Download EECS 560 Notes Chapter 8 Disjoint Set ADT

Chapter 8--Disjoint Set ADT
The disjoint set ADT is an efficient data structure to solve the
equivalence problem.
An equivalence relation R on a set S satisfies three properties:
1.
2.
3.
Section 8.2 The Dynamic Equivalence Problem
equivalence class
data structure -- disjoint set ADT.
operations
The union/find algorithm
The algorithm must operate on-line
Two strategies
A fast find operation
Section 8.3
Figure 1
:
1
2
3
4
5
6
7
8
3
8
5
7
5
8
Union operations
5
2
Figure 2
Union operations
Result of union(6, 4) on the set in figure 2:
Figure 3
Section 8.4 Smart Union Algorithms
union by height
Section 8.5 Path Compression
Example of path compression
Now, after find(2) the result is:
Other operations:
Skip lists
Motivation:
Skip Lists (continued)
Inserting 30 into the list
Generating nodes for skip lists
Algorithm is taken from Lewis and Denenberg's Data Structures &
Their Algorithms
function RandomLevel(): integer
// Produce a random level between 0 and MaxLevel
v0
while Rand() < 1/2 and v < MaxLevel do v  v+ 1
return v
The find operation on a skip list
Advantages and disadvantages of skip lists
Huffman Codes
Suppose the following bit strings are the codes to w, x, y and z:
w
00
x y
101 10
z
110
Consider the string 10110.
Consider instead the following code for the same four characters:
w
0
x
y z
100 101 11
Then the message 01100110101100 decodes as
Building Huffman Codes
Methodology:
Huffman Code Example
letter:
frequency:
E
50
T
40
A
30
O
25
I
20
N
15
I
N
Then we have the following codes:
letter:
code:
E
T
A
O
Huffman(C)
n  |C|
QC
for i  1 to n – 1
create a new node z
left[z]  x where x is obtained by a deletemin(Q)
right[z]  y where y is obtained by a deletmin(Q)
freq[z]  freq[x] + freq[y]
insert(Q, z)
return(deletemin(Q)) // the root of the tree