This document discusses algorithm design and analysis. It introduces sorting as an example problem and compares the insertion sort and merge sort algorithms. Insertion sort runs in O(n2) time in the worst case, while merge sort runs in O(nlogn) time. It provides pseudocode for insertion sort and merge sort and analyzes their time complexities. It also covers algorithm analysis techniques like recursion trees and asymptotic notation.
Whenever we want to perform analysis of an algorithm, we need to calculate the complexity of that algorithm. But when we calculate complexity of an algorithm it does not provide exact amount of resource required. So instead of taking exact amount of resource we represent that complexity in a general form (Notation) which produces the basic nature of that algorithm. We use that general form (Notation) for the analysis process.
Whenever we want to perform analysis of an algorithm, we need to calculate the complexity of that algorithm. But when we calculate complexity of an algorithm it does not provide exact amount of resource required. So instead of taking exact amount of resource we represent that complexity in a general form (Notation) which produces the basic nature of that algorithm. We use that general form (Notation) for the analysis process.
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From the perspective of Design and Analysis of Algorithm. I made these slide by collecting data from many sites.
I am Danish Javed. Student of BSCS Hons. at ITU Information Technology University Lahore, Punjab, Pakistan.
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.
Analysis and design of algorithms part2Deepak John
Analysis of searching and sorting. Insertion sort, Quick sort, Merge sort and Heap sort. Binomial Heaps and Fibonacci Heaps, Lower bounds for sorting by comparison of keys. Comparison of sorting algorithms. Amortized Time Analysis. Red-Black Trees – Insertion & Deletion.
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.
The Art Pastor's Guide to Sabbath | Steve ThomasonSteve Thomason
What is the purpose of the Sabbath Law in the Torah. It is interesting to compare how the context of the law shifts from Exodus to Deuteronomy. Who gets to rest, and why?
This is a presentation by Dada Robert in a Your Skill Boost masterclass organised by the Excellence Foundation for South Sudan (EFSS) on Saturday, the 25th and Sunday, the 26th of May 2024.
He discussed the concept of quality improvement, emphasizing its applicability to various aspects of life, including personal, project, and program improvements. He defined quality as doing the right thing at the right time in the right way to achieve the best possible results and discussed the concept of the "gap" between what we know and what we do, and how this gap represents the areas we need to improve. He explained the scientific approach to quality improvement, which involves systematic performance analysis, testing and learning, and implementing change ideas. He also highlighted the importance of client focus and a team approach to quality improvement.
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
Ethnobotany and Ethnopharmacology:
Ethnobotany in herbal drug evaluation,
Impact of Ethnobotany in traditional medicine,
New development in herbals,
Bio-prospecting tools for drug discovery,
Role of Ethnopharmacology in drug evaluation,
Reverse Pharmacology.
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
The French Revolution Class 9 Study Material pdf free download
Introduction
1. 1
Introduction to Algorithm design and analysis
Example: sorting problem.
Input: a sequence of n number <a1, a2, …,an>
Output: a permutation (reordering) <a1', a2', …,an'>
such that a1'≤ a2' ≤ … ≤ an '.
Different sorting algorithms:
Insertion sort and Mergesort.
2. 2
Efficiency comparison of two algorithms
• Suppose n=106
numbers:
– Insertion sort: c1n2
– Merge sort: c2n (lg n)
– Best programmer (c1=2), machine language, one billion/second computer A.
– Bad programmer (c2=50), high-language, ten million/second computer B.
– 2 (106
)2
instructions/109
instructions per second = 2000 seconds.
– 50 (106
lg 106
) instructions/107
instructions per second ≈ 100 seconds.
– Thus, merge sort on B is 20 times faster than insertion sort on A!
– If sorting ten million numbers, 2.3 days VS. 20 minutes.
• Conclusions:
– Algorithms for solving the same problem can differ dramatically in their
efficiency.
– much more significant than the differences due to hardware and software.
3. 3
Algorithm Design and Analysis
• Design an algorithm
– Prove the algorithm is correct.
• Loop invariant.
• Recursive function.
• Formal (mathematical) proof.
• Analyze the algorithm
– Time
• Worse case, best case, average case.
• For some algorithms, worst case occurs often, average case is often roughly as bad as
the worst case. So generally, worse case running time.
– Space
• Sequential and parallel algorithms
– Random-Access-Model (RAM)
– Parallel multi-processor access model: PRAM
4. 4
Insertion Sort Algorithm (cont.)
INSERTION-SORT(A)
1. for j = 2 to length[A]
2. do key ← A[j]
3. //insert A[j] to sorted sequence A[1..j-1]
4. i ← j-1
5. while i >0 and A[i]>key
6. do A[i+1] ← A[i] //move A[i] one position right
7. i ← i-1
8. A[i+1] ← key
5. 5
Correctness of Insertion Sort Algorithm
• Loop invariant
– At the start of each iteration of the for loop, the
subarray A[1..j-1] contains original A[1..j-1] but in
sorted order.
• Proof:
– Initialization : j=2, A[1..j-1]=A[1..1]=A[1], sorted.
– Maintenance: each iteration maintains loop invariant.
– Termination: j=n+1, so A[1..j-1]=A[1..n] in sorted
order.
6. 6
Analysis of Insertion Sort
INSERTION-SORT(A) cost times
1. for j = 2 to length[A] c1 n
2. do key ← A[j] c2 n-1
3. //insert A[j] to sorted sequence A[1..j-1] 0 n-1
4. i ← j-1 c4 n-1
5. while i >0 and A[i]>key c5 ∑j=2
n
tj
6. do A[i+1] ← A[i] c6 ∑j=2
n
(tj–1)
7. i ← i-1 c7 ∑j=2
n
(tj–1)
8. A[i+1] ← key c8 n–1
(tj is the number of times the while loop test in line 5 is executed for that value of j)
The total time cost T(n) = sum of cost × times in each line
=c1n + c2(n-1) + c4(n-1) + c5∑j=2
n
tj+ c6∑j=2
n
(tj-1)+ c7∑j=2
n
(tj-1)+ c8(n-1)
7. 7
Analysis of Insertion Sort (cont.)
• Best case cost: already ordered numbers
– tj=1, and line 6 and 7 will be executed 0 times
– T(n) = c1n + c2(n-1) + c4(n-1) + c5(n-1) + c8(n-1)
=(c1 + c2 + c4 + c5 + c8)n – (c2 + c4 + c5 + c8) = cn + c‘
• Worst case cost: reverse ordered numbers
– tj=j,
– so ∑j=2
n
tj= ∑j=2
n
j =n(n+1)/2-1, and ∑j=2
n
(tj–1) = ∑j=2
n
(j–1) = n(n-1)/2, and
– T(n) = c1n + c2(n-1) + c4(n-1) + c5(n(n+1)/2 -1) + + c6(n(n-1)/2 -1) + c7(n(n-
1)/2)+ c8(n-1) =((c5+ c6 + c7)/2)n2 +(c1 + c2 + c4 +c5/2-c6/2-c7/2+c8)n-(c2 + c4 + c5
+ c8) =an2
+bn+c
• Average case cost: random numbers
– in average, tj = j/2. T(n) will still be in the order of n2
, same as the worst case.
8. 8
Merge Sort—divide-and-conquer
• Divide: divide the n-element sequence into
two subproblems of n/2 elements each.
• Conquer: sort the two subsequences
recursively using merge sort. If the length
of a sequence is 1, do nothing since it is
already in order.
• Combine: merge the two sorted
subsequences to produce the sorted answer.
9. 9
Merge Sort –merge function
• Merge is the key operation in merge sort.
• Suppose the (sub)sequence(s) are stored in
the array A. moreover, A[p..q] and
A[q+1..r] are two sorted subsequences.
• MERGE(A,p,q,r) will merge the two
subsequences into sorted sequence A[p..r]
– MERGE(A,p,q,r) takes Θ(r-p+1).
10. 10
MERGE-SORT(A,p,r)
1. if p < r
2. then q ← (p+r)/2
3. MERGE-SORT(A,p,q)
4. MERGE-SORT(A,q+1,r)
5. MERGE(A,p,q,r)
Call to MERGE-SORT(A,1,n) (suppose n=length(A))
11. 11
Analysis of Divide-and-Conquer
• Described by recursive equation
• Suppose T(n) is the running time on a problem of
size n.
• T(n) = Θ(1) if n≤nc
aT(n/b)+D(n)+C(n) if n> nc
Where a: number of subproblems
n/b: size of each subproblem
D(n): cost of divide operation
C(n): cost of combination operation
12. 12
Analysis of MERGE-SORT
• Divide: D(n) = Θ(1)
• Conquer: a=2,b=2, so 2T(n/2)
• Combine: C(n) = Θ(n)
• T(n) = Θ(1) if n=1
2T(n/2)+ Θ(n) if n>1
• T(n) = c if n=1
2T(n/2)+ cn if n>1
13. 13
Compute T(n) by Recursive Tree
• The recursive equation can be solved by recursive
tree.
• T(n) = 2T(n/2)+ cn, (See its Recursive Tree).
• lg n+1 levels, cn at each level, thus
• Total cost for merge sort is
– T(n) =cnlg n +cn = Θ(nlg n).
– Question: best, worst, average?
• In contrast, insertion sort is
– T(n) = Θ(n2
).
15. 15
Order of growth
• Lower order item(s) are ignored, just keep the
highest order item.
• The constant coefficient(s) are ignored.
• The rate of growth, or the order of growth,
possesses the highest significance.
• Use Θ(n2
) to represent the worst case running time
for insertion sort.
• Typical order of growth: Θ(1), Θ(lg n),
Θ(√n),Θ(n), Θ(nlg n), Θ(n2
), Θ(n3
), Θ(2n
), Θ(n!)
• Asymptotic notations: Θ, O, Ω, o, ω.
16. 16
f(n) = Θ(g(n))
n
n0
c1g(n)
c2g(n)
f(n)
f(n) = O(g(n))
n
n0
f(n)
cg(n)
f(n) = Ω(g(n))
nn0
cg(n)
f(n)
There exist positive constants c1 and c2
such that there is a positive constant n0
such that …
There exist positive constants c
such that there is a positive constant n0
such that …
There exist positive constants c
such that there is a positive constant n0
such that …
17. Prove f(n)=an2
+bn+c=Θ(n2
)
• a, b, c are constants and a>0.
• Find c1, and c2 (and n0) such that
– c1n2
≤f(n) ≤c2n2
for all n≥ n0.
• It turns out: c1=a/4, c2=7a/4 and
– n0= 2⋅max(|b|/a, sqrt(|c|/a))
• Here we also can see that lower terms and
constant coefficient can be ignored.
• How about f(n)=an3
+bn2
+cn+d? 17
18. 18
o-notation
• For a given function g(n),
– o(g(n))={f(n): for any positive constant c,there exists a
positive n0 such that 0≤ f(n) ≤ cg(n) for all n≥ n0}
– Write f(n) ∈ o(g(n)), or simply f(n) = o(g(n)).
2g(n)
f(n) = o(g(n))
nn0
g(n)
f(n)
1/2g(n)
n0 n0
19. 19
Notes on o-notation
• O-notation may or may not be asymptotically tight
for upper bound.
– 2n2
= O(n2
) is tight, but 2n = O(n2
) is not tight.
• o-notition is used to denote an upper bound that is
not tight.
– 2n = o(n2
), but 2n2
≠ o(n2
).
• Difference: for some positive constant c in O-
notation, but all positive constants c in o-notation.
• In o-notation, f(n) becomes insignificant relative to
g(n) as n approaches infinitely: i.e.,
– lim = 0.f(n)
g(n)n→∝
20. 20
ω-notation
• For a given function g(n),
ω(g(n))={f(n): for any positive constant c, there exists a
positive n0 such that 0 ≤ cg(n) ≤ f(n) for all n≥ n0}
– Write f(n) ∈ ω(g(n)), or simply f(n) = ω(g(n)).
∀ ω-notation, similar to o-notation, denotes lower
bound that is not asymptotically tight.
– n2
/2 = ω(n), but n2
/2 ≠ ω(n2
)
• f(n) = ω(g(n)) if and only if g(n)=o(f(n)).
• lim = ∝
f(n)
g(n)n→∝
21. 21
Techniques for Algorithm Design and Analysis
• Data structure: the way to store and organize data.
– Disjoint sets
– Balanced search trees (red-black tree, AVL tree, 2-3 tree).
• Design techniques:
– divide-and-conquer, dynamic programming, prune-and-search,
laze evaluation, linear programming, …
• Analysis techniques:
– Analysis: recurrence, decision tree, adversary argument,
amortized analysis,…
22. 22
NP-complete problem
• Hard problem:
– Most problems discussed are efficient (poly time)
– An interesting set of hard problems: NP-complete.
• Why interesting:
– Not known whether efficient algorithms exist for them.
– If exist for one, then exist for all.
– A small change may cause big change.
• Why important:
– Arise surprisingly often in real world.
– Not waste time on trying to find an efficient algorithm to get best
solution, instead find approximate or near-optimal solution.
• Example: traveling-salesman problem.