From f91ca0e30d907722b8ee233fc9006db744503567 Mon Sep 17 00:00:00 2001 From: AngryCracker Date: Tue, 13 Mar 2018 02:24:27 +0530 Subject: [PATCH 1/3] Added section on min-conflicts --- csp.ipynb | 608 ++++++++++++++++++++++++++++++++++++++++++++++++++++-- 1 file changed, 590 insertions(+), 18 deletions(-) diff --git a/csp.ipynb b/csp.ipynb index 1de9e1312..3eeed2a81 100644 --- a/csp.ipynb +++ b/csp.ipynb @@ -52,7 +52,9 @@ { "cell_type": "code", "execution_count": null, - "metadata": {}, + "metadata": { + "collapsed": true + }, "outputs": [], "source": [ "psource(CSP)" @@ -105,7 +107,9 @@ { "cell_type": "code", "execution_count": null, - "metadata": {}, + "metadata": { + "collapsed": true + }, "outputs": [], "source": [ "psource(different_values_constraint)" @@ -139,7 +143,9 @@ { "cell_type": "code", "execution_count": null, - "metadata": {}, + "metadata": { + "collapsed": true + }, "outputs": [], "source": [ "psource(MapColoringCSP)" @@ -178,9 +184,114 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 4, "metadata": {}, - "outputs": [], + "outputs": [ + { + "data": { + "text/html": [ + "\n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "

\n", + "\n", + "
def queen_constraint(A, a, B, b):\n",
+       "    """Constraint is satisfied (true) if A, B are really the same variable,\n",
+       "    or if they are not in the same row, down diagonal, or up diagonal."""\n",
+       "    return A == B or (a != b and A + a != B + b and A - a != B - b)\n",
+       "
\n", + "\n", + "\n" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], "source": [ "psource(queen_constraint)" ] @@ -194,9 +305,191 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 5, "metadata": {}, - "outputs": [], + "outputs": [ + { + "data": { + "text/html": [ + "\n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "

\n", + "\n", + "
class NQueensCSP(CSP):\n",
+       "    """Make a CSP for the nQueens problem for search with min_conflicts.\n",
+       "    Suitable for large n, it uses only data structures of size O(n).\n",
+       "    Think of placing queens one per column, from left to right.\n",
+       "    That means position (x, y) represents (var, val) in the CSP.\n",
+       "    The main structures are three arrays to count queens that could conflict:\n",
+       "        rows[i]      Number of queens in the ith row (i.e val == i)\n",
+       "        downs[i]     Number of queens in the \\ diagonal\n",
+       "                     such that their (x, y) coordinates sum to i\n",
+       "        ups[i]       Number of queens in the / diagonal\n",
+       "                     such that their (x, y) coordinates have x-y+n-1 = i\n",
+       "    We increment/decrement these counts each time a queen is placed/moved from\n",
+       "    a row/diagonal. So moving is O(1), as is nconflicts.  But choosing\n",
+       "    a variable, and a best value for the variable, are each O(n).\n",
+       "    If you want, you can keep track of conflicted variables, then variable\n",
+       "    selection will also be O(1).\n",
+       "    >>> len(backtracking_search(NQueensCSP(8)))\n",
+       "    8\n",
+       "    """\n",
+       "\n",
+       "    def __init__(self, n):\n",
+       "        """Initialize data structures for n Queens."""\n",
+       "        CSP.__init__(self, list(range(n)), UniversalDict(list(range(n))),\n",
+       "                     UniversalDict(list(range(n))), queen_constraint)\n",
+       "\n",
+       "        self.rows = [0]*n\n",
+       "        self.ups = [0]*(2*n - 1)\n",
+       "        self.downs = [0]*(2*n - 1)\n",
+       "\n",
+       "    def nconflicts(self, var, val, assignment):\n",
+       "        """The number of conflicts, as recorded with each assignment.\n",
+       "        Count conflicts in row and in up, down diagonals. If there\n",
+       "        is a queen there, it can't conflict with itself, so subtract 3."""\n",
+       "        n = len(self.variables)\n",
+       "        c = self.rows[val] + self.downs[var+val] + self.ups[var-val+n-1]\n",
+       "        if assignment.get(var, None) == val:\n",
+       "            c -= 3\n",
+       "        return c\n",
+       "\n",
+       "    def assign(self, var, val, assignment):\n",
+       "        """Assign var, and keep track of conflicts."""\n",
+       "        oldval = assignment.get(var, None)\n",
+       "        if val != oldval:\n",
+       "            if oldval is not None:  # Remove old val if there was one\n",
+       "                self.record_conflict(assignment, var, oldval, -1)\n",
+       "            self.record_conflict(assignment, var, val, +1)\n",
+       "            CSP.assign(self, var, val, assignment)\n",
+       "\n",
+       "    def unassign(self, var, assignment):\n",
+       "        """Remove var from assignment (if it is there) and track conflicts."""\n",
+       "        if var in assignment:\n",
+       "            self.record_conflict(assignment, var, assignment[var], -1)\n",
+       "        CSP.unassign(self, var, assignment)\n",
+       "\n",
+       "    def record_conflict(self, assignment, var, val, delta):\n",
+       "        """Record conflicts caused by addition or deletion of a Queen."""\n",
+       "        n = len(self.variables)\n",
+       "        self.rows[val] += delta\n",
+       "        self.downs[var + val] += delta\n",
+       "        self.ups[var - val + n - 1] += delta\n",
+       "\n",
+       "    def display(self, assignment):\n",
+       "        """Print the queens and the nconflicts values (for debugging)."""\n",
+       "        n = len(self.variables)\n",
+       "        for val in range(n):\n",
+       "            for var in range(n):\n",
+       "                if assignment.get(var, '') == val:\n",
+       "                    ch = 'Q'\n",
+       "                elif (var + val) % 2 == 0:\n",
+       "                    ch = '.'\n",
+       "                else:\n",
+       "                    ch = '-'\n",
+       "                print(ch, end=' ')\n",
+       "            print('    ', end=' ')\n",
+       "            for var in range(n):\n",
+       "                if assignment.get(var, '') == val:\n",
+       "                    ch = '*'\n",
+       "                else:\n",
+       "                    ch = ' '\n",
+       "                print(str(self.nconflicts(var, val, assignment)) + ch, end=' ')\n",
+       "            print()\n",
+       "
\n", + "\n", + "\n" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], "source": [ "psource(NQueensCSP)" ] @@ -210,7 +503,7 @@ }, { "cell_type": "code", - "execution_count": 4, + "execution_count": 6, "metadata": { "collapsed": true }, @@ -219,6 +512,279 @@ "eight_queens = NQueensCSP(8)" ] }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We have defined our CSP. \n", + "We now need to solve this.\n", + "\n", + "### Min-conflicts\n", + "As stated above, the `min_conflicts` algorithm is an efficient method to solve such a problem.\n", + "
\n", + "To begin with, all the variables of the CSP are _randomly_ initialized. \n", + "
\n", + "The algorithm then randomly selects a variable that has conflicts and violates some constraints of the CSP.\n", + "
\n", + "The selected variable is then assigned a value that _minimizes_ the number of conflicts.\n", + "
\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", + "
\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": 7, + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "\n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "

\n", + "\n", + "
def min_conflicts(csp, max_steps=100000):\n",
+       "    """Solve a CSP by stochastic hillclimbing on the number of conflicts."""\n",
+       "    # Generate a complete assignment for all variables (probably with conflicts)\n",
+       "    csp.current = current = {}\n",
+       "    for var in csp.variables:\n",
+       "        val = min_conflicts_value(csp, var, current)\n",
+       "        csp.assign(var, val, current)\n",
+       "    # Now repeatedly choose a random conflicted variable and change it\n",
+       "    for i in range(max_steps):\n",
+       "        conflicted = csp.conflicted_vars(current)\n",
+       "        if not conflicted:\n",
+       "            return current\n",
+       "        var = random.choice(conflicted)\n",
+       "        val = min_conflicts_value(csp, var, current)\n",
+       "        csp.assign(var, val, current)\n",
+       "    return None\n",
+       "
\n", + "\n", + "\n" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "psource(min_conflicts)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Let's use this algorithm to solve the `eight_queens` CSP." + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "solution = min_conflicts(eight_queens)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "This is indeed a valid solution. \n", + "Let's write a helper function to visualize the solution space." + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "import matplotlib.pyplot as plt\n", + "import numpy as np\n", + "%matplotlib inline\n", + "\n", + "def display_NQueensCSP(solution):\n", + " n = len(solution)\n", + " board = np.zeros((n, n))\n", + " for i in range(n):\n", + " for j in range(n):\n", + " if (i + j) % 2 == 0:\n", + " board[i][j] = 2\n", + " \n", + " for (k, v) in solution.items():\n", + " board[k][v] = 1\n", + " \n", + " fig = plt.figure(figsize=(7, 7))\n", + " ax = fig.add_subplot(111)\n", + " ax.set_title(f'{n} Queens')\n", + " plt.imshow(board, cmap='binary', interpolation='nearest')\n", + " ax.set_aspect('equal')\n", + " fig.tight_layout()\n", + " plt.show()" + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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+ "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "display_NQueensCSP(solution)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The gray cells indicate the positions of the queens." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Lets' see if we can find a different solution." + ] + }, + { + "cell_type": "code", + "execution_count": 11, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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+ "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "eight_queens = NQueensCSP(8)\n", + "solution = min_conflicts(eight_queens)\n", + "display_NQueensCSP(solution)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The solution is a bit different this time. \n", + "Running the above cell several times should give you various valid solutions.\n", + "
\n", + "In the `search.ipynb` notebook, we will see how NQueensProblem can be solved using a heuristic search method such as `uniform_cost_search` and `astar_search`." + ] + }, { "cell_type": "markdown", "metadata": {}, @@ -466,7 +1032,9 @@ { "cell_type": "code", "execution_count": null, - "metadata": {}, + "metadata": { + "collapsed": true + }, "outputs": [], "source": [ "psource(mrv)" @@ -475,7 +1043,9 @@ { "cell_type": "code", "execution_count": null, - "metadata": {}, + "metadata": { + "collapsed": true + }, "outputs": [], "source": [ "psource(num_legal_values)" @@ -484,7 +1054,9 @@ { "cell_type": "code", "execution_count": null, - "metadata": {}, + "metadata": { + "collapsed": true + }, "outputs": [], "source": [ "psource(CSP.nconflicts)" @@ -500,7 +1072,9 @@ { "cell_type": "code", "execution_count": null, - "metadata": {}, + "metadata": { + "collapsed": true + }, "outputs": [], "source": [ "psource(lcv)" @@ -663,7 +1237,9 @@ { "cell_type": "code", "execution_count": null, - "metadata": {}, + "metadata": { + "collapsed": true + }, "outputs": [], "source": [ "psource(tree_csp_solver)" @@ -1162,11 +1738,7 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.5.3" - }, - "widgets": { - "state": {}, - "version": "1.1.1" + "version": "3.6.1" } }, "nbformat": 4, From 57ddf35ec7571044212e437395263711d67bb695 Mon Sep 17 00:00:00 2001 From: AngryCracker Date: Tue, 13 Mar 2018 02:37:33 +0530 Subject: [PATCH 2/3] Refactor one-liner for loop --- csp.ipynb | 6 +----- 1 file changed, 1 insertion(+), 5 deletions(-) diff --git a/csp.ipynb b/csp.ipynb index 3eeed2a81..be3882387 100644 --- a/csp.ipynb +++ b/csp.ipynb @@ -701,11 +701,7 @@ "\n", "def display_NQueensCSP(solution):\n", " n = len(solution)\n", - " board = np.zeros((n, n))\n", - " for i in range(n):\n", - " for j in range(n):\n", - " if (i + j) % 2 == 0:\n", - " board[i][j] = 2\n", + " board = np.array([2 * int((i + j) % 2) for j in range(n) for i in range(n)]).reshape((n, n))\n", " \n", " for (k, v) in solution.items():\n", " board[k][v] = 1\n", From 02449df1e1e1715d91f23944407fbb7e22ee157a Mon Sep 17 00:00:00 2001 From: AngryCracker Date: Tue, 13 Mar 2018 04:18:47 +0530 Subject: [PATCH 3/3] Added tests for min_conflicts and NQueensCSP --- tests/test_csp.py | 55 +++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 55 insertions(+) diff --git a/tests/test_csp.py b/tests/test_csp.py index f63e657aa..0f282e3fe 100644 --- a/tests/test_csp.py +++ b/tests/test_csp.py @@ -351,6 +351,61 @@ def test_min_conflicts(): australia_impossible = MapColoringCSP(list('RG'), 'SA: WA NT Q NSW V; NT: WA Q; NSW: Q V; T: ') assert min_conflicts(australia_impossible, 1000) is None + assert min_conflicts(NQueensCSP(2), 1000) is None + assert min_conflicts(NQueensCSP(3), 1000) is None + + +def test_nqueens_csp(): + csp = NQueensCSP(8) + + assignment = {0: 0, 1: 1, 2: 2, 3: 3, 4: 4} + csp.assign(5, 5, assignment) + assert len(assignment) == 6 + csp.assign(6, 6, assignment) + assert len(assignment) == 7 + csp.assign(7, 7, assignment) + assert len(assignment) == 8 + assert assignment[5] == 5 + assert assignment[6] == 6 + assert assignment[7] == 7 + assert csp.nconflicts(3, 2, assignment) == 0 + assert csp.nconflicts(3, 3, assignment) == 0 + assert csp.nconflicts(1, 5, assignment) == 1 + assert csp.nconflicts(7, 5, assignment) == 2 + csp.unassign(1, assignment) + csp.unassign(2, assignment) + csp.unassign(3, assignment) + assert 1 not in assignment + assert 2 not in assignment + assert 3 not in assignment + + assignment = {} + assignment = {0: 0, 1: 1, 2: 4, 3: 1, 4: 6} + csp.assign(5, 7, assignment) + assert len(assignment) == 6 + csp.assign(6, 6, assignment) + assert len(assignment) == 7 + csp.assign(7, 2, assignment) + assert len(assignment) == 8 + assert assignment[5] == 7 + assert assignment[6] == 6 + assert assignment[7] == 2 + assignment = {0: 0, 1: 1, 2: 4, 3: 1, 4: 6, 5: 7, 6: 6, 7: 2} + assert csp.nconflicts(7, 7, assignment) == 4 + assert csp.nconflicts(3, 4, assignment) == 0 + assert csp.nconflicts(2, 6, assignment) == 2 + assert csp.nconflicts(5, 5, assignment) == 3 + csp.unassign(4, assignment) + csp.unassign(5, assignment) + csp.unassign(6, assignment) + assert 4 not in assignment + assert 5 not in assignment + assert 6 not in assignment + + for n in range(5, 9): + csp = NQueensCSP(n) + solution = min_conflicts(csp) + assert not solution or sorted(solution.values()) == list(range(n)) def test_universal_dict():