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Min-Max Heaps. A double-ended priority queue is a data structure that supports the following operations: inserting an element with an arbitrary key deleting an element with the largest key deleting an element with the smallest key A Min-Max Heap supports all of the above operations.

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Min-Max Heaps

- A double-ended priority queue is a data structure that supports the following operations:
- inserting an element with an arbitrary key
- deleting an element with the largest key
- deleting an element with the smallest key

- A Min-Max Heap supports all of the above operations.

Min-Max Heap (Cont.)

- Definition: A min-max heap is a complete binary tree such that if it is not empty, each element has a data member called key. Alternating levels of this tree are min levels and max levels, respectively. The root is on a min level. Let x be any node in a min-max heap. If x is on a min (max) level then the element in x has the minimum (maximum) key from among all elements in the subtree with root x. A node on a min (max) level is called a min (max) node.

Min-Max Heap (Cont.)

- The min-max heap stores in a one-dimension array h.
- Insert a key 5 into this min-max heap.
- Initially key 5 is inserted at j. Now since 5 < 10 (which is j’s parent), 5 is guaranteed to be smaller than all keys in nodes that are both on max levels and on the path from j to root. Only need to check nodes on min levels.
- When inserting 80 into this min-max heap, since 80 > 10, and 10 is on the min level, we are assured that 80 is larger than all keys in the nodes that are both on min levels and on the path from j to the root. Only need to check nodes on max levels.

Program 9.3 Insertion Into A Min-Max Heap

template <class KeyType>

void MinMaxHeap<KeyType>::Insert(const Element<KeyType>& x)

// inset x into the min-max heap

{

if (n==MaxSize) {MinMaxFull(); return;}

n++;

int p =n/2; // p is the parent of the new node

if(!p) {h[1] = x; return;} // insert into an empty heap

switch(level(p)) {

case MIN:

if (x.key < h[p].key) { // follow min levels

h[n] = h[p];

VerifyMin(p, x);

}

else { VerifyMax(n, x); } // follow max levels

break;

case MAX:

if (x.key > h[p].key) { // follow max levels

h[n] = h[p];

VerifyMax(p, x);

}

else { VerifyMin(n, x); } // follow min levels

break;

}

}

Program 9.4 Searching For The Correct Max Node For Insertion

template <class KeyType>

void MinMaxHeap<KeyType>::VerifyMax(int i, const Elemetn<KeyType>& x)

// Follow max nodes from the max node I to the root and insert x at proper place

{

for (int gp = i/4; // grandparent of i

gp && (x.key > h[gp].key);

gp /= 4)

{ // move h[gp] to h[i]

h[i] = h[gp];

i = gp;

}

h[i] = x; // x is to be inserted into node i

}

O(logn)

Deletion of the Min Element (Cont.)

- When delete the smallest key from the min-max heap, the root has the smallest key (key 7). So the root is deleted.
- The last element with key 12 is also deleted from the min-max heap and then reinsert into the min-max heap. Two steps to follow:
- The root has no children. In this case x is to be inserted into the root.
- The root has at least one child. Now the smallest key in the min-max heap is in one of the children or grandchildren of the root. Assume node k has the smallest key, then following conditions must be considered:
- x.key ≤ h[k].key. x may be inserted into the root.
- x.key >h[k].key and k is a child of the root. Since k is a max node, it has not descendents with key larger than h[k].key. So, node k has no descendents with key larger than x.key. So the element h[k] may be moved to the root, and x can be inserted into node k.
- x.key> h[k] and k is a grandchild of the root. h[k] is moved to the root. Let p the parent of k. If x.key > h[p].key, then h[p] and x are to be interchanged.

Deaps

- A deap is a double-ended heap that supports the double-ended priority operations of insert, delet-min, and delete-max.
- Similar to min-max heap but deap is faster on these operations by a constant factor, and the algorithms are simpler.

Deaps (Cont.)

- Definition: A deap is a complete binary tree that is either empty or satisfies the following properties:
(1) The root contains no element

(2) The left subtree is a min heap.

(3) The right subtree is a max heap.

(4) If the right subtree is not empty, then let i be any node in the left subtree. Let j be the corresponding node in the right subtree. If such a j does not exist, then let j be the node in the right subtree that corresponds to the parent of i. The key in node i is less than or equal to that of j.

Deap (Cont.)

- From Deap’s definition, it’s obvious that for an n-element deap, the min element is the root of min heap and the max element is the root of the max heap.
- If n = 1, then the min and max elements are the same and are in the root of the min heap.
- Since deap is a complete binary tree, it may be stored as an implicit data structure in a one-dimension array similar to min, max, min-max heaps.
- In the case of deap, the position 1 of the array is not used. For an n-element deap, it occupied n+1 element of an array.
- If i is a node in the min heap of the deap, its corresponding node in the max heap is .
- Then j defined in property (4) of definition is given by
if (j > n+1) j /= 2;

Program 9.7: Inserting Into A Deap

template <class KeyType>

void Deap<KeyType>::Insert(const Element<KeyType>& x) {

//Insert x into the deap

int I;

if (n==MaxSize) {DeapFull(); return;}

n++;

if (n==1) {d[2]=x; return;} // insert into an empty deap

int p = n+1; // p is the new last position of the deap

switch(MaxHead(p)) {

case TRUE: // p is a position in the max heap

i = MinPartner(p);

if (x.key < d[i].key) {

d[p] = d[i];

MinInsert(i, x);

}

else MaxInsert(p, x);

break;

case FALSE: // p is a position in the min heap

i = MaxPartner(p);

if (x.key > d[i].key) {

d[p] = d[i];

MaxInsert(i, x);

}

else MinInsert(p, x);

}

}

O(log n)

Deletion of Min Element

- Suppose we want to remove the minimum element from the deap.
- We first place the last element into a temporary element t.
- Vacancy created by the deletion of the min element is filled by moving from position 2 to a leaf.
- Each move is preceded by a step that moves the smaller of the elements in the children of the current node up.
- Then move to the position previously occupied by the moved element.
- Repeat the process until we move the empty node to a leaf position.
- Compare the key put in the temporary element with the max partner.
- If <, no exchange is needed. The temporary element is inserted into the empty leaf node.
- If >, exchange them.

Leftist Trees

- To combine two priority queues into a single priority queue, the combine operation takes O(n) time if heaps are used.
- The complexity of combine operation can be reduced to O(log n) if a leftist tree is used.
- Leftist trees are defined using the concept of an extended binary tree. An extended binary tree is a binary tree in which all empty binary subtrees have been replaced by a square node.
- The square nodes in an extended binary tree are called external nodes. The original nodes of the binary tree are called internal nodes.

Leftist Trees (Cont.)

- Let x be a node in an extended binary tree. Let LeftChild(x) and RightChild(x), respectively, denote the left and right children of the internal node x.
- Define shortest(x) to be the length of a shortest path from x to an external node. It is easy to see that shortest(x) satisfies the following recurrence:
shortest(x) =

0 if x is an external node

1 + min{shortest(LeftChild(x)), RightChild(x))} otherwise

Leftist Tree Definition

Definition: A leftist tree is a binary tree such that if it is not empty, then shortest(LeftChild(x)) ≥ shortest(RightChild(x)) for every internal node x.

Lemma 9.1: Let x be the root of a leftist tree that has n (internal) nodes

- n ≥ 2shortest(x) – 1
- The rightmost root to external node path is the shortest root to external node path. Its length is shortest(x).

Class Definition of A Leftist Tree

template<class KeyType>class MinLeftistTree; // forward declaration

template<class KeyType>

class LeftistNode {

friend class MinLeftistTree<KeyType>;

private:

Element<KeyType>data;

LeftistNode *LeftChild, *RightChild;

int shortest;

}

template<class KeyType>

class MinLeftistTree:public MinPQ<KeyType> {

public:

// constructor

MinLeftistTree(LeftistNode<KeyType> *int = 0) root(int) {};

// the three min-leftist tree operations

void Insert(const Element<KeyType>&);

Element<KeyType>* DeleteMin(Element<KeyType>&);

void MinCombine(MinLeftistTree<KeyType>*);

private:

LeftistNode<KeyType>* MinUnion(LeftistNode<KeyType>*, LeftistNode<KeyType>*);

LeftistNode<KeyType>* root;

};

Definition of A Min (Max) Leftist Tree

- Definition: A min leftist tree (max leftist tree) is a leftist tree in which the key value in each node is no larger (smaller) than the key values in its children (if any). In other words, a min (max) leftist tree is a leftist tree that is also a min (max) tree.

Min (Max) Leftist Tree (Cont.)

- Like any other tree structures, the popular operations on the min (max) leftist trees are insert, delete, and combine.
- The insert and delete-min operations can both be done by using the combine operation.
- e.g., to insert an element x into a min leftist tree, we first create a min leftist tree that contains the single element x. Then we combine the two min leftist trees.
- To delete the min element from a nonempty min leftist tree, we combine the min leftist trees root->LeftChild and root->RightChild and delete the node root.

Combine Leftist Trees

- To combine two leftist trees:
- First, a new binary tree containing all elements in both trees is obtained by following the rightmost paths in one or both trees.
- Next, the left and right subtrees of nodes are interchanged as necessary to convert this binary tree into a leftist tree.

- The complexity of combining two leftist trees is O(log n)

Combining The Min Leftist Tree

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Binomial Heaps

- A binomial heap is a data structure that supports the same functions as those supported by leftist trees.
- Unlike leftist trees, where an individual operation can be performed in O(log n) time, it is possible that certain individual operations performed on a binomial heap may take O(n) time.
- By amortizing part of the cost of expensive operations over the inexpensive ones, then the amortized complexity of an individual operation is either O(1) or O(log n) depending on the type of operations.

Cost Amortization

- Given a sequence of operations I1, I2, D1, I3, I4, I5, I6, D2, I7. Assume each insert operation costs one time unit and D1 and D2 operations take 8 and 10 time units, respectively.
- The total cost to perform the sequence of operations is 25.
- If we charge some actual cost of an operation to other operations, this is called cost amortization.
- In this example, the amortized cost of I1 – I6 each has 2 time units, I7 has one, and D1 and D2 each has 6.
- Now suppose we can prove that no matter what sequence of insert and delete-min operations is performed, we can charge costs in such a way that the amortized cost of each insertion is no more than 2 and that of each deletion is no more than 6. We can claim that the sequence of insert/delete-min operations has cost no more than 2*i + 6*d.
- With the actual cost, we conclude that the sequence cost is no more than i+10*d.
- Combining the above two bounds, we obtain min{2*i+6*d, i+10*d}.

Binomial Heaps

- Binomial heaps have min binomial heap and max binomial heap.
- We refer to the min binomial heap as B-heap.
- B-heap can perform an insert and a combine operation in O(1) actual and amortized time and a delete-min operation with O(log n) amortized time.
- A node in a B-heap has the following data members:
- degree: is the number of children it has
- child: is a pointer points to any one of its children. All children forms a circular list.
- link: is a singly link used to maintain a circular list with its siblings.
- data

- The roots of the min trees that comprise a B-heap are linked to form a singly linked circular list. The B-heap is then pointed at by a single pointer min to the min tree root with smallest key.

Insertion Into A B-Heap

- An element x can be inserted into a B-heap by first putting x into a new node and then inserting this node into the circular list pointed at by min. The operation is done in O(1) Time.
- To combine two nonempty B-heaps, combine the top circular lists of each into a single circular list.
- The new combined B-heap pointer is the min pointer of one of the two trees, depending on which has the smaller key.
- Again the combine operation can be done in O(1) time.

Deletion of Min Element

- If min is 0, then the B-heap is empty. No delete operations can be performed.
- If min is not 0, the node is pointed by min. Delete-min operation deletes this node from the circular list. The new B-heap consists of the remaining min trees and the submin trees of the delete root.
- To form the new B-heap, min trees with the same degrees are joined in pairs. The min tree whose root has the larger key becomes the subtree of the other min tree.

Joining Two Degree-Two Min Trees

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Since no two min trees have the same degree, the min join process stops.

Program 9.12 Steps In A Delete-Min Operation

template<class KeyType>

Element<KeyType>*Binomial<KeyType>::DeleteMin(Element<KeyType>&x)

Step 1: [Handle empty B-heap] if (!min){ DeletionError(); return();}

Step 2: [Deletion from nonempty B-heap] x=min->data; y=min->child; delete min from its circular list; following this deletion, min points to any remaining node in the resulting list; if there is no such node, then min = 0;

Step 3: [Min-tree joining] Consider the min trees in the lists min and y; join together pairs of min trees of the same degree until all remaining min trees have different degrees;

Step 4: [Form min tree root list] Link the roots of the remaining min trees (if any) together to form a circular list; set min to point to the root (if any) with minimum key; return &x;

Binomial Tree Definition

- Definition: The binomial tree Bk, of degree k is a tree such that if k = 0, then the tree has exactly one node, and if k > 0, then the tree consists of a root whose degree is k and whose subtrees are B0, B1, …, Bk-1.
- Lemma 9.2: Let a be a B-heap with n elements that results from a sequence of insert, combine, and delete-min operations performed on a collection of initially empty B-heaps. Each min tree in a has degree ≤ log2n. Consequently, MaxDegree ≤ , and the actual cost of a delete-min operation is O(log n + s).

B-Heap Costs

- Theorem 9.1: If a sequence of n insert, combine, and delete-min operations is performed on initially empty B-heaps, then we can amortize costs such that the amortized time complexity of each insert and combine is O(1), and that of each delete-min operation is O(log n).

Fibonacci Heaps

- A Fibonacci heap is a data structure that supports the three binomial heap operations: insert, delete-min (or delete-max), and combine, as well as the following additional operations:
- decrease key: Decrease the key of a specified node by a given positive amount
- delete: Delete the element in a specified node.

- The first of these can be done in O(1) amortized time and the second in O(log n) amortized time.
- The binomial heap operations can be performed in the same asymptotic times using a Fibonacci heap as they can be using a binomial heap.

Fibonacci Heaps (Cont.)

- Two varieties of Fibonacci heaps:
- Min Fibonacci heap: is a collection of min trees
- Max Fibonacci heap: is a collection of max trees.

- Refers to min Fibonacci heap as F-heaps.
- B-heaps are a special case of F-heaps.
- A node in F-heap data structure contains additional data members other than those in B-heaps:
- parent: is used to point to the node’s parent (if any).
- ChildCut: to support cascading cut described later.
- LeftLink and RightLink: replace the link data member in B-heap node. These links form a doubly linked circular list.

- In F-heaps, singly linked circular list is replaced by doubly linked circular list.

Deletion From An F-Heap

- The basic operations insert, delete-min, and combine are performed exactly as for the case of B-heaps.
- Follow the below steps to delete a node b from an F-heap:
- If min = b, then do a delete-min; otherwise do Steps 2, 3, and 4 below
- Delete b from its doubly linked list
- Combine the doubly linked list of b’s children with the doubly linked list pointed at by min into a single doubly linked list. Trees of equal degree are not joined as in a delete-min operation.
- Dispose of node b.

Decrease Key

- To decrease the key in node b, do the following:
- Reduce the key in b
- If b is not a min tree root and its key is smaller than that in its parent, then delete b from its doubly linked list and insert it into the doubly linked list of min tree roots.
- Change min to point to b if the key in b is smaller than that in min.

F-Heap After The Reduction of 15 by 4

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Cascading Cut

- Because the new delete and decrease-key operations, the F-heap is not necessary a Binomial tree. Therefore, the analysis of theorem 9.1 is no longer true for F-heaps if no restructuring is done.
- To ensure that each min tree of degree k has at least ck nodes, for some c, c> 1, each delete and decrease-key operations must be followed by a particular step called cascading cut.
- The data member ChildCut is used to assist the cascading cut step.
- ChildCut data member is only used for non-root node.

Cascading Cut (Cont.)

- ChildCut of node x is TRUE iff one of the children of node x was cut off after the most recent time x was made the child of its current parent.
- Whenever a delete or decrease-key operation deletes a node q that is not a min tree root from its doubly linked list, then the cascading cut step is invoked.
- During the steps, we examine the nodes on the path from the parent p of the deleted node q up the nearest ancestor of the deleted node with ChildCut = FALSE.
- If there is no such ancestor, then the path goes from p to the root of the min tree containing p.
- All nonroot nodes on this path with ChildCut data member TRUE are deleted from their respective doubly linked list and added to the doubly linked list of min tree root nodes of the F-heap.
- If the path has a node with ChildCut set to FALSE, then it is changed to TRUE.

A Cascading Cut Example

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F-Heap Analysis

- Lemma 9.3: Let a be an F-heap with n elements that results from a sequence of insert, combine, delete-min, delete, and decrease-key operations performed on initially empty F-heaps.
- Let b be any node in any of the min trees of a. The degree of b is at most logΦ m, where and, m is the number elements in the subtree with root b.
- MaxDegree ≤ , and the actual cost of a delete-min operation is O(log n + s).

Theorem 9.2

Theorem 9.2: If a sequence of n insert, combine, delete, delete-min, and decrease-key operations is performed on an initially empty F-heap, then we can amortize costs such that the amortized time complexity of each insert, combine, and decrease-key operation is O(1) and that of each delete-min and delete operation is O(log n). The total time complexity of the entire sequence is the sum of the amortized complexities of the individual operations in the sequence.

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