I've been doing a lot of studying from many different resources on algorithm analysis lately, and one thing I'm currently confused about is why time complexity is often defined in terms of the number of steps/operations an algorithm performs.
For instance, in Introduction to Algorithms, 3rd Edition by Cormen, he states:
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.
I've seen other resources define the time complexity as such as well. I have a problem with this because, for one, it's called TIME complexity, not "step complexity" or "operations complexity." Secondly, while it's not a definitive source, an answer to a post here on Stackoverflow states "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." Further, on the Wikipedia page for time complexity it states "In computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm." Again, these are definitive sources, things makes logical sense using these definitions.
When analyzing an algorithm and deriving its time complexity function, such as in Figure 1 below, you get an equation that is in units of time. It CAN represent the amount of operations the algorithm performs, but only if those constant factors (C_1, C_2, C_3, etc.) are each a value of 1.
Figure 1
So with all that said, I'm just wondering how it's possible for this to be defined as the number of steps when that's not really what it represents. I'm trying to clear things up and make the connection between time and number of operations. I feel like there is a lot of information that hasn't been explicitly stated in the resources I've studied. Hoping someone can help clear things up for me, and without going into discussion about Big-O because that shouldn't be needed and misses the point of the question, in my opinion.
Thank you everyone for your time and help.
why time complexity is often defined in terms of the number of steps/operations an algorithm performs?
TL;DR: because that is how the asymptotic analysis work; also, do not forget, that time is a relative thing.
Longer story:
Measuring the performance in time, as we, humans understand the time in a daily use, doesn't make much sense, as it is not always that trivial task to do.. furthermore - it even makes no sense in a broader perspective.
How would you measure what is the space and time your algorithm takes? what will be the conditional and predefined unit of the measurement you're going to apply to see the running time/space complexity of your algorithm?
You can measure it on your clock, or use some libraries/API to see exactly how many seconds/minutes/megabytes your algorithm took.. or etc.
However, this all will be VERY much variable! because, the time/space your algorithm took, will depend on:
Particular hardware you're using (architecture, CPU, RAM, etc.);
Particular programming language;
Operating System;
Compiler, you used to compile your high-level code into lower abstraction;
Other environment-specific details (sometimes, even on the temperature.. as CPUs might be scaling operating frequency dynamically)..
therefore, it is not the good thing to measure your complexity in the precise timing (again, as we understand the timing on this planet).
So, if you want to know the complexity (let's say time complexity) of your algorithm, why would it make sense to have a different time for different machines, OSes, and etc.? Algorithm Complexity Analysis is not about measuring runtime on a particular machine, but about having a clear and mathematically defined precise boundaries for the best, average and worst cases.
I hope this makes sense.
Fine, we finally get to the point, that algorithm analysis should be done as a standalone, mathematical complexity analysis.. which would not care what is the machine, OS, system architecture, or anything else (apart from algorithm itself), as we need to observe the logical abstraction, without caring about whether you're running it on Windows 10, Intel Core2Duo, or Arch Linux, Intel i7, or your mobile phone.
What's left?
Best (so far) way for the algorithm analysis, is to do the Asymptotic Analysis, which is an abstract analysis calculated on the basis of input.. and that is counting almost all the steps and operations performed in the algorithm, proportionally to your input.
This way you can speak about the Algorithm, per se, instead of being dependent on the surrounding circumstances.
Moreover; not only we shouldn't care about machine or peripheral factors, we also shouldn't care about Lower Order Terms and Constant Factors in the mathematical expression of the Asymptotic Analysis.
Constant Factors:
Constant Factors are instructions which are independent from the Input data. i.e. which are NOT dependent on the input argument data.
Few reasons why you should ignore them are:
Different programming language syntaxes, as well as their compiled files, will have different number of constant operations/factors;
Different Hardware will give different run-time for the same constant factors.
So, you should eliminate thinking about analyzing constant factors and overrule/ignore them. Only focus on only input-related important factors; therefore:
O(2n) == O(5n) and all these are O(n);
6n2 == 10n2 and all these are n2.
One more reason why we won't care about constant factors is that they we usually want to measure the complexity for sufficiently large inputs.. and when the input grows to the + infinity, it really makes no sense whether you have n or 2n.
Lower order terms:
Similar concept applies in this point:
Lower order terms, by definition, become increasingly irrelevant as you focus on large inputs.
When you have 5x4+24x2+5, you will never care much on exponent that is less than 4.
Time complexity is not about measuring how long an algorithm takes in terms of seconds. It's about comparing different algorithms, how they will perform with a certain amount if input data. And how this performance develops when the input data gets bigger.
In this context, the "number of steps" is an abstract concept for time, that can be compared independently from any hardware. Ie you can't tell how long it will take to execute 1000 steps, without exact specifications of your hardware (and how long one step will take). But you can always tell, that executing 2000 steps will take about twice as long as executing 1000 steps.
And you can't really discuss time complexity without going into Big-O, because that's what it is.
You should note that Algorithms are more abstract than programs. You check two algorithms on a paper or book and you want to analyze which works faster for an input data of size N. So you must analyze them with logic and statements. You can also run them on a computer and measure the time, but that's not proof.
Moreover, different computers execute programs at different speeds. It depends on CPU speed, RAM, and many other conditions. Even a program on a single computer may be run at different speeds depending on available resources at a time.
So, time for algorithms must be independent of how long a single atomic instruction takes to be executed on a specific computer. It's considered just one step or O(1). Also, we aren't interested in constants. For example, it doesn't matter if a program has two or 10 instructions. Both will be run on a fraction of microseconds. Usually, the number of instructions is limited and they are all run fast on computers. What is important are instructions or loops whose execution depends on a variable, which could be the size of the input to the program.
Related
I'm currently reading this book called CLRS 2.2 page 25. In which the author describes the Running time of an algorithm as
The running time of an algorithm on a particular input is the number of primitive
operations or “steps” executed.
Also the author uses the running time to analyze algorithms. Then I referred a book called Data Structures and Algorithms made easy by Narasimha Karumanchi.
In which he describes the following.
1.7 Goal of the Analysis of Algorithms
The goal of the analysis of algorithms is to compare algorithms (or solutions) mainly in terms of
running time but also in terms of other factors (e.g., memory, developer effort, etc.)
1.9 How to Compare Algorithms:
To compare algorithms, let us define a few objective measures:
Execution times? Not a good measure as execution times are specific to a particular computer.
Number of statements executed? Not a good measure, since the number of statements varies
with the programming language as well as the style of the individual programmer.
Ideal solution? Let us assume that we express the running time of a given algorithm as a function
of the input size n (i.e., f(n)) and compare these different functions corresponding to running
times. This kind of comparison is independent of machine time, programming style, etc.
As you can see from CLRS the author describes the running time as the number of steps executed whereas in the second book the author says its not a good measure to use Number of step executed to analyze the algorithms. Also the running time depends on the computer (my assumption) but the author from the second book says that we cannot consider the Execution time to analyze algorithms as it totally depends on the computer.
I thought the execution time and the running time are same!
So,
What is the real meaning or definition of running time and execution time? Are they the same of different?
Does running time describe the number of steps executed or not?
Does running time depend on the computer or not?
thanks in advance.
What is the real meaning or definition of running time and execution time? Are they the same of different?
The definition of "running time" in 'Introduction to Algorithms' by C,L,R,S [CLRS] is actually not a time, but a number of steps. This is not what you would intuitively use as a definition. Most would agree that "runnning" and "executing" are the same concept, and that "time" is expressed in a unit of time (like milliseconds). So while we would normally consider these two terms to have the same meaning, in CLRS they have deviated from that, and gave a different meaning to "running time".
Does running time describe the number of steps executed or not?
It does mean that in CLRS. But the definition that CLRS uses for "running time" is particular, and not the same as you might encounter in other resources.
CLRS assumes here that a primitive operation (i.e. a step) takes O(1) time.
This is typically true for CPU instructions, which take up to a fixed maximum number of cycles (where each cycle represents a unit of time), but it may not be true in higher level languages. For instance, some languages have a sort instruction. Counting that as a single "step" would give useless results in an analysis.
Breaking down an algorithm into its O(1) steps does help to analyse the complexity of an algorithm. Counting the steps for different inputs may only give a hint about the complexity though. Ultimately, the complexity of an algorithm requires a (mathematical) proof, based on the loops and the known complexity of the steps used in an algorithm.
Does running time depend on the computer or not?
Certainly the execution time may differ. This is one of the reasons we want to by a new computer once in a while.
The number of steps may depend on the computer. If both support the same programming language, and you count steps in that language, then: yes. But if you would do the counting more thoroughly and would count the CPU instructions that are actually ran by the compiled program, then it might be different. For instance, a C compiler on one computer may generate different machine code than a different C compiler on another computer, and so the number of CPU instructions may be less on the one than the other, even though they result from the same C program code.
Practically however, this counting at CPU instruction level is not relevant for determining the complexity of an algorithm. We generally know the time complexity of each instruction in the higher level language, and that is what counts for determining the overall complexity of an algorithm.
Space and time are considered as barometers of analyzing the complexity of an algorithm. But these days with the presence of GPU on mobile devices, there are numerous possible applications which can use that high-performance to run complex algorithms on a mobile device. For eg: iOS's Metal framework can be used for GPGPU operations. But needless to say it consumes a lot of power. So, my question is, if I am developing/implementing, say, a graph search algorithm, on a mobile device, should't I also consider the "power" complexity of my algorithm along with space-time? Now, I know the argument could be that power is something that the algorithm doesn't consume itself directly and I completely agree with that. So, maybe my grammar here is incorrect in saying that power is another dimension of measuring an algorithm's efficiency. But shouldn't power be seen as a performance measure of an algorithm?
No.
Complexity explains how the algorithm scales in time / memory. Power will be a function of time and memory.
Say you have algorithm A - O(N^2) and B - O(N^3) and they both solve the same problem. For n = 1000 B uses 1 unit of power while A uses 20. Now as you scale it up to n=10k B will need 1000 units of power while A will need only 2000. At n = 100k B will need 1'000'000 while A will need 200'000. And so on.
This assumes that the energy consumption is constant for the execution of the algorithm.
By the way the same thing happens with time. For example for short arrays nothing beats linear search.
For a specific case (rendering UI on fixed resolution) it makes sense to measure power usage and optimize it. But what works for the resolution today will not necessarily be the right thing tomorrow.
For this to be possible, you need a model of energy consumption that you can relate to the atomic operations in your algorithms.
Like "a multiply consumes one unit of energy" and "a memory slot uses two units of energy per unit of time". Maybe the relation Energy = Time x Space could make sense.
Anyway, such a "naïve" model may suffer the same phenomenon as the model of time complexity: it doesn't bear any similarity with the behavior of modern architectures and can be wrong by orders of magnitude.
Using more accurate models would be just intractable analytically.
I am a new developer. Please help me understand what a posteriori and a priori analyses of algorithm operations are. I googled it, but I did not get any proper answers.
Apriori analysis of algorithms : it means we do analysis (space and time) of an algorithm prior to running it on specific system - that is, we determine time and space complexity of algorithm by just seeing the algorithm rather than running it on particular system (with different processor and compiler).
Apostiari analysis of algorithms : it means we do analysis of algorithm only after running it on system. It directly depends on system and changes from system to system.
In industry we cannot do Apostiari analysis as software is generally made for an anonymous user which runs it on system different (in processor like Pentium 3 or Pentium 4) from those present in the industry.
In Apriory it is the reason we use asymptotic notations to determine time and space complexity as they changes from computer to computer but asymptotically they are same.
A Priori Analysis − This is all about the theoretical analysis of an algorithm. theWhere efficiency of an algorithm is measured by assuming that all other factors, for example, processor speed, are constant and have no effect on the implementation.
A Posterior Analysis − This is more of an empirical analysis of an algorithm. The selected algorithm is implemented using programming language and then executed on target computer machine. In this analysis, actual statistics like running time and space required, are collected.
In short
In an priory analysis, we obtain a function which bounds the algorithm
computing time.
In a posteriori analysis, we collect actual statistics
about the algorithms consumption of time and space, while it is
executing.
Here is the book.
Somewhat longer:
Wikipedia definition
Ans another article citation
By far the most important reason to analyze an algorithm is to make
sure it will correctly solve your problem. If our algorithm doesn't
work, nothing else matters. So we must analyze it to prove that it
will always work as expected.
We must also look at the efficiency of our algorithm. If it solves our
problem, but does so in O(nn) time (or space!), then we should
probably look at a redesign.
In posterior analysis we run the algo on a system to check its original statics so that we can calculate its time and space complexity. but since it changes from system to system so it is not that effective. we calculate its time complexity in terms of a specific system requirements.
In prior analysis we only see the algo and give its analysis in terms of asymptotic notations. It doesn't change system to system in terms of asymptotic notation.
From section 12.7 of Manage Software Testing (by Peter Farrell-Vinay), a priori analysis is a stage where a function is defined using some theoretical model (like a Finite State Machine). This model is then used to determine various characteristics of that function (like time and space usage).
In the a posteriori stage, evidence of the function's characteristics (like time and space usage) are collected and compared with those calculated during the a priori analysis.
Posteriori analysis depends upon hardware algo and programming language algo ,it gives exact answers
Priori analysis is hardware independent ,it depends on no of times statement is executed
i have learned that a program is measured by it's complexity - i mean by Big O Notation.
why don't we measure it by it's absolute running time?
thanks :)
You use the complexity of an algorithm instead of absolute running times to reason about algorithms, because the absolute running time of a program does not only depend on the algorithm used and the size of the input. It also depends on the machine it's running on, various implementations detail and what other programs are currently using system resources. Even if you run the same application twice with the same input on the same machine, you won't get exactly the same time.
Consequently when given a program you can't just make a statement like "this program will take 20*n seconds when run with an input of size n" because the program's running time depends on a lot more factors than the input size. You can however make a statement like "this program's running time is in O(n)", so that's a lot more useful.
Absolute running time is not an indicator of how the algorithm grows with different input sets. It's possible for a O(n*log(n)) algorithm to be far slower than an O(n^2) algorithm for all practical datasets.
Running time does not measure complexity, it only measures performance, or the time required to perform the task. An MP3 player will run for the length of the time require to play the song. The elapsed CPU time may be more useful in this case.
One measure of complexity is how it scales to larger inputs. This is useful for planning the require hardware. All things being equal, something that scales relatively linearly is preferable to one which scales poorly. Things are rarely equal.
The other measure of complexity is a measure of how simple the code is. The code complexity is usually higher for programs with relatively linear performance complexity. Complex code can be costly maintain, and changes are more likely to introduce errors.
All three (or four) measures are useful, and none of them are highly useful by themselves. The three together can be quite useful.
The question could use a little more context.
In programming a real program, we are likely to measure the program's running time. There are multiple potential issues with this though
1. What hardware is the program running on? Comparing two programs running on different hardware really doesn't give a meaningful comparison.
2. What other software is running? If anything else running, it's going to steal CPU cycles (or whatever other resource your program is running on).
3. What is the input? As already said, for a small set, a solution might look very fast, but scalability goes out the door. Also, some inputs are easier than others. If as a person, you hand me a dictionary and ask me to sort, I'll hand it right back and say done. Giving me a set of 50 cards (much smaller than a dictionary) in random order will take me a lot longer to do.
4. What is the starting conditions? If your program runs for the first time, chances are, spinning it off the hard disk will take up the largest chunk of time on modern systems. Comparing two implementations with small inputs will likely have their differences masked by this.
Big O notation covers a lot of these issues.
1. Hardware doesn't matter, as everything is normalized by the speed of 1 operation O(1).
2. Big O talks about the algorithm free of other algorithms around it.
3. Big O talks about how the input will change the running time, not how long one input takes. It tells you the worse the algorithm will perform, not how it performs on an average or easy input.
4. Again, Big O handles algorithms, not programs running in a physical system.
I was interested to know about parameters other than space and time during analysing the effectiveness of an algorithms. For example, we can focus on the effective trap function while developing encryption algorithms. What other things can you think of ?
First and foremost there's correctness. Make sure your algorithm always works, no matter what the input. Even for input that the algorithm is not designed to handle, you should print an error mesage, not crash the entire application. If you use greedy algorithms, make sure they truly work in every case, not just a few cases you tried by hand.
Then there's practical efficiency. An O(N2) algorithm can be a lot faster than an O(N) algorithm in practice. Do actual tests and don't rely on theoretical results too much.
Then there's ease of implementation. You usually don't need the best intro sort implementation to sort an array of 100 integers once, so don't bother.
Look for worst cases in your algorithms and if possible, try to avoid them. If you have a generally fast algorithm but with a very bad worst case, consider detecting that worst case and solving it using another algorithm that is generally slower but better for that single case.
Consider space and time tradeoffs. If you can afford the memory in order to get better speeds, there's probably no reason not to do it, especially if you really need the speed. If you can't afford the memory but can afford to be slower, do that.
If you can, use existing libraries. Don't roll your own multiprecision library if you can use GMP for example. For C++, stuff like boost and even the STL containers and algorithms have been worked on for years by an army of people and are most likely better than you can do alone.
Stability (sorting) - Does the algorithm maintain the relative order of equal elements?
Numeric Stability - Is the algorithm prone to error when very large or small real numbers are used?
Correctness - Does the algorithm always give the correct answer? If not, what is the margin of error?
Generality - Does the algorithm work in many situation (e.g. with many different data types)?
Compactness - Is the program for the algorithm concise?
Parallelizability - How well does performance scale when the number of concurrent threads of execution are increased?
Cache Awareness - Is the algorithm designed to maximize use of the computer's cache?
Cache Obliviousness - Is the algorithm tuned for particulary cache-sizes / cache-line-sizes or does it perform well regardless of the parameters of the cache?
Complexity. 2 algorithms being the same in all other respects, the one that's much simpler is going to be a much better candidate for future customization and use.
Ease of parallelization. Depending on your use case, it might not make any difference or, on the other hand, make the algorithm useless because it can't use 10000 cores.
Stability - some algorithms may "blow up" with certain test conditions, e.g. take an inordinately long time to execute, or use an inordinately large amount of memory, or perhaps not even terminate.
For algorithms that perform floating point operations, the accumulation of round-off error is often a consideration.
Power consumption, for embedded algorithms (think smartcards).
One important parameter that is frequently measure in the analysis of algorithms is that of Cache hits and cache misses. While this is a very implementation and architecture dependent issue, it is possible to generalise somewhat. One particularly interesting property of the algorithm is being Cache-oblivious, which means that the algorithm will use the cache optimally on multiple machines with different cache sizes and structures without modification.
Time and space are the big ones, and they seem so plain and definitive, whereby they should often be qualified (1). The fact that the OP uses the word "parameter" rather than say "criteria" or "properties" is somewhat indicative of this (as if a big O value on time and on space was sufficient to frame the underlying algorithm).
Other criteria include:
domain of applicability
complexity
mathematical tractability
definitiveness of outcome
ease of tuning (may be tied to "complexity" and "tactability" afore mentioned)
ability of running the algorithm in a parallel fashion
(1) "qualified": As hinted in other answers, a -technically- O(n^2) algorithm may be found to be faster than say an O(n) algorithm, in 90% of the cases (which, btw, may turn out to be 100% of the practical cases)
worst case and best case are also interesting, especially when linked to some conditions in the input. if your input data shows some properties, an algorithm, by taking advantage of this property, may perform better that another algorithm which performs the same task but does not use that property.
for example, many sorting algorithm perform very efficiently when input are partially ordered in a specific way which minimizes the number of operations the algorithm has to execute.
(if your input is mostly sorted, an insertion sort will fit nicely, while you would never use that algorithm otherwise)
If we're talking about algorithms in general, then (in the real world) you might have to think about CPU/filesystem(read/write operations)/bandwidth usage.
True they are way down there in the list of things you need worry about these days, but given a massive enough volume of data and cheap enough infrastructure you might have to tweak your code to ease up on one or the other.
What you are interested aren’t parameters, rather they are intrinsic properties of an algorithm.
Anyway, another property you might be interested in, and analyse an algorithm for, concerns heuristics (or rather, approximation algorithms), i.e. algorithms which don’t find an exact solution but rather one that is (hopefully) good enough.
You can analyze how far a solution is from the theoretical optimal solution in the worst case. For example, an existing algorithm (forgot which one) approximates the optimal travelling salesman tour by a factor of two, i.e. in the worst case it’s twice as long as the optimal tour.
Another metric concerns randomized algorithms where randomization is used to prevent unwanted worst-case behaviours. One example is randomized quicksort; quicksort has a worst-case running time of O(n2) which we want to avoid. By shuffling the array beforehand we can avoid the worst-case (i.e. an already sorted array) with a very high probability. Just how high this probability is can be important to know; this is another intrinsic property of the algorithm that can be analyzed using stochastic.
For numeric algorithms, there's also the property of continuity: that is, whether if you change input slightly, output also changes only slightly. See also Continuity analysis of programs on Lambda The Ultimate for a discussion and a link to an academical paper.
For lazy languages, there's also strictness: f is called strict if f _|_ = _|_ (where _|_ denotes the bottom (in the sense of domain theory), a computation that can't produce a result due to non-termination, errors etc.), otherwise it is non-strict. For example, the function \x -> 5 is non-strict, because (\x -> 5) _|_ = 5, whereas \x -> x + 1 is strict.
Another property is determinicity: whether the result of the algorithm (or its other properties, such as running time or space consumption) depends solely on its input.
All these things in the other answers about the quality of various algorithms are important and should be considered.
But time and space are two things that vary at some rate compared to the size of the input (n). So what else can vary according to n?
There are several that are related to I/O. For example, the number of writes to a disk is an important one, which may not be directly shown by space and time estimates alone. This becomes particularly important with flash memory, where the number of writes to the same memory location is the significant metric in some algorithms.
Another I/O metric would be "chattiness". A networking protocol might send shorter messages more often adding up to the same space and time as another networking protocol, but some aspect of the system (perhaps billing?) might make minimizing either the size or number of the messages desireable.
And that brings us to Cost, which is a very important algorithmic consideration sometimes. The cost of an algorithm may be affected by both space and time in different amounts (consider the separate costing of server storage space and gigabits of data transfer), but the cost is the thing that you wish to minimize overall, so it may have its own big-O estimations.