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This Presentation Elliptical Curve Cryptography give a brief explain about this topic, it will use to enrich your knowledge on this topic. Use this ppt for your reference purpose and if you have any queries you'll ask questions.
CMSC 56 | Lecture 12: Recursive Definition & Algorithms, and Program Correctnessallyn joy calcaben
Recursive Definition & Algorithms, and Program Correctness
CMSC 56 | Discrete Mathematical Structure for Computer Science
October 23, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
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This Presentation Elliptical Curve Cryptography give a brief explain about this topic, it will use to enrich your knowledge on this topic. Use this ppt for your reference purpose and if you have any queries you'll ask questions.
CMSC 56 | Lecture 12: Recursive Definition & Algorithms, and Program Correctnessallyn joy calcaben
Recursive Definition & Algorithms, and Program Correctness
CMSC 56 | Discrete Mathematical Structure for Computer Science
October 23, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
Elliptic Curve Cryptography for those who are afraid of mathsMartijn Grooten
A low level introduction into elliptic curve cryptography, as presented at BSides San Francisco 2016.
NB don't be put off by the 100 slides; every transition is on its own slide.
DSA (Data Structure and Algorithm) QuestionsRESHAN FARAZ
DSA (Data Structure and Algorithm) Questions.
University Assignment/Homework Questions.
Stack, Queue , Binary Tree, Heap, AVL, Linked List.
Important DSA Questions to boost the concept.
Backtracking is a general algorithm for finding all (or some) solutions to some computational problems, notably constraint satisfaction problems, that incrementally builds candidates to the solutions, and abandons each partial candidate c ("backtracks") as soon as it determines that c cannot possibly be completed to a valid solution.
Error control codes are necessary for transmission and storage of large volumes of date sensitive to errors. BCH codes and Reed Solomon codes are the most important class of multiple error correcting codes for binary and non-binary channels respectively. Peterson and later Berlekamp and Massey discovered powerful algorithms which became viable with the help of new digital technology. Use of Galois fields gave a structured approach to designing of these codes. This presentation deals with above in a very structured and systematic manner.
At the end of this lecture students should be able to;
Define the C standard functions for managing file input output.
Apply taught concepts for writing programs.
At the end of this lecture students should be able to;
Describe the looping structures in C programming language.
Practice the control flow of different looping structures in C programming language.
Practice the variants in control flow of different looping structures in C programming language.
Apply taught concepts for writing programs.
Elliptic Curve Cryptography for those who are afraid of mathsMartijn Grooten
A low level introduction into elliptic curve cryptography, as presented at BSides San Francisco 2016.
NB don't be put off by the 100 slides; every transition is on its own slide.
DSA (Data Structure and Algorithm) QuestionsRESHAN FARAZ
DSA (Data Structure and Algorithm) Questions.
University Assignment/Homework Questions.
Stack, Queue , Binary Tree, Heap, AVL, Linked List.
Important DSA Questions to boost the concept.
Backtracking is a general algorithm for finding all (or some) solutions to some computational problems, notably constraint satisfaction problems, that incrementally builds candidates to the solutions, and abandons each partial candidate c ("backtracks") as soon as it determines that c cannot possibly be completed to a valid solution.
Error control codes are necessary for transmission and storage of large volumes of date sensitive to errors. BCH codes and Reed Solomon codes are the most important class of multiple error correcting codes for binary and non-binary channels respectively. Peterson and later Berlekamp and Massey discovered powerful algorithms which became viable with the help of new digital technology. Use of Galois fields gave a structured approach to designing of these codes. This presentation deals with above in a very structured and systematic manner.
At the end of this lecture students should be able to;
Define the C standard functions for managing file input output.
Apply taught concepts for writing programs.
At the end of this lecture students should be able to;
Describe the looping structures in C programming language.
Practice the control flow of different looping structures in C programming language.
Practice the variants in control flow of different looping structures in C programming language.
Apply taught concepts for writing programs.
COM1407: Program Control Structures – Decision Making & BranchingHemantha Kulathilake
At the end of this lecture students should be able to;
Define the operation of if, if-else, nested if-else, switch and conditional operator.
Justify the control flow of the program under the aforementioned C language constructs.
Apply taught concepts for writing programs.
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Similar to Algorithms presentation on Path Matrix, Bell Number and Sorting (20)
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June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
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unwillingness to rectify this violation through action requires accountability.
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Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
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2. Agenda Items and Presenters
Bell Numbers
All Pairs Shortest Path
Shell Sort and Radix Sort Psuedocode
3. Bell Numbers
A Use Case
In how many ways, counting ties, can 8 horses cross a finish line?
4. Bell Numbers
Horse Racing Example
Let’s consider just four horses for now and develop a recurrence
relation, a systematic way of counting the possible outcomes.
In mathematics, a recurrence relation is an equation that recursively
defines a sequence, one or more initial terms are given: each further
term of the sequence is defined as a function of the preceding terms.
Four horses may finish in one, two, three or four “blocks”.
Define “blocks”…
A block in this case is a pattern of possible ways the horses can
finish.
If we have fours horses as follows…
9. Bell Numbers
More Background Info
Three block example:
(Alpha) (Boxtrot) (Yisele, Xray) have finished separately.
Yisele and Xray have tied.
10. Bell Numbers
Blocks
Three blocks – (Alpha), (Beta), (Yisele, Xray) may be arranged in 3!
= 6 ways.
The arrangements for a block of three ways in which the horses can
finish.
Note: This does not take into account places (first, second, third,
fourth).
(Alpha) (Boxtrot) (Yisele, Xray), (Alpha) (Yisele) (Boxtrot, Xray),
(Alpha) (Xray) (Boxtro, Yisele), (Boxtrot) (Yisele) (Alpha, Xray),
(Boxtrot) (Xray) (Alpha, Yisele), (Yisele) (Xray) (Alpha, Beta)
11. Bell Numbers
Blocks, Partitions
Arrangements and Outcomes
# Blocks Partitions
1
(Alpha, Boxtrot, Yisele, Xray)
Total Partitions
1
Arrangements per Partition
1!
Outcomes Possible
1
2
(Alpha) (Boxtrot, Yisele, Xray) , (Boxtrot) (Alpha, Yisele, Xray),
(Yisele) (Alpha, Boxtrot, Xray), (Xray) (Alpha, Boxtrot, Yisele).
(Alpha, Boxtrot) (Yisele, Xray), (Alpha, Yisele) (Boxtrot, Xray),
(Alpha, Xray) (Boxtrot, Yisele)
7
2!
14
3
(Alpha) (Boxtrot) (Yisele, Xray), (Alpha) (Yisele) (Boxtrot, Xray),
(Alpha) (Xray) (Boxtro, Yisele), (Boxtrot) (Yisele) (Alpha, Xray),
(Boxtrot) (Xray) (Alpha, Yisele), (Yisele) (Xray) (Alpha, Beta)
6
3!
36
4
(Alpha) (Boxtrot) (Yisele) (Xray)
1
15
4!
24
Bell Number
The number of ways a set of n elements can be partitioned into m
nonempty subsets S(n,m) is the Bell number. In this case 4
horses (elements) can be partitioned in 15 different ways. In this
case we added it up manually but we can determine a formula
and also a graphical way to calculate as follows….
12. Bell Numbers Calculate Graphically
The numbers can be constructed by using the Bell Triangle. Start
with a row with the number one. Afterward each row begins with the
last number of the previous row and continues to the right adding
each number to the number above it to get the next number in the
row..
13. Bell Numbers
The Formula
S(n+1) = S(n,m-1) + m * S(n,m)
n = elements
m = blocks
B(5) = S(5,1) + S(5,2) + S(5,3) + S(5,4) + S(5,5)
14. Answer to the Problem
H8 =
Where S(n,k) denotes the # ways n horses can cross in k blocks.
H8 = 1×1! + 127×2! + 966×3! + 1701×4! +
1050×5! + 266×6! + 28×7! + 1×8!
= 545835.
H8 denotes the number of ways 8 horses can cross the finish line.
15. Bell Numbers
Psuedocode
How do you represent a partitioning of a set of n elements?
(This example is somewhat more challenging that the previous ones. Feel free to skim it, and
go on.) The n’th Bell number Bn is the number of ways of partitioning n (distinct) objects. One
way of computing the Bell numbers is by using the following double recursive algorithm. This
algorithm computes numbers with two arguments: B(i,j). The n’th Bell number, Bn is computed
as
B(n,n). For example to find B3 compute B(3,3).1
B(1,1) = 1.
B(n,1) = B(n-1,n-1) for n > 1.
B(i,j) = B(i-1,j-1) + B(i,j-1) for n > 1 and 1 < j # i.
Technical Information:
Language:
Objects:
16. All Pairs Shortest Path
Given a weighted graph G(V,E,w), the all-pairs shortest paths
problem is to find the shortest paths between all pairs of vertices vi,
vj ∈ V.
A number of algorithms are known for solving this problem.
17. All Pairs Shortest Path
Consider the multiplication of the weighted adjacency matrix with itself
- except, in this case, we replace the multiplication operation in matrix
multiplication by addition, and the addition operation by minimization.
Notice that the product of weighted adjacency matrix with itself
returns a matrix that contains shortest paths of length 2 between any
pair of nodes.
It follows from this argument that An contains all shortest paths.
Transitive Closure : of R is the smallest transitive relation containing R.
18. All Pairs Shortest Path
0
0
1
0
0
0
1
0
0
1
0
1
0
1
0
1
0
0
0
1
0
0
0
1
0
1
0
0
0
1
0
0
R
R[i,j] = {
P1
1
0
Pk [i,j] = {
1
0
If there is a path of
exactly K from i-> j
20. All Pairs Shortest Path
An is computed by doubling powers - i.e., as A, A2, A4, A8, and so on.
We need log n matrix multiplications, each taking time O(n3).
The serial complexity of this procedure is O(n3log n).
This algorithm is not optimal, since the best known algorithms have
complexity O(n3).
21. All Pairs Shortest Path
Parallel Formulation
Each of the log n matrix multiplications can be performed in parallel.
We can use n3/log n processors to compute each matrix-matrix
product in time log n.
The entire process takes O(log2n) time.
22. All Pairs Shortest Path
Parallel Formulation
def adj2(i,j):
if adj(i,j) == 1:
return 1
else:
for k in range(0,n): # where n is the number of vertices in G
if adj(i,k) == 1 and adj(k,j) == 1:
return 1
return 0
26. Shell Sort
Psuedocode
# Sort an array a[0...n-1].
gaps = [701, 301, 132, 57, 23, 10, 4, 1]
foreach (gap in gaps)
{
# Do an insertion sort for each gap size.
for (i = gap; i < n; i += 1)
{
temp = a[i]
for (j = i; j >= gap and a[j - gap] > temp; j -= gap)
{
a[j] = a[j - gap]
}
a[j] = temp
}
}
Technical Information:
Language:
Programming Constructs:
Efficiency:
27. Radix Sort
For simplicity, say you want to use the decimal radix (=10) for sorting. Then you would start
by separating the numbers by units and then putting them together again; next you would
separate the numbers by tens and then put them together again; then by hundreds and so on
until all the numbers are sorted. Each time you loop, just read the list from left to right. You
can also imagine you are separating the numbers into buckets. Here is an illustration using
28. Radix Sort
5, 213, 55, 21, 2334, 31, 20, 430
Separate by units:
zeros: 20, 430
ones: 21, 31
twos:
threes: 213
fours: 2334
fives: 5, 55
Back together: 20, 430, 21, 31, 213, 2334, 5, 55
To put them back together, first read the zeroes bucket, then the ones bucket, then so on,
until you read the nines bucket.
29. Radix Sort
Separate by tens:
zeros: 05
ones: 213
twos: 20, 21
threes: 430, 2334,
fours:
fives: 55
Back together: 5, 213, 20, 21, 430, 2334, 55
30. Radix Sort
Separate by hundreds:
zeros: 005, 020, 021, 055
ones:
twos: 213
threes: 2334
fours: 430
fives:
Back together: 5, 20, 21, 55, 213, 2334, 430
31. Radix Sort
Separate by thousands:
zeros: 0005, 0020, 0021, 0055,0213, 0430
ones:
twos: 2334
threes:
fours:
fives:
Back together: 5, 20, 21, 55, 213, 430, 2334