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386 votes
12 answers
348k 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 $O(n \log n)$ ...
Janoma's user avatar
  • 5,545
329 votes
7 answers
159k 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 ...
Mirrana's user avatar
  • 4,309
218 votes
32 answers
53k 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 ...
user56834's user avatar
  • 3,882
195 votes
3 answers
26k 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 ...
Raphael's user avatar
  • 72.5k
194 votes
10 answers
55k 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, ...
StrugglingProgrammer's user avatar
182 votes
13 answers
67k 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 ...
Brent's user avatar
  • 2,553
146 votes
3 answers
20k views

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

We were given the following exercise. Let $\qquad \displaystyle f(n) = \begin{cases} 1 & 0^n \text{ occurs in the decimal representation of } \pi \\ 0 & \text{else}\end{cases}$ ...
Raphael's user avatar
  • 72.5k
145 votes
12 answers
43k 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 ...
Suresh's user avatar
  • 5,387
141 votes
5 answers
206k 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 ...
Raphael's user avatar
  • 72.5k
137 votes
3 answers
48k 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 ...
Nikunj Banka's user avatar
  • 1,545
137 votes
7 answers
41k 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 ...
user avatar
132 votes
14 answers
21k 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, ...
Ari's user avatar
  • 1,417
128 votes
15 answers
16k 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 ...
D.W.'s user avatar
  • 160k
128 votes
5 answers
153k 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?
Piper's user avatar
  • 1,769
127 votes
6 answers
16k 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. ...
Ken Li's user avatar
  • 3,078
111 votes
9 answers
107k 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 ...
Victor's user avatar
  • 1,289
110 votes
2 answers
81k views

Quicksort Partitioning: Hoare vs. Lomuto

There are two quicksort partition methods mentioned in Cormen: (the argument A is the array, and [p, r] is the range, inclusive,...
Robert S. Barnes's user avatar
107 votes
7 answers
88k views

Why is it best to use a prime number as a mod in a hashing function?

If I have a list of key values from 1 to 100 and I want to organize them in an array of 11 buckets, I've been taught to form a mod function $$ H = k \bmod \ 11$$ Now all the values will be placed ...
CodyBugstein's user avatar
  • 2,957
106 votes
5 answers
18k 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....
Raphael's user avatar
  • 72.5k
103 votes
7 answers
23k views

Why is deep learning hyped despite bad VC dimension?

The Vapnik–Chervonenkis (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. ...
yters's user avatar
  • 1,427
101 votes
5 answers
10k 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 ...
shole's user avatar
  • 1,200
101 votes
3 answers
35k 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 ...
Miles's user avatar
  • 1,323
99 votes
5 answers
106k 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....
Raphael's user avatar
  • 72.5k
98 votes
10 answers
181k 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 ...
Raphael's user avatar
  • 72.5k
97 votes
11 answers
29k 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 ...
Raphael's user avatar
  • 72.5k
96 votes
13 answers
26k views

What are GPUs bad at?

I understand that GPUs are generally used to do LOTS of calculations in parallel. I understand why we would want to parallelize processes in order to speed things up. However, GPUs aren't always ...
ChocolateOverflow's user avatar
94 votes
7 answers
24k 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 ...
evil_potato's user avatar
  • 1,372
93 votes
6 answers
18k 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 ...
Marc van Leeuwen's user avatar
90 votes
11 answers
27k 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 ...
JFFIGK's user avatar
  • 1,017
87 votes
6 answers
31k views

Formal program verification in practice

As a software engineer, I write a lot of code for industrial products. Relatively complicated stuff with classes, threads, some design efforts, but also some compromises for performance. I do a lot of ...
Frank's user avatar
  • 1,642
86 votes
8 answers
60k views

Graph searching: Breadth-first vs. depth-first

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 ...
malexmave's user avatar
  • 985
86 votes
6 answers
22k 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 ...
user avatar
80 votes
3 answers
51k 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 ...
Soham's user avatar
  • 937
80 votes
3 answers
55k views

Express boolean logic operations in zero-one integer linear programming (ILP)

I have an integer linear program (ILP) with some variables $x_i$ that are intended to represent boolean values. The $x_i$'s are constrained to be integers and to hold either 0 or 1 ($0 \le x_i \le 1$)...
D.W.'s user avatar
  • 160k
79 votes
2 answers
8k 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?...
Klangen's user avatar
  • 1,100
79 votes
2 answers
13k 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 ...
Dave Clarke's user avatar
  • 20.2k
78 votes
9 answers
23k 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 ...
Teodor Dyakov's user avatar
78 votes
8 answers
67k views

Floyd's Cycle detection algorithm | Determining the starting point of cycle

I am seeking help understanding Floyd's cycle detection algorithm. I have gone through the explanation on wikipedia (http://en.wikipedia.org/wiki/Cycle_detection#Tortoise_and_hare) I can see how the ...
Anurag Kapur's user avatar
78 votes
5 answers
15k 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 ...
rshah's user avatar
  • 863
77 votes
6 answers
17k 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 ...
Alex's user avatar
  • 903
77 votes
4 answers
17k views

Dependent types vs refinement types

Could somebody explain the difference between dependent types and refinement types? As I understand it, a refinement type contains all values of a type fulfilling a predicate. Is there a feature of ...
Joey Eremondi's user avatar
76 votes
4 answers
73k views

What is tail recursion?

I know the general concept of recursion. I came across the concept of tail recursion while studying the quicksort algorithm. In this video of quick sort algorithm from MIT at 18:30 seconds the ...
Geek's user avatar
  • 1,191
76 votes
9 answers
11k 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 ...
Anderson Nascimento Nunes's user avatar
76 votes
1 answer
16k 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 ...
Raphael's user avatar
  • 72.5k
74 votes
4 answers
34k 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 ...
Gilles 'SO- stop being evil''s user avatar
73 votes
4 answers
119k views

Find median of unsorted array in $O(n)$ time

To find the median of an unsorted array, we can make a min-heap in $O(n\log n)$ time for $n$ elements, and then we can extract one by one $n/2$ elements to get the median. But this approach would take ...
Luv's user avatar
  • 851
72 votes
10 answers
117k views

Minimum spanning tree vs Shortest path

What is the difference between minimum spanning tree algorithm and a shortest path algorithm? In my data structures class we covered two minimum spanning tree algorithms (Prim's and Kruskal's) and ...
flashburn's user avatar
  • 1,223
72 votes
4 answers
17k 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 ...
Raphael's user avatar
  • 72.5k
71 votes
6 answers
21k views

Are there minimum criteria for a programming language being Turing complete?

Does there exist a set of programming language constructs in a programming language in order for it to be considered Turing Complete? From what I can tell from wikipedia, the language needs to ...
Khanzor's user avatar
  • 1,451
69 votes
10 answers
12k 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, ...
Ichiro's user avatar
  • 815

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