- Complexity and Big-O Notation
Performance: how much time/memory/disk/
Complexity: how do the resource requirements of a program or algorithm scale, i.e., what happens as the size of the problem being solved gets larger.
Complexity affects performance but not the other way around
Big-O Notation
We express complexity using big-O notation. For a problem of size N:
constant-time method is "order 1": O
linear-time method is "order N": O
quadratic-time method is "order N squared": O
How to Determine Complexities
1-Sequence of statements
statement 1;
statement 2;
...
statement
If each statement is "simple" (only involves basic operations) then the time for each statement is constant and the total time is also constant: O
2-if-then-else statements
Here, either sequence 1 will execute, or sequence 2 will execute. Therefore, the worst-case time is the slowest of the two possibilities: max
3-for loops
for (
sequence of statements
}
The loop executes N times, so the sequence of statements also executes N times. Since we assume the statements are O
4-Nested loops
for (
for (
sequence of statements
}
}
The outer loop executes N times. Every time the outer loop executes, the inner loop executes M times. As a result, the statements in the inner loop execute
Best-case and Average-case Complexity
Some methods may require different amounts of time on different calls, even when the problem size is the same for both calls. For example, consider the add method that adds an item to the end of the list. In the worst case (the array is full), that method requires time proportional to the number of items in the list (because it has to copy all of them into the new, larger array). However, when the array is not full,
When do Constants Matter?
Recall that when we use big-O notation, we drop constants and low-order terms. This is because when the problem size gets sufficiently large, those terms don't matter. However, this means that two algorithms can have the same big-O time complexity, even though one is always faster than the other. For example, suppose algorithm 1 requires N2 time, and algorithm 2 requires 10 * N2 + N time. For both algorithms, the time is O
http://pages.cs.wisc.edu/~
- Algorithmic Complexity
Asymptotic Notations
The goal of computational complexity is to classify algorithms according to their performances. We will represent the time function T
Constant Time: O
array: accessing any element
fixed-size stack: push and pop methods
fixed-size queue:
Linear Time: O
array: linear search, traversing, find minimum
queue: contains method
Logarithmic Time: O
binary search
Quadratic Time: O
bubble sort, selection sort, insertion sort
Definition of "big Omega"
We need the notation for the lower bound.
Definition of "big Theta"
To measure the complexity of a particular algorithm, means to find the upper and lower bounds
Analysis of Algorithms
the worst-case
the best-case
the average case
the amortized
Example. Let us consider an algorithm of sequential searching in an array
Its worst-case
Its best-case
Its average case
http://www.cs.cmu.edu/~
- Sorting: Ep 07a - Big O Notation
- Big-O notation.mp4
http://www.youtube.com/watch?v=WoKlmaP3dto
- Algorithms: Big O, Growth Functions, and Worst Case Analysis (1/3)
http://www.youtube.com/watch?v=aPxauBqARGU&feature=relmfu
- Big-Oh, Omega and Theta notation
http://www.youtube.com/watch?v=pe250F4ttmE
- When solving a computer science problem there will usually be more than just one solution. These solutions will often be in the form of different algorithms, and you will
generally want to compare the algorithms to see which one is more efficient.
This is where Big O analysis helps
it measures the efficiency of an algorithm based on the time it takes for the algorithm to run as a function of the input size.
Basically, Big-O will want to express how many times the '
The word 'touched' can mean different things in different algorithms - in some algorithms it may mean the number of times a constant
http://www.programmerinterview.com/index.php/data-structures/big-o-notation/
- Big Omega
Big O describes the upper (worst case) bound, and Big Omega describes the lower (best case) bound.
Insertion Sort has an upper bound of O
http://www.
- asymptotic bounds
Big O, Big Omega, and Big Theta
http://www.youtube.com/watch?v=6Ol2JbwoJp0
http://xoax.net/comp_sci/
- Algorithms Lesson 7: Analyzing
Bubblesort
http://www.youtube.com/watch?v=ni_zk257Nqo
http://www.cs.auckland.ac.nz/compsci220s1t/lectures/lecturenotes/GG-lectures/BigOhexamples.pdf
http://www.programmerinterview.com/index.php/data-structures/big-o-notation/
http://www.cforcoding.com/2009/07/plain-english-explanation-of-big-o.html
This measure of efficiency is called asymptotic complexity,
or when calculating a function is difficult or impossible and
The most commonly used notation for specifying asymptotic complexity—that is, for
estimating the rate of function growth—is the big-O notation
Given two positive-valued functions
This definition reads:
not larger than
The relationship between
an upper bound on the value of
as
Algorithms: Big O, Growth Functions, and Worst Case Analysis (1/3)
Big O Part 1 – Linear Complexity
describes the impact of increasing the input data on the time taken for a program to run
big o space complexity
describes the impact of increasing the input data on the extra memory needed by the program
big o describes how the time is taken or how memory is used
big o describes complexity
common sense tells us that a program takes longer when there's more data to work on
linear search or sequential search
brute force approach
time is directly proportional to the amount of data
linear search complexity
big o time complexity is linear, O( n )
Big O Part 2 – Constant Complexity
peek operations, peek over top item wo removing it
LIFO data structure
no matter how much data is in the stack the time taken remains same
increasing the amount of data makes no difference to the time taken by push or pop operations
stack operations complexity
big o complexity is constant
O( 1)
Big O Part 3 – Quadratic Complexity
the dominant term
n^3 is dominant to n^2
bubble sort
bubble sort complexity
for n data items, a simple implementation performs (n -1)*( n -1) operations
the dominant term is n^2
big o time complexity is quadratic, O( n^2)
enhanced bubble sort
for n data items, enhanced bubble sort algorithm performs (n -1) +( n -2) +( n -3)..... +2+1
(n^2-n )/2
the dominant term is n^2, still O( n^2)
the big o complexity of enhanced bubble sort is still O( n^2)
alternative enhanced bubble sort
best vs worst case scenario
- DAA - Analysis of Algorithms
In theoretical analysis of algorithms, it is common to estimate their complexity in the asymptotic sense, i.e., to estimate the complexity function for arbitrarily large input. The term "analysis of algorithms" was coined by Donald Knuth.
Algorithm analysis is an important part of computational complexity theory, which provides theoretical estimation for the required resources of an algorithm to solve a specific computational problem. Most algorithms are designed to work with inputs of arbitrary length. Analysis of algorithms is the determination of the amount of time and space resources required to execute it.
Usually, the efficiency or running time of an algorithm is stated as a function relating the input length to the number of steps, known as time complexity, or volume of memory, known as space complexity.
The Need for Analysis
In this chapter, we will discuss the need for analysis of algorithms and how to choose a better algorithm for a particular problem as one computational problem can be solved by different algorithms.
By considering an algorithm for a specific problem, we can begin to develop pattern recognition so that similar types of problems can be solved by the help of this algorithm.
Algorithms are often quite different from one another, though the objective of these algorithms are the same. For example, we know that a set of numbers can be sorted using different algorithms. Number of comparisons performed by one algorithm may vary with others for the same input. Hence, time complexity of those algorithms may differ. At the same time, we need to calculate the memory space required by each algorithm.
Analysis of algorithm is the process of analyzing the problem-solving capability of the algorithm in terms of the time and size required (the size of memory for storage while implementation). However, the main concern of analysis of algorithms is the required time or performance. Generally, we perform the following types of analysis −
Worst-case − The maximum number of steps taken on any instance of size a.
Best-case − The minimum number of steps taken on any instance of size a.
Average case − An average number of steps taken on any instance of size a.
Amortized − A sequence of operations applied to the input of size a averaged over time.
In this context, if we compare bubble sort and merge sort. Bubble sort does not require additional memory, but merge sort requires additional space. Though time complexity of bubble sort is higher compared to merge sort, we may need to apply bubble sort if the program needs to run in an environment, where memory is very limited.
https://www.tutorialspoint.com/design_and_analysis_of_algorithms/analysis_of_algorithms.htm
- Asymptotic Notations and Apriori Analysis
Complexity of an algorithm is analyzed in two perspectives: Time and Space.
Time Complexity
It’s a function describing the amount of time required to run an algorithm in terms of the size of the input. "Time" can mean the number of memory accesses performed, the number of comparisons between integers, the number of times some inner loop is executed, or some other natural unit related to the amount of real time the algorithm will take.
Space Complexity
It’s a function describing the amount of memory an algorithm takes in terms of the size of input to the algorithm. We often speak of "extra" memory needed, not counting the memory needed to store the input itself. Again, we use natural (but fixed-length) units to measure this.
Space complexity is sometimes ignored because the space used is minimal and/or obvious, however sometimes it becomes as important an issue as time.
Asymptotic Notations
Execution time of an algorithm depends on the instruction set, processor speed, disk I/O speed, etc. Hence, we estimate the efficiency of an algorithm asymptotically.
Different types of asymptotic notations are used to represent the complexity of an algorithm. Following asymptotic notations are used to calculate the running time complexity of an algorithm.
O − Big Oh
Ω − Big omega
θ − Big theta
o − Little Oh
ω − Little omega
https://www.tutorialspoint.com/design_and_analysis_of_algorithms/design_and_analysis_of_algorithms_asymptotic_notations_apriori.htm
