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‎The_Knight's_independence (2).ipynb

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{
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"nbformat": 4,
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"nbformat_minor": 0,
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"metadata": {
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"colab": {
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"provenance": []
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},
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"kernelspec": {
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"name": "python3",
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"display_name": "Python 3"
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},
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"language_info": {
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"name": "python"
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}
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},
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"cells": [
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{
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"cell_type": "code",
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"execution_count": 9,
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"metadata": {
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"id": "cN8DZ70tBImW"
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},
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"outputs": [],
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"source": [
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"# The Knight's Independence Problem (The k Knight's problem)\n",
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"# Matheus Ribeiro Alencar https://github.com/matheusriale\n",
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"import random\n",
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"import copy\n",
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"\n",
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"#initialize the chessboard\n",
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"def initialize_chessboard(dim):\n",
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" matrix = []\n",
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" row = [0]*dim\n",
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" for i in range(dim):\n",
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" matrix.append(row[:]) #copy of row, no indexing problems later on\n",
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" return matrix"
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]
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},
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{
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"cell_type": "code",
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"source": [
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"#Knight's move restrictions, we restrict the squares attacked\n",
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"def place_attacks(pos_x,pos_y,chessboard):\n",
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" if pos_x+2<dim and pos_y +1<dim:\n",
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" chessboard[pos_x+2][pos_y +1] = 'X'\n",
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" if pos_x+2<dim and pos_y -1 >=0:\n",
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" chessboard[pos_x+2][pos_y -1] = 'X'\n",
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" if pos_x+1<dim and pos_y +2<dim:\n",
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" chessboard[pos_x+1][pos_y +2] = 'X'\n",
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" if pos_x+1<dim and pos_y -2>=0:\n",
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" chessboard[pos_x+1][pos_y -2] = 'X'\n",
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" if pos_x-2>=0 and pos_y+1<dim:\n",
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" chessboard[pos_x-2][pos_y +1] = 'X'\n",
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" if pos_x-2>=0 and pos_y-1>=0:\n",
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" chessboard[pos_x-2][pos_y -1] = 'X'\n",
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" if pos_x-1>=0 and pos_y-2>=0:\n",
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" chessboard[pos_x-1][pos_y -2] = 'X'\n",
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" if pos_x-1>=0 and pos_y+2<dim:\n",
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" chessboard[pos_x-1][pos_y +2] = 'X'"
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],
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"metadata": {
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"id": "EmvnRNUNEEDT"
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},
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"execution_count": 10,
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"outputs": []
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},
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{
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"cell_type": "code",
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"source": [
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"def check_attacks(chessboard):\n",
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" pos_x = None\n",
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" pos_y = None\n",
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" less_restrictions = -9 # Highest number of squares restricted by the knight's moves is 8\n",
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" A = copy.deepcopy(chessboard) # copy of chessboard\n",
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" # Insert copies of original object, source: https://docs.python.org/dev/library/copy.html#module-copy\n",
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" for i in range(0,len(A)):\n",
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" for j in range(0,len(A)):\n",
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" restricted_fields = 0\n",
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"\n",
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" if A[i][j] != 'X' and A[i][j] == 0: #square availabel, we will check if it gets us a good solution\n",
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" if i+2<(len(A)) and j+1<(len(A)):\n",
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" if A[i+2][j +1] != 'X' and A[i+2][j +1] <= 0:\n",
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" restricted_fields = restricted_fields - 1\n",
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"\n",
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" if i+2<(len(A)) and j -1 >=0:\n",
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" if A[i+2][j -1] != 'X' and A[i+2][j -1] <= 0:\n",
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" restricted_fields = restricted_fields - 1\n",
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"\n",
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" if i+1<(len(A)) and j +2<(len(A)):\n",
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" if A[i+1][j +2] != 'X' and A[i+1][j +2] <= 0:\n",
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" restricted_fields = restricted_fields - 1\n",
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"\n",
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" if i+1<(len(A)) and j -2>=0:\n",
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" if A[i+1][j -2] != 'X' and A[i+1][j -2] <= 0:\n",
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" restricted_fields = restricted_fields - 1\n",
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"\n",
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" if i-2>=0 and j+1<(len(A)):\n",
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" if A[i-2][j +1] != 'X' and A[i-2][j +1] <= 0:\n",
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" restricted_fields = restricted_fields - 1\n",
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"\n",
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" if i-2>=0 and j-1>=0:\n",
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" if A[i-2][j -1]!= 'X' and A[i-2][j -1] <= 0:\n",
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" restricted_fields = restricted_fields - 1\n",
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"\n",
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" if i-1>=0 and j-2>=0:\n",
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" if A[i-1][j -2]!= 'X' and A[i-1][j -2] <= 0:\n",
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" restricted_fields = restricted_fields - 1\n",
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"\n",
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" if i-1>=0 and j+2<len(A):\n",
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" if A[i-1][j +2]!= 'X' and A[i-1][j +2] <= 0:\n",
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" restricted_fields = restricted_fields - 1\n",
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"\n",
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"\n",
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" A[i][j] = restricted_fields\n",
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"\n",
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" if less_restrictions <= restricted_fields: #we keep our best solution (highest number, minimum = -8, highest = 0(best scenario))\n",
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" less_restrictions = restricted_fields\n",
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" pos_x = i\n",
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" pos_y = j\n",
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"\n",
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"\n",
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" return (pos_x,pos_y,A)"
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],
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"metadata": {
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"id": "M815-gViEBDJ"
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},
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"execution_count": 11,
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"outputs": []
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},
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{
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"cell_type": "code",
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"source": [
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"#Placing the knights\n",
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"def place_knights(chessboard):\n",
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" global current_knight\n",
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"\n",
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" if current_knight == 1:\n",
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" first_move = True\n",
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" else:\n",
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" first_move = False\n",
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" dim = len(chessboard)\n",
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"\n",
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" if first_move: #On first move, we got no restrictions yet. By our heuristic, we will start in one of the 4 corners since it only blocks 2 squares\n",
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"\n",
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" #We will choose randomly the first square to be occupied\n",
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"\n",
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" corners = [0,dim-1]\n",
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" pos_x = random.choice(corners)\n",
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" pos_y = random.choice(corners)\n",
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" chessboard[pos_x][pos_y] = current_knight\n",
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"\n",
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"\n",
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" #We restrict positions on our chessboard based on the knight that was placed\n",
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" place_attacks(pos_x,pos_y,chessboard)\n",
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"\n",
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" current_knight = current_knight + 1\n",
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"\n",
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"\n",
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" #After first move\n",
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" else:\n",
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" for i in range(dim*dim - 1):\n",
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"\n",
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" pos = check_attacks(chessboard)\n",
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" if pos[0] == None:\n",
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" break\n",
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" chessboard[pos[0]][pos[1]] = current_knight #Alocamos o i-ésimo cavaleiro a melhor posição possível (segundo nossa heurística).\n",
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" place_attacks(pos[0],pos[1],chessboard)\n",
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"\n",
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" current_knight = current_knight + 1\n",
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" #return chessboard\n",
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"\n",
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" return reset_chessboard()\n",
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" return"
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],
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"metadata": {
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"id": "9GR6nY_uD6YS"
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},
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"execution_count": 12,
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"outputs": []
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},
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{
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"cell_type": "code",
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"source": [
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"#We keep our chessboard and the number of knights on that board\n",
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"#We reset the board to start another one, if necessary\n",
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"def reset_chessboard():\n",
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" global chessboard_solutions\n",
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" global solutions\n",
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" global current_knight\n",
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" global chessboard\n",
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"\n",
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" chessboard_solutions.append(chessboard)\n",
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" n_knights = current_knight - 1\n",
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" solutions.append(n_knights)\n",
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" current_knight = 1\n",
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"\n",
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" dim= len(chessboard)\n",
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" chessboard = []\n",
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" row = [0]*dim\n",
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" for i in range(dim):\n",
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" chessboard.append(row[:])\n",
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"\n",
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" return chessboard_solutions,solutions"
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],
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"metadata": {
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"id": "jdA6fqVeD11V"
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},
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"execution_count": 13,
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"outputs": []
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},
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{
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"cell_type": "code",
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"source": [
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"def branch(P): #we will branch our board P in P0 (without putting a knight on the next position) and in P1(having a knight on the next position)\n",
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" P1 = copy.deepcopy(P)\n",
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"#----------------------------------------PRUNE-------------------------------------------------------\n",
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" next = 1 #variable to allow the continuation of our branches\n",
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" #check if we can prune P and consequently P1\n",
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" #verify prune comparing the best solution (putting knights in all remaining squares) with the heuristic solution\n",
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" max_knights = 0\n",
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" for i in range(len(P)):\n",
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" for j in range(len(P)):\n",
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" if P[i][j] != 'X' and (P[i][j] == 'K' or P[i][j] == 0):\n",
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" max_knights = max_knights + 1\n",
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" if max_knights < solutions[0]: #if the highest number is lower, it is not worth it, we return None\n",
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" P = None\n",
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" P1 = None\n",
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" return (P,P1,next)\n",
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"#-----------------------------------------------------------------------------------------------------\n",
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" global final_solutions #vector where we will add our solutions\n",
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"\n",
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" next = 0\n",
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" for i in range(len(P)):\n",
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" for j in range(len(P)):\n",
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" if P[i][j] == 0: #if exists at least one 0 element (square available), we will put our markings X -> no Knight, K -> Knight\n",
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" next = 1\n",
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" P[i][j] = 'X'\n",
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" P1[i][j] = 'K'\n",
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" place_attacks(i,j,P1)\n",
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" break\n",
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" if next == 1:\n",
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" break\n",
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" else:\n",
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" if P not in final_solutions: #Discarding repeated solutions\n",
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" final_solutions.append(P)\n",
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" return (P,P1,next)"
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],
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"metadata": {
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"id": "Pkxcmu7-Dvws"
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},
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"execution_count": 14,
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"outputs": []
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},
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{
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"cell_type": "code",
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"source": [
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"#--------------------------------------------------------- MAIN -------------------------------------------------------------------\n",
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"#initializing variables and global variables (our chessboard)\n",
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"dim = int(input(\"Chessboard dimension: \\n\"))\n",
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"chessboard = initialize_chessboard(dim)\n",
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"current_knight = 1\n",
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"chessboard_solutions = []\n",
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"solutions = [] #number of knights on that board\n",
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"first_knight = True #for heuristic's use in first move\n",
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"\n",
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"#--------------------------------------------------------- MAIN HEURISTIC ---------------------------------------------------------\n",
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"#In this section we use our heuristic to estabilish a \"good\" solution\n",
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"print(\"\\n-------------------------------------------------------\\n\")\n",
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"test_solutions = []\n",
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"while len(solutions) < 1:\n",
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" test_solutions.append(place_knights(chessboard)) #keep every board solution and it's number of knights\n",
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"\n",
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"print(\"By our heuristc we have:\")\n",
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"for i in chessboard_solutions[0]:\n",
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" print(i)\n",
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"print('Highest number of knights by our heuristic:',solutions[0],'\\n')\n",
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"\n",
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"#----------------------------------------------------------------------------------------------------------------------------------\n",
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"#Initializing our queue, and solutions vector\n",
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"queue = [initialize_chessboard(dim)]\n",
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"q_solutions = []\n",
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"final_solutions = []\n",
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"\n",
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"while len(queue) > 0:\n",
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" P = queue.pop(0)\n",
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" v = branch(P)\n",
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" if v[2] == 1:\n",
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" if v[0]!= None:\n",
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" queue.append(v[0])\n",
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" if v[1]!= None:\n",
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" queue.append(v[1])\n",
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"\n",
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"solutions_size = len(final_solutions) #keep our number of possible solutions\n",
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"for i in range(solutions_size):\n",
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" count = 0\n",
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" for j in range(dim):\n",
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" for k in range(dim):\n",
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" if final_solutions[i][j][k] == 'K':\n",
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" count = count + 1\n",
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" q_solutions.append(count) #count the number of knights per solution\n",
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" #No need to remove all lists from our queue, in the moment we got no more squares available we stop the process\n",
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"\n",
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"print(\"Solutions possiblle: \", solutions_size)\n",
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"max = idx = 0\n",
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"for i in range(solutions_size):\n",
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" if q_solutions[i] >= max:\n",
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" max = q_solutions[i]\n",
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" idx = i\n",
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"\n",
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"print(\"Highest number of knights possible: \",max)\n",
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"print(\"Possible chessboard configuration: \")\n",
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"for i in final_solutions[idx]:\n",
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" print(i)"
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],
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"metadata": {
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"colab": {
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"base_uri": "https://localhost:8080/"
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},
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"id": "rNrH87sMDJMm",
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"outputId": "809fc858-880b-4817-cf2a-08d89ea3af95"
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},
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"execution_count": 15,
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"outputs": [
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{
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"output_type": "stream",
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"name": "stdout",
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"text": [
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"Chessboard dimension: \n",
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"3\n",
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"\n",
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"-------------------------------------------------------\n",
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"\n",
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"By our heuristc we have:\n",
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"[5, 'X', 4]\n",
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"['X', 2, 'X']\n",
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"[3, 'X', 1]\n",
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"Highest number of knights by our heuristic: 5 \n",
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"\n",
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"Solutions possiblle: 16\n",
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"Highest number of knights possible: 5\n",
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"Possible chessboard configuration: \n",
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"['X', 'K', 'X']\n",
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"['K', 'K', 'K']\n",
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"['X', 'K', 'X']\n"
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]
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}
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]
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},
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{
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"cell_type": "code",
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"source": [],
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"metadata": {
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"id": "h5daznWRwytg"
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},
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"execution_count": 15,
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"outputs": []
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}
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]
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}

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