authors/mapgccv/miscellaneous/sudoku.ipynb

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      "source": [
        "# Simple sudoku solver using logical constraints (with GUI)\n",
        "[![sudoku.ipynb](https://img.shields.io/badge/github-%23121011.svg?logo=github)](https://github.com/ampl/colab.ampl.com/blob/master/authors/mapgccv/miscellaneous/sudoku.ipynb) [![Open In Colab](https://colab.research.google.com/assets/colab-badge.svg)](https://colab.research.google.com/github/ampl/colab.ampl.com/blob/master/authors/mapgccv/miscellaneous/sudoku.ipynb) [![Kaggle](https://kaggle.com/static/images/open-in-kaggle.svg)](https://kaggle.com/kernels/welcome?src=https://github.com/ampl/colab.ampl.com/blob/master/authors/mapgccv/miscellaneous/sudoku.ipynb) [![Gradient](https://assets.paperspace.io/img/gradient-badge.svg)](https://console.paperspace.com/github/ampl/colab.ampl.com/blob/master/authors/mapgccv/miscellaneous/sudoku.ipynb) [![Open In SageMaker Studio Lab](https://studiolab.sagemaker.aws/studiolab.svg)](https://studiolab.sagemaker.aws/import/github/ampl/colab.ampl.com/blob/master/authors/mapgccv/miscellaneous/sudoku.ipynb) [![Hits](https://h.ampl.com/https://github.com/ampl/colab.ampl.com/blob/master/authors/mapgccv/miscellaneous/sudoku.ipynb)](https://colab.ampl.com)\n",
        "\n",
        "Description: Simple sudoku model with two formulations: as a Constraint Programming problem using the *alldiff* operator and as a MIP. Note that the CP formulation is more natural but it needs a solver supporting logical constraints or a MIP solver with automatic reformulation support (see [here](https://mp.ampl.com/) for more information).\n",
        "A simple GUI implemented using ipywidgets helps with data visualization and specification.\n",
        "\n",
        "Tags: amplpy, constraint-programming, GUI, highlights\n",
        "\n",
        "Notebook author: Christian Valente <<ccv@ampl.com>>\n",
        "\n",
        "Model author: Christian Valente"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 1,
      "metadata": {
        "id": "jxi-HR24j7jh"
      },
      "outputs": [],
      "source": [
        "# Install dependencies (using ipywidgets for the simple GUI)\n",
        "%pip install -q amplpy ipywidgets"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 2,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "eG1Wv8Uej7ji",
        "outputId": "017637d3-3e61-4c44-c0e3-fd89f70b0fc3"
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      "source": [
        "# Google Colab & Kaggle integration\n",
        "from amplpy import AMPL, ampl_notebook\n",
        "\n",
        "ampl = ampl_notebook(\n",
        "    modules=[\"highs\"],  # modules to install\n",
        "    license_uuid=\"default\",  # license to use\n",
        ")  # instantiate AMPL object and register magics"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "FS5h7tJSj7jj"
      },
      "source": [
        "## Define the AMPL model of a sudoku game.\n",
        "\n",
        "In this example, we will show two models to solve a sudoku.\n",
        "\n",
        "1. **MIP formulation** where we will use a binary variable to indicate if a cell is occupied with any of the possible numbers.\n",
        "2. **Constraint Programming formulation** using the logical operator `alldiff` to avoid the explicit use of binary variables.\n",
        "\n",
        "That the model is parametric in terms of size: **BaseNumber** defines the size of the (square) subgrids making up the game. For a normal sudoku game, where the numbers go from 1 to 9, this has to be set to 3."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "dB9xZiSrz8V-"
      },
      "source": [
        "### Common infrastructure\n",
        "The entities defined in the next cell are shared by both the MIP and the CP formulations"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 3,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "hz6EykEs9w3O",
        "outputId": "a15bab7f-f605-4904-8272-0500055ed291"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "Overwriting common_sudoku.mod\n"
          ]
        }
      ],
      "source": [
        "%%writefile common_sudoku.mod\n",
        "# The base number of this sudoku; 3 is the default (9 numbers game)\n",
        "param BaseNumber default 3;\n",
        "\n",
        "# The length of each line/column, derived from BaseNumber\n",
        "param GridSize := BaseNumber * BaseNumber;\n",
        "\n",
        "# Set of all Rows\n",
        "set Rows := {1..GridSize};\n",
        "\n",
        "# Set of all Columns\n",
        "set Columns := {1..GridSize};\n",
        "\n",
        "# This indexed set memorizes the tuples of coordinates for each\n",
        "# sub-grid making up the sudoku grid\n",
        "set SubGridCoordinates{SubGridRow in 1..BaseNumber, SubGridCol in 1..BaseNumber}\n",
        "    within {Rows, Columns}\n",
        "    = {(SubGridRow-1)*BaseNumber+1..SubGridRow*BaseNumber,\n",
        "        (SubGridCol-1)*BaseNumber+1..SubGridCol*BaseNumber};\n",
        "\n",
        "# The variables representing the numbers at all positions in the sudoku grid\n",
        "var SudokuGrid{Rows, Columns} >= 1, <= GridSize integer;\n",
        "\n",
        "# Set this parameter to non-zero to fix a position to have that value\n",
        "param FixedValues{Rows, Columns} default 0;\n",
        "\n",
        "# Dummy objective, just to \"encourage\" the solver to have a defined objective\n",
        "maximize DummyObjective: SudokuGrid[1,1];\n",
        "\n",
        "# Fix input data (forces the variable at the corresponding location to have\n",
        "# the same value as the parameter)\n",
        "subject to FixFixedValues{row in Rows, col in Columns : FixedValues[row, col] > 0}:\n",
        "    SudokuGrid[row, col] = FixedValues[row, col];"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "1SNd7kTm0Yj8"
      },
      "source": [
        "### MIP formulation\n",
        "In the MIP formulation, we will use the binary variable `NumberPresence`, defined for all the cells and all the possible numbers, to indicate if a specific number is present in the related cell.\n",
        "A set of constraints will then be needed to ensure that:\n",
        "\n",
        "1. `MIPUniqueNumberPerPosition` each cell contains only one number\n",
        "2. `MIPUniqueNumberPerRow` each row contains all the possible numbers\n",
        "3. `MIPUniqueNumberPerColumn` each column contains all the possible numbers\n",
        "4. `MIPUniqueNumberPerSubGrid` each sub-grid contains all the possible numbers\n",
        "5. `MIPLinkToSudokuGrid` the variable `NumberPresence` is linked to the variable `SudokuGrid` above. Note that this is not strictly necessary for the model itself, but it is useful when sharing the same base entities with the CP model."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 4,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "XCNdTmCa96qE",
        "outputId": "3a5e93cc-701c-466b-a693-9890144efd01"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "Overwriting mip_sudoku.mod\n"
          ]
        }
      ],
      "source": [
        "%%writefile mip_sudoku.mod\n",
        "# Binary decision variable that represents the presence or absence of a specific\n",
        "# number at a particular position in the Sudoku grid.\n",
        "var NumberPresence{Number in 1..GridSize, Columns, Rows} binary;\n",
        "\n",
        "# Each position in the grid must have only one number\n",
        "MIPUniqueNumberPerPosition{row in Rows, col in Columns}:\n",
        "    sum{num in 1..GridSize} NumberPresence[num, col, row] = 1;\n",
        "\n",
        "# Each number must appear exactly once in each row\n",
        "MIPUniqueNumberPerRow{row in Rows, num in 1..GridSize}:\n",
        "    sum{col in Columns} NumberPresence[num, col, row] = 1;\n",
        "\n",
        "# Each number must appear exactly once in each column\n",
        "MIPUniqueNumberPerColumn{col in Columns, num in 1..GridSize}:\n",
        "    sum{row in Rows} NumberPresence[num, col, row] = 1;\n",
        "\n",
        "# Each number must appear exactly once in each sub-grid\n",
        "MIPUniqueNumberPerSubGrid{num in 1..GridSize, SubGridRow in 1..BaseNumber, SubGridCol in 1..BaseNumber}:\n",
        "    sum{(row, col) in SubGridCoordinates[SubGridRow, SubGridCol]}\n",
        "    NumberPresence[num, col, row] = 1;\n",
        "\n",
        "# Link to the SudokuGrid variable\n",
        "MIPLinkToSudokuGrid{row in Rows, col in Columns}:\n",
        "    sum{num in 1..GridSize} NumberPresence[num, col, row] * num = SudokuGrid[row, col];"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Hd7rFNjc1hjD"
      },
      "source": [
        "### CP formulation\n",
        "The Constraint Programming formulation is much more readable and compact, and it follows the human intuitive understanding of the game.\n",
        "Using the operator `alldiff`, which forces all variables passed as operands to assume different values, we simply need the following constraints:\n",
        "\n",
        "1. `RowAllDifferent` each `SudokuGrid` instance in a row must contain a different number\n",
        "2. `ColumnAllDifferent` each `SudokuGrid` instance in a column must contain a different number\n",
        "3. `SubGridAllDifferent` each `SudokuGrid` instance in a subsquare must contain a different number"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 5,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "n-WeoaUD-AEA",
        "outputId": "3b8d116e-e0a2-49c9-b6a3-9978ed4f3220"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "Overwriting cp_sudoku.mod\n"
          ]
        }
      ],
      "source": [
        "%%writefile cp_sudoku.mod\n",
        "# All numbers in one row must be different\n",
        "subject to RowAllDifferent{row in Rows}:\n",
        "    alldiff {col in Columns} SudokuGrid[row, col];\n",
        "\n",
        "# All numbers in one column must be different\n",
        "subject to ColumnAllDifferent{col in Columns}:\n",
        "    alldiff {row in Rows} SudokuGrid[row, col];\n",
        "\n",
        "# All numbers within each sub-grid must be different\n",
        "subject to SubGridAllDifferent{SubGridRow in 1..BaseNumber, SubGridCol in 1..BaseNumber}:\n",
        "    alldiff {(row, col) in SubGridCoordinates[SubGridRow, SubGridCol]} SudokuGrid[row, col];"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "qcvCg2Dg2zJe"
      },
      "source": [
        "### Data definition\n",
        "This is an example used to populate the sudoku below. It has the standard size (9x9), so we specify a `BASE` of 3."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 6,
      "metadata": {
        "id": "dU47NyUs22TS"
      },
      "outputs": [],
      "source": [
        "from random import seed, random\n",
        "\n",
        "seed(1234)\n",
        "\n",
        "\n",
        "def random_state(base=3):\n",
        "    if base != 3:\n",
        "        return {}\n",
        "    solution = [\n",
        "        [2, 5, 7, 8, 6, 3, 1, 4, 9],\n",
        "        [4, 9, 6, 5, 7, 1, 8, 3, 2],\n",
        "        [8, 1, 3, 9, 4, 2, 7, 6, 5],\n",
        "        [1, 6, 5, 2, 9, 4, 3, 7, 8],\n",
        "        [9, 8, 4, 1, 3, 7, 5, 2, 6],\n",
        "        [3, 7, 2, 6, 5, 8, 4, 9, 1],\n",
        "        [7, 2, 9, 4, 8, 5, 6, 1, 3],\n",
        "        [5, 3, 1, 7, 2, 6, 9, 8, 4],\n",
        "        [6, 4, 8, 3, 1, 9, 2, 5, 7],\n",
        "    ]\n",
        "    return {\n",
        "        (i + 1, j + 1): solution[i][j] if random() <= 1 / 3.0 else 0\n",
        "        for i in range(9)\n",
        "        for j in range(9)\n",
        "    }"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "vx8BBvXHj7jk"
      },
      "source": [
        "### GUI-related code\n",
        "The following code creates the Class **SudokuSchema**, a convenient wrapper around ipywidgets to create and display a grid that resembles the game board. It provides functions to get and set the values in the schema itself."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 7,
      "metadata": {
        "id": "cpTbYwr8j7jk"
      },
      "outputs": [],
      "source": [
        "import ipywidgets as widgets\n",
        "from IPython.display import display\n",
        "\n",
        "\n",
        "class SudokuSchema:\n",
        "    def _create_one_grid(self, start_row: int, start_col: int):\n",
        "        gridItems = [\n",
        "            widgets.VBox(\n",
        "                [\n",
        "                    self.items[row, col]\n",
        "                    for row in range(start_row, start_row + self.BASE)\n",
        "                ]\n",
        "            )\n",
        "            for col in range(start_col, start_col + self.BASE)\n",
        "        ]\n",
        "        return widgets.HBox(\n",
        "            gridItems, layout=widgets.Layout(border=\"solid 2px\", width=\"140px\")\n",
        "        )\n",
        "\n",
        "    def __init__(self, base):\n",
        "        \"\"\"Initializes a sudoku schema with base dimension BASE\"\"\"\n",
        "        self.BASE = base\n",
        "        self.BSQUARED = base**2\n",
        "        # Create all widgets\n",
        "        self.items = {\n",
        "            (r, c): widgets.BoundedIntText(\n",
        "                value=0,\n",
        "                min=0,\n",
        "                max=self.BASE**4,\n",
        "                step=1,\n",
        "                description=\"\",\n",
        "                layout=widgets.Layout(width=\"40px\", height=\"40px\"),\n",
        "            )\n",
        "            for r in range(self.BSQUARED)\n",
        "            for c in range(self.BSQUARED)\n",
        "        }\n",
        "        self.sudoku = widgets.HBox(\n",
        "            [\n",
        "                widgets.VBox(\n",
        "                    [\n",
        "                        self._create_one_grid(r, c)\n",
        "                        for r in range(0, self.BSQUARED, self.BASE)\n",
        "                    ]\n",
        "                )\n",
        "                for c in range(0, self.BSQUARED, self.BASE)\n",
        "            ]\n",
        "        )\n",
        "        self.create_selection_button()\n",
        "        self.create_buttons()\n",
        "\n",
        "    def display(self):\n",
        "        \"\"\"Display the current schema on the notebook\"\"\"\n",
        "        display(self.sudoku)\n",
        "        display(self.selector)\n",
        "        if self.BASE == 3:\n",
        "            display(widgets.HBox([self.random_button, self.solve_button]))\n",
        "        else:\n",
        "            display(widgets.HBox([self.solve_button]))\n",
        "\n",
        "    def get_values(self):\n",
        "        \"\"\"Get the current non zero values as a (r,c) : value dictionary\"\"\"\n",
        "        return {\n",
        "            (r + 1, c + 1): self.items[r, c].value\n",
        "            for r in range(self.BSQUARED)\n",
        "            for c in range(self.BSQUARED)\n",
        "            if self.items[r, c].value != 0\n",
        "        }\n",
        "\n",
        "    def set_values(self, values: dict):\n",
        "        \"\"\"Set the values in the schema from the specified (r,c) : value dictionary\"\"\"\n",
        "        for (r, c), v in values.items():\n",
        "            self.items[r - 1, c - 1].value = round(v)\n",
        "\n",
        "    def create_selection_button(self):\n",
        "        self.selector = widgets.RadioButtons(\n",
        "            options=[\"Constraint Programming\", \"MIP\"],\n",
        "            value=\"Constraint Programming\",\n",
        "            layout={\"width\": \"max-content\"},\n",
        "            description=\"Formulation:\",\n",
        "            disabled=False,\n",
        "        )\n",
        "\n",
        "    def get_selected_formulation(self):\n",
        "        return self.selector.value\n",
        "\n",
        "    def set_random_board(self):\n",
        "        state = random_state(self.BASE)\n",
        "        self.set_values(state)\n",
        "\n",
        "    def create_buttons(self):\n",
        "        self.random_button = widgets.Button(\n",
        "            description=\"Random board\",\n",
        "            disabled=False,\n",
        "            button_style=\"info\",\n",
        "            tooltip=\"Random board\",\n",
        "            icon=\"bolt\",\n",
        "        )\n",
        "        self.random_button.on_click(lambda btn: self.set_random_board())\n",
        "        self.solve_button = widgets.Button(\n",
        "            description=\"Solve\",\n",
        "            disabled=False,\n",
        "            button_style=\"success\",\n",
        "            tooltip=\"Solve model\",\n",
        "            icon=\"bolt\",\n",
        "        )\n",
        "        self.solve_button.on_click(solve_and_display)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "fRfYdBe0j7jm"
      },
      "source": [
        "### Solve and display\n",
        "Pressing the *Solve model* button below the schema the sudoku will be solver by means of the function below. At first, we get which formulation is selected, and then use the AMPL to solve the instance using the selected formulation."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 8,
      "metadata": {
        "id": "5CRFkWkij7jn"
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      "outputs": [],
      "source": [
        "# Solve and display the solution\n",
        "def solve_and_display(button):\n",
        "    # Get the selected formulation from the radio button\n",
        "    formulation = sudoku.get_selected_formulation()\n",
        "    ampl = AMPL()\n",
        "    ampl.read(\"common_sudoku.mod\")\n",
        "    ampl.param[\"FixedValues\"] = sudoku.get_values()\n",
        "    if formulation == \"Constraint Programming\":\n",
        "        ampl.read(\"cp_sudoku.mod\")\n",
        "    else:\n",
        "        ampl.read(\"mip_sudoku.mod\")\n",
        "\n",
        "    print(f\"Solving the {formulation} formulation!\")\n",
        "    # Solve the selected model\n",
        "    ampl.solve(solver=\"highs\")\n",
        "    assert ampl.solve_result == \"solved\", ampl.solve_result\n",
        "\n",
        "    # Get the data from AMPL and assign them to the entities making up the grid above\n",
        "    sudoku.set_values(ampl.get_data(\"SudokuGrid\").to_dict())"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "7F8uyalbj7jl"
      },
      "source": [
        "### Show the sudoku schema\n",
        "The following cell creates the sudoku schema - of the size specified and visualizes it. It also shows a radio button that allows you to choose between the two formulations and a button to begin the solution process. The button will call the function `solve_and_display` defined above."
      ]
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        },
        "id": "W_YRCZjHj7jm",
        "outputId": "99030bac-901d-449f-9ae4-48cc747f1df0"
      },
      "outputs": [
        {
          "data": {
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              "model_id": "10bdba0d4ecf4a29a4f330e4acecc0a7",
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              "HBox(children=(VBox(children=(HBox(children=(VBox(children=(BoundedIntText(value=0, layout=Layout(height='40px…"
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            },
            "text/plain": [
              "RadioButtons(description='Formulation:', layout=Layout(width='max-content'), options=('Constraint Programming'…"
            ]
          },
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        {
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              "HBox(children=(Button(button_style='info', description='Random board', icon='bolt', style=ButtonStyle(), toolt…"
            ]
          },
          "metadata": {},
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        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "Solving the Constraint Programming formulation!\n",
            "HiGHS 1.7.1: \b\b\b\b\b\b\b\b\b\b\b\b\bHiGHS 1.7.1: optimal solution; objective 6\n",
            "123 simplex iterations\n",
            "1 branching nodes\n"
          ]
        }
      ],
      "source": [
        "# Create and display the grid\n",
        "sudoku = SudokuSchema(base=3)\n",
        "sudoku.display()"
      ]
    }
  ],
  "metadata": {
    "colab": {
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    },
    "kernelspec": {
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