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    TERM PAPER

    ON

    BINOMIAL HEAP

    SUBJECT: ALGORITHM ANALYSIS AND DESIGN

    SUBMITTED TO SUBMITTED BY

    MissShivani Malhotra Name : Nivedita kalia

    Roll no : RK1R10B43

    Reg no :10803407

    B.TECH(CSE)

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    ACKNOWLEDGMENT

    Words are the dress of thoughts, appreciating andacknowledging those, who are responsible for the

    successful completion of the project.

    Our sincerity gratitude goes to LECT. Miss Shivani

    Malhotra who assigned us responsibility to work on this

    project and provided us all the help, guidance and

    encouragement to complete this project. The encouragementand guidance given by her have made this a personally

    rewarding experience. We thank her for her support and

    inspiration, without which, understanding the details of the

    project would have been exponentially difficult.

    Finally, yet importantly, I would like to express my heartfelt

    thanks to my beloved parents for their blessings, my

    friends/classmates for their help and wishes for the successful

    completion of this term paper.

    With Sincere Thanks,(Nivedita Kalia)

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    TABLE OF CONTENTS

    1. Abstract

    2. Introduction

    3. Binomial trees

    4. The Binomial Heap Properties

    5. Implementation of a Binomial Heap

    6. Operations on Binomial Heaps

    7. Search for the Minimum Key

    8. Find Minimum Key

    9. Unite Two Binomial Heaps

    10. Binomial Heap Union11. Insert New Node

    12. Delete a Node

    13. Performance

    14. Applications

    15. Bibliography

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    ABSTRACT

    A binomial heap is a collection of binomial

    trees that satisfies the followingbinomial-heap properties:

    No two binomial trees in the collectionhave the same size.

    Each node in each tree has a key.

    Each binomial tree in the collection isheap-ordered in the sense that each

    non-root has a key strictly less than the

    key of its parent.

    By the first property we have the following:

    For all n >= 1 and k >=0, Bk appears in an

    n-node binary heap if and only if the (k+1)st

    bit of the binary representation of n is a 1.

    This means that the number of trees in a

    binomial heap is O(log n).

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    INTRODUCTION

    Binomial heaps were invented in 1978 by J.

    Vuillemin

    The binary heap data structure is fine for the simple

    operations of inserting, deleting and extracting

    elements, but other operations aren't so well

    supported.

    One such operation is the Union operation, which

    joins two heaps together.

    If the heaps are binary heaps then this requires

    building up a new heap from scratch, using the

    elements of the old heaps, which is expensive for

    large heaps.

    Binomial heap presents the data structure, which

    supports Union operations more efficiently.

    Binomial heaps can be minimumheaps or maximum heaps, and in this case, the focus

    is only on minimum heaps.

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    BINOMIAL TREES

    The

    binomial tree is the building block for the

    binomial heap. A binomial tree is an ordered

    tree that is, a tree where the children of each

    node are ordered.

    Binomial trees are defined recursively,

    building up from single nodes. A single tree

    of degree k is constructed from two trees ofdegree k - 1 by making the root of one tree

    the leftmost child of the root of the other

    tree.

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    The Binomial Heap Properties

    A binomial heap is a collection of binomial trees

    that satisfies the following binomial-heap

    properties:

    1. No two binomial trees in the collection have

    the same size.

    2. Each node in each tree has a key.

    3. Each binomial tree in the collection is heap-

    ordered in the sense that each non-root has a key

    strictly less than the key of its parent.

    The number of trees in a binomial heap is

    O(log n).

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    Implementation of a Binomial Heap

    A field key for its key

    A field degree for the number of children

    A pointer child, which points to the leftmost-child

    A pointer sibling, which points to the right-sibling

    A pointerp, which points to the parent

    The roots of the trees are connected so that the sizes

    of the connected trees are in decreasing order. Also,

    for a heap H, head [H] points to the head of the list

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    Operations on Binomial Heaps

    Creation of a new heap.

    Search for the minimum key.

    Uniting two binomial heaps.

    Insertion of a node.

    Removal of the root of a tree.

    Decreasing a key.

    Removal of a node.

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    Search for the Minimum Key

    To do this we find the smallest key among those

    stored at the roots connected to the head of H.

    What's the cost of minimum-search?

    The cost is O(log n) because there are O(log n)

    heaps, in each tree the minimum is located at the

    root, and the roots are linked.

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    Find Minimum Key

    Algorithm

    1.If heap is empty, simply stop execution.

    2.Else, assign the key of the first root in the Root list to Min

    variable.

    3.Move till the end of Root list by comparing keys with Minand update Min if needed.

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    Insert New Node

    Let us assume that H is an existing Binomial Heap and we want

    to insert x in it.

    Algorithm,

    1.Create a single node Binomial Heap H consisting of x.

    2.Union( H, H),

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    Unite Two Binomial Heaps

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    Binomial Heap Union

    Create heap H that is union of heaps H and H.

    Mergeable heaps

    Easy if H and H are each orderkbinomial trees

    Connect roots of H and H

    Choose smaller key to be root of H

    Running Time O(log N)

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    Performance

    All of the following operations work in O(log n)

    time on a binomial heap with n elements:

    Insert a new element to the heap Find the element with minimum key

    Delete the element with minimum key from the

    heap

    Decrease key of a given element

    Delete given element from the heap

    Merge two given heaps to one heap

    Finding the element with minimum key can also be

    done in O(1) by using an additional pointer to the

    minimum.

    http://en.wikipedia.org/wiki/Big_O_notationhttp://en.wikipedia.org/wiki/Big_O_notation
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    Applications

    1. Discrete event simulation :

    Discrete-event simulation(DES) where the operation of

    a system is represented as a chronological sequence of

    events

    2. Priority queues :

    A priority queue is an abstract data type which is

    like a regular queue or stackdata structure, but

    additionally, each element is associated with a"priority"

    http://en.wikipedia.org/wiki/Discrete_event_simulationhttp://en.wikipedia.org/wiki/Discrete_timehttp://en.wikipedia.org/wiki/Simulationhttp://en.wikipedia.org/wiki/Simulationhttp://en.wikipedia.org/wiki/Systemhttp://en.wikipedia.org/wiki/Sequence_of_eventshttp://en.wikipedia.org/wiki/Sequence_of_eventshttp://en.wikipedia.org/wiki/Priority_queuehttp://en.wikipedia.org/wiki/Abstract_data_typehttp://en.wikipedia.org/wiki/Queue_%28data_structure%29http://en.wikipedia.org/wiki/Stack_%28data_structure%29http://en.wikipedia.org/wiki/Stack_%28data_structure%29http://en.wikipedia.org/wiki/Queue_%28data_structure%29http://en.wikipedia.org/wiki/Abstract_data_typehttp://en.wikipedia.org/wiki/Priority_queuehttp://en.wikipedia.org/wiki/Sequence_of_eventshttp://en.wikipedia.org/wiki/Sequence_of_eventshttp://en.wikipedia.org/wiki/Systemhttp://en.wikipedia.org/wiki/Simulationhttp://en.wikipedia.org/wiki/Discrete_timehttp://en.wikipedia.org/wiki/Discrete_event_simulation
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    BIBLIOGRAPHY

    www.cs.tau.ac.il/~dannyf/ds09/fibo-ds2008.ppt

    cs.anu.edu.au/people/Warren.Armstrong/apac

    gradalgo.wordpress.com/tag/binomial-heaps

    oops.math.spbu.ru/projects/BinomialHeap/

    www.cs.arizona.edu/classes/cs545/fall09/binomial.prn.pdf

    www.cs.cornell.edu/courses/CS6820/2012sp/Handouts

    activities.tjhsst.edu/sct/lectures/1112/binomheap

    activities.tjhsst.edu/sct/lectures/1112/binomheap

    cs.hubfs.net/topic/None/56608

    tmue.edu.tw/~lai/af-teach/af-algorithm-DS/.../BinomialHeaps

    http://www.cs.tau.ac.il/~dannyf/ds09/fibo-ds2008.ppthttp://www.cs.arizona.edu/classes/cs545/fall09/binomial.prn.pdfhttp://www.cs.arizona.edu/classes/cs545/fall09/binomial.prn.pdfhttp://www.cs.arizona.edu/classes/cs545/fall09/binomial.prn.pdfhttp://www.cs.arizona.edu/classes/cs545/fall09/binomial.prn.pdfhttp://www.cs.cornell.edu/courses/CS6820/2012sp/Handoutshttp://www.cs.cornell.edu/courses/CS6820/2012sp/Handoutshttp://www.cs.cornell.edu/courses/CS6820/2012sp/Handoutshttp://www.cs.arizona.edu/classes/cs545/fall09/binomial.prn.pdfhttp://www.cs.tau.ac.il/~dannyf/ds09/fibo-ds2008.ppt