Download Class Notes for Week 3

CSE 5311 Advanced Algorithms
Instructor: Dr. Gautam Das
Week 3 Class Notes
Submitted By:
Abhishek Hemrajani & Pinky Dewani
CSE 5311 Advanced Algorithms
Topics Covered in Week 3:
 Median Finding Algorithm (Description and
Analysis)
 Binary Search Tree
CSE 5311 Advanced Algorithms
Given a set of “n” numbers we can say that,
 Mean: Average of the “n” numbers
 Median: Having sorted the “n” numbers, the
value which lies in the middle of the list such
that half the numbers are higher than it and
half the numbers are lower than it.
Thus the problem of finding the median can
be generalized to finding the kth largest
number where k = n/2.
CSE 5311 Advanced Algorithms
kth largest number can be found using:
 Scan Approach with a time complexity
T(n) = kn
 Sort Approach with a time complexity
T(n) = nlogn
CSE 5311 Advanced Algorithms
Linear Time Median Finding Algorithm
 Input: S = {a1,a2,…….,an}
k such that 1 ≤ k ≤ n
 Output: kth largest number
 Algorithm: kth_largest (S, k)
CSE 5311 Advanced Algorithms
Steps:
1)
Group the numbers into sets of 5
2)
Sort individual groups and find the median of each
group
3)
Let “M” be set of medians and find median of “M”
using MedianOfMedian
(MOM) = kth_largest(M,M/2)
4)
Partition original data around the MOM such that
values less than it are in set “L” and values greater
than it are in set “R”
CSE 5311 Advanced Algorithms
Steps Continued:
5) If |L| = k-1, then return MOM else
If |L| > k-1, then return kth_largest(L,k) else
return kth_largest(R,k-|L|)
CSE 5311 Advanced Algorithms
Example:
(……..2,5,9,19,24,54,5,87,9,10,44,32,21,13,24,
18,26,16,19,25,39,47,56,71,91,61,44,28……
…) is a set of “n” numbers
CSE 5311 Advanced Algorithms
Step1: Group numbers in sets of 5 (Vertically)
………………..
………………..
2
5
54
44
4
25
5
32
18
39
21
26
47
………………..
………………..
………………..
9
87
19
9
13
16
56
………………..
……………….. 24
10
2
19
71
………………..
………………..
………………..
CSE 5311 Advanced Algorithms
Step2: Find Median of each group
………………..
………………..
2
5
5
2
4
9
13
16
25
39
………………..
………………..
………………..
9
10
21
18
47
19
54
32
19
56
………………..
……………….. 24
87
44
26
71
………………..
………………..
………………..
Median of each group
CSE 5311 Advanced Algorithms
Step3: Find the MedianOfMedians
………………..
3.n/10
………………..
2
5
5
2
4
25
9
13
16
39
………………..
………………..
………………..
9
10
21
18
47
19
54
32
19
56
………………..
……………….. 24
87
44
26
71
………………..
………………..
………………..
Find m ,the median of medians
CSE 5311 Advanced Algorithms
Step4: Partition original data around the MOM
M
L
3n/10<L<7n/10
R
3n/10<R<7n/10
Step5: If |L| = k-1, then return MOM else
If |L| > k-1, then return kth_largest(L,k) else
return kth_largest(R,k-|L|)
CSE 5311 Advanced Algorithms
Time Analysis:
Step
1
2
3
4
5
Task
Group into sets
of 5
Find Median of
each group
Find MOM
Partition around
MOM
Condition
Complexity
O (n)
O (n)
T (n/5)
O (n)
T (7n/10)
{Worst Case}
CSE 5311 Advanced Algorithms
Time Complexity of Algorithm:
T(n) = O (n) + T (n/5) + T (7n/10)
T(1) = 1
Assume T (n) ≤ Cn (For it to be linear time)
L.H.S = Cn
R.H.S = C1n + Cn/5 + 7Cn/10 = (C1 + 9/10C)n
Hence, L.H.S = R.H.S if C = 10C1
Thus it is a Linear Time Algorithm
CSE 5311 Advanced Algorithms
Why sets of 5?
Assuming sets of 3,
T(n) = O (n) + T (n/3) + T (2n/3)
T(1) = 1
Assume T (n) ≤ Cn (For it to be linear time)
L.H.S = Cn
R.H.S = C1n + Cn/3 + 2Cn/3 = (C1 + C) n
Hence, L.H.S = R.H.S if C1 = 0
Thus this is invalid assumption. Hence we do
not use groups of 3!
CSE 5311 Advanced Algorithms
Assuming sets of 7,
T(n) = O (n) + T (n/7) + T (5n/7)
T(1) = 1
Assume T (n) ≤ Cn (For it to be linear time)
L.H.S = Cn
R.H.S = C1n + Cn/7 + 5Cn/7 = (C1 + 6C/7) n
Hence, L.H.S = R.H.S if C = 7C1
However, constant factor of O (n) term increases to
a sub-optimal value as size of set increases!
CSE 5311 Advanced Algorithms
Points to remember:
 This algorithm uses Divide and Conquer
strategy
 The technique is Recursion
 Most realistic approaches randomly select
the MOM
CSE 5311 Advanced Algorithms
Maintenance of Dynamic Sets:
Given a Set “S” and a value “X”, we need to
perform the following three operations:
 Insert (X, S)
 Delete (X,S)
 Find (X, S)
Example:
Customer Database, Yellow Pages etc.
CSE 5311 Advanced Algorithms
Data Structures used for Dynamic Sets:
Operation
Unordered
Array
Sorted
Array
Unordered
List
Goal
Insert (X, S)
O (1)
O (n)
O (1)
O (logn)
Delete (X,S)
O (n)
O (n)
O (n)
O (logn)
Find (X, S)
O (n)
O (logn)
O (n)
O (logn)
CSE 5311 Advanced Algorithms
Binary Search Tree:
A Binary Search Tree is a data structure such that at any node, the
values in the left subtree are smaller than the node and the values in
the right subtree are greater than the node.
Example: Given S = {3, 2, 5, 4, 7, 6, 1}, the Binary Search Tree is:
3
2
1
5
7
4
6
CSE 5311 Advanced Algorithms
All operations performed on the Binary Search
Tree are of the order O (path from root to leaf)
which can be O (logn) in a favorable case or O
(n) in the worst case (Sorted Input).
CSE 5311 Advanced Algorithms
Sorting using In-Order Traversal:
Binary Search Tree can be sorted using In-Order
Traversal as follows:
Sort (T)
{
Sort (LeftChild (T))
Write (T)
Sort (RightChild (T))
}
Here, T (n) = T (m) + T (n-m-1) + O (1)
CSE 5311 Advanced Algorithms
For a Balanced Binary Search Tree:
T(n) = T (L.H.S) + T (R.H.S) + O (1)
= 2 T (n/2) + 1
= O (n)
CSE 5311 Advanced Algorithms
Points to remember:
 Pointers do not give advantage of Random
Access but make the data structure dynamic.
 B Tree and B+ Tree are optimizations of
Binary Search Tree to give guaranteed
O (logn) performance.