All Questions

29,669 questions
273k views

Why is quicksort better than other sorting algorithms in practice?

In a standard algorithms course we are taught that quicksort is $O(n \log n)$ on average and $O(n^2)$ in the worst case. At the same time, other sorting algorithms are studied which are $O(n \log n)$ ...
104k views

What is the definition of $P$, $NP$, $NP$-complete and $NP$-hard?

I'm in a course about computing and complexity, and am unable to understand what these terms mean. All I know is that NP is a subset of NP-complete, which is a subset of NP-hard, but I have no idea ...
46k views

Why is writing down mathematical proofs more fault-proof than writing computer code?

I have noticed that I find it far easier to write down mathematical proofs without making any mistakes, than to write down a computer program without bugs. It seems that this is something more ...
48k views

How can a language whose compiler is written in C ever be faster than C?

Taking a look at Julia's webpage, you can see some benchmarks of several languages across several algorithms (timings shown below). How can a language with a compiler originally written in C, ...
15k views

Is there a system behind the magic of algorithm analysis?

There are lots of questions about how to analyze the running time of algorithms (see, e.g., runtime-analysis and algorithm-analysis). Many are similar, for instance those asking for a cost analysis ...
46k views

Why, really, is the Halting Problem so important?

I don't understand why the Halting Problem is so often used to dismiss the possibility of determining whether a program halts. The Wikipedia article correctly explains that a deterministic machine ...
37k views

Why are there so many programming languages?

I'm pretty fluent in C/C++, and can make my way around the various scripting languages (awk/sed/perl). I've started using python a lot more because it combines some of the nifty aspects of C++ with ...
13k views

How can it be decidable whether $\pi$ has some sequence of digits?

We were given the following exercise. Let $\qquad \displaystyle f(n) = \begin{cases} 1 & 0^n \text{ occurs in the decimal representation of } \pi \\ 0 & \text{else}\end{cases}$ ...
18k views

Why can I look at a graph and immediately find the closest point to another point, but it takes me O(n) time through programming?

Let me clarify: Given a scatterplot of some given number of points n, if I want to find the closest point to any point in the plot mentally, I can immediately ignore most points in the graph, ...
27k views

Is Category Theory useful for learning functional programming?

I'm learning Haskell and I'm fascinated by the language. However I have no serious math or CS background. But I am an experienced software programmer. I want to learn category theory so I can become ...
145k views

How to convert finite automata to regular expressions?

Converting regular expressions into (minimal) NFA that accept the same language is easy with standard algorithms, e.g. Thompson's algorithm. The other direction seems to be more tedious, though, and ...
10k views

Why hasn't there been an encryption algorithm that is based on the known NP-Hard problems?

Most of today's encryption, such as the RSA, relies on the integer factorization, which is not believed to be a NP-hard problem, but it belongs to BQP, which makes it vulnerable to quantum computers. ...
13k views

How not to solve P=NP?

There are lots of attempts at proving either $\mathsf{P} = \mathsf{NP}$ or $\mathsf{P} \neq \mathsf{NP}$, and naturally many people think about the question, having ideas for proving either direction....
28k views

BIT: What is the intuition behind a binary indexed tree and how was it thought about?

A binary indexed tree has very less or relatively no literature as compared to other data structures. The only place where it is taught is the topcoder tutorial. Although the tutorial is complete in ...
8k views

What are the reasons to learn different algorithms / data structures serving the same purpose?

I have been wondering about this question since I was an undergraduate student. It is a general question but I will elaborate with examples below. I have seen a lot of algorithms - for example, for ...
50k views

How/when is calculus used in Computer Science?

Many computer science programs require two or three calculus classes. I'm wondering, how and when is calculus used in computer science? The CS content of a degree in computer science tends to focus ...
8k views

How to fool the “try some test cases” heuristic: Algorithms that appear correct, but are actually incorrect

To try to test whether an algorithm for some problem is correct, the usual starting point is to try running the algorithm by hand on a number of simple test cases -- try it on a few example problem ...
22k views

How does one know which notation of time complexity analysis to use?

In most introductory algorithm classes, notations like $O$ (Big O) and $\Theta$ are introduced, and a student would typically learn to use one of these to find the time complexity. However, there are ...
57k views

How to prove that a language is not context-free?

We learned about the class of context-free languages $\mathrm{CFL}$. It is characterised by both context-free grammars and pushdown automata so it is easy to show that a given language is context-free....
13k views

Solving or approximating recurrence relations for sequences of numbers

In computer science, we have often have to solve recurrence relations, that is find a closed form for a recursively defined sequence of numbers. When considering runtimes, we are often interested ...
93k views

What's the difference between a binary search tree and a binary heap?

These two seem very similar and have almost an identical structure. What's the difference? What are the time complexities for different operations of each?
23k views

Why is the unit of image size not Pixel²?

If you calculate the area of a rectangle, you just multiply the height and the width and get back the unit squared. Example: 5cm * 10cm = 50cm² In contrast, if you calculate the size of an image, you ...
56k views

Quicksort Partitioning: Hoare vs. Lomuto

There are two quicksort partition methods mentioned in Cormen: ...
17k views

Why are some programming languages “faster” or “slower” than others?

I have noticed that some applications or algorithms that are built on a programming language, say C++/Rust run faster or snappier than those built on say, Java/Node.js, running on the same machine. I ...
16k views

Why is deep learning hyped despite bad VC dimension?

The VC-dimension formula for neural networks ranges from $O(E)$ to $O(E^2)$, with $O(E^2V^2)$ in the worst case, where $E$ is the number of edges and $V$ is the number of nodes. The number of training ...
52k views

When searching graphs, there are two easy algorithms: breadth-first and depth-first (Usually done by adding all adjactent graph nodes to a queue (breadth-first) or stack (depth-first)). Now, are ...
14k views

How can we assume that basic operations on numbers take constant time?

Normally in algorithms we do not care about comparison, addition, or subtraction of numbers -- we assume they run in time $O(1)$. For example, we assume this when we say that comparison-based sorting ...
7k views

What does the “Lambda” in “Lambda calculus” stand for?

I've been reading about Lambda calculus recently but strangely I can't find an explanation for why it is called "Lambda" or where the expression comes from. Can anyone explain the origins of the term?...
96k views

How to prove that a language is not regular?

We learned about the class of regular languages $\mathrm{REG}$. It is characterised by any one concept among regular expressions, finite automata and left-linear grammars, so it is easy to show that a ...
10k views

What properties of a programming language make compilation impossible?

Question: "Certain properties of a programming language may require that the only way to get the code written in it be executed is by interpretation. In other words, compilation to a native machine ...
24k views

(When) is hash table lookup O(1)?

It is often said that hash table lookup operates in constant time: you compute the hash value, which gives you an index for an array lookup. Yet this ignores collisions; in the worst case, every item ...
11k views

Is there any concrete relation between Gödel's incompleteness theorem, the halting problem and universal Turing machines?

I've always thought vaguely that the answer to the above question was affirmative along the following lines. Gödel's incompleteness theorem and the undecidability of the halting problem both being ...
15k views

Why is addition as fast as bit-wise operations in modern processors?

I know that bit-wise operations are so fast on modern processors, because they can operate on 32 or 64 bits on parallel, so bit-wise operations take only one clock cycle. However addition is a complex ...
13k views

What important/crucial real-world applications use blockchain?

As part of some blockchain-related research I am currently undertaking, the notion of using blockchains for a variety of real-world applications are thrown about loosely. Therefore, I propose the ...
16k views

What is the novelty in MapReduce?

A few years ago, MapReduce was hailed as revolution of distributed programming. There have also been critics but by and large there was an enthusiastic hype. It even got patented! [1] The name is ...
14k views

Why is the Turing Machine a popular model of computation?

I am a CS undergraduate. I understand how Turing came up with his abstract machine (modeling a person doing a computation), but it seems to me to be an awkward, inelegant abstraction. Why do we ...
21k views

How do computers keep track of time?

How are computers able to tell the correct time and date every time? Whenever I close the computer (shut it down) all connections and processes inside stop. How is it that when I open the computer ...
15k views

Is legislation NP-complete?

I would like to know if there has been any work relating legal code to complexity. In particular, suppose we have the decision problem "Given this law book and this particular set of circumstances, is ...
17k views

How can I explain to my parents that I study programming languages?

I am currently finishing my MSc in computer science. I am interested in programming languages, especially in type systems. I got interested in research in this field and next semester I will start a ...
11k views

Why don't compilers automatically insert deallocations?

In languages like C, the programmer is expected to insert calls to free. Why doesn't the compiler do this automatically? Humans do it in a reasonable amount of time(ignoring bugs), so it is not ...
11k views

Can a dynamic language like Ruby/Python reach C/C++ like performance?

I wonder if it is possible to build compilers for dynamic languages like Ruby to have similar and comparable performance to C/C++? From what I understand about compilers, take Ruby for instance, ...
6k views

Are there any problems that get easier as they increase in size?

This may be a ridiculous question, but is it possible to have a problem that actually gets easier as the inputs grow in size? I doubt any practical problems are like this, but maybe we can invent a ...
7k views

What is coinduction?

I've heard of (structural) induction. It allows you to build up finite structures from smaller ones and gives you proof principles for reasoning about such structures. The idea is clear enough. But ...
7k views

Why do we not combine random number generators?

There are many applications where a pseudo random number generator is used. So people implement one that they think is great only to find later that it's flawed. Something like this happened with ...
4k views

Algorithmic intuition for logarithmic complexity

I believe I have a reasonable grasp of complexities like $\mathcal{O}(1)$, $\Theta(n)$ and $\Theta(n^2)$. In terms of a list, $\mathcal{O}(1)$ is a constant lookup, so it's just getting the head of ...
10k views

In-place algorithm for interleaving an array

You are given an array of $2n$ elements $$a_1, a_2, \dots, a_n, b_1, b_2, \dots b_n$$ The task is to interleave the array, using an in-place algorithm such that the resulting array looks like ...
10k views

Language theoretic comparison of LL and LR grammars

People often say that LR(k) parsers are more powerful than LL(k) parsers. These statements are vague most of the time; in particular, should we compare the classes for a fixed $k$ or the union over ...
9k views

Human computing power: Can humans decide the halting problem on Turing Machines?

We know the halting problem (on Turing Machines) is undecidable for Turing Machines. Is there some research into how well the human mind can deal with this problem, possibly aided by Turing Machines ...