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4a3f117 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 | BACKTRACKING
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Backtracking explores all possible solutions by building candidates incrementally.
If a candidate fails, it backtracks and tries the next option.
Template:
def backtrack(state, choices):
if is_solution(state):
record(state)
return
for choice in choices:
if is_valid(state, choice):
make_choice(state, choice)
backtrack(state, remaining_choices)
undo_choice(state, choice) # BACKTRACK
--- Generate All Subsets ---
def subsets(nums):
result = []
def backtrack(start, current):
result.append(current[:]) # record every state
for i in range(start, len(nums)):
current.append(nums[i])
backtrack(i + 1, current)
current.pop() # backtrack
backtrack(0, [])
return result
print(subsets([1, 2, 3]))
# [[], [1], [1,2], [1,2,3], [1,3], [2], [2,3], [3]]
--- Generate All Permutations ---
def permute(nums):
result = []
def backtrack(current):
if len(current) == len(nums):
result.append(current[:])
return
for num in nums:
if num not in current:
current.append(num)
backtrack(current)
current.pop() # backtrack
backtrack([])
return result
print(permute([1, 2, 3]))
# [[1,2,3], [1,3,2], [2,1,3], [2,3,1], [3,1,2], [3,2,1]]
--- Combination Sum ---
Find all combinations that sum to target. Numbers can be reused.
def combination_sum(candidates, target):
result = []
candidates.sort()
def backtrack(start, current, remaining):
if remaining == 0:
result.append(current[:])
return
for i in range(start, len(candidates)):
if candidates[i] > remaining:
break
current.append(candidates[i])
backtrack(i, current, remaining - candidates[i]) # i (not i+1) allows reuse
current.pop()
backtrack(0, [], target)
return result
print(combination_sum([2, 3, 6, 7], 7))
# [[2, 2, 3], [7]]
--- N-Queens Problem ---
Place N queens on NxN board so no two queens attack each other.
def solve_n_queens(n):
result = []
cols = set()
diag1 = set() # row - col
diag2 = set() # row + col
board = [['.' for _ in range(n)] for _ in range(n)]
def backtrack(row):
if row == n:
result.append([''.join(r) for r in board])
return
for col in range(n):
if col in cols or (row - col) in diag1 or (row + col) in diag2:
continue
cols.add(col)
diag1.add(row - col)
diag2.add(row + col)
board[row][col] = 'Q'
backtrack(row + 1)
cols.remove(col) # backtrack
diag1.remove(row - col)
diag2.remove(row + col)
board[row][col] = '.'
backtrack(0)
return result
solutions = solve_n_queens(4)
print(f"Number of solutions for 4-Queens: {len(solutions)}") # 2
for sol in solutions:
for row in sol:
print(row)
print()
--- Sudoku Solver ---
Fill 9x9 grid so each row, column, and 3x3 box has digits 1-9.
def solve_sudoku(board):
def is_valid(board, row, col, num):
num = str(num)
if num in board[row]:
return False
if num in [board[r][col] for r in range(9)]:
return False
box_r, box_c = 3 * (row // 3), 3 * (col // 3)
for r in range(box_r, box_r + 3):
for c in range(box_c, box_c + 3):
if board[r][c] == num:
return False
return True
def backtrack():
for r in range(9):
for c in range(9):
if board[r][c] == '.':
for num in range(1, 10):
if is_valid(board, r, c, num):
board[r][c] = str(num)
if backtrack():
return True
board[r][c] = '.' # backtrack
return False
return True
backtrack()
--- Word Search in Grid ---
Find if word exists in 2D grid by moving up/down/left/right.
def word_search(board, word):
rows, cols = len(board), len(board[0])
def backtrack(r, c, index):
if index == len(word):
return True
if r < 0 or r >= rows or c < 0 or c >= cols:
return False
if board[r][c] != word[index]:
return False
temp = board[r][c]
board[r][c] = '#' # mark visited
found = (backtrack(r+1, c, index+1) or
backtrack(r-1, c, index+1) or
backtrack(r, c+1, index+1) or
backtrack(r, c-1, index+1))
board[r][c] = temp # backtrack (restore)
return found
for r in range(rows):
for c in range(cols):
if backtrack(r, c, 0):
return True
return False
board = [["A","B","C","E"],["S","F","C","S"],["A","D","E","E"]]
print(word_search(board, "ABCCED")) # True
print(word_search(board, "SEE")) # True
print(word_search(board, "ABCB")) # False
--- Rat in a Maze ---
Find path from (0,0) to (n-1,n-1) in a grid (1=open, 0=blocked).
def rat_in_maze(maze):
n = len(maze)
solution = [[0]*n for _ in range(n)]
def is_safe(r, c):
return 0 <= r < n and 0 <= c < n and maze[r][c] == 1
def solve(r, c):
if r == n-1 and c == n-1:
solution[r][c] = 1
return True
if is_safe(r, c):
solution[r][c] = 1
if solve(r+1, c): return True
if solve(r, c+1): return True
solution[r][c] = 0 # backtrack
return False
return False
if solve(0, 0):
return solution
return []
maze = [[1,0,0,0],[1,1,0,1],[0,1,0,0],[1,1,1,1]]
path = rat_in_maze(maze)
for row in path:
print(row)
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