CSE233 Course Review

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CSE233 Course Review

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Title: CSE233 Course Review

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 Review and Final exam

Trees

5
Tree Terminology

A tree t 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
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
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 n
nodes
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
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 n1 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

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

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
Insertion into a Max Heap

New element is 20
Are we finished?

36
Insertion into a Max Heap

Exchange the positions with 7
Are we finished?

37
Insertion into a Max Heap

Exchange the positions with 8
Are we finished?

38
Insertion into a Max Heap

Exchange the positions with 9
Are we finished?

39
Deletion from a Max Heap

Max element is in the root
What happens when we
delete an element?

40
Deletion from a Max Heap

After the max element is removed.
Are we finished?

41
Deletion from a Max Heap

Heap with 10 nodes.
Reinsert 8 into the heap.

42
Deletion from a Max Heap

Reinsert 8 into the heap.
Are we finished?

43
Deletion from a Max Heap

Exchange the position with 15
Are we finished?

44
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
51
Max Heap Initialization

Are we finished?
Done!

52
Exercise 12.7

Do Exercise 12.7
theHeap -, 10, 2, 7, 6, 5, 9, 12, 35, 22, 15,
1, 3, 4

12.7 (a) complete binary tree

53
Exercise 12.7 (b)

12.7 (b) The heapified tree

54
Exercise 12.7 (c)

12.7 (c) The heap after 15 is inserted is

55
Exercise 12.7 (c)

12.7 (c) The heap after 20 is inserted is

56
Exercise 12.7 (c)

12.7 (c) The heap after 45 is inserted is

57
Exercise 12.7 (d)

12.7 (d) The heap after the first remove max
operation is

58
Exercise 12.7 (d)

12.7 (d) The heap after the second remove max
operation is

59
Exercise 12.7 (d)

12.7 (d) The heap after the third remove max
operation is

Leftist Trees

61
Leftist Trees

Leftist tree is a tree which tends to lean to the
left
Leftist tree structures are useful for
applications
to meld (i.e., combine) pairs of priority queues
using multiple queues of varying size
External node a special node that replaces each
empty subtree
Internal node a node with non-empty subtrees
Extended binary tree a binary tree with
external nodes added

62
Height-Biased Leftist Tree (HBLT)

Let s(x) be the length (height) of a shortest
path from node x to an external node in its
subtree
If x is an external node, s(x) 0
If x is an internal node, s(x) min s(L), s(R)
1, where L and R are left and right children of
x
A binary tree is a height-biased leftist tree
(HBLT) iff at every internal node, the s value of
the left child is greater than or equal to the s
value of the right child
A max HBLT is an HBLT that is also a max tree
A min HBLT is an HBLT that is also a min tree

63
Weight-Biased Leftist Tree (WBLT)

Let the weight, w(x), of node x to be the number
of internal nodes in the subtree with root x
If x is an external node, w(x) 0
If x is an internal node, its weight is one more
than the sum of the weights of its children
A binary tree is a weight-biased leftist tree
(WBLT) iff at every internal node, the w value of
the left child is greater than or equal to the w
value of the right child
A max (min) WBLT is a max (min) tree that is also
a WBLT

64
Extended Binary Tree
65
Melding max HBLTs
66
Applications of Heaps

Sort (heap sort)
Machine scheduling
Huffman codes

67
Heap Sort

use element key as priority
Algorithm
put elements to be sorted into a priority queue
(i.e., initialize a heap)
extract (delete) elements from the priority queue
if a min priority queue is used, elements are
extracted in non-decreasing order of priority
if a max priority queue is used, elements are
extracted in non-increasing order of priority

68
Machine Scheduling Problem

m identical machines
n jobs/tasks to be performed
The machine scheduling problem is to assign jobs
to machines so that the time at which the last
job completes is minimum

69
NP-hard Problems

The class of problems for which no one has
developed a polynomial time algorithm.
No algorithm whose complexity is O(nk ml) is
known for any NP-hard problem (for any constants
k and l)
NP stands for Nondeterministic Polynomial
NP-hard problems are often solved by heuristics
(or approximation algorithms), which do not
guarantee optimal solutions
Longest Processing Time (LPT) rule is a good
heuristic for minimum finish time scheduling.

70
Huffman Codes

For text compression, the LZW method relies on
the recurrence of substrings in a text
Huffman codes is another text compression method,
which relies on the relative frequency (i.e., the
number of occurrences of a symbol) with which
different symbols appear in a text
Uses extended binary trees
Variable-length codes that satisfy the property,
where no code is a prefix of another
Huffman tree is a binary tree with minimum
weighted external path length for a given set of
frequencies (weights)

Tournament Trees

72
Tournament Trees

Like the heap, a tournament tree is a complete
binary tree that is most efficiently stored using
array-based binary tree
Used to obtain efficient implementations of two
approximation algorithms for the bin packing
problem (another NP-hard problem)
Types of tournament trees winner loser trees
The tournament is played in the sudden-death mode
Tournament trees are also called selection trees

73
Winner Trees

A winner tree for n players is a complete binary
tree with n external nodes and n-1 internal
nodes. Each internal node records the winner of
the match.
To determine the winner of a match, we assume
that each player has a value
In a min (max) winner tree, the player with the
smaller (larger) value wins

74
Loser Trees

A loser tree for n players is also a complete
binary tree with n external nodes and n-1
internal nodes. Each internal node records the
loser of the match.
The overall winner is recorded in tree0

Bin Packing Problems

76
Bin Packing Problem

We have bins that have a capacity binCapacity and
n objects that need to be packed into these bins
Object i requires objSizei, where 0 lt
objSizei ? binCapacity, units of capacity
Feasible packing - an assignment of objects to
bins so that no bins capacity is exceeded
Optimal packing - a feasible packing that uses
the fewest number of bins
Goal pack objects with the minimum number of
bins
The bin packing problem is an NP-hard problem

77
Truck Loading Problem

Have parcels to pack into trucks
Each parcel has a weight
Each truck has a load limit
Goal Minimize the number of trucks needed
Equivalent to the bin packing problem

78
Bin Packing Approximation Algorithms

First Fit (FF)
First Fit Decreasing (FFD)
Best Fit (BF)
Best Fit Decreasing (BFD)

79
First Fit (FF) Bin Packing

Bins are arranged in left to right order.
Objects are packed one at a time in a given
order.
Current object is packed into the leftmost
bininto which it fits.
If there is no bin into which current object
fits,start a new bin.

80
Best Fit (BF) Bin Packing

Let binj.unusedCapacity denote the capacity
available in bin j
Initially, the available capacity is binCapacity
for all bins
Object i is packed into the bin with the least
unusedCapacity that is at least objSizei
If there is no bin into which current object
fits,start a new bin.

81
First Fit Decreasing (FFD) Bin Packing

This method is the same as FF except that the
objects reordered in a decreasing size so that
objSizei ? objSizei1, 1 ? i lt n

82
Best Fit Decreasing (BFD) Bin Packing

This method is the same as BF except that the
objects are reordered as for FFD

Binary Search Trees

84
Binary Search Tree

A binary tree that may be empty. A nonempty
binary search tree satisfies the following
properties
Each node has a key (or value), and no two nodes
have the same key (i.e., all keys are distinct).
For every node x, all keys in the left subtree of
x are smaller than that in x.
For every node x, all keys in the right subtree
of x are larger than that in x.
The left and right subtrees of the root are also
binary search trees

85
Examples of Binary Trees

Which of the above trees are binary search trees?
? (b) and (c)
Why isnt (a) a binary search tree?
? It violates the property 3

86
Indexed Binary Search Trees

Binary search tree.
Each node has an additional field LeftSize.
Support search and delete operations by rankas
well as all the binary search tree operations.
LeftSize
the number of elements in its left subtree
the rank of an element with respect to the
elements in its subtree (e.g., the fourth element
in sorted order)

87
Indexed Binary Search Tree Example

LeftSize values are in red.

88
The Operation Ascend()

How can we output all elements in ascending
order of keys?
Do an inorder traversal (left, root, right).
What would be the output?
2, 6, 8, 10, 15, 20, 25, 30, 40

89
The Operation Search(key, e)

Search begins at the root
If the root is NULL, the search tree is empty and
the search fails.
If key is less than the root, then left subtree
is searched
If key is greater than the root, then right
subtree is searched
If key equals the root, then the search
terminates successfully
The time complexity for search is O(height)
See Program 11.4 for the search operation code

90
The Operation Insert(key, e)

To insert a new element e into a binary search
tree, we must first verify that its key does not
already exist by performing a search in the tree
If the search is successful, we do not insert
If the search is unsuccessful, then the element
is inserted at the point the search terminated
Why insert it at that point?
The time complexity for insert is O(height)
See Figure 14.3 for examples
See Program 14.5 for the insert operation code

91
Insert Example
We wish to insert an element with the key
35. Where should it be inserted?
92
Insert Example
Insert an element with the key 7.
93
Insert Example
Insert an element with the key 18.
94
The Operation Delete(key, e)

For deletion, there are three cases for the
element to be deleted
Element is in a leaf.
Element is in a degree 1 node (i.e., has exactly
one nonempty subtree).
Element is in a degree 2 node (i.e., has exactly
two nonempty subtrees).

95
Case 1 Delete from a Leaf

For case 1, we can simply discard the leaf node.
Example, delete a leaf element. key7

96
Case 2 Delete from a Degree 1 Node

Example Delete key40

97
Case 3 Delete from a Degree 2 Node

Example Delete key10

98
Case 3 Delete from a Degree 2 Node

Replace with the largest key in the left
subtree(or the smallest in the right subtree)
Which node is the largest key in the left
subtree?

99
Case 3 Delete from a Degree 2 Node
The largest key must be in a leaf or degree 1
node.
100

AVL Trees

101
Balanced Binary Search Trees

Trees with a worst-case height of O(log n) are
called balanced trees
An example of a balanced tree is AVL
(Adelson-Velsky and Landis) tree

102
AVL Tree

Definition
Binary tree.
If T is a nonempty binary tree with TL and TR as
its left and right subtrees, then T is an AVL
tree iff
TL and TR are AVL trees, and
hL hR ? 1 where hL and hR are the heights of
TL and TR, respectively

103
AVL Search Trees

An AVL search tree is a binary search tree that
is also an AVL tree

104
Indexed AVL Search Trees

An indexed AVL search tree is an indexed binary
search tree that is also an AVL tree

105
Balance Factor

To facilitate insertion and deletion, a balance
factor (bf) is associated with each node
The balance factor bf(x) of a node x is defined
as height(x?leftChild) height(x?rightChild)
Balance factor of each node in an AVL tree must
be 1, 0, or 1

106
AVL Tree with Balance Factors

Is this an AVL tree?
What is the balance factor for each node in this
AVL tree?
Is this an AVL search tree?

107
Inserting into an AVL Search Trees

If we use the strategy of Program 14.5 to insert
an element into an AVL search tree, the result
may not be an AVL tree
That is, the tree may become unbalanced
If the tree becomes unbalanced, we must adjust
the tree to restore balance - this adjustment is
called rotation

108
Inserting into an AVL Search Tree
Insert(9)

Where is 9 going to be inserted into?
After the insertion, is the tree still an AVL
search tree? (i.e., still balanced?)

109
Imbalance Types

After an insertion, when the balance factor of
node A is 2 or 2, the node A is one of the
following four imbalance types
LL new node is in the left subtree of the left
subtree of A
LR new node is in the right subtree of the left
subtree of A
RR new node is in the right subtree of the right
subtree of A
RL new node is in the left subtree of the right
subtree of A

110
Rotation

Definition
To switch children and parents among two or three
adjacent nodes to restore balance of a tree.
A rotation may change the depth of some nodes,
but does not change their relative ordering.

111
Left Rotation

Definition
In a binary search tree, pushing a node A down
and to the left to balance the tree.
A's right child replaces A, and the right child's
left child becomes A's right child.

112
Right Rotation

Definition
In a binary search tree, pushing a node A down
and to the right to balance the tree.
A's left child replaces A, and the left child's
right child becomes A's left child.

113
AVL Rotations

To balance an unbalanced AVL tree (after an
insertion), we may need to perform one of the
following rotations LL, RR, LR, RL

Figure 15.3 Inserting into an AVL search tree
114
An LL Rotation
115
An LR Rotation
116
Single and Double Rotations

Single rotations the transformations done to
correct LL and RR imbalances
Double rotations the transformations done to
correct LR and RL imbalances
The transformation to correct LR imbalance can be
achieved by an RR rotation followed by an LL
rotation
The transformation to correct RL imbalance can be
achieved by an LL rotation followed by an RR
rotation (do Exercise 15.13)
See Figure 15.6 for the AVL-search-tree-insertion
algorithm

117
Deletion from an AVL Search Tree

To delete an element from an AVL search tree, we
can use Program 14.6
Deletion of a node may also produce an imbalance
Imbalance incurred by deletion is classified
intothe types R0, R1, R-1, L0, L1, and L-1
Rotation is also needed for rebalancing

118
An R0 Rotation
Figure 15.7 An R0 rotation (single rotation)
119
An R1 Rotation
Figure 15.8 An R1 rotation (single rotation)
120
An R-1 Rotation
121

Graphs

122
Topics related to Graphs

Graph terminology vertex, edge, adjacent,
incident, degree, cycle, path, connected
component, spanning tree
Types of graphs undirected, directed, weighted
Graph representations adjacency matrix, array
adjacency lists, linked adjacency lists
Graph search methods breath-first, depth-first
search
Algorithms
to find a path in a graph
to find the connected components of an undirected
graph
to find a spanning tree of a connected undirected
graph

123
Graphs

G (V,E)
V is the vertex set.
Vertices are also called nodes and points.
E is the edge set.
Each edge connects two vertices.
Edges are also called arcs and lines.
Vertices i and j are adjacent vertices iff (i, j)
is an edge in the graph
The edge (i, j) is incident on the vertices i and
j

124
Graphs

Undirected edge has no orientation (no arrow
head)
Directed edge has an orientation (has an arrow
head)
Undirected graph all edges are undirected
Directed graph all edges are directed

125
Path and Simple Path

A sequence of vertices P i1, i2, , ik is an i1
to ik path in the graph G(V, E) iff the edge
(ij, ij1) is in E for every j, 1 j lt k
A simple path is a path in which all vertices,
except possibly in the first and last, are
different

126
Length (Cost) of a Path

Each edge in a graph may have an associated
length (or cost). The length of a path is the sum
of the lengths of the edges on the path
What is the length of the path 5, 9, 11, 10?

127
Subgraph Cycle

Let G (V, E) be an undirected graph
A graph H is a subgraph of graph G iff its vertex
and edge sets are subsets of those of G
A cycle is a simple path with the same start and
end vertex

128
Spanning Tree

Let G (V, E) be an undirected graph
A connected undirected graph that contains no
cycles is a tree
A subgraph of G that contains all the vertices of
G and is a tree is a spanning tree
A spanning tree has n vertices and n-1 edges
The spanning tree that costs the least is called
the minimum-cost spanning tree

129
Bipartite Graph

A bipartite graph is a special graph where the
set of vertices can be divided into two disjoint
sets U and V such that no edge has both
end-points in the same set.
A simple undirected graph G (V, E) is called
bipartite if there exists a partition of the
vertex set V V1 U V2 so that both V1 and V2 are
independent sets.

130
Graph Properties

The degree of vertex i is the no. of edges
incident on vertex i.
The sum of vertex degrees 2e (where e is the
number of edges)
In-degree of vertex i is the number of edges
incident to i
Out-degree of vertex i is the number of edges
incident from i

131
Complete Undirected/Directed Graphs

A complete undirected graph has n(n-1)/2 edges
(i.e., all possible edges) and is denoted by Kn

A complete directed graph (also denoted by Kn) on
n vertices contains exactly n(n-1) edges

132
Path Finding

Path between 1 and 8

What is a possible path its length?
A path is 1, 2, 5, 9, 8 and its length is 20.

133
Connected Graph

Let G (V, E) be an undirected graph
G is connected iff there is a path between every
pair of vertices in G

134
Representation of Unweighted Graphs

The most frequently used representations for
unweighted graphs are
Adjacency Matrix
Linked adjacency lists
Array adjacency lists

135
Adjacency Matrix

0/1 n x n matrix, where n of vertices
A(i, j) 1 iff (i, j) is an edge.

136
Adjacency Matrix Properties

Diagonal entries are zero.
Adjacency matrix of an undirected graph is
symmetric (A(i,j) A(j,i) for all i and j).

137
Adjacency Matrix for Digraph

Diagonal entries are zero.
Adjacency matrix of a digraph need not be
symmetric.
See Figure 16.9 for more adjacency matrices

138
Adjacency Lists

Adjacency list for vertex i is a linear list of
vertices adjacent from vertex i.
An array of n adjacency lists for each vertex of
the graph.

139
Linked Adjacency Lists

Each adjacency list is a chain.Array length
n. of chain nodes 2e (undirected graph) of
chain nodes e (digraph)

See Figure 16.11 for more linked adjacency lists

140
Array Adjacency Lists

Each adjacency list is an array list.Array
length n. of chain nodes 2e (undirected
graph) of chain nodes e (digraph)

See Figure 16.12 for more array adjacency lists

141
Representation of Weighted Graphs

Weighted graphs are represented with simple
extensions of those used for unweighted graphs
The cost-adjacency-matrix representation uses a
matrix C just like the adjacency-matrix
representation does
Cost-adjacency matrix C(i, j) cost of edge (i,
j)
Adjacency lists each list element is a
pair(adjacent vertex, edge weight)

142
Tips on the Final Exam