span><span class=\"p\">,</span> <span class=\"n\">val</span><span class=\"p\">,</span> <span class=\"n\">assignment</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""Assign var, and keep track of conflicts."""</span>\n",
" <span class=\"n\">oldval</span> <span class=\"o\">=</span> <span class=\"n\">assignment</span><span class=\"o\">.</span><span class=\"n\">get</span><span class=\"p\">(</span><span class=\"n\">var</span><span class=\"p\">,</span> <span class=\"bp\">None</span><span class=\"p\">)</span>\n",
" <span class=\"k\">if</span> <span class=\"n\">val</span> <span class=\"o\">!=</span> <span class=\"n\">oldval</span><span class=\"p\">:</span>\n",
" <span class=\"k\">if</span> <span class=\"n\">oldval</span> <span class=\"ow\">is</span> <span class=\"ow\">not</span> <span class=\"bp\">None</span><span class=\"p\">:</span> <span class=\"c1\"># Remove old val if there was one</span>\n",
" <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">record_conflict</span><span class=\"p\">(</span><span class=\"n\">assignment</span><span class=\"p\">,</span> <span class=\"n\">var</span><span class=\"p\">,</span> <span class=\"n\">oldval</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n",
" <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">record_conflict</span><span class=\"p\">(</span><span class=\"n\">assignment</span><span class=\"p\">,</span> <span class=\"n\">var</span><span class=\"p\">,</span> <span class=\"n\">val</span><span class=\"p\">,</span> <span class=\"o\">+</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n",
" <span class=\"n\">CSP</span><span class=\"o\">.</span><span class=\"n\">assign</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"p\">,</span> <span class=\"n\">var</span><span class=\"p\">,</span> <span class=\"n\">val</span><span class=\"p\">,</span> <span class=\"n\">assignment</span><span class=\"p\">)</span>\n",
"\n",
" <span class=\"k\">def</span> <span class=\"nf\">unassign</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"p\">,</span> <span class=\"n\">var</span><span class=\"p\">,</span> <span class=\"n\">assignment</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""Remove var from assignment (if it is there) and track conflicts."""</span>\n",
" <span class=\"k\">if</span> <span class=\"n\">var</span> <span class=\"ow\">in</span> <span class=\"n\">assignment</span><span class=\"p\">:</span>\n",
" <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">record_conflict</span><span class=\"p\">(</span><span class=\"n\">assignment</span><span class=\"p\">,</span> <span class=\"n\">var</span><span class=\"p\">,</span> <span class=\"n\">assignment</span><span class=\"p\">[</span><span class=\"n\">var</span><span class=\"p\">],</span> <span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n",
" <span class=\"n\">CSP</span><span class=\"o\">.</span><span class=\"n\">unassign</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"p\">,</span> <span class=\"n\">var</span><span class=\"p\">,</span> <span class=\"n\">assignment</span><span class=\"p\">)</span>\n",
"\n",
" <span class=\"k\">def</span> <span class=\"nf\">record_conflict</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"p\">,</span> <span class=\"n\">assignment</span><span class=\"p\">,</span> <span class=\"n\">var</span><span class=\"p\">,</span> <span class=\"n\">val</span><span class=\"p\">,</span> <span class=\"n\">delta</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""Record conflicts caused by addition or deletion of a Queen."""</span>\n",
" <span class=\"n\">n</span> <span class=\"o\">=</span> <span class=\"nb\">len</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">variables</span><span class=\"p\">)</span>\n",
" <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">rows</span><span class=\"p\">[</span><span class=\"n\">val</span><span class=\"p\">]</span> <span class=\"o\">+=</span> <span class=\"n\">delta</span>\n",
" <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">downs</span><span class=\"p\">[</span><span class=\"n\">var</span> <span class=\"o\">+</span> <span class=\"n\">val</span><span class=\"p\">]</span> <span class=\"o\">+=</span> <span class=\"n\">delta</span>\n",
" <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">ups</span><span class=\"p\">[</span><span class=\"n\">var</span> <span class=\"o\">-</span> <span class=\"n\">val</span> <span class=\"o\">+</span> <span class=\"n\">n</span> <span class=\"o\">-</span> <span class=\"mi\">1</span><span class=\"p\">]</span> <span class=\"o\">+=</span> <span class=\"n\">delta</span>\n",
"\n",
" <span class=\"k\">def</span> <span class=\"nf\">display</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"p\">,</span> <span class=\"n\">assignment</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""Print the queens and the nconflicts values (for debugging)."""</span>\n",
" <span class=\"n\">n</span> <span class=\"o\">=</span> <span class=\"nb\">len</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">variables</span><span class=\"p\">)</span>\n",
" <span class=\"k\">for</span> <span class=\"n\">val</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">):</span>\n",
" <span class=\"k\">for</span> <span class=\"n\">var</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">):</span>\n",
" <span class=\"k\">if</span> <span class=\"n\">assignment</span><span class=\"o\">.</span><span class=\"n\">get</span><span class=\"p\">(</span><span class=\"n\">var</span><span class=\"p\">,</span> <span class=\"s1\">''</span><span class=\"p\">)</span> <span class=\"o\">==</span> <span class=\"n\">val</span><span class=\"p\">:</span>\n",
" <span class=\"n\">ch</span> <span class=\"o\">=</span> <span class=\"s1\">'Q'</span>\n",
" <span class=\"k\">elif</span> <span class=\"p\">(</span><span class=\"n\">var</span> <span class=\"o\">+</span> <span class=\"n\">val</span><span class=\"p\">)</span> <span class=\"o\">%</span> <span class=\"mi\">2</span> <span class=\"o\">==</span> <span class=\"mi\">0</span><span class=\"p\">:</span>\n",
" <span class=\"n\">ch</span> <span class=\"o\">=</span> <span class=\"s1\">'.'</span>\n",
" <span class=\"k\">else</span><span class=\"p\">:</span>\n",
" <span class=\"n\">ch</span> <span class=\"o\">=</span> <span class=\"s1\">'-'</span>\n",
" <span class=\"k\">print</span><span class=\"p\">(</span><span class=\"n\">ch</span><span class=\"p\">,</span> <span class=\"n\">end</span><span class=\"o\">=</span><span class=\"s1\">' '</span><span class=\"p\">)</span>\n",
" <span class=\"k\">print</span><span class=\"p\">(</span><span class=\"s1\">' '</span><span class=\"p\">,</span> <span class=\"n\">end</span><span class=\"o\">=</span><span class=\"s1\">' '</span><span class=\"p\">)</span>\n",
" <span class=\"k\">for</span> <span class=\"n\">var</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">):</span>\n",
" <span class=\"k\">if</span> <span class=\"n\">assignment</span><span class=\"o\">.</span><span class=\"n\">get</span><span class=\"p\">(</span><span class=\"n\">var</span><span class=\"p\">,</span> <span class=\"s1\">''</span><span class=\"p\">)</span> <span class=\"o\">==</span> <span class=\"n\">val</span><span class=\"p\">:</span>\n",
" <span class=\"n\">ch</span> <span class=\"o\">=</span> <span class=\"s1\">'*'</span>\n",
" <span class=\"k\">else</span><span class=\"p\">:</span>\n",
" <span class=\"n\">ch</span> <span class=\"o\">=</span> <span class=\"s1\">' '</span>\n",
" <span class=\"k\">print</span><span class=\"p\">(</span><span class=\"nb\">str</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">nconflicts</span><span class=\"p\">(</span><span class=\"n\">var</span><span class=\"p\">,</span> <span class=\"n\">val</span><span class=\"p\">,</span> <span class=\"n\">assignment</span><span class=\"p\">))</span> <span class=\"o\">+</span> <span class=\"n\">ch</span><span class=\"p\">,</span> <span class=\"n\">end</span><span class=\"o\">=</span><span class=\"s1\">' '</span><span class=\"p\">)</span>\n",
" <span class=\"k\">print</span><span class=\"p\">()</span>\n",
"</pre></div>\n",
"</body>\n",
"</html>\n"
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"<IPython.core.display.HTML object>"
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{
"cell_type": "markdown",
"The _ ___init___ _ method takes only one parameter **n** i.e. the size of the problem. To create an instance, we just pass the required value of n into the constructor."
Tarun Kumar Vangani
a validé
]
},
{
"cell_type": "code",
Tarun Kumar Vangani
a validé
"outputs": [],
"source": [
]
},
{
"cell_type": "markdown",
"### Min-conflicts\n",
"As stated above, the `min_conflicts` algorithm is an efficient method to solve such a problem.\n",
"<br>\n",
"In the start, all the variables of the CSP are _randomly_ initialized. \n",
"<br>\n",
"The algorithm then randomly selects a variable that has conflicts and violates some constraints of the CSP.\n",
"<br>\n",
"The selected variable is then assigned a value that _minimizes_ the number of conflicts.\n",
"<br>\n",
"This is a simple **stochastic algorithm** which works on a principle similar to **Hill-climbing**.\n",
"The conflicting state is repeatedly changed into a state with fewer conflicts in an attempt to reach an approximate solution.\n",
"<br>\n",
"This algorithm sometimes benefits from having a good initial assignment.\n",
"Using greedy techniques to get a good initial assignment and then using `min_conflicts` to solve the CSP can speed up the procedure dramatically, especially for CSPs with a large state space."
]
},
{
"cell_type": "code",
"execution_count": 11,
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{
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