|
| 1 | +""" |
| 2 | + Binomial Heap |
| 3 | + |
| 4 | + Reference: Advanced Data Structures, Peter Brass |
| 5 | +""" |
| 6 | + |
| 7 | + |
| 8 | +class Node: |
| 9 | + """ |
| 10 | + Node in a doubly-linked binomial tree, containing: |
| 11 | + - value |
| 12 | + - size of left subtree |
| 13 | + - link to left, right and parent nodes |
| 14 | + """ |
| 15 | + |
| 16 | + def __init__(self, val): |
| 17 | + self.val = val |
| 18 | + # Number of nodes in left subtree |
| 19 | + self.left_tree_size = 0 |
| 20 | + self.left = None |
| 21 | + self.right = None |
| 22 | + self.parent = None |
| 23 | + |
| 24 | + def mergeTrees(self, other): |
| 25 | + """ |
| 26 | + In-place merge of two binomial trees of equal size. |
| 27 | + Returns the root of the resulting tree |
| 28 | + """ |
| 29 | + assert ( |
| 30 | + self.left_tree_size == other.left_tree_size |
| 31 | + ), "Unequal Sizes of Blocks" |
| 32 | + |
| 33 | + if self.val < other.val: |
| 34 | + other.left = self.right |
| 35 | + other.parent = None |
| 36 | + if self.right: |
| 37 | + self.right.parent = other |
| 38 | + self.right = other |
| 39 | + self.left_tree_size = ( |
| 40 | + self.left_tree_size * 2 + 1 |
| 41 | + ) |
| 42 | + return self |
| 43 | + else: |
| 44 | + self.left = other.right |
| 45 | + self.parent = None |
| 46 | + if other.right: |
| 47 | + other.right.parent = self |
| 48 | + other.right = self |
| 49 | + other.left_tree_size = ( |
| 50 | + other.left_tree_size * 2 + 1 |
| 51 | + ) |
| 52 | + return other |
| 53 | + |
| 54 | + |
| 55 | +class BinomialHeap: |
| 56 | + """ |
| 57 | + Min-oriented priority queue implemented with the Binomial Heap data |
| 58 | + structure implemented with the BinomialHeap class. It supports: |
| 59 | + |
| 60 | + - Insert element in a heap with n elemnts: Guaranteed logn, amoratized 1 |
| 61 | + - Merge (meld) heaps of size m and n: O(logn + logm) |
| 62 | + - Delete Min: O(logn) |
| 63 | + - Peek (return min without deleting it): O(1) |
| 64 | + |
| 65 | + Example: |
| 66 | + |
| 67 | + Create a random permutation of 30 integers to be inserted and |
| 68 | + 19 of them deleted |
| 69 | + >>> import numpy as np |
| 70 | + >>> permutation = np.random.permutation(list(range(30))) |
| 71 | +
|
| 72 | + Create a Heap and insert the 30 integers |
| 73 | + |
| 74 | + __init__() test |
| 75 | + >>> first_heap = BinomialHeap() |
| 76 | +
|
| 77 | + 30 inserts - insert() test |
| 78 | + >>> for number in permutation: |
| 79 | + ... first_heap.insert(number) |
| 80 | + |
| 81 | + Size test |
| 82 | + >>> print(first_heap.size) |
| 83 | + 30 |
| 84 | + |
| 85 | + Deleting - delete() test |
| 86 | + >>> for i in range(25): |
| 87 | + ... print(first_heap.deleteMin(), end=" ") |
| 88 | + 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 |
| 89 | +
|
| 90 | + Create a new Heap |
| 91 | + >>> second_heap = BinomialHeap() |
| 92 | + >>> vals = [17, 20, 31, 34] |
| 93 | + >>> for value in vals: |
| 94 | + ... second_heap.insert(value) |
| 95 | + |
| 96 | + |
| 97 | + The heap should have the following structure: |
| 98 | + |
| 99 | + 17 |
| 100 | + / \ |
| 101 | + # 31 |
| 102 | + / \ |
| 103 | + 20 34 |
| 104 | + / \ / \ |
| 105 | + # # # # |
| 106 | + |
| 107 | + preOrder() test |
| 108 | + >>> print(second_heap.preOrder()) |
| 109 | + [(17, 0), ('#', 1), (31, 1), (20, 2), ('#', 3), ('#', 3), (34, 2), ('#', 3), ('#', 3)] |
| 110 | + |
| 111 | + printing Heap - __str__() test |
| 112 | + >>> print(second_heap) |
| 113 | + 17 |
| 114 | + -# |
| 115 | + -31 |
| 116 | + --20 |
| 117 | + ---# |
| 118 | + ---# |
| 119 | + --34 |
| 120 | + ---# |
| 121 | + ---# |
| 122 | +
|
| 123 | + mergeHeaps() test |
| 124 | + >>> merged = second_heap.mergeHeaps(first_heap) |
| 125 | + >>> merged.peek() |
| 126 | + 17 |
| 127 | + |
| 128 | + values in merged heap; (merge is inplace) |
| 129 | + >>> while not first_heap.isEmpty(): |
| 130 | + ... print(first_heap.deleteMin(), end=" ") |
| 131 | + 17 20 25 26 27 28 29 31 34 |
| 132 | + |
| 133 | + """ |
| 134 | + |
| 135 | + def __init__( |
| 136 | + self, bottom_root=None, min_node=None, heap_size=0 |
| 137 | + ): |
| 138 | + self.size = heap_size |
| 139 | + self.bottom_root = bottom_root |
| 140 | + self.min_node = min_node |
| 141 | + |
| 142 | + def mergeHeaps(self, other): |
| 143 | + """ |
| 144 | + In-place merge of two binomial heaps. |
| 145 | + Both of them become the resulting merged heap |
| 146 | + """ |
| 147 | + |
| 148 | + # Empty heaps corner cases |
| 149 | + if other.size == 0: |
| 150 | + return |
| 151 | + if self.size == 0: |
| 152 | + self.size = other.size |
| 153 | + self.bottom_root = other.bottom_root |
| 154 | + self.min_node = other.min_node |
| 155 | + return |
| 156 | + # Update size |
| 157 | + self.size = self.size + other.size |
| 158 | + |
| 159 | + # Update min.node |
| 160 | + if self.min_node.val > other.min_node.val: |
| 161 | + self.min_node = other.min_node |
| 162 | + # Merge |
| 163 | + |
| 164 | + # Order roots by left_subtree_size |
| 165 | + combined_roots_list = [] |
| 166 | + i, j = self.bottom_root, other.bottom_root |
| 167 | + while i or j: |
| 168 | + if i and ( |
| 169 | + (not j) |
| 170 | + or i.left_tree_size < j.left_tree_size |
| 171 | + ): |
| 172 | + combined_roots_list.append((i, True)) |
| 173 | + i = i.parent |
| 174 | + else: |
| 175 | + combined_roots_list.append((j, False)) |
| 176 | + j = j.parent |
| 177 | + # Insert links between them |
| 178 | + for i in range(len(combined_roots_list) - 1): |
| 179 | + if ( |
| 180 | + combined_roots_list[i][1] |
| 181 | + != combined_roots_list[i + 1][1] |
| 182 | + ): |
| 183 | + combined_roots_list[i][ |
| 184 | + 0 |
| 185 | + ].parent = combined_roots_list[i + 1][0] |
| 186 | + combined_roots_list[i + 1][ |
| 187 | + 0 |
| 188 | + ].left = combined_roots_list[i][0] |
| 189 | + # Consecutively merge roots with same left_tree_size |
| 190 | + i = combined_roots_list[0][0] |
| 191 | + while i.parent: |
| 192 | + if ( |
| 193 | + ( |
| 194 | + i.left_tree_size |
| 195 | + == i.parent.left_tree_size |
| 196 | + ) |
| 197 | + and (not i.parent.parent) |
| 198 | + ) or ( |
| 199 | + i.left_tree_size == i.parent.left_tree_size |
| 200 | + and i.left_tree_size |
| 201 | + != i.parent.parent.left_tree_size |
| 202 | + ): |
| 203 | + |
| 204 | + # Neighbouring Nodes |
| 205 | + previous_node = i.left |
| 206 | + next_node = i.parent.parent |
| 207 | + |
| 208 | + # Merging trees |
| 209 | + i = i.mergeTrees(i.parent) |
| 210 | + |
| 211 | + # Updating links |
| 212 | + i.left = previous_node |
| 213 | + i.parent = next_node |
| 214 | + if previous_node: |
| 215 | + previous_node.parent = i |
| 216 | + if next_node: |
| 217 | + next_node.left = i |
| 218 | + else: |
| 219 | + i = i.parent |
| 220 | + # Updating self.bottom_root |
| 221 | + while i.left: |
| 222 | + i = i.left |
| 223 | + self.bottom_root = i |
| 224 | + |
| 225 | + # Update other |
| 226 | + other.size = self.size |
| 227 | + other.bottom_root = self.bottom_root |
| 228 | + other.min_node = self.min_node |
| 229 | + |
| 230 | + # Return the merged heap |
| 231 | + return self |
| 232 | + |
| 233 | + def insert(self, val): |
| 234 | + """ |
| 235 | + insert a value in the heap |
| 236 | + """ |
| 237 | + if self.size == 0: |
| 238 | + self.bottom_root = Node(val) |
| 239 | + self.size = 1 |
| 240 | + self.min_node = self.bottom_root |
| 241 | + else: |
| 242 | + # Create new node |
| 243 | + new_node = Node(val) |
| 244 | + |
| 245 | + # Update size |
| 246 | + self.size += 1 |
| 247 | + |
| 248 | + # update min_node |
| 249 | + if val < self.min_node.val: |
| 250 | + self.min_node = new_node |
| 251 | + # Put new_node as a bottom_root in heap |
| 252 | + self.bottom_root.left = new_node |
| 253 | + new_node.parent = self.bottom_root |
| 254 | + self.bottom_root = new_node |
| 255 | + |
| 256 | + # Consecutively merge roots with same left_tree_size |
| 257 | + while ( |
| 258 | + self.bottom_root.parent |
| 259 | + and self.bottom_root.left_tree_size |
| 260 | + == self.bottom_root.parent.left_tree_size |
| 261 | + ): |
| 262 | + |
| 263 | + # Next node |
| 264 | + next_node = self.bottom_root.parent.parent |
| 265 | + |
| 266 | + # Merge |
| 267 | + self.bottom_root = self.bottom_root.mergeTrees( |
| 268 | + self.bottom_root.parent |
| 269 | + ) |
| 270 | + |
| 271 | + # Update Links |
| 272 | + self.bottom_root.parent = next_node |
| 273 | + self.bottom_root.left = None |
| 274 | + if next_node: |
| 275 | + next_node.left = self.bottom_root |
| 276 | + |
| 277 | + def peek(self): |
| 278 | + """ |
| 279 | + return min element without deleting it |
| 280 | + """ |
| 281 | + return self.min_node.val |
| 282 | + |
| 283 | + def isEmpty(self): |
| 284 | + return self.size == 0 |
| 285 | + |
| 286 | + def deleteMin(self): |
| 287 | + """ |
| 288 | + delete min element and return it |
| 289 | + """ |
| 290 | + # assert not self.isEmpty(), "Empty Heap" |
| 291 | + |
| 292 | + # Save minimal value |
| 293 | + min_value = self.min_node.val |
| 294 | + |
| 295 | + # Last element in heap corner case |
| 296 | + if self.size == 1: |
| 297 | + # Update size |
| 298 | + self.size = 0 |
| 299 | + |
| 300 | + # Update bottom root |
| 301 | + self.bottom_root = None |
| 302 | + |
| 303 | + # Update min_node |
| 304 | + self.min_node = None |
| 305 | + |
| 306 | + return min_value |
| 307 | + # No right subtree corner case |
| 308 | + # The structure of the tree implies that this should be the bottom root |
| 309 | + # and there is at least one other root |
| 310 | + if self.min_node.right == None: |
| 311 | + # Update size |
| 312 | + self.size -= 1 |
| 313 | + |
| 314 | + # Update bottom root |
| 315 | + self.bottom_root = self.bottom_root.parent |
| 316 | + self.bottom_root.left = None |
| 317 | + |
| 318 | + # Update min_node |
| 319 | + self.min_node = self.bottom_root |
| 320 | + i = self.bottom_root.parent |
| 321 | + while i: |
| 322 | + if i.val < self.min_node.val: |
| 323 | + self.min_node = i |
| 324 | + i = i.parent |
| 325 | + return min_value |
| 326 | + # General case |
| 327 | + # Find the BinomialHeap of the right subtree of min_node |
| 328 | + bottom_of_new = self.min_node.right |
| 329 | + bottom_of_new.parent = None |
| 330 | + min_of_new = bottom_of_new |
| 331 | + size_of_new = 1 |
| 332 | + |
| 333 | + # Size, min_node and bottom_root |
| 334 | + while bottom_of_new.left: |
| 335 | + size_of_new = size_of_new * 2 + 1 |
| 336 | + bottom_of_new = bottom_of_new.left |
| 337 | + if bottom_of_new.val < min_of_new.val: |
| 338 | + min_of_new = bottom_of_new |
| 339 | + # Corner case of single root on top left path |
| 340 | + if (not self.min_node.left) and ( |
| 341 | + not self.min_node.parent |
| 342 | + ): |
| 343 | + self.size = size_of_new |
| 344 | + self.bottom_root = bottom_of_new |
| 345 | + self.min_node = min_of_new |
| 346 | + # print("Single root, multiple nodes case") |
| 347 | + return min_value |
| 348 | + # Remaining cases |
| 349 | + # Construct heap of right subtree |
| 350 | + newHeap = BinomialHeap( |
| 351 | + bottom_root=bottom_of_new, |
| 352 | + min_node=min_of_new, |
| 353 | + heap_size=size_of_new, |
| 354 | + ) |
| 355 | + |
| 356 | + # Update size |
| 357 | + self.size = self.size - 1 - size_of_new |
| 358 | + |
| 359 | + # Neighbour nodes |
| 360 | + previous_node = self.min_node.left |
| 361 | + next_node = self.min_node.parent |
| 362 | + |
| 363 | + # Initialize new bottom_root and min_node |
| 364 | + self.min_node = previous_node or next_node |
| 365 | + self.bottom_root = next_node |
| 366 | + |
| 367 | + # Update links of previous_node and search below for new min_node and |
| 368 | + # bottom_root |
| 369 | + if previous_node: |
| 370 | + previous_node.parent = next_node |
| 371 | + |
| 372 | + # Update bottom_root and search for min_node below |
| 373 | + self.bottom_root = previous_node |
| 374 | + self.min_node = previous_node |
| 375 | + while self.bottom_root.left: |
| 376 | + self.bottom_root = self.bottom_root.left |
| 377 | + if self.bottom_root.val < self.min_node.val: |
| 378 | + self.min_node = self.bottom_root |
| 379 | + if next_node: |
| 380 | + next_node.left = previous_node |
| 381 | + |
| 382 | + # Search for new min_node above min_node |
| 383 | + i = next_node |
| 384 | + while i: |
| 385 | + if i.val < self.min_node.val: |
| 386 | + self.min_node = i |
| 387 | + i = i.parent |
| 388 | + # Merge heaps |
| 389 | + self.mergeHeaps(newHeap) |
| 390 | + |
| 391 | + return min_value |
| 392 | + |
| 393 | + def preOrder(self): |
| 394 | + """ |
| 395 | + Returns the Pre-order representation of the heap including |
| 396 | + values of nodes plus their level distance from the root; |
| 397 | + Empty nodes appear as # |
| 398 | + """ |
| 399 | + # Find top root |
| 400 | + top_root = self.bottom_root |
| 401 | + while top_root.parent: |
| 402 | + top_root = top_root.parent |
| 403 | + # preorder |
| 404 | + heap_preOrder = [] |
| 405 | + self.__traversal(top_root, heap_preOrder) |
| 406 | + return heap_preOrder |
| 407 | + |
| 408 | + def __traversal(self, curr_node, preorder, level=0): |
| 409 | + """ |
| 410 | + Pre-order traversal of nodes |
| 411 | + """ |
| 412 | + if curr_node: |
| 413 | + preorder.append((curr_node.val, level)) |
| 414 | + self.__traversal( |
| 415 | + curr_node.left, preorder, level + 1 |
| 416 | + ) |
| 417 | + self.__traversal( |
| 418 | + curr_node.right, preorder, level + 1 |
| 419 | + ) |
| 420 | + else: |
| 421 | + preorder.append(("#", level)) |
| 422 | + |
| 423 | + def __str__(self): |
| 424 | + """ |
| 425 | + Overwriting str for a pre-order print of nodes in heap; |
| 426 | + Performance is poor, so use only for small examples |
| 427 | + """ |
| 428 | + if self.isEmpty(): |
| 429 | + return "" |
| 430 | + preorder_heap = self.preOrder() |
| 431 | + |
| 432 | + return "\n".join( |
| 433 | + ("-" * level + str(value)) |
| 434 | + for value, level in preorder_heap |
| 435 | + ) |
| 436 | + |
| 437 | + |
| 438 | +# Unit Tests |
| 439 | +if __name__ == "__main__": |
| 440 | + import doctest |
| 441 | + |
| 442 | + doctest.testmod() |
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