Algorithm Analysis Advanced Data Structuremi.cau.ac.kr/teaching/lecture_alg/W05P.pdfAlgorithm Analysis / Chung-Ang University / Professor Jaesung Lee Disjoint Set Operations •Relation

Algorithm Analysis

Advanced Data Structure

Chung-Ang University, Jaesung Lee

Algorithm Analysis / Chung-Ang University / Professor Jaesung Lee

Priority Queue, Heap and Heap Sort

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Max Heap data structure

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Representation of Heap Tree

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Representation of Heap Tree

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Representation of Heap Tree• Let us consider the following elements arranged in the form of array as follows:

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Representation of Heap Tree• A heap tree represented using a single array looks as follows:

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Operations on heap tree• Insertion

• Deletion and

• Merging

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Insertion into a heap tree

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Insertion into a heap tree

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Insertion into a heap tree• The algorithm Max_heap_insert to insert a data into a max heap tree is as follows:

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Insertion into a heap tree• Insert 40, 80, 35, 90, 45, 50, 70

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Deletion of a node from heap tree• The algorithm for deleting root node is as follows:

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Deletion of a node from heap tree• And the algorithm for “adjust” function is as follows:

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Deletion of a node from heap tree

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Merging two heap trees

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Applications of heap tree

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Heap Sort

• (Step 1) Build a heap tree with the given set of data.

• (Step 2-a) Remove the top most item (the largest) and replace it with the last element in the heap.

• (Step 2-b) Re-heapify the complete binary tree.

• (Step 2-c) Place the deleted node in the output.

• (Step 3) Continue Step 2 until the heap tree is empty.

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Heap Sort• A Heap Sort Algorithm

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Heap Sort• Time complexity

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Heap Sort• Example

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Heap Sort• Priority queue implementation using heap tree

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Heap Sort• Priority queue implementation using heap tree

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Priority Queue• Binary Search Trees

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Priority Queue• Binary Tree Searching

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Priority Queue• Why use binary search trees?

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Priority Queue• Inserting nodes into a Binary Search Tree

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Priority Queue• Deleting nodes from a Binary Search Tree

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Dictionary• Dictionary

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Dictionary• Dictionary (Example)

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Dictionary• Dictionary (Example)

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Dictionary

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• Get operation

• Put operation

• Remove operation


Disjoint Set Operations• Relation between Dictionary and Set

• A set is an unordered collection (possibly empty) of distinct items.

• We can implement a set as a bit vector over the universal set.

• We can implement a set with a list structure (with insertion constraints).

• A multiset or bag is a set without the uniqueness constraint.

• Basic operations of a multiset: Search, Insert, Delete.

• A basic data structure that accomplishes these operations is dictionary.

• Sometimes we need to dynamically partition some n-element set into a collection of disjoint sets.

• Sometimes we need to take the union or intersection of sets.

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Disjoint Set Operations• Set and Disjoint Set

• Disjoint Set Operations: Union and Find

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Disjoint Set Operations• Disjoint Set Union (example)

• �� = {1,7,8,9}

• �� = {2,5,10}

• �� ∪ S� = {1,2,5,7,8,9,10}

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Disjoint Set Operations• Find (example)

• �� = {1,7,8,9}

• �� = {2,5,10}

• �� = 3,4,6

• ��(4) = ��• ��(5) = ��• ��(7) = ��

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Disjoint Set Operations• Set Representation (example)

• �� = {1,7,8,9}

• �� = {2,5,10}

• �� = 3,4,6

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Disjoint Set Operations• Disjoint Union

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Disjoint Set Operations• Find

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Disjoint Set Operations• Union and Find Algorithms (example)

• For the following sets, the array representation is as shown below.

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Disjoint Set Operations• Algorithm for Union operation

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Disjoint Set Operations• Algorithm for Find operation

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Disjoint Set Operations• Analysis of SimpleUnion(i,j) and SimpleFind(i)

• For example, consider the sets:

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Disjoint Set Operations• Analysis of SimpleUnion(i,j) and SimpleFind(i)

• The sequence of Union operations results the degenerate tree as below.

• Time complexity is � ∑ �� = � �� .

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Documents

Algorithm Analysis Advanced Data Structuremi.cau.ac.kr/teaching/lecture_alg/W05P.pdfAlgorithm Analysis / Chung-Ang University / Professor Jaesung Lee Disjoint Set Operations •Relation