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CSE233 Course Review

CSE233 Course Review. CSE, POSTECH. What we studied in the 1 st half. Week 1 : Overview of C++, Program performance Week 2 : Array-based and linked representations of Lists Week 3 : Arrays and Matrices Week 4 : Performance Measurement Week 5 : Stacks Week 6 : Queues

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CSE233 Course Review

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  1. CSE233 Course Review CSE, POSTECH

  2. What we studied in the 1st half Week 1 : Overview of C++, Program performance Week 2 : Array-based and linked representations of Lists Week 3 : Arrays and Matrices Week 4 : Performance Measurement Week 5 : Stacks Week 6 : Queues Week 7 : Skip lists and Hashing Week 8 : Review and Midterm Exam

  3. What we studied in the 2nd half Week 9 : Binary and other trees Week 10 : Priority queues, Heaps, Leftist trees Week 11 : Tournament trees and Bin packing Week 12 : Binary search trees Week 13 : AVL trees Week 14 : Graphs Week 15 : Graph Search Methods Week 16 : Review and Final exam

  4. Trees

  5. Tree Terminology • A treet is a finite nonempty set of elements • The element at the top is called the root • The remaining elements, if any, are partitioned into trees, which are called the subtrees of t. • Elements next in the hierarchy are the children of the root. • Elements next in the hierarchy are the grandchildren of the root, and so on. • Elements at the lowest level of the hierarchy are the leaves.

  6. Leaves, Parent, Grandparent, Siblings, Ancestors, Descendents Leaves = {Mike,AI,Sue,Chris} Parent(Mary) = Joe Grandparent(Sue) = Mary Siblings(Mary) = {Ann,John} Ancestors(Mike) = {Ann,Joe} Descendents(Mary)={Mark,Sue}

  7. level 1 level 2 level 3 level 4 Levels and Height • Root is at level 1 and its children are at level 2. • Height = depth = number of levels

  8. Node Degree • Node degree is the number of children it has

  9. Tree Degree • Tree degree is the maximum of node degrees tree degree = 3

  10. Binary Trees

  11. Binary Tree • A finite (possibly empty) collection of elements • A nonempty binary tree has a root element and the remaining elements (if any) are partitioned into two binary trees • They are called the left and right subtrees of the binary tree

  12. - different when viewed as a binary tree a a - same when viewed as a tree b c c b Difference Between a Tree & a Binary Tree • A binary tree may be empty; a tree cannot be empty. • No node in a binary tree may have a degree more than 2, whereas there is no limit on the degree of a node in a tree. • The subtrees of a binary tree are ordered; those of a tree are not ordered.

  13. Binary Tree for Mathematical Expressions Figure 11.5 Expression Trees

  14. Full Binary Tree • A full binary tree of height h has exactly 2h-1 nodes. • Numbering the nodes in a full binary tree • Number the nodes 1 through 2h-1 • Number by levels from top to bottom • Within a level, number from left to right

  15. Complete Binary Tree with N Nodes • Start with a full binary tree that has at least nnodes • Number the nodes as described earlier. • The binary tree defined by the nodes numbered 1 through n is the n-node complete binary tree. • A full binary tree is a special case of a complete binary tree

  16. Example of Complete Binary Tree • Complete binary tree with 10 nodes.

  17. Fig. 11.8 Incomplete binary trees Incomplete Binary Trees • Complete binary tree with some missing elements

  18. Binary Tree Representation • Array representation • Linked representation

  19. Array Representation of Binary Tree • The binary tree is represented in an array by storing each element at the array position corresponding to the number assigned to it.

  20. Right-Skewed Binary Tree • An n node binary tree needs an array whose length is between n+1 and 2n. • Right-skewed binary tree wastes the most space • What about left-skewed binary tree?

  21. Linked Representation of Binary Tree • The most popular way to present a binary tree • Each element is represented by a node that has two link fields (leftChild and rightChild) and an element field

  22. Priority Queues

  23. Priority Queues • A priority queue is a collection of zero or more elements  each element has a priority or value • Unlike the FIFO queues, the order of deletion from a priority queue (e.g., who gets served next) is determined by the element priority • Elements are deleted by increasing or decreasing order of priority rather than by the order in which they arrived in the queue

  24. Priority Queues • Operations performed on priority queues 1) Find an element, 2) insert a new element, 3) delete an element, etc. • In a Min priority queue, find/delete operation finds/deletes the element with minimum priority • In a Max priority queue, find/delete operation finds/deletes the element with maximum priority • Two or more elements can have the same priority

  25. Implementation of Priority Queues • Implemented using heaps and leftist trees • Heap is a complete binary tree that is efficiently stored using the array-based representation • Leftist tree is a linked data structure suitable for the implementation of a priority queue

  26. Min (Max) Trees

  27. Max (Min) Tree • A max tree (min tree) is a tree in which the value in each node is greater (less)than or equal to those in its children (if any) • Nodes of a max or min tree may have more than two children (i.e., may not be binary tree)

  28. Max Tree Example

  29. Min Tree Example

  30. Heaps

  31. Heaps - Definitions • A max heap (min heap) is a max (min) tree that is also a complete binary tree

  32. Max Heap with 9 Nodes

  33. Min Heap with 9 Nodes

  34. Array Representation of Heap • A heap is efficiently represented as an array.

  35. 9 8 7 6 7 2 6 5 1 20 Insertion into a Max Heap • New element is 20 • Are we finished?

  36. 9 8 7 6 20 2 6 5 1 7 Insertion into a Max Heap • Exchange the positions with 7 • Are we finished?

  37. 9 20 7 6 8 2 6 5 1 7 Insertion into a Max Heap • Exchange the positions with 8 • Are we finished?

  38. 20 9 7 6 8 2 6 5 1 7 Insertion into a Max Heap • Exchange the positions with 9 • Are we finished?

  39. 20 15 7 6 9 2 6 5 1 7 8 Deletion from a Max Heap • Max element is in the root • What happens when we • delete an element?

  40. 15 7 6 9 2 6 5 1 7 8 Deletion from a Max Heap • After the max element is removed. • Are we finished?

  41. 15 7 6 9 2 6 5 1 7 8 Deletion from a Max Heap • Heap with 10 nodes. • Reinsert 8 into the heap.

  42. 8 15 7 6 9 2 6 5 1 7 Deletion from a Max Heap • Reinsert 8 into the heap. • Are we finished?

  43. 15 8 7 6 9 2 6 5 1 7 Deletion from a Max Heap • Exchange the position with 15 • Are we finished?

  44. 15 9 7 6 8 2 6 5 1 7 Deletion from a Max Heap • Exchange the position with 9 • Are we finished?

  45. Max Heap Initialization • Heap initialization means to construct a heap by adjusting the tree if necessary • Example: input array = [-,1,2,3,4,5,6,7,8,9,10,11]

  46. Max Heap Initialization - Start at rightmost array position that has a child.

  47. Max Heap Initialization

  48. Max Heap Initialization

  49. Max Heap Initialization

  50. Max Heap Initialization

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