The document discusses the dynamic programming approach to solving the matrix chain multiplication problem. It explains that dynamic programming breaks problems down into overlapping subproblems, solves each subproblem once, and stores the solutions in a table to avoid recomputing them. It then presents the algorithm MATRIX-CHAIN-ORDER that uses dynamic programming to solve the matrix chain multiplication problem in O(n^3) time, as opposed to a brute force approach that would take exponential time.
Dynamic Programming is an algorithmic paradigm that solves a given complex problem by breaking it into subproblems and stores the results of subproblems to avoid computing the same results again.
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
Dynamic Programming is one of the most interesting design techniques. The concise idea is to avoid recomputations. Matrix Chain Multiplication and All Pairs Shortest Paths are two interesting applications of this design technique
Dynamic Programming design technique is one of the fundamental algorithm design techniques, and possibly one of the ones that are hardest to master for those who did not study it formally. In these slides (which are continuation of part 1 slides), we cover two problems: maximum value contiguous subarray, and maximum increasing subsequence.
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).
Dynamic Programming is an algorithmic paradigm that solves a given complex problem by breaking it into subproblems and stores the results of subproblems to avoid computing the same results again.
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
Dynamic Programming is one of the most interesting design techniques. The concise idea is to avoid recomputations. Matrix Chain Multiplication and All Pairs Shortest Paths are two interesting applications of this design technique
Dynamic Programming design technique is one of the fundamental algorithm design techniques, and possibly one of the ones that are hardest to master for those who did not study it formally. In these slides (which are continuation of part 1 slides), we cover two problems: maximum value contiguous subarray, and maximum increasing subsequence.
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).
Enumeration methods are very important in a variety of settings, both mathematical and applications. For many problems there is actually no real hope to do the enumeration in reasonable time since the number of solutions is so big. This talk is about how to compute at the limit.
The talk is decomposed into:
(a) Regular enumeration procedure where one uses computerized case distinction.
(b) Use of symmetry groups for isomorphism checks.
(c) The augmentation scheme that allows to enumerate object up to isomorphism without keeping the full list in memory.
(d) The homomorphism principle that allows to map a complex problem to a simpler one.
adminSticky Note- the process of inversion is shown in .docxbobbywlane695641
admin
Sticky Note
- the process of inversion is shown in the figure on next page 1-A and in Fig. 2.1 (GeomathBook.pdf, p. 22/153)
- ie., it is an artform with subjective and quantitative objective components
admin
Sticky Note
In math, science, and engineering, the 'inverse problem' of relating data observations to unknown elements or coefficients of a model is a 'process' commonly called 'inversion'.
However, other names may be used depending on the application - eg., it may be called
=> blackbox analysis in physics
=> factor analysis in social sciences
=> forward and inverse modeling in geophysics
=> or simply modeling in any dicipline
=> trial-and-error analysis
=> deconvolution in signal processing and engineering
=> ????? in geodesy?
=> etc.
admin
Sticky Note
AX is the linear or non-linear forward model,
so that AX = B + 'errors', where
B is the observed data set,
admin
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A is the system input or design matrix,
admin
Sticky Note
and X is the impluse response or set of unknown blackbox coefficients that the inversion seeks to determine
admin
Sticky Note
The essence of any inverse problem is the forward model = AX,
which in practice is always processed by linear digital operations.
admin
Sticky Note
SENSITIVITY ANALYSIS - pretty much where most modern developments in inversion analysis are concentrated.
admin
Sticky Note
the most meaningful set of solution coefficients
admin
Sticky Note
translating (ie., rationalizing) the best set of solution coefficients into 'new' information on the conceptual model - usually augmented with digital graphics.
admin
Sticky Note
Inversion examples of conceptual and mathematical models are shown on page 2-A below and in Fig. 2.2 (GeomathBook.pdf, p. 23/153)
admin
Sticky Note
Conceptual models are imagined simplifications of reality
admin
Sticky Note
Mathematical models are the further simplified expressions of conceptual models
admin
Sticky Note
joint inversions would obtain density and velocity solutions that satisfy both gravity and seismic observations
admin
Sticky Note
always
admin
Sticky Note
always
admin
Sticky Note
always
admin
Sticky Note
=> eg., the data set is improperly 'scaled' relative to computer's working precision
=> for example, the inverse problem is improperly posed to find angstrom scale variations in observations that are known only to nearest kilometer, etc.
admin
Sticky Note
Figure on page 8-A below shows examples of model 'appropriateness' relative to inversion 'objectives'
admin
Sticky Note
Inversion (ie., modeling) objectives dictate the mathematical model that should be invoked - or model 'appropriateness'
In other words, the forward model should
- not be overly complex numerically so as to waste computational resources,
- or too simple numerically so as not to achieve the inversion objective.
- Rather, it should be just right or 'optimal' so that the inversion objective is achieved with 'minimum' resource expenditu.
adminSticky Note- the process of inversion is shown in .docxgalerussel59292
admin
Sticky Note
- the process of inversion is shown in the figure on next page 1-A and in Fig. 2.1 (GeomathBook.pdf, p. 22/153)
- ie., it is an artform with subjective and quantitative objective components
admin
Sticky Note
In math, science, and engineering, the 'inverse problem' of relating data observations to unknown elements or coefficients of a model is a 'process' commonly called 'inversion'.
However, other names may be used depending on the application - eg., it may be called
=> blackbox analysis in physics
=> factor analysis in social sciences
=> forward and inverse modeling in geophysics
=> or simply modeling in any dicipline
=> trial-and-error analysis
=> deconvolution in signal processing and engineering
=> ????? in geodesy?
=> etc.
admin
Sticky Note
AX is the linear or non-linear forward model,
so that AX = B + 'errors', where
B is the observed data set,
admin
Sticky Note
A is the system input or design matrix,
admin
Sticky Note
and X is the impluse response or set of unknown blackbox coefficients that the inversion seeks to determine
admin
Sticky Note
The essence of any inverse problem is the forward model = AX,
which in practice is always processed by linear digital operations.
admin
Sticky Note
SENSITIVITY ANALYSIS - pretty much where most modern developments in inversion analysis are concentrated.
admin
Sticky Note
the most meaningful set of solution coefficients
admin
Sticky Note
translating (ie., rationalizing) the best set of solution coefficients into 'new' information on the conceptual model - usually augmented with digital graphics.
admin
Sticky Note
Inversion examples of conceptual and mathematical models are shown on page 2-A below and in Fig. 2.2 (GeomathBook.pdf, p. 23/153)
admin
Sticky Note
Conceptual models are imagined simplifications of reality
admin
Sticky Note
Mathematical models are the further simplified expressions of conceptual models
admin
Sticky Note
joint inversions would obtain density and velocity solutions that satisfy both gravity and seismic observations
admin
Sticky Note
always
admin
Sticky Note
always
admin
Sticky Note
always
admin
Sticky Note
=> eg., the data set is improperly 'scaled' relative to computer's working precision
=> for example, the inverse problem is improperly posed to find angstrom scale variations in observations that are known only to nearest kilometer, etc.
admin
Sticky Note
Figure on page 8-A below shows examples of model 'appropriateness' relative to inversion 'objectives'
admin
Sticky Note
Inversion (ie., modeling) objectives dictate the mathematical model that should be invoked - or model 'appropriateness'
In other words, the forward model should
- not be overly complex numerically so as to waste computational resources,
- or too simple numerically so as not to achieve the inversion objective.
- Rather, it should be just right or 'optimal' so that the inversion objective is achieved with 'minimum' resource expenditu.
We have compiled the most important slides from each speaker's presentation. This year’s compilation, available for free, captures the key insights and contributions shared during the DfMAy 2024 conference.
Final project report on grocery store management system..pdfKamal Acharya
In today’s fast-changing business environment, it’s extremely important to be able to respond to client needs in the most effective and timely manner. If your customers wish to see your business online and have instant access to your products or services.
Online Grocery Store is an e-commerce website, which retails various grocery products. This project allows viewing various products available enables registered users to purchase desired products instantly using Paytm, UPI payment processor (Instant Pay) and also can place order by using Cash on Delivery (Pay Later) option. This project provides an easy access to Administrators and Managers to view orders placed using Pay Later and Instant Pay options.
In order to develop an e-commerce website, a number of Technologies must be studied and understood. These include multi-tiered architecture, server and client-side scripting techniques, implementation technologies, programming language (such as PHP, HTML, CSS, JavaScript) and MySQL relational databases. This is a project with the objective to develop a basic website where a consumer is provided with a shopping cart website and also to know about the technologies used to develop such a website.
This document will discuss each of the underlying technologies to create and implement an e- commerce website.
Industrial Training at Shahjalal Fertilizer Company Limited (SFCL)MdTanvirMahtab2
This presentation is about the working procedure of Shahjalal Fertilizer Company Limited (SFCL). A Govt. owned Company of Bangladesh Chemical Industries Corporation under Ministry of Industries.
Understanding Inductive Bias in Machine LearningSUTEJAS
This presentation explores the concept of inductive bias in machine learning. It explains how algorithms come with built-in assumptions and preferences that guide the learning process. You'll learn about the different types of inductive bias and how they can impact the performance and generalizability of machine learning models.
The presentation also covers the positive and negative aspects of inductive bias, along with strategies for mitigating potential drawbacks. We'll explore examples of how bias manifests in algorithms like neural networks and decision trees.
By understanding inductive bias, you can gain valuable insights into how machine learning models work and make informed decisions when building and deploying them.
Student information management system project report ii.pdfKamal Acharya
Our project explains about the student management. This project mainly explains the various actions related to student details. This project shows some ease in adding, editing and deleting the student details. It also provides a less time consuming process for viewing, adding, editing and deleting the marks of the students.
Hierarchical Digital Twin of a Naval Power SystemKerry Sado
A hierarchical digital twin of a Naval DC power system has been developed and experimentally verified. Similar to other state-of-the-art digital twins, this technology creates a digital replica of the physical system executed in real-time or faster, which can modify hardware controls. However, its advantage stems from distributing computational efforts by utilizing a hierarchical structure composed of lower-level digital twin blocks and a higher-level system digital twin. Each digital twin block is associated with a physical subsystem of the hardware and communicates with a singular system digital twin, which creates a system-level response. By extracting information from each level of the hierarchy, power system controls of the hardware were reconfigured autonomously. This hierarchical digital twin development offers several advantages over other digital twins, particularly in the field of naval power systems. The hierarchical structure allows for greater computational efficiency and scalability while the ability to autonomously reconfigure hardware controls offers increased flexibility and responsiveness. The hierarchical decomposition and models utilized were well aligned with the physical twin, as indicated by the maximum deviations between the developed digital twin hierarchy and the hardware.
The Internet of Things (IoT) is a revolutionary concept that connects everyday objects and devices to the internet, enabling them to communicate, collect, and exchange data. Imagine a world where your refrigerator notifies you when you’re running low on groceries, or streetlights adjust their brightness based on traffic patterns – that’s the power of IoT. In essence, IoT transforms ordinary objects into smart, interconnected devices, creating a network of endless possibilities.
Here is a blog on the role of electrical and electronics engineers in IOT. Let's dig in!!!!
For more such content visit: https://nttftrg.com/
Water billing management system project report.pdfKamal Acharya
Our project entitled “Water Billing Management System” aims is to generate Water bill with all the charges and penalty. Manual system that is employed is extremely laborious and quite inadequate. It only makes the process more difficult and hard.
The aim of our project is to develop a system that is meant to partially computerize the work performed in the Water Board like generating monthly Water bill, record of consuming unit of water, store record of the customer and previous unpaid record.
We used HTML/PHP as front end and MYSQL as back end for developing our project. HTML is primarily a visual design environment. We can create a android application by designing the form and that make up the user interface. Adding android application code to the form and the objects such as buttons and text boxes on them and adding any required support code in additional modular.
MySQL is free open source database that facilitates the effective management of the databases by connecting them to the software. It is a stable ,reliable and the powerful solution with the advanced features and advantages which are as follows: Data Security.MySQL is free open source database that facilitates the effective management of the databases by connecting them to the software.
6th International Conference on Machine Learning & Applications (CMLA 2024)ClaraZara1
6th International Conference on Machine Learning & Applications (CMLA 2024) will provide an excellent international forum for sharing knowledge and results in theory, methodology and applications of on Machine Learning & Applications.
2. 1. Dynamic Programming
2. Comparison with divide n concuer
3. Where used?
4. Matrix Multiplication
5. Algo to multiply 2 matrices
6. #scaler multiplication used in matrix multiplication
7. Costs &parenthisization
8. Matrix multiplication Problem
9. Counting the #parenthisization
10. Brute force Method
11. Computing optimal cost
12. Matrix chain Order
13. Overlapping Subproblem
14. Memorization
3. dynamic programming is a method for
solving complex problems by breaking them
down into simpler subproblems.and the
subproblems are dependent.
programming is an expression of algorithm.i
dynamic programing the languag,used does
not refer to any computer language,rather it
refers to formal language.
4. Dynamic programming, like the divide-and-conquer method,
solves problems by combining the solutions to subproblems.
divide-and-conquer algorithms partition the problem into
disjoint subproblems, solve the subproblems recursively, and
then combine their solutions to solve the original problem. In
contrast, dynamic programming applies when the subproblems
overlap—that is, when subproblems share subsubproblems.
a divide-and-conquer algorithm does more work than necessary,
repeatedly solving the common subsubproblems. A dynamic-
programming algorithm solves each subsubproblem just once
and then saves its answer in a table, thereby avoiding the work
of recomputing the answer every time it solves each
subsubproblem
5. We typically apply dynamic programming to
optimization problems.
Such problems can have many possible
solutions. Each solution has a value, and we
wish to find a solution with the optimal
(minimum or maximum) value. We call such a
solution an optimal solution to the problem,
as opposed to the optimal solution, since
there may be several solutions that achieve
the optimal value.
6. Say,there are 2 matrices A & B.
We can multiply 2 matrices only if they are
compatible.
Condition to be multiplied:#collumns of A
=#rows of B.
If the resultant matrix is C,then
Am*n * B n*p= Cm*p
7. MATRIX-MULTIPLY(A,B)
1 .if columns[A] ≠rows[B]
2 . Then error “incompatible dimensions”
3 . let C be a new A:rows B:columns matrix
4 .for i <-1 to A.rows
5 .for j<-1 to B.columns
6 .cij =0
7 .for k =1 to A.columns
8 .cij =cij + aik .bkj
9 .return C
8. In the above example of matrix multiplication
if we consider,
m=2,n=3,p=4,the total number of scaler
multiplication used is 24,i.e m*n*p.
We shall express costs in terms of the
number of scaler multiplications.
10. matrix-chain multiplication problem can be
stated as follows: given a chain
<A1,A2,…,An> of n matrices, where for
i=1,2, …,n, matrix Ai has dimension pi-1*pi ,
fully parenthesize the product A1A2…An in a
way that minimizes the number of scalar
multiplications.
Our goal is only to determine an order for
multiplying matrices that has the lowest cost.
11. P(n) is the number of alternative
parenthesizations of a sequence of n
matrices.
When n=1,P(n)=1,as there is only 1 matrix,so
only one way to put parenthesis.
n=2,P(n)=1,only one way to put parenthesis.
n=3,P(n)=2, (A1*A2)*A3 & A1*(A2*A3)
N=4,P(n)=6
(((A1*A2)*A3)*A4) (A1*((A2*A3)*A4)
((A1*A2)*(A3*A4)) (A1*(A2*(A3*A4)))
(A1*(A2*A3))*A4 ((A1*A2)*(A3*A4))
12. Thus we obtain the sequence:
◦ P(n)= 1 if n=1
∑k=1to n-1 P(k)P(n-k) if n≥2 a fully
parenthasize matrix product is a
prodct of 2 fully parenthasize matrix
sbproducts.split between two
sbprducts may occr between the k th
& k+1 th matrix for any k=1…n-1
Rate of growth: Ω((4^n)/(n^(3/2)))
13. general problem-solving technique that
consists of systematically enumerating all
possible candidates for the solution and
checking whether each candidate satisfies the
problem's statement.
Lower bound of this problem using brute
force method is Ω((4^n)/(n^(3/2)))
That means brute force method as the solution of
the problem is not so good.
14. At this point, we could easily write a recursive
algorithm based on recurrence to compute the
minimum cost m[1,n]for multiplying A1,A2,…, An.
Aswe sawfor the rod-cutting problem, and this
recursive algorithm takes exponential time, which is
no better than the brute-force method of checking
each way of parenthesizing the product.
we have relatively few distinct subproblems: one
subproblem for each choice of i and j satisfying 1 ≤i≤
j≤n, or (nc2 +n)=Θ(n^2) in all. A recursive algorithm
may encounter each subproblem many times in
different branches of its recursion tree. This property
of overlapping subproblems is the second hallmark of
when dynamic programming applies.
15. Instead of computing the solution to recurrence
recursively, we compute the optimal cost by using a
tabular, bottom-up approach.
We shall implement the tabular, bottom-up method
in the procedure MATRIXCHAIN-ORDER, which
appears below. This procedure assumes that matrix
Ai has dimensions pi-1 *pi for i=1,2,…,n. Its input is
a sequence p=<p0; p1; : : : ;pni> where length[p] =
n+1. The procedure uses an auxiliary table
m[1…n,1…n] for storing the m[I,j] costs and another
auxiliary table s[1…n,1…n]that records which index
of k achieved the optimal cost in computing m[i; j
].We shall use the table s to construct an optimal
solution.
16. In order to implement the bottom-up approach, we
must determine which entries of the table we refer to
when computing m[i,j].the cost m[i ,j] of computing a
matrix-chain product of j-i+1matrices depends only
on the costs of computing matrix-chain products of
fewer than j-i+1 matrices. That is, for k =I,i +1,…,j-
1, the matrix Ai…k is a product of k-i+1< j-i+1
matrices and the matrix Ak+1…j is a product of j-k<
j-i +1 matrices. Thus, the algorithm should fill in the
table min a manner that corresponds to solving the
parenthesization problem on matrix chains of
increasing length. For the subproblem of optimally
parenthesizing the chain Ai,Ai+1,… Aj,we consider
the subproblem size to be the length j-i+1 of the
chain.
17. MATRIX-CHAIN-ORDER.p/
1. n D p:length 1
2.Let m[1,… n.1,… n] and s[1,… n-1. 2…n] be new tables
3. for i =1 to n
4.Do m[i,j] <-0
5. for l =2 to n // l is the chain length
6. dofor i =1 to n –l+1
7. Do j<-i+l-1
8. m[i, j]<- infinity
9.for k <-i to j-1
10. Do q <-m[i, k]+m[k+1,j]+pi-1.pk.pj
11. if q <m[i,j]
12. m[i, j]<-q
13. s[i,j]<-k
14. return m and s
18. running time of this algorithm is in fact also
Ω(n^3) The algorithm requiresΘ(n^2) space
to store the m and s tables. Thus, MATRIX-
CHAINORDER is much more efficient than the
exponential-time method of enumerating all
possible parenthesizations and checking each
one
19. In dynamic problem
subproblems are
dependent.
Dynamic-programming
algorithms typically take
advantage of overlapping
subproblems by solving
each subproblem once and
then storing the solution in
a table where it can be
looked up when needed,
using constant time per
lookup.
Cost is reduced.
(A1 *A2) (A3*(A4*A5))
(A1*A2) (A3*A4)*A5
^
|
Overlapping
subproblem
20. A memoized recursive algorithm maintains an
entry in a table for the solution to each
subproblem. Each table entry initially
contains a special value to indicate that the
entry has yet to be filled in. When the
subproblem is first encountered as the
recursive algorithm unfolds, its solution is
computed and then stored in the table. Each
subsequent time that we encounter this
subproblem, we simply look up the value
stored in the table and return it