binomial_heap.py

"""
Binomial Heap
Reference: Advanced Data Structures, Peter Brass
"""


class Node:
    """
    Node in a doubly-linked binomial tree, containing:
        - value
        - size of left subtree
        - link to left, right and parent nodes
    """

    def __init__(self, val):
        self.val = val
        # Number of nodes in left subtree
        self.left_tree_size = 0
        self.left = None
        self.right = None
        self.parent = None

    def merge_trees(self, other):
        """
        In-place merge of two binomial trees of equal size.
        Returns the root of the resulting tree
        """
        assert self.left_tree_size == other.left_tree_size, "Unequal Sizes of Blocks"

        if self.val < other.val:
            other.left = self.right
            other.parent = None
            if self.right:
                self.right.parent = other
            self.right = other
            self.left_tree_size = self.left_tree_size * 2 + 1
            return self
        else:
            self.left = other.right
            self.parent = None
            if other.right:
                other.right.parent = self
            other.right = self
            other.left_tree_size = other.left_tree_size * 2 + 1
            return other


class BinomialHeap:
    r"""
    Min-oriented priority queue implemented with the Binomial Heap data
    structure implemented with the BinomialHeap class. It supports:
        - Insert element in a heap with n elements: Guaranteed logn, amoratized 1
        - Merge (meld) heaps of size m and n: O(logn + logm)
        - Delete Min: O(logn)
        - Peek (return min without deleting it): O(1)

    Example:

    Create a random permutation of 30 integers to be inserted and 19 of them deleted
    >>> import numpy as np
    >>> permutation = np.random.permutation(list(range(30)))

    Create a Heap and insert the 30 integers
    __init__() test
    >>> first_heap = BinomialHeap()

    30 inserts - insert() test
    >>> for number in permutation:
    ...     first_heap.insert(number)

    Size test
    >>> first_heap.size
    30

    Deleting - delete() test
    >>> [first_heap.delete_min() for _ in range(20)]
    [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]

    Create a new Heap
    >>> second_heap = BinomialHeap()
    >>> vals = [17, 20, 31, 34]
    >>> for value in vals:
    ...     second_heap.insert(value)


    The heap should have the following structure:

                    17
                   /  \
                  #    31
                      /  \
                    20    34
                   /  \  /  \
                  #    # #   #

    preOrder() test
    >>> " ".join(str(x) for x in second_heap.pre_order())
    "(17, 0) ('#', 1) (31, 1) (20, 2) ('#', 3) ('#', 3) (34, 2) ('#', 3) ('#', 3)"

    printing Heap - __str__() test
    >>> print(second_heap)
    17
    -#
    -31
    --20
    ---#
    ---#
    --34
    ---#
    ---#

    mergeHeaps() test
    >>>
    >>> merged = second_heap.merge_heaps(first_heap)
    >>> merged.peek()
    17

    values in merged heap; (merge is inplace)
    >>> results = []
    >>> while not first_heap.is_empty():
    ...     results.append(first_heap.delete_min())
    >>> results
    [17, 20, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 34]
    """

    def __init__(self, bottom_root=None, min_node=None, heap_size=0):
        self.size = heap_size
        self.bottom_root = bottom_root
        self.min_node = min_node

    def merge_heaps(self, other):
        """
        In-place merge of two binomial heaps.
        Both of them become the resulting merged heap
        """

        # Empty heaps corner cases
        if other.size == 0:
            return
        if self.size == 0:
            self.size = other.size
            self.bottom_root = other.bottom_root
            self.min_node = other.min_node
            return
        # Update size
        self.size = self.size + other.size

        # Update min.node
        if self.min_node.val > other.min_node.val:
            self.min_node = other.min_node
        # Merge

        # Order roots by left_subtree_size
        combined_roots_list = []
        i, j = self.bottom_root, other.bottom_root
        while i or j:
            if i and ((not j) or i.left_tree_size < j.left_tree_size):
                combined_roots_list.append((i, True))
                i = i.parent
            else:
                combined_roots_list.append((j, False))
                j = j.parent
        # Insert links between them
        for i in range(len(combined_roots_list) - 1):
            if combined_roots_list[i][1] != combined_roots_list[i + 1][1]:
                combined_roots_list[i][0].parent = combined_roots_list[i + 1][0]
                combined_roots_list[i + 1][0].left = combined_roots_list[i][0]
        # Consecutively merge roots with same left_tree_size
        i = combined_roots_list[0][0]
        while i.parent:
            if (
                (i.left_tree_size == i.parent.left_tree_size) and (not i.parent.parent)
            ) or (
                i.left_tree_size == i.parent.left_tree_size
                and i.left_tree_size != i.parent.parent.left_tree_size
            ):

                # Neighbouring Nodes
                previous_node = i.left
                next_node = i.parent.parent

                # Merging trees
                i = i.merge_trees(i.parent)

                # Updating links
                i.left = previous_node
                i.parent = next_node
                if previous_node:
                    previous_node.parent = i
                if next_node:
                    next_node.left = i
            else:
                i = i.parent
        # Updating self.bottom_root
        while i.left:
            i = i.left
        self.bottom_root = i

        # Update other
        other.size = self.size
        other.bottom_root = self.bottom_root
        other.min_node = self.min_node

        # Return the merged heap
        return self

    def insert(self, val):
        """
        insert a value in the heap
        """
        if self.size == 0:
            self.bottom_root = Node(val)
            self.size = 1
            self.min_node = self.bottom_root
        else:
            # Create new node
            new_node = Node(val)

            # Update size
            self.size += 1

            # update min_node
            if val < self.min_node.val:
                self.min_node = new_node
            # Put new_node as a bottom_root in heap
            self.bottom_root.left = new_node
            new_node.parent = self.bottom_root
            self.bottom_root = new_node

            # Consecutively merge roots with same left_tree_size
            while (
                self.bottom_root.parent
                and self.bottom_root.left_tree_size
                == self.bottom_root.parent.left_tree_size
            ):

                # Next node
                next_node = self.bottom_root.parent.parent

                # Merge
                self.bottom_root = self.bottom_root.merge_trees(self.bottom_root.parent)

                # Update Links
                self.bottom_root.parent = next_node
                self.bottom_root.left = None
                if next_node:
                    next_node.left = self.bottom_root

    def peek(self):
        """
        return min element without deleting it
        """
        return self.min_node.val

    def is_empty(self):
        return self.size == 0

    def delete_min(self):
        """
        delete min element and return it
        """
        # assert not self.isEmpty(), "Empty Heap"

        # Save minimal value
        min_value = self.min_node.val

        # Last element in heap corner case
        if self.size == 1:
            # Update size
            self.size = 0

            # Update bottom root
            self.bottom_root = None

            # Update min_node
            self.min_node = None

            return min_value
        # No right subtree corner case
        # The structure of the tree implies that this should be the bottom root
        # and there is at least one other root
        if self.min_node.right is None:
            # Update size
            self.size -= 1

            # Update bottom root
            self.bottom_root = self.bottom_root.parent
            self.bottom_root.left = None

            # Update min_node
            self.min_node = self.bottom_root
            i = self.bottom_root.parent
            while i:
                if i.val < self.min_node.val:
                    self.min_node = i
                i = i.parent
            return min_value
        # General case
        # Find the BinomialHeap of the right subtree of min_node
        bottom_of_new = self.min_node.right
        bottom_of_new.parent = None
        min_of_new = bottom_of_new
        size_of_new = 1

        # Size, min_node and bottom_root
        while bottom_of_new.left:
            size_of_new = size_of_new * 2 + 1
            bottom_of_new = bottom_of_new.left
            if bottom_of_new.val < min_of_new.val:
                min_of_new = bottom_of_new
        # Corner case of single root on top left path
        if (not self.min_node.left) and (not self.min_node.parent):
            self.size = size_of_new
            self.bottom_root = bottom_of_new
            self.min_node = min_of_new
            # print("Single root, multiple nodes case")
            return min_value
        # Remaining cases
        # Construct heap of right subtree
        new_heap = BinomialHeap(
            bottom_root=bottom_of_new, min_node=min_of_new, heap_size=size_of_new
        )

        # Update size
        self.size = self.size - 1 - size_of_new

        # Neighbour nodes
        previous_node = self.min_node.left
        next_node = self.min_node.parent

        # Initialize new bottom_root and min_node
        self.min_node = previous_node or next_node
        self.bottom_root = next_node

        # Update links of previous_node and search below for new min_node and
        # bottom_root
        if previous_node:
            previous_node.parent = next_node

            # Update bottom_root and search for min_node below
            self.bottom_root = previous_node
            self.min_node = previous_node
            while self.bottom_root.left:
                self.bottom_root = self.bottom_root.left
                if self.bottom_root.val < self.min_node.val:
                    self.min_node = self.bottom_root
        if next_node:
            next_node.left = previous_node

            # Search for new min_node above min_node
            i = next_node
            while i:
                if i.val < self.min_node.val:
                    self.min_node = i
                i = i.parent
        # Merge heaps
        self.merge_heaps(new_heap)

        return min_value

    def pre_order(self):
        """
        Returns the Pre-order representation of the heap including
        values of nodes plus their level distance from the root;
        Empty nodes appear as #
        """
        # Find top root
        top_root = self.bottom_root
        while top_root.parent:
            top_root = top_root.parent
        # preorder
        heap_pre_order = []
        self.__traversal(top_root, heap_pre_order)
        return heap_pre_order

    def __traversal(self, curr_node, preorder, level=0):
        """
        Pre-order traversal of nodes
        """
        if curr_node:
            preorder.append((curr_node.val, level))
            self.__traversal(curr_node.left, preorder, level + 1)
            self.__traversal(curr_node.right, preorder, level + 1)
        else:
            preorder.append(("#", level))

    def __str__(self):
        """
        Overwriting str for a pre-order print of nodes in heap;
        Performance is poor, so use only for small examples
        """
        if self.is_empty():
            return ""
        preorder_heap = self.pre_order()

        return "\n".join(("-" * level + str(value)) for value, level in preorder_heap)


# Unit Tests
if __name__ == "__main__":
    import doctest

    doctest.testmod()