The document discusses the divide and conquer algorithm design paradigm. It begins by defining divide and conquer as recursively breaking down a problem into smaller sub-problems, solving the sub-problems, and then combining the solutions to solve the original problem. Some examples of problems that can be solved using divide and conquer include binary search, quicksort, merge sort, and the fast Fourier transform algorithm. The document then discusses control abstraction, efficiency analysis, and uses divide and conquer to provide algorithms for large integer multiplication and merge sort. It concludes by defining the convex hull problem and providing an example input and output.
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.
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.
Design and Analysis of Algorithm help to design the algorithms for solving different types of problems in Computer Science. It also helps to design and analyze the logic of how the program will work before developing the actual code for a program.
This presentation contains information about the divide and conquer algorithm. It includes discussion regarding its part, technique, skill, advantages and implementation issues.
Design and Analysis of Algorithm help to design the algorithms for solving different types of problems in Computer Science. It also helps to design and analyze the logic of how the program will work before developing the actual code for a program.
This presentation contains information about the divide and conquer algorithm. It includes discussion regarding its part, technique, skill, advantages and implementation issues.
270-1/02-divide-and-conquer_handout.pdf
CS 270
Algorithms
Oliver
Kullmann
Growth of
Functions
Divide-and-
Conquer
Min-Max-
Problem
Tutorial
Week 2
Divide and Conquer
1 Growth of Functions
2 Divide-and-Conquer
Min-Max-Problem
3 Tutorial
CS 270
Algorithms
Oliver
Kullmann
Growth of
Functions
Divide-and-
Conquer
Min-Max-
Problem
Tutorial
General remarks
First we consider an important tool for the analysis of
algorithms: Big-Oh.
Then we introduce an important algorithmic paradigm:
Divide-and-Conquer.
We conclude by presenting and analysing a simple example.
Reading from CLRS for week 2
Chapter 2, Section 3
Chapter 3
CS 270
Algorithms
Oliver
Kullmann
Growth of
Functions
Divide-and-
Conquer
Min-Max-
Problem
Tutorial
Growth of Functions
A way to describe behaviour of functions in the limit. We
are studying asymptotic efficiency.
Describe growth of functions.
Focus on what’s important by abstracting away low-order
terms and constant factors.
How we indicate running times of algorithms.
A way to compare “sizes” of functions:
O corresponds to ≤
Ω corresponds to ≥
Θ corresponds to =
We consider only functions f , g : N → R≥0.
CS 270
Algorithms
Oliver
Kullmann
Growth of
Functions
Divide-and-
Conquer
Min-Max-
Problem
Tutorial
O-Notation
O
(
g(n)
)
is the set of all functions f (n) for which there are
positive constants c and n0 such that
f (n) ≤ cg(n) for all n ≥ n0.
cg(n)
f (n)
n
n0
g(n) is an asymptotic upper bound for f (n).
If f (n) ∈ O(g(n)), we write f (n) = O(g(n)) (we will precisely
explain this soon)
CS 270
Algorithms
Oliver
Kullmann
Growth of
Functions
Divide-and-
Conquer
Min-Max-
Problem
Tutorial
O-Notation Examples
2n2 = O(n3), with c = 1 and n0 = 2.
Example of functions in O(n2):
n2
n2 + n
n2 + 1000n
1000n2 + 1000n
Also
n
n/1000
n1.999999
n2/ lg lg lg n
CS 270
Algorithms
Oliver
Kullmann
Growth of
Functions
Divide-and-
Conquer
Min-Max-
Problem
Tutorial
Ω-Notation
Ω
(
g(n)
)
is the set of all functions f (n) for which there are
positive constants c and n0 such that
f (n) ≥ cg(n) for all n ≥ n0.
cg(n)
f (n)
n
n0
g(n) is an asymptotic lower bound for f (n).
CS 270
Algorithms
Oliver
Kullmann
Growth of
Functions
Divide-and-
Conquer
Min-Max-
Problem
Tutorial
Ω-Notation Examples
√
n = Ω(lg n), with c = 1 and n0 = 16.
Example of functions in Ω(n2):
n2
n2 + n
n2 − n
1000n2 + 1000n
1000n2 − 1000n
Also
n3
n2.0000001
n2 lg lg lg n
22
n
CS 270
Algorithms
Oliver
Kullmann
Growth of
Functions
Divide-and-
Conquer
Min-Max-
Problem
Tutorial
Θ-Notation
Θ
(
g(n)
)
is the set of all functions f (n) for which there are
positive constants c1, c2 and n0 such that
c1g(n) ≤ f (n) ≤ c2g(n) for all n ≥ n0.
c2g(n)
c1g(n)
f (n)
n
n0
g(n) is an asymptotic tight bound for f (n).
CS 270
Algorithms
Oliver
Kullmann
Growth of
Functions
Divide-and-
Conquer
Min-Max-
Problem
Tutorial
Θ-Notation (cont’d)
E.
This power point presentation will give you the knowledge of merge sort algorithm how it works with a given problem solving example. It also describe about the time complexity of merge sort algorithm, and the program in c .
Overview of the fundamental roles in Hydropower generation and the components involved in wider Electrical Engineering.
This paper presents the design and construction of hydroelectric dams from the hydrologist’s survey of the valley before construction, all aspects and involved disciplines, fluid dynamics, structural engineering, generation and mains frequency regulation to the very transmission of power through the network in the United Kingdom.
Author: Robbie Edward Sayers
Collaborators and co editors: Charlie Sims and Connor Healey.
(C) 2024 Robbie E. Sayers
Hybrid optimization of pumped hydro system and solar- Engr. Abdul-Azeez.pdffxintegritypublishin
Advancements in technology unveil a myriad of electrical and electronic breakthroughs geared towards efficiently harnessing limited resources to meet human energy demands. The optimization of hybrid solar PV panels and pumped hydro energy supply systems plays a pivotal role in utilizing natural resources effectively. This initiative not only benefits humanity but also fosters environmental sustainability. The study investigated the design optimization of these hybrid systems, focusing on understanding solar radiation patterns, identifying geographical influences on solar radiation, formulating a mathematical model for system optimization, and determining the optimal configuration of PV panels and pumped hydro storage. Through a comparative analysis approach and eight weeks of data collection, the study addressed key research questions related to solar radiation patterns and optimal system design. The findings highlighted regions with heightened solar radiation levels, showcasing substantial potential for power generation and emphasizing the system's efficiency. Optimizing system design significantly boosted power generation, promoted renewable energy utilization, and enhanced energy storage capacity. The study underscored the benefits of optimizing hybrid solar PV panels and pumped hydro energy supply systems for sustainable energy usage. Optimizing the design of solar PV panels and pumped hydro energy supply systems as examined across diverse climatic conditions in a developing country, not only enhances power generation but also improves the integration of renewable energy sources and boosts energy storage capacities, particularly beneficial for less economically prosperous regions. Additionally, the study provides valuable insights for advancing energy research in economically viable areas. Recommendations included conducting site-specific assessments, utilizing advanced modeling tools, implementing regular maintenance protocols, and enhancing communication among system components.
Courier management system project report.pdfKamal Acharya
It is now-a-days very important for the people to send or receive articles like imported furniture, electronic items, gifts, business goods and the like. People depend vastly on different transport systems which mostly use the manual way of receiving and delivering the articles. There is no way to track the articles till they are received and there is no way to let the customer know what happened in transit, once he booked some articles. In such a situation, we need a system which completely computerizes the cargo activities including time to time tracking of the articles sent. This need is fulfilled by Courier Management System software which is online software for the cargo management people that enables them to receive the goods from a source and send them to a required destination and track their status from time to time.
Cosmetic shop management system project report.pdfKamal Acharya
Buying new cosmetic products is difficult. It can even be scary for those who have sensitive skin and are prone to skin trouble. The information needed to alleviate this problem is on the back of each product, but it's thought to interpret those ingredient lists unless you have a background in chemistry.
Instead of buying and hoping for the best, we can use data science to help us predict which products may be good fits for us. It includes various function programs to do the above mentioned tasks.
Data file handling has been effectively used in the program.
The automated cosmetic shop management system should deal with the automation of general workflow and administration process of the shop. The main processes of the system focus on customer's request where the system is able to search the most appropriate products and deliver it to the customers. It should help the employees to quickly identify the list of cosmetic product that have reached the minimum quantity and also keep a track of expired date for each cosmetic product. It should help the employees to find the rack number in which the product is placed.It is also Faster and more efficient way.
About
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Technical Specifications
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
Key Features
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface
• Compatible with MAFI CCR system
• Copatiable with IDM8000 CCR
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
Application
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Saudi Arabia stands as a titan in the global energy landscape, renowned for its abundant oil and gas resources. It's the largest exporter of petroleum and holds some of the world's most significant reserves. Let's delve into the top 10 oil and gas projects shaping Saudi Arabia's energy future in 2024.
CFD Simulation of By-pass Flow in a HRSG module by R&R Consult.pptxR&R Consult
CFD analysis is incredibly effective at solving mysteries and improving the performance of complex systems!
Here's a great example: At a large natural gas-fired power plant, where they use waste heat to generate steam and energy, they were puzzled that their boiler wasn't producing as much steam as expected.
R&R and Tetra Engineering Group Inc. were asked to solve the issue with reduced steam production.
An inspection had shown that a significant amount of hot flue gas was bypassing the boiler tubes, where the heat was supposed to be transferred.
R&R Consult conducted a CFD analysis, which revealed that 6.3% of the flue gas was bypassing the boiler tubes without transferring heat. The analysis also showed that the flue gas was instead being directed along the sides of the boiler and between the modules that were supposed to capture the heat. This was the cause of the reduced performance.
Based on our results, Tetra Engineering installed covering plates to reduce the bypass flow. This improved the boiler's performance and increased electricity production.
It is always satisfying when we can help solve complex challenges like this. Do your systems also need a check-up or optimization? Give us a call!
Work done in cooperation with James Malloy and David Moelling from Tetra Engineering.
More examples of our work https://www.r-r-consult.dk/en/cases-en/
Forklift Classes Overview by Intella PartsIntella Parts
Discover the different forklift classes and their specific applications. Learn how to choose the right forklift for your needs to ensure safety, efficiency, and compliance in your operations.
For more technical information, visit our website https://intellaparts.com
Immunizing Image Classifiers Against Localized Adversary Attacksgerogepatton
This paper addresses the vulnerability of deep learning models, particularly convolutional neural networks
(CNN)s, to adversarial attacks and presents a proactive training technique designed to counter them. We
introduce a novel volumization algorithm, which transforms 2D images into 3D volumetric representations.
When combined with 3D convolution and deep curriculum learning optimization (CLO), itsignificantly improves
the immunity of models against localized universal attacks by up to 40%. We evaluate our proposed approach
using contemporary CNN architectures and the modified Canadian Institute for Advanced Research (CIFAR-10
and CIFAR-100) and ImageNet Large Scale Visual Recognition Challenge (ILSVRC12) datasets, showcasing
accuracy improvements over previous techniques. The results indicate that the combination of the volumetric
input and curriculum learning holds significant promise for mitigating adversarial attacks without necessitating
adversary training.
Planning Of Procurement o different goods and services
Divide and Conquer
1. Divide and Conquer
Prof. Shashikant V. Athawale
Assistant Professor | Computer Engineering
Department | AISSMS College of Engineering,
Kennedy Road, Pune , MH, India - 411001
3. Introduction
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.
The solutions to the sub-problems are then combined
to give a solution to the original problem.
4. Applications
Binary Search
Quicksort
Merge Sort
Closest Pair of Points
Strassen’s Algorithm
Cooley–Tukey Fast Fourier Transform (FFT)
algorithm
Karatsuba algorithm for fast multiplication
5. Control Abstraction
The meaning of abstractions is preserving
information that is relevant in a given context, and
forgetting information that is irrelevant in that
context.
Control abstraction: the ability to procedural
data where in the program procedural data is
irrelevant.
A control abstraction is a procedure whose flow
of control is clear but whose basic operations are
defined by other processes whose
meanings/implementations are left undefined.
6. Efficiency Analysis
• Divide and Conquer algorithm solves a problem
using following three steps.
• 1. Divide: Break the given problem into sub-
problems of same type.
2. Conquer: Recursively solve these sub-problems.
3. Combine: Appropriately combine the answers.
• Divide and Conquer algorithm is a very efficient
algorithm.
7. Master’s Theorem
Master’s Theorem is a popular method for solving the
recurrence relations.
Here, a >= 1, b > 1, k >= 0 and p is a real number.
8. Master Theorem Cases-
To solve recurrence relations using Master’s theorem, we compare a with
bk.
Case-1:
If a > bk, then T(n) = θ (nlog
b
a)
Case-02:
If a = bk and
If p < -1, then T(n) = θ (nlog
b
a)
If p = -1, then T(n) = θ (nlog
b
a.log2n)
If p > -1, then T(n) = θ (nlog
b
a.logp+1n)
9. Case-03:
If a < bk and
If p < 0, then T(n) = O (nk)
If p >= 0, then T(n) = θ (nklogpn)
10. Example:
Solve the following recurrence relation using
Master’s theorem-
T(n) = 2T(n/2) + nlogn
Solution-
We compare the given recurrence relation
with T(n) = aT(n/b) + θ (nklogpn).
Then, we have-
a = 2, b = 2, k = 1, p = 1
Now, a = 2 and bk = 21 = 2.
Clearly, a = bk.
So, we follow case-02.
11. Since p = 1, so we have-
T(n) = θ (nlog
b
a.logp+1n)
T(n) = θ (nlog
2
2.log1+1n)
Thus,
T(n) = θ (nlog2n)
12. Large Integer Multiplication:
Using Divide and Conquer, we can multiply two
integers in less time complexity. We divide the
given numbers in two halves. Let the given
numbers be X and Y.
If a positional numeral system is used, a natural
way of multiplying numbers is taught in schools as
long multiplication, sometimes called grade-
school multiplication, sometimes called Standard
Algorithm: multiply the multiplicand by each digit
of the multiplier and then add up all the properly
shifted results.
13. 1. Grade School Multiplication:
Given two binary strings that represent value of
two integers, find the product of two strings. For
example, if the first bit string is “1100” and second
bit string is “1010”, output should be 120.
For simplicity, let the length of two strings be
same and be n.
A Naive Approach is to follow the process we
study in school. One by one take all bits of second
number and multiply it with all bits of first
number. Finally add all multiplications. This
algorithm takes O(n^2) time.
14. 2. Divide and Conquer
Using Divide and Conquer, we can multiply two
integers in less time complexity. We divide the
given numbers in two halves. Let the given
numbers be X and Y.
For simplicity let us assume that n is even
The product XY can be written as following.
X = Xl*2n/2 + Xr [Xl and Xr contain leftmost and rightmost n/2 bits of X]
Y = Yl*2n/2 + Yr [Yl and Yr contain leftmost and rightmost n/2 bits of Y]
XY = (Xl*2n/2 + Xr)(Yl*2n/2 + Yr)
= 2n XlYl + 2n/2(XlYr + XrYl) + XrYr
15. If we take a look at the above formula, there are four
multiplications of size n/2, so we basically divided the
problem of size n into four sub-problems of size n/2.
But that doesn’t help because solution of recurrence
T(n) = 4T(n/2) + O(n) is O(n^2). The tricky part of this
algorithm is to change the middle two terms to some
other form so that only one extra multiplication would
be sufficient. The following is tricky expression for
middle two terms.
So the final value of XY becomes :
XY = 2n XlYl + 2n/2 * [(Xl + Xr)(Yl + Yr) - XlYl - XrYr] + XrYr
16. Merge Sort
The merge() function is used for merging two
halves. The merge(arr, l, m, r) is key process that
assumes that arr[l..m] and arr[m+1..r] are sorted
and merges the two sorted sub-arrays into one.
MergeSort(arr[], l, r) If r > l
1. Find the middle point to divide the array into two
halves: middle m = (l+r)/2
2. Call mergeSort for first half: Call mergeSort(arr, l, m)
3. Call mergeSort for second half: Call mergeSort(arr, m+1,
r)
4. Merge the two halves sorted in step 2 and 3:
Call merge(arr, l, m, r)
17.
18. Convex Hull
A convex hull is the smallest convex polygon containing all
the given points.
Input is an array of points specified by their x and y
coordinates. The output is the convex hull of this set of
points.
Input : points[] = {(0, 0), (0, 4), (-4, 0), (5, 0), (0, -6), (1, 0)};
Output : (-4, 0), (5, 0), (0, -6), (0, 4)