In the Amortized analysis.
Average case running time: average over all possible inputs for one algorithm (operation).
If using probability, called expected running time
What is the different between expected time and Amortized time?
Expected time:
We make some assumptions and, based on these assumptions, we make statements about the running time.
Hash tables is one such example. We assume that the data is well-distributed, and claim that the running time of operations are O(1), whereas the worst-case running time for an operation is actually O(n).
Amortized time:
Even though one operation may take longer than some given time, the time across multiple operations will balance out to give the mentioned running time.
(Well-implemented) self-resizing arrays is one such example. When you insert, it takes O(n) to resize the array, but, across many inserts, each will take O(1) on average.
Related
When I first took a class on algorithms, I was confused as to what was actually being measured when talking about asymptotic time complexity, since it sure wasn't the time the computer took to run a program. Instead, my mental model was that we were measuring the asymptotic step complexity, that is the asymptotic number of steps the CPU would take to run the algorithm.
Any reason why we reason about time complexity as opposed to step complexity and talk about how much time an algorithm takes as opposed to how many steps (asymptotically) a CPU takes to execute the algorithm?
Indeed, the number of steps is the determining factor, with the condition that the duration of a step is not dependent on the input -- it should never take more time than some chosen constant time.
What exactly that constant time is, will depend on the system you run it on. Some CPUs are just faster than others, and some CPUs are more specialised in one kind of operation, and less in another. Two different steps may therefore represent different times: on one CPU step A may execute with a shorter delay than step B, while on another it may be the inverse. It might even be, that on the same CPU step A sometimes can execute faster than other times (for example, because of some favourable condition in the pipe of that CPU).
All that makes it impossible to say something useful by just measuring the time to run a step. Instead, we consider that there is a maximum time (for a given CPU) for all the different kinds of "steps" we have identified in the algorithm, such that the individual execution of one step will never exceed that maximum time.
So when we talk about time complexity we do say something about the time an algorithm will take. If an algorithm has O(n²) time complexity, it means we can find a value minN and a constant time C (we may freely choose those), such that for every n >= minN, the total time T it takes to run the algorithm is bounded by T < Cn². Note especially that T and C are not a number of steps, but really measures of time (e.g. milliseconds). However the choice of C will depend on the CPU and the maximum we found for its step execution. So we don't actually know which value C will have in general, we just prove that such a C exists for every CPU (or whatever executes the algorithm).
In short, we make an equivalence between a step and a unit of time, such that the execution of a step is guaranteed to be bounded by that unit of time.
You are right, we measure the computational steps that an algorithm uses to run on a Turing machine. However, we do not count every single step. Instead, we are typically interested in the runtime differences of algorithms ignoring constant factors as we do when using the O-notation.
Also, I believe the term is quite intuitive to grasp. Everybody has a basic understanding of what you mean when you talk about how much time an algorithm takes (I can even explain that to my mother). However, if you talk about how many steps an algorithm needs, you may find yourself in a discussion about the computational model (what kind of CPU).
The term time complexity isn't wrong (in fact, I believe it is quite what we are looking for). The term step complexity would be misleading.
Kindly explain with a simple problem
I have been going through a book where it is said that ArrayList will double after it reaches its limit and if the ArrayList is full it takes O(N) insertion time for inserting N element.Kindly explain by taking an ArrayList of few elements
Amortised time explained in simple terms:
If you do an operation say a million times, you don't really care about the worst-case or the best-case of that operation - what you care about is how much time is taken in total when you repeat the operation a million times.
So it doesn't matter if the operation is very slow once in a while, as long as "once in a while" is rare enough for the slowness to be diluted away. Essentially amortised time means "average time taken per operation, if you do many operations". Amortised time doesn't have to be constant; you can have linear and logarithmic amortised time or whatever else.
Let's take the example of a dynamic array, to which you repeatedly add new items. Normally adding an item takes constant time (that is, O(1)). But each time the array is full, you allocate twice as much space, copy your data into the new region, and free the old space. Assuming allocates and frees run in constant time, this enlargement process takes O(n) time where n is the current size of the array.
So each time you enlarge, you take about twice as much time as the last enlarge. But you've also waited twice as long before doing it! The cost of each enlargement can thus be "spread out" among the insertions. This means that in the long term, the total time taken for adding m items to the array is O(m), and so the amortised time (i.e. time per insertion) is O(1).
I am trying to do homework with a friend and one question asks the average running time of search, add, and delete for the linear probing method. I think it's O(n) because it has to check at certain number of nodes until it finds an open one to add. And when searching it starts at the original index and moves up until it finds the desired number. But my friends says it's O(1). Which one is right?
When we talk about Asymptotic complexities we generally take into account very large n. Now for collision handling in a Hash Table some of the methods are chained hashing & linear probing. In both the cases two things may happen (that will help in answering your question): 1. You may require resizing of the hash table due to it getting full 2. Collisions may happen.
In the worst case it will depend on how you have implemented your hash table, say in linear probing you dont find the number,you keep on moving and the number you were looking for was at the end. Here comes the O(n) worst case running time. Coming to chained hashing technique, when a collision happens, to handle them say we have stored the keys in a balanced binary tree so the worst case running time would be O(log n).
Now coming to best case running time, I think there is no confusion, in either case it would be O(1).
O(n) would happen in worst case and not in an average case of a good designed hash table. If that starts happening in average case hash tables wont find a place in Data Structures because then balanced trees on an average would give you O(log n) always and ON TOP OF THAT will preserve the order too.
Sorry to say this but unfortunately your friend is right. Your case would happen in worst case scenarios.
Also look here for more informative stuff i.e. the amortized running time: Time complexity of Hash table
I am working on a Project that improves the Quick-sort algorithms worst case time complexity. I modified the algorithm by choosing the median pivot instead of the left most selection and introduced insertion sort after a certain number of iterations. The results are as follows:
For an input of unsorted data of length 5000 to 100000:
The number of Comparison made in my modified Quick-sort are very less than the number of comparisons made in regular Quick-sort.
Elapsed time for both is zero secs for all length if data.
For an input of already sorted data of length 5000 to 100000:
The number of Comparison made in my modified Quick-sort are still very less than the number of comparisons made in regular Quick-sort.
Elapsed time for my modified Quick-sort is very less than the elapsed time of the regular Quick-sort for all length of data.
How can I now prove that the time complexity O(n^2) for already sorted data has been improved? I have all the above data but dont know how to theoretically show it? No direct answers but hints will be fine.
The usual way to demonstrate algorithmic improvements in sorting algorithms is to instrument the code to count the number of comparisons and then run different algorithms over several different datasets, each with different characteristics (random, already sorted, reverse sorted, mostly sorted, etc).
A good model for this kind of analysis is Tim Peter's write-up for his Timsort algorithm: http://hg.python.org/cpython/file/2.7/Objects/listsort.txt
What is the difference between time complexity and running time? Are they the same?
Running time is how long it takes a program to run. Time complexity is a description of the asymptotic behavior of running time as input size tends to infinity.
You can say that the running time "is" O(n^2) or whatever, because that's the idiomatic way to describe complexity classes and big-O notation. In fact the running time is not a complexity class, it's either a duration, or a function which gives you the duration. "Being O(n^2)" is a mathematical property of that function, not a full characterisation of it. The exact running time might be 2036*n^2 + 17453*n + 18464 CPU cycles, or whatever. Not that you very often need to know it in that much detail, and anyway it might well depend on the actual input as well as the size of the input.
The time complexity and running time are two different things altogether.
Time complexity is a complete theoretical concept related to algorithms, while running time is the time a code would take to run, not at all theoretical.
Two algorithms may have the same time complexity, say O(n^2), but one may take twice as much running time as the other one.
From CLRS 2.2 pg. 25
The running time of an algorithm on a particular input is the number
of primitive operations or “steps” executed. It is convenient to
define the notion of step so that it is as machine-independent as
possible.
Now from Wikipedia
... time complexity of an algorithm quantifies the amount of time taken by an algorithm to run as a function of the length of the string
representing the input.
Time complexity is commonly estimated by counting the number of
elementary operations performed by the algorithm, where an elementary
operation takes a fixed amount of time to perform.
Notice that both descriptions emphasize the relationship of the size of the input to the number of primitive/elementary operations.
I believe this makes it clear both refer to the same concept.
In practice though you'll find that enterprisey jargon rarely matches academic terminology, e.g., tons of people work doing code optimization but rarely solve optimization problems.
"Running time" refers to the algorithm under consideration:
Another algorithm might be able solve the same problem asymptotically faster, that is, with less running time.
"Time complexity" on the other hand is inherent to the problem under consideration.
It is defined as the least running time of any algorithm solving said problem.
The same distincting applies to other measures of algorithmic cost such as memory, #processors, communication volume etc.
(Blum's Speedup Theorem demonstrates that the "least" time may in general not be attainable...)
To analyze an algorithm is to determine the amount of resources (such as time and storage) necessary to execute it. Most algorithms are designed to work with inputs of arbitrary length. Usually the efficiency or running time of an algorithm is stated as a function relating the input length to the number of steps (time complexity) or storage locations (space complexity).
Running time measures the number of operations it takes to complete a code or program. the keyword here is "operations" and "complete", the time taken for every single operation to complete can be affected by the processor, memory, etc.
With running time, If we have 2 different algorithms solving the same problem, the optimized algorithm might take a longer time to complete than the non-optimized one because of varying factors like ram, the current state of the PC (serving other programs) etc. or even the function for calculating the runtime itself.
For this reason, it is not enough to measure the efficiency of an algorithm based on operations it takes to complete but rather time against input, that way all the external factors are eliminated and that's exactly what time complexity does.
Time complexity is the measurement of an algorithm's time behavior as input size increases.
Time complexity can also be calculated from the logic behind the algorithm/code.
On the other hand, running time can be calculated when the code is completed.