it contains the detail information about Dynamic programming, Knapsack problem, Forward / backward knapsack, Optimal Binary Search Tree (OBST), Traveling sales person problem(TSP) using dynamic programming
This presentation contains information about the divide and conquer algorithm. It includes discussion regarding its part, technique, skill, advantages and implementation issues.
In computer science, divide and conquer is an algorithm design paradigm based on multi-branched recursion. A divide-and-conquer algorithm works by recursively breaking down a problem into two or more sub-problems of the same or related type until these become simple enough to be solved directly.
Divide and Conquer Algorithms - D&C forms a distinct algorithm design technique in computer science, wherein a problem is solved by repeatedly invoking the algorithm on smaller occurrences of the same problem. Binary search, merge sort, Euclid's algorithm can all be formulated as examples of divide and conquer algorithms. Strassen's algorithm and Nearest Neighbor algorithm are two other examples.
it contains the detail information about Dynamic programming, Knapsack problem, Forward / backward knapsack, Optimal Binary Search Tree (OBST), Traveling sales person problem(TSP) using dynamic programming
This presentation contains information about the divide and conquer algorithm. It includes discussion regarding its part, technique, skill, advantages and implementation issues.
In computer science, divide and conquer is an algorithm design paradigm based on multi-branched recursion. A divide-and-conquer algorithm works by recursively breaking down a problem into two or more sub-problems of the same or related type until these become simple enough to be solved directly.
Divide and Conquer Algorithms - D&C forms a distinct algorithm design technique in computer science, wherein a problem is solved by repeatedly invoking the algorithm on smaller occurrences of the same problem. Binary search, merge sort, Euclid's algorithm can all be formulated as examples of divide and conquer algorithms. Strassen's algorithm and Nearest Neighbor algorithm are two other examples.
In computer science, divide and conquer (D&C) is an algorithm design paradigm based on multi-branched recursion. A divide and conquer algorithm works by recursively breaking down a problem into two or more sub-problems of the same (or related) type, until these become simple enough to be solved directly. The solutions to the sub-problems are then combined to give a solution to the original problem.
In computer science, merge sort (also commonly spelled mergesort) is an O(n log n) comparison-based sorting algorithm. Most implementations produce a stable sort, which means that the implementation preserves the input order of equal elements in the sorted output. Mergesort is a divide and conquer algorithm that was invented by John von Neumann in 1945. A detailed description and analysis of bottom-up mergesort appeared in a report by Goldstine and Neumann as early as 1948.
BackTracking Algorithm: Technique and ExamplesFahim Ferdous
This slides gives a strong overview of backtracking algorithm. How it came and general approaches of the techniques. Also some well-known problem and solution of backtracking algorithm.
In computer science, divide and conquer (D&C) is an algorithm design paradigm based on multi-branched recursion. A divide and conquer algorithm works by recursively breaking down a problem into two or more sub-problems of the same (or related) type, until these become simple enough to be solved directly. The solutions to the sub-problems are then combined to give a solution to the original problem.
In computer science, merge sort (also commonly spelled mergesort) is an O(n log n) comparison-based sorting algorithm. Most implementations produce a stable sort, which means that the implementation preserves the input order of equal elements in the sorted output. Mergesort is a divide and conquer algorithm that was invented by John von Neumann in 1945. A detailed description and analysis of bottom-up mergesort appeared in a report by Goldstine and Neumann as early as 1948.
BackTracking Algorithm: Technique and ExamplesFahim Ferdous
This slides gives a strong overview of backtracking algorithm. How it came and general approaches of the techniques. Also some well-known problem and solution of backtracking algorithm.
Case study of Divide and Conquer approach contains information about-merge sort and quick sort algorithms, closest pair of points, binary search, la-russe multiplicaton, min-max problems and also strassen multiplication.
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Explore our comprehensive data analysis project presentation on predicting product ad campaign performance. Learn how data-driven insights can optimize your marketing strategies and enhance campaign effectiveness. Perfect for professionals and students looking to understand the power of data analysis in advertising. for more details visit: https://bostoninstituteofanalytics.org/data-science-and-artificial-intelligence/
Levelwise PageRank with Loop-Based Dead End Handling Strategy : SHORT REPORT ...Subhajit Sahu
Abstract — Levelwise PageRank is an alternative method of PageRank computation which decomposes the input graph into a directed acyclic block-graph of strongly connected components, and processes them in topological order, one level at a time. This enables calculation for ranks in a distributed fashion without per-iteration communication, unlike the standard method where all vertices are processed in each iteration. It however comes with a precondition of the absence of dead ends in the input graph. Here, the native non-distributed performance of Levelwise PageRank was compared against Monolithic PageRank on a CPU as well as a GPU. To ensure a fair comparison, Monolithic PageRank was also performed on a graph where vertices were split by components. Results indicate that Levelwise PageRank is about as fast as Monolithic PageRank on the CPU, but quite a bit slower on the GPU. Slowdown on the GPU is likely caused by a large submission of small workloads, and expected to be non-issue when the computation is performed on massive graphs.
Opendatabay - Open Data Marketplace.pptxOpendatabay
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As Europe's leading economic powerhouse and the fourth-largest hashtag#economy globally, Germany stands at the forefront of innovation and industrial might. Renowned for its precision engineering and high-tech sectors, Germany's economic structure is heavily supported by a robust service industry, accounting for approximately 68% of its GDP. This economic clout and strategic geopolitical stance position Germany as a focal point in the global cyber threat landscape.
In the face of escalating global tensions, particularly those emanating from geopolitical disputes with nations like hashtag#Russia and hashtag#China, hashtag#Germany has witnessed a significant uptick in targeted cyber operations. Our analysis indicates a marked increase in hashtag#cyberattack sophistication aimed at critical infrastructure and key industrial sectors. These attacks range from ransomware campaigns to hashtag#AdvancedPersistentThreats (hashtag#APTs), threatening national security and business integrity.
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Techniques to optimize the pagerank algorithm usually fall in two categories. One is to try reducing the work per iteration, and the other is to try reducing the number of iterations. These goals are often at odds with one another. Skipping computation on vertices which have already converged has the potential to save iteration time. Skipping in-identical vertices, with the same in-links, helps reduce duplicate computations and thus could help reduce iteration time. Road networks often have chains which can be short-circuited before pagerank computation to improve performance. Final ranks of chain nodes can be easily calculated. This could reduce both the iteration time, and the number of iterations. If a graph has no dangling nodes, pagerank of each strongly connected component can be computed in topological order. This could help reduce the iteration time, no. of iterations, and also enable multi-iteration concurrency in pagerank computation. The combination of all of the above methods is the STICD algorithm. [sticd] For dynamic graphs, unchanged components whose ranks are unaffected can be skipped altogether.
2. Introduction
Divide-and-conquer is a top-down
technique for designing algorithms
that consists of dividing the problem
into smaller sub problems hoping
that the solutions of the sub
problems are easier to find and then
composing the partial solutions into
the solution of the original problem.
3. Divide-and-conquer
1. Divide: Break the problem into
sub-problems of same type.
2. Conquer: Recursively solve
these sub-problems.
3. Combine: Combine the solution
sub-problems.
8. Divide-and-conquer Detail
Divide/Break
This step involves breaking the problem
into smaller sub-problems.
Sub-problems should represent a part of
the original problem.
This step generally takes a recursive
approach to divide the problem until no
sub-problem is further divisible.
At this stage, sub-problems become
atomic in nature but still represent some
part of the actual problem.
10. Divide-and-conquer Detail
Merge/Combine
When the smaller sub-problems are
solved, this stage recursively combines
them until they formulate a solution of
the original problem.
This algorithmic approach works
recursively and conquer & merge steps
works so close that they appear as one.
11. Standard Algorithms
based on D & C
The following algorithms are based on
divide-and-conquer algorithm design
paradigm.
Merge Sort
Quick Sort
Binary Search
Strassen's Matrix Multiplication
Closest pair (points)
Cooley–Tukey Fast Fourier Transform
(FFT) algorithm
12. Advantages of D & C
1. Solving difficult problems:
Divide and conquer is a powerful tool for solving
conceptually difficult problems: all it requires is a
way of breaking the problem into sub-problems,
of solving the trivial cases and of combining sub-
problems to the original problem.
2. Parallelism:
Divide and conquer algorithms are naturally
adapted for execution in multi-processor
machines, especially shared-memory systems
where the communication of data between
processors does not need to be planned in
advance, because distinct sub-problems can be
executed on different processors.
13. Advantages of D & C
3. Memory Access:
Divide-and-conquer algorithms naturally tend to
make efficient use of memory caches. The reason
is that once a sub-problem is small enough, it and
all its sub-problems can, in principle, be solved
within the cache, without accessing the slower
main memory.
4. Roundoff control:
In computations with rounded arithmetic, e.g.
with floating point numbers, a divide-and-conquer
algorithm may yield more accurate results than a
superficially equivalent iterative method.
14. Advantages of D & C
For solving difficult problems like Tower
Of Hanoi, divide & conquer is a powerful
tool
Results in efficient algorithms.
Divide & Conquer algorithms are adapted
foe execution in multi-processor
machines
Results in algorithms that use memory
cache efficiently.
15. Limitations of D & C
Recursion is slow.
Very simple problem may be more
complicated than an iterative approach.
Example: adding n numbers etc
16. Example of Multiplication of
N digit integers.
Multiplication can be perform using divide
and conquer technique.
First we know the decimal system of
number which are shown as under.
Number is 3754
3*103 + 7*102 + 5*101 + 4*100
17. Example of Multiplication of
N digit integers.
2345 * 6789
2*103 + 3*102 + 4*101 + 5*100
6*103 + 7*102 + 8*101 + 9*100
4 3 2 1
18. Example of Multiplication of
N digit integers.
2345 * 6789
2*103 + 3*102 + 4*101 + 5*100
6*103 + 7*102 + 8*101 + 9*100
4 3 2 1
19. Example of Multiplication of
N digit integers.
2345 * 6789
2*103 + 3*102 + 4*101 + 5*100
6*103 + 7*102 + 8*101 + 9*100
4 3 2 1
20. Closest-Pair Problem:
Divide and Conquer
Brute force approach requires comparing every point
with every other point
Given n points, we must perform 1 + 2 + 3 + … + n-
2 + n-1 comparisons.
Brute force O(n2)
The Divide and Conquer algorithm yields O(n log
n)
Reminder: if n = 1,000,000 then
n2 = 1,000,000,000,000 whereas
n log n = 20,000,000
2
)1(1
1
nn
k
n
k
23. Lets sort based on the X-axis
O(n log n) using quicksort or mergesort
1
2
3
4
5
6
7
8
9
1
0
1
1
1
2
1
3 1
4
Closest-Pair Algorithm
24. Step 2: Split the points, i.e.,
Draw a line at the mid-point between 7
and 8
1
2
3
4
5
6
7
8
9
1
0
1
1
1
2
1
3 1
4
Closest-Pair Algorithm
Sub-Problem
1
Sub-Problem
2
25. Advantage: Normally, we’d have to
compare each of the 14 points with every
other point.
(n-1)n/2 = 13*14/2 = 91 comparisons
1
2
3
4
5
6
7
8
9
1
0
1
1
1
2
1
3 1
4
Closest-Pair Algorithm
Sub-Problem
1
Sub-Problem
2
26. Advantage: Now, we have two sub-
problems of half the size. Thus, we have to
do 6*7/2 comparisons twice, which is 42
comparisons
1
2
3
4
5
6
7
8
9
1
0
1
1
1
2
1
3 1
4
Closest-Pair Algorithm
d1
d2
Sub-Problem
1
Sub-Problem
2
solution d = min(d1, d2)
27. Advantage: With just one split we cut the
number of comparisons in half. Obviously,
we gain an even greater advantage if we
split the sub-problems.
1
2
3
4
5
6
7
8
9
1
0
1
1
1
2
1
3 1
4
Closest-Pair Algorithm
d1
d2
Sub-Problem
1
Sub-Problem
2
d = min(d1, d2)
28. Problem: However, what if the closest two
points are each from different sub-
problems?
1
2
3
4
5
6
7
8
9
1
0
1
1
1
2
1
3 1
4
Closest-Pair Algorithm
d1
d2
Sub-Problem
1
Sub-Problem
2
29. Here is an example where we have to
compare points from sub-problem 1 to the
points in sub-problem 2.
1
2
3
4
5
6
7
8
9
1
0
1
1
1
2
1
3 1
4
Closest-Pair Algorithm
d1
d2
Sub-Problem
1
Sub-Problem
2
30. However, we only have to compare points
inside the following “strip.”
1
2
3
4
5
6
7
8
9
1
0
1
1
1
2
1
3 1
4
Closest-Pair Algorithm
d1
d2
Sub-Problem
1
Sub-Problem
2
dd
d = min(d1, d2)
31. Step 3: In fact we can continue to split
until each sub-problem is trivial, i.e., takes
one comparison.
1
2
3
4
5
6
7
8
9
1
0
1
1
1
2
1
3 1
4
Closest-Pair Algorithm
32. Finally: The solution to each sub-problem
is combined until the final solution is
obtained
1
2
3
4
5
6
7
8
9
1
0
1
1
1
2
1
3 1
4
Closest-Pair Algorithm
33. Finally: On the last step the ‘strip’ will
likely be very small. Thus, combining the
two largest sub-problems won’t require
much work.
1
2
3
4
5
6
7
8
9
1
0
1
1
1
2
1
3 1
4
Closest-Pair Algorithm
34. 1
2
3
4
5
6
7
8
9
1
0
1
1
1
2
1
3 1
4
Closest-Pair Algorithm
In this example, it takes 22 comparisons to
find the closets-pair.
The brute force algorithm would have taken
91 comparisons.
But, the real advantage occurs when there are
millions of points.
35. Closest-Pair Algo
Float DELTA-LEFT, DELTA-RIGHT;
Float DELTA;
If n= 2 then
return distance form p(1) to p(2);
Else
P-LEFT <- (p(1),p(2),p(n/2));
P-RIGHT <- (p(n/2 + 1),p(n/2 + 2),p(n));
DELTA-LEFT <- Closest_pair(P-LEFT,n2);
DELTA-RIGHT <- Closest_pair(P-RIGHT,n2);
DELTA<- minimum(DELTA-LEFT,DELTA-RIGHT)
36. Closest-Pair Algo
For I in 1---s do
for j in i+1..s do
if(|x[i] – x[j] | > DELTA and
|y[i] – y[j]|> DELTA ) then
exit;
end
if(distance(q[I],q[j] <DELTA)) then
DELTA <- distance (q[i],q[j]);
end
end
41. Timing Analysis
D&C algorithm running time in
mainly affected by 3 factors
The number of sub-instance(α)
into which a problem is split.
The ratio of initial problem size to
sub problem size(ß)
The number of steps required to
divide the initial instance and to
combine sub-solutions expressed
as a function of the input size n.
42. Timing Analysis
P is D & C Where α sub instance each of
size n/ß
Let Tp(n) donete the number of steps
taken by P on instances of size n.
Tp(n0) = constant (recursive-base);
Tp(n)= αTp (n/ß)+y(n);
α is number
of sub
instance
ß is number
of sub size
y is constant