Merge 071ce716e4 into f3f32ae3ca

data_structures/heap/fibonacci_heap.py

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+"""
+Fibonacci Heap
+A more efficient priority queue implementation that provides amortized time bounds
+that are better than those of the binary and binomial heaps.
+reference: https://en.wikipedia.org/wiki/Fibonacci_heap
+
+Operations supported:
+- Insert: O(1) amortized
+- Find minimum: O(1)
+- Delete minimum: O(log n) amortized
+- Decrease key: O(1) amortized
+- Merge: O(1)
+"""
+
+
+class Node:
+    """
+    A node in a Fibonacci heap.
+
+    Args:
+        val: The value stored in the node.
+
+    Attributes:
+        val: The value stored in the node.
+        parent: Reference to parent node.
+        child: Reference to one child node.
+        left: Reference to left sibling.
+        right: Reference to right sibling.
+        degree: Number of children.
+        mark: Boolean indicating if node has lost a child.
+    """
+
+    def __init__(self, val):
+        self.val = val
+        self.parent = None
+        self.child = None
+        self.left = self
+        self.right = self
+        self.degree = 0
+        self.mark = False
+
+    def add_sibling(self, node):
+        """
+        Adds a node as a sibling to the current node.
+
+        Args:
+            node: The node to add as a sibling.
+        """
+        node.left = self
+        node.right = self.right
+        self.right.left = node
+        self.right = node
+
+    def add_child(self, node):
+        """
+        Adds a node as a child of the current node.
+
+        Args:
+            node: The node to add as a child.
+        """
+        node.parent = self
+        if not self.child:
+            self.child = node
+        else:
+            self.child.add_sibling(node)
+        self.degree += 1
+
+    def remove(self):
+        """Removes this node from its sibling list."""
+        self.left.right = self.right
+        self.right.left = self.left
+
+
+class FibonacciHeap:
+    """
+    A Fibonacci heap implementation providing
+    amortized efficient priority queue operations.
+
+    This implementation provides the following time complexities:
+        - Insert: O(1) amortized
+        - Find minimum: O(1)
+        - Delete minimum: O(log n) amortized
+        - Decrease key: O(1) amortized
+        - Merge: O(1)
+
+    Example:
+    >>> heap = FibonacciHeap()
+    >>> node1 = heap.insert(3)
+    >>> node2 = heap.insert(2)
+    >>> node3 = heap.insert(15)
+    >>> heap.peek()
+    2
+    >>> heap.delete_min()
+    2
+    >>> heap.peek()
+    3
+    >>> other_heap = FibonacciHeap()
+    >>> node4 = other_heap.insert(1)
+    >>> heap.merge_heaps(other_heap)
+    >>> heap.peek()
+    1
+    """
+
+    def __init__(self):
+        """Initializes an empty Fibonacci heap."""
+        self.min_node = None
+        self.size = 0
+
+    def is_empty(self):
+        """
+        Checks if the heap is empty.
+
+        Returns:
+            bool: True if heap is empty, False otherwise.
+        """
+        return self.min_node is None
+
+    def insert(self, val):
+        """
+        Inserts a new value into the heap.
+
+        Args:
+            val: Value to insert.
+
+        Returns:
+            Node: The newly created node.
+        """
+        node = Node(val)
+        if not self.min_node:
+            self.min_node = node
+        else:
+            self.min_node.add_sibling(node)
+            if node.val < self.min_node.val:
+                self.min_node = node
+        self.size += 1
+        return node
+
+    def peek(self):
+        """
+        Returns the minimum value without removing it.
+
+        Returns:
+            The minimum value in the heap.
+
+        Raises:
+            IndexError: If the heap is empty.
+        """
+        if not self.min_node:
+            raise IndexError("Heap is empty")
+        return self.min_node.val
+
+    def merge_heaps(self, other):
+        """
+        Merges another Fibonacci heap into this one.
+
+        Args:
+            other: Another FibonacciHeap instance to merge with this one.
+        """
+        if not other.min_node:
+            return
+        if not self.min_node:
+            self.min_node = other.min_node
+        else:
+            # Merge root lists
+            self.min_node.right.left = other.min_node.left
+            other.min_node.left.right = self.min_node.right
+            self.min_node.right = other.min_node
+            other.min_node.left = self.min_node
+
+            if other.min_node.val < self.min_node.val:
+                self.min_node = other.min_node
+
+        self.size += other.size
+
+    def __link_trees(self, node1, node2):
+        """
+        Links two trees of same degree.
+
+        Args:
+            node1: First tree's root node.
+            node2: Second tree's root node.
+        """
+        node1.remove()
+        if node2.child:
+            node2.child.add_sibling(node1)
+        else:
+            node2.child = node1
+        node1.parent = node2
+        node2.degree += 1
+        node1.mark = False
+
+    def delete_min(self):
+        """
+        Removes and returns the minimum value from the heap.
+
+        Returns:
+            The minimum value that was removed.
+
+        Raises:
+            IndexError: If the heap is empty.
+        """
+        if not self.min_node:
+            raise IndexError("Heap is empty")
+
+        min_val = self.min_node.val
+
+        # Add all children to root list
+        if self.min_node.child:
+            curr = self.min_node.child
+            while True:
+                next_node = curr.right
+                curr.parent = None
+                curr.mark = False
+                self.min_node.add_sibling(curr)
+                if curr.right == self.min_node.child:
+                    break
+                curr = next_node
+
+        # Remove minimum node
+        if self.min_node.right == self.min_node:
+            self.min_node = None
+        else:
+            self.min_node.remove()
+            self.min_node = self.min_node.right
+            self.__consolidate()
+
+        self.size -= 1
+        return min_val
+
+    def __consolidate(self):
+        """
+        Consolidates the trees in the heap after a delete_min operation.
+
+        This is an internal method that maintains the heap's structure.
+        """
+        max_degree = int(self.size**0.5) + 1
+        degree_table = [None] * max_degree
+
+        # Collect all roots
+        roots = []
+        curr = self.min_node
+        while True:
+            roots.append(curr)
+            curr = curr.right
+            if curr == self.min_node:
+                break
+
+        # Consolidate trees
+        for root in roots:
+            degree = root.degree
+            while degree_table[degree]:
+                other = degree_table[degree]
+                if root.val > other.val:
+                    root, other = other, root
+                self.__link_trees(other, root)
+                degree_table[degree] = None
+                degree += 1
+            degree_table[degree] = root
+
+        # Find new minimum
+        self.min_node = None
+        for degree in range(max_degree):
+            if degree_table[degree]:
+                if not self.min_node:
+                    self.min_node = degree_table[degree]
+                    self.min_node.left = self.min_node
+                    self.min_node.right = self.min_node
+                else:
+                    self.min_node.add_sibling(degree_table[degree])
+                    if degree_table[degree].val < self.min_node.val:
+                        self.min_node = degree_table[degree]
+
+    def decrease_key(self, node, new_val):
+        """
+        Decreases the value of a node.
+
+        Args:
+            node: The node whose value should be decreased.
+            new_val: The new value for the node.
+
+        Raises:
+            ValueError: If new value is greater than current value.
+        """
+        if new_val > node.val:
+            raise ValueError("New value is greater than current value")
+
+        node.val = new_val
+        parent = node.parent
+
+        if parent and node.val < parent.val:
+            self.__cut(node, parent)
+            self.__cascading_cut(parent)
+
+        if node.val < self.min_node.val:
+            self.min_node = node
+
+    def __cut(self, node, parent):
+        """
+        Cuts a node from its parent.
+
+        Args:
+            node: Node to be cut.
+            parent: Parent of the node to be cut.
+        """ """
+        Performs cascading cut operation.
+
+        Args:
+            node: Starting node for cascading cut.
+        """
+
+        parent.degree -= 1
+        if parent.child == node:
+            parent.child = node.right if node.right != node else None
+        node.remove()
+        node.left = node
+        node.right = node
+        node.parent = None
+        node.mark = False
+        self.min_node.add_sibling(node)
+
+    def __cascading_cut(self, node):
+        """
+        Performs cascading cut operation.
+
+        Args:
+            node: Starting node for cascading cut.
+        """
+
+        if parent := node.parent:
+            if not node.mark:
+                node.mark = True
+            else:
+                self.__cut(node, parent)
+                self.__cascading_cut(parent)
+
+    def __str__(self):
+        """
+        Returns a string representation of the heap.
+
+        Returns:
+            str: A string showing the heap structure.
+        """
+        if not self.min_node:
+            return "Empty heap"
+
+        def print_tree(node, level=0):
+            result = []
+            curr = node
+            while True:
+                result.append("-" * level + str(curr.val))
+                if curr.child:
+                    result.extend(print_tree(curr.child, level + 1))
+                curr = curr.right
+                if curr == node:
+                    break
+            return result
+
+        return "\n".join(print_tree(self.min_node))
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()