The document discusses greedy algorithms, which attempt to find optimal solutions to optimization problems by making locally optimal choices at each step that are also globally optimal. It provides examples of problems that greedy algorithms can solve optimally, such as minimum spanning trees and change making, as well as problems they can provide approximations for, like the knapsack problem. Specific greedy algorithms covered include Kruskal's and Prim's for minimum spanning trees.
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.
P, NP, NP-Complete, and NP-Hard
Reductionism in Algorithms
NP-Completeness and Cooks Theorem
NP-Complete and NP-Hard Problems
Travelling Salesman Problem (TSP)
Travelling Salesman Problem (TSP) - Approximation Algorithms
PRIMES is in P - (A hope for NP problems in P)
Millennium Problems
Conclusions
Mastering Greedy Algorithms: Optimizing Solutions for Efficiency"22bcs058
Greedy algorithms are fundamental techniques used in computer science and optimization problems. They belong to a class of algorithms that make decisions based on the current best option without considering the overall future consequences. Despite their simplicity and intuitive appeal, greedy algorithms can provide efficient solutions to a wide range of problems across various domains.
At the core of greedy algorithms lies a simple principle: at each step, choose the locally optimal solution that seems best at the moment, with the hope that it will lead to a globally optimal solution. This principle makes greedy algorithms easy to understand and implement, as they typically involve iterating through a set of choices and making decisions based on some criteria.
One of the key characteristics of greedy algorithms is their greedy choice property, which states that at each step, the locally optimal choice leads to an optimal solution overall. This property allows greedy algorithms to make decisions without needing to backtrack or reconsider previous choices, resulting in efficient solutions for many problems.
Greedy algorithms are commonly used in problems involving optimization, scheduling, and combinatorial optimization. Examples include finding the minimum spanning tree in a graph (Prim's and Kruskal's algorithms), finding the shortest path in a weighted graph (Dijkstra's algorithm), and scheduling tasks to minimize completion time (interval scheduling).
Despite their effectiveness in many situations, greedy algorithms may not always produce the optimal solution for a given problem. In some cases, a greedy approach can lead to suboptimal solutions that are not globally optimal. This occurs when the greedy choice property does not guarantee an optimal solution at each step, or when there are conflicting objectives that cannot be resolved by a greedy strategy alone.
To mitigate these limitations, it is essential to carefully analyze the problem at hand and determine whether a greedy approach is appropriate. In some cases, greedy algorithms can be augmented with additional techniques or heuristics to improve their performance or guarantee optimality. Alternatively, other algorithmic paradigms such as dynamic programming or divide and conquer may be better suited for certain problems.
Overall, greedy algorithms offer a powerful and versatile tool for solving optimization problems efficiently. By understanding their principles and characteristics, programmers and researchers can leverage greedy algorithms to tackle a wide range of computational challenges and design elegant solutions that balance simplicity and effectiveness.
P, NP, NP-Complete, and NP-Hard
Reductionism in Algorithms
NP-Completeness and Cooks Theorem
NP-Complete and NP-Hard Problems
Travelling Salesman Problem (TSP)
Travelling Salesman Problem (TSP) - Approximation Algorithms
PRIMES is in P - (A hope for NP problems in P)
Millennium Problems
Conclusions
Mastering Greedy Algorithms: Optimizing Solutions for Efficiency"22bcs058
Greedy algorithms are fundamental techniques used in computer science and optimization problems. They belong to a class of algorithms that make decisions based on the current best option without considering the overall future consequences. Despite their simplicity and intuitive appeal, greedy algorithms can provide efficient solutions to a wide range of problems across various domains.
At the core of greedy algorithms lies a simple principle: at each step, choose the locally optimal solution that seems best at the moment, with the hope that it will lead to a globally optimal solution. This principle makes greedy algorithms easy to understand and implement, as they typically involve iterating through a set of choices and making decisions based on some criteria.
One of the key characteristics of greedy algorithms is their greedy choice property, which states that at each step, the locally optimal choice leads to an optimal solution overall. This property allows greedy algorithms to make decisions without needing to backtrack or reconsider previous choices, resulting in efficient solutions for many problems.
Greedy algorithms are commonly used in problems involving optimization, scheduling, and combinatorial optimization. Examples include finding the minimum spanning tree in a graph (Prim's and Kruskal's algorithms), finding the shortest path in a weighted graph (Dijkstra's algorithm), and scheduling tasks to minimize completion time (interval scheduling).
Despite their effectiveness in many situations, greedy algorithms may not always produce the optimal solution for a given problem. In some cases, a greedy approach can lead to suboptimal solutions that are not globally optimal. This occurs when the greedy choice property does not guarantee an optimal solution at each step, or when there are conflicting objectives that cannot be resolved by a greedy strategy alone.
To mitigate these limitations, it is essential to carefully analyze the problem at hand and determine whether a greedy approach is appropriate. In some cases, greedy algorithms can be augmented with additional techniques or heuristics to improve their performance or guarantee optimality. Alternatively, other algorithmic paradigms such as dynamic programming or divide and conquer may be better suited for certain problems.
Overall, greedy algorithms offer a powerful and versatile tool for solving optimization problems efficiently. By understanding their principles and characteristics, programmers and researchers can leverage greedy algorithms to tackle a wide range of computational challenges and design elegant solutions that balance simplicity and effectiveness.
Optimization problems can be divided into two categories, depending on whether the variables are continuous or discrete:
An optimization problem with discrete variables is known as a discrete optimization, in which an object such as an integer, permutation or graph must be found from a countable set.
A problem with continuous variables is known as a continuous optimization, in which an optimal value from a continuous function must be found. They can include constrained problems and multimodal problems.
Dynamic Programming is a method for solving a complex problem by breaking it down into a collection of simpler subproblems, solving each of those subproblems just once, and storing their solutions using a memory-based data structure (array, map,etc).
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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.
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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
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• 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.
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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.
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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.
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Automobile Management System Project Report.pdfKamal Acharya
The proposed project is developed to manage the automobile in the automobile dealer company. The main module in this project is login, automobile management, customer management, sales, complaints and reports. The first module is the login. The automobile showroom owner should login to the project for usage. The username and password are verified and if it is correct, next form opens. If the username and password are not correct, it shows the error message.
When a customer search for a automobile, if the automobile is available, they will be taken to a page that shows the details of the automobile including automobile name, automobile ID, quantity, price etc. “Automobile Management System” is useful for maintaining automobiles, customers effectively and hence helps for establishing good relation between customer and automobile organization. It contains various customized modules for effectively maintaining automobiles and stock information accurately and safely.
When the automobile is sold to the customer, stock will be reduced automatically. When a new purchase is made, stock will be increased automatically. While selecting automobiles for sale, the proposed software will automatically check for total number of available stock of that particular item, if the total stock of that particular item is less than 5, software will notify the user to purchase the particular item.
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COLLEGE BUS MANAGEMENT SYSTEM PROJECT REPORT.pdfKamal Acharya
The College Bus Management system is completely developed by Visual Basic .NET Version. The application is connect with most secured database language MS SQL Server. The application is develop by using best combination of front-end and back-end languages. The application is totally design like flat user interface. This flat user interface is more attractive user interface in 2017. The application is gives more important to the system functionality. The application is to manage the student’s details, driver’s details, bus details, bus route details, bus fees details and more. The application has only one unit for admin. The admin can manage the entire application. The admin can login into the application by using username and password of the admin. The application is develop for big and small colleges. It is more user friendly for non-computer person. Even they can easily learn how to manage the application within hours. The application is more secure by the admin. The system will give an effective output for the VB.Net and SQL Server given as input to the system. The compiled java program given as input to the system, after scanning the program will generate different reports. The application generates the report for users. The admin can view and download the report of the data. The application deliver the excel format reports. Because, excel formatted reports is very easy to understand the income and expense of the college bus. This application is mainly develop for windows operating system users. In 2017, 73% of people enterprises are using windows operating system. So the application will easily install for all the windows operating system users. The application-developed size is very low. The application consumes very low space in disk. Therefore, the user can allocate very minimum local disk space for this application.
Event Management System Vb Net Project Report.pdfKamal Acharya
In present era, the scopes of information technology growing with a very fast .We do not see any are untouched from this industry. The scope of information technology has become wider includes: Business and industry. Household Business, Communication, Education, Entertainment, Science, Medicine, Engineering, Distance Learning, Weather Forecasting. Carrier Searching and so on.
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1. Analysis and Design of Algorithms
(2150703)
Presented by :
Jay Patel (130110107036)
Gujarat Technological University
G.H Patel College of Engineering and Technology
Department of Computer Engineering
Greedy Algorithms
Guided by:
Namrta Dave
2. Greedy Algorithms:
• Many real-world problems are optimization problems in that they attempt to find an
optimal solution among many possible candidate solutions.
• An optimization problem is one in which you want to find, not just a solution, but the best
solution
• A “greedy algorithm” sometimes works well for optimization problems
• A greedy algorithm works in phases. At each phase: You take the best you can get right
now, without regard for future consequences.You hope that by choosing a local optimum at
each step, you will end up at a global optimum
• A familiar scenario is the change-making problem that we often encounter at a cash
register: receiving the fewest numbers of coins to make change after paying the bill for a
purchase.
3. • Constructs a solution to an optimization problem piece by
• piece through a sequence of choices that are:
1.feasible, i.e. satisfying the constraints
2.locally optimal (with respect to some neighborhood definition)
3.greedy (in terms of some measure), and irrevocable
• For some problems, it yields a globally optimal solution for every instance. For most, does
not but can be useful for fast approximations. We are mostly interested in the former case
in this class.
Greedy Technique:
4. Greedy Techniques:
• Optimal solutions:
• change making for “normal” coin denominations
• minimum spanning tree (MST)
• Prim’s MST
• Kruskal’s MST
• simple scheduling problems
• Dijkstra’s algo
• Huffman codes
• Approximations/heuristics:
• traveling salesman problem (TSP)
• knapsack problem
• other combinatorial optimization problems
5. Greedy Scenario:
• Feasible
• Has to satisfy the problem’s constraints
• Locally Optimal
• Has to make the best local choice among all feasible choices available on that step
• If this local choice results in a global optimum then the problem has optimal
substructure
• Irrevocable
• Once a choice is made it can’t be un-done on subsequent steps of the algorithm
• Simple examples:
• Playing chess by making best move without look-ahead
• Giving fewest number of coins as change
• Simple and appealing, but don’t always give the best solution
6. Change-Making Problem:
Given unlimited amounts of coins of denominations , give change for amount n with the least
number of coins
Example: d1 = 25 INR, d2 =10 INR, d3 = 5 INR, d4 = 1 INR and n = 48 INR
Greedy solution: <1, 2, 0, 3>
So one 25 INR coin
Two 10 INR coin
Zero 5 INR coin
Three 1 INR coin
But it doesn’t give optimal solution everytime.
7. Failure of Greedy algorithm
Example:
• In some (fictional) monetary system, “Coin” come in 1 INR, 7 INR, and 10 INR coins
Using a greedy algorithm to count out 15 INR, you would get
A 10 INR coin
Five 1 INR coin, for a total of 15 INR
This requires six coins
A better solution would be to use two 7 INR coin and one 1 INR coin
This only requires three coins
The greedy algorithm results in a solution, but not in an optimal solution
8. Knapsack Problem:
• Given n objects each have a weight wi and a value vi , and given a knapsack of total
capacity W. The problem is to pack the knapsack with these objects in order to maximize
the total value of those objects packed without exceeding the knapsack’s capacity.
• More formally, let xi denote the fraction of the object i to be included in the knapsack, 0
xi 1, for 1 i n. The problem is to find values for the xi such that
• Note that we may assume because otherwise, we would choose xi = 1 for each i
which would be an obvious optimal solution.
n
i
ii
n
i
ii vxWwx
11
maximized.isand
n
i
i Ww
1
9. The optimal Knapsack Algorithm:
This algorithm is for time complexity O(n lgn))
(1) Sort the n objects from large to small based on the ratios vi/wi . We assume the
arrays w[1..n] and v[1..n] store the respective weights and values after sorting.
(2) initialize array x[1..n] to zeros.
(3) weight = 0; i = 1
(4) while (i n and weight < W) do
(I) if weight + w[i] W then x[i] = 1
(II) else x[i] = (W – weight) / w[i]
(III) weight = weight + x[i] * w[i]
(IV) i++
10. There seem to be 3 obvious greedy strategies:
(Max value) Sort the objects from the highest value to the lowest, then pick them in that order.
(Min weight) Sort the objects from the lowest weight to the highest, then pick them in that
order.
(Max value/weight ratio) Sort the objects based on the value to weight ratios, from the highest
to the lowest, then select.
Example: Given n = 5 objects and a knapsack capacity W = 100 as in Table I. The three
solutions are given in Table II.
Knapsack Problem:
W
V
V/W
10 20 30 40 50
20 30 66 40 60
2.0 1.5 2.2 1.0 1.2
Max Vi
Min Wi
Max Vi/Wi
SELECT Xi
0 0 1 0.5 1
1 1 1 1 0
1 1 1 0 0.8
Value
146
156
164
12. A cable company want to connect five villages to their network which currently
extends to the market town of Avonford.
What is the minimum length of cable needed?
A F
B C
D
E
2
7
4
5
8 6
4
5
3
8
Example
Solution for MST:
13. Kruskal’s Algorithm:
A F
B
C
D
E
2
7
4
5
8 6
4
5
3
8
List the edges in order of size:
ED 2 AB 3
AE 4 CD 4
BC 5 EF 5
CF 6 AF 7
BF 8 CF 8
MST-KRUSKAL(G, w)
1. A ← Ø
2. for each vertex v V[G]
3. do MAKE-SET(v)
4. sort the edges of E into nondecreasing order
by weight w
5. for each edge (u, v) E, taken in
nondecreasing
order by weight
6. do if FIND-SET(u) ≠ FIND-SET(v)
7. then A ← A {(u, v)}
8. UNION(u, v)
9. return A
14. Select the shortest
edge in the network
ED 2
A F
B C
D
E
2
7
4
5
8 6 4
5
3
8
Select the next shortest
edge which does not
create a cycle
ED 2
AB 3
A F
B C
D
E
2
7
4
5
8 6 4
5
3
8
1
43
2
Select the next shortest
edge which does not
create a cycle
ED 2
AB 3
CD 4 (or AE 4)
A F
B
C
D
E
2
7
4 5
8 6 4
5
3
8
Select the next shortest
edge which does not
create a cycle
ED 2
AB 3
CD 4
AE 4
A F
B C
D
E
2
7
4
5
8 6 4
5
3
8
15. Select the next shortest
edge which does not
create a cycle
ED 2
AB 3
CD 4
AE 4
BC 5 – forms a cycle
EF 5
A F
B
C
D
E
2
7
4
5
8 6 4
5
3
8
All vertices have been conn
The solution is
ED 2
AB 3
CD 4
AE 4
EF 5
A F
B C
D
E
2
7
4
5
8 6 4
5
3
8
5
6
Total weight of tree: 18
Kruskal’s Algorithm:
16. Prim’s Algorithm:
MST-PRIM(G, w, r)
1. for each u V [G]
2. do key[u] ← ∞
3. π[u] ← NIL
4. key[r] ← 0
5. Q ← V [G]
6. while Q ≠ Ø
7. do u ← EXTRACT-MIN(Q)
8. for each v Adj[u]
9. do if v Q and w(u, v) < key[v]
10. then π[v] ← u
11. key[v] ← w(u, v)
17. A F
B C
D
E
2
7
4
5
8 6 4
5
3
8
Select any vertex
A
Select the shortest edge connected to that vertex
AB 3
Prim’s Algorithm:
18. A F
B C
D
E
2
7
4
5
8 6 4
5
3
8
Select the shortest
edge connected to
any vertex already
connected.
AE 4
1
43
2
Select the shortest
edge connected to
any vertex already
connected.
ED 2
A F
B
C
D
E
2
7
4
5
8 6 4
5
3
8
Select the shortest
edge connected to
any vertex already
connected.
DC 4
A F
B
C
D
E
2
7
4
5
8 6 4
5
3
8
Select the shortest
edge connected to
any vertex already
connected.
EF 5
A F
B
C
D
E
2
7
4
5
8 6 4
5
3
8
20. There are some methods left:
• Dijkstra’s algorithm
• Huffman’s Algorithm
• Task scheduling
• Travelling salesman Problem etc.
• Dynamic Greedy Problems
Greedy Algorithms:
We can find the optimized solution with Greedy method which may be optimal sometime.