All Questions

153
votes
3answers
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 ...
82
votes
10answers
12k 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 ...
35
votes
6answers
7k views

Sorting functions by asymptotic growth

Assume I have a list of functions, for example $\qquad n^{\log \log(n)}, 2^n, n!, n^3, n \ln n, \dots$ How do I sort them asymptotically, i.e. after the relation defined by $\qquad f \leq_O g \...
72
votes
7answers
92k 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 ...
234
votes
6answers
102k 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 ...
87
votes
5answers
55k 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....
46
votes
8answers
60k views

How to prove a language is regular?

There are many methods to prove that a language is not regular, but what do I need to do to prove that some language is regular? For instance, if I am given that $L$ is regular, how can I prove that ...
89
votes
3answers
21k 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 ...
25
votes
2answers
19k views

How to prove that a language is context-free?

There are many techniques to prove that a language is not context-free, but how do I prove that a language is context-free? What techniques are there to prove this? Obviously, one way is to exhibit ...
106
votes
4answers
140k 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 ...
39
votes
4answers
10k views

What are common techniques for reducing problems to each other?

In computability and complexity theory (and maybe other fields), reductions are ubiquitous. There are many kinds, but the principle remains the same: show that one problem $L_1$ is at least as hard as ...
21
votes
1answer
4k views

How to show that L = L(G)?

Specifying formal languages by giving formal grammars is a frequent task: we need grammars not only to describe languages, but also to parse them, or even do proper science. In all cases, it is ...
41
votes
2answers
12k views

How to show that a function is not computable?

I know that there exist a Turing Machine, if a function is computable. Then how to show that the function is not computable or there aren't any Turing Machine for that. Is there anything like a ...
50
votes
3answers
30k 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 ($0 \le x_i \le 1$)...
37
votes
5answers
90k views

How to come up with the runtime of algorithms? [duplicate]

I've not gone much deep into CS. So, please forgive me if the question is not good or out of scope for this site. I've seen in many sites and books, the big-O notations like $O(n)$ which tell the ...
36
votes
1answer
6k views

What is the difference between an algorithm, a language and a problem?

It seems that on this site, people will often correct others for confusing "algorithms" and "problems." What are the difference between these? How do I know when I should be considering algorithms and ...
128
votes
3answers
13k views

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

We were given the following exercise. Let $\qquad \displaystyle f(n) = \begin{cases} 1 & 0^n \text{ occurs in the decimal representation of } \pi \\ 0 & \text{else}\end{cases}$ ...
72
votes
6answers
13k 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 ...
42
votes
6answers
4k views

Dealing with intractability: NP-complete problems

Assume that I am a programmer and I have an NP-complete problem that I need to solve it. What methods are available to deal with NPC problems? Is there a survey or something similar on this topic?
30
votes
4answers
34k views

How do O and Ω relate to worst and best case?

Today we discussed in a lecture a very simple algorithm for finding an element in a sorted array using binary search. We were asked to determine its asymptotic complexity for an array of $n$ elements. ...
42
votes
1answer
9k views

Show that { xy ∣ |x| = |y|, x ≠ y } is context-free

I remember coming across the following question about a language that supposedly is context-free, but I was unable to find a proof of the fact. Have I perhaps misremembered the question? Anyway, here'...
23
votes
4answers
1k views

How to fool the plot inspection heuristic?

Over here, Dave Clarke proposed that in order to compare asymptotic growth you should plot the functions at hand. As a theoretically inclined computer scientist, I call(ed) this vodoo as a plot is ...
26
votes
2answers
5k views

How do I construct reductions between problems to prove a problem is NP-complete?

I am taking a complexity course and I am having trouble with coming up with reductions between NPC problems. How can I find reductions between problems? Is there a general trick that I can use? How ...
50
votes
6answers
9k 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 ...
27
votes
2answers
4k views

Why are the total functions not enumerable?

We learned about the concept of enumerations of functions. In practice, they correspond to programming languages. In a passing remark, the professor mentioned that the class of all total functions (i....
137
votes
11answers
45k 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 ...
24
votes
2answers
17k views

How to prove greedy algorithm is correct

I have a greedy algorithm that I suspect might be correct, but I'm not sure. How do I check whether it is correct? What are the techniques to use for proving a greedy algorithm correct? Are there ...
43
votes
2answers
2k views

What is the meaning of $O(m+n)$?

This is a basic question, but I'm thinking that $O(m+n)$ is the same as $O(\max(m,n))$, since the larger term should dominate as we go to infinity? Also, that would be different from $O(\min(m,n))$. ...
70
votes
4answers
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 ...
39
votes
7answers
3k views

Explaining the relevance of asymptotic complexity of algorithms to practice of designing algorithms

In algorithms and complexity we focus on the asymptotic complexity of algorithms, i.e. the amount of resources an algorithm uses as the size of the input goes to infinity. In practice, what is ...
36
votes
3answers
2k views

Decision problems vs “real” problems that aren't yes-or-no

I read in many places that some problems are difficult to approximate (it is NP-hard to approximate them). But approximation is not a decision problem: the answer is a real number and not Yes or No. ...
23
votes
2answers
7k views

Optimization version of decision problems

It is known that each optimization/search problem has an equivalent decision problem. For example the shortest path problem optimization/search version: Given an undirected unweighted graph $G ...
33
votes
2answers
6k views

Perplexed by Rice's theorem

Summary: According to Rice's theorem, everything is impossible. And yet, I do this supposedly impossible stuff all the time! Of course, Rice's theorem doesn't simply say "everything is impossible". ...
289
votes
11answers
271k 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 $O(n \log n)$ ...
48
votes
2answers
6k views

Are there subexponential-time algorithms for NP-complete problems?

Are there NP-complete problems which have proven subexponential-time algorithms? I am asking for the general case inputs, I am not talking about tractable special cases here. By sub-exponential, I ...
29
votes
6answers
30k views

Why Do Computers Use the Binary Number System (0,1)?

Why Do Computers Use the Binary Number System (0,1)? Why don't they use Ternary Number System (0,1,2) or any other number system instead?
29
votes
6answers
8k views

Differences and relationships between randomized and nondeterministic algorithms?

What differences and relationships are between randomized algorithms and nondeterministic algorithms? From Wikipedia A randomized algorithm is an algorithm which employs a degree of randomness ...
20
votes
2answers
3k views

Encoding 1-out-of-n constraint for SAT solvers

I'm using a SAT solver to encode a problem, and as part of the SAT instance, I have boolean variables $x_1,x_2,\dots,x_n$ where it is intended that exactly one of these should be true and the rest ...
59
votes
1answer
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 ...
50
votes
4answers
4k views

Why polynomial time is called “efficient”?

Why in computer science any complexity which is at most polynomial is considered efficient? For any practical application(a), algorithms with complexity $n^{\log n}$ are way faster than algorithms ...
23
votes
4answers
18k views

How to prove that a grammar is unambiguous?

My problem is how can I prove that a grammar is unambiguous? I have the following grammar: $$S → statement ∣ \mbox{if } expression \mbox{ then } S ∣ \mbox{if } expression \mbox{ then } S \mbox{ else } ...
20
votes
5answers
8k views

Is O(mn) considered “linear” or “quadratic” growth?

If I have some function whose time complexity is O(mn), where m and n are the sizes of its two inputs, would we call its time complexity "linear" (since it's linear in both m and n) or "quadratic" (...
18
votes
8answers
2k views

Justification for neglecting constants in Big O

Many a times if the complexities are having constants such as 3n, we neglect this constant and say O(n) and not O(3n). I am unable to understand how can we neglect such three fold change? Some thing ...
95
votes
5answers
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....
79
votes
1answer
54k views

Quicksort Partitioning: Hoare vs. Lomuto

There are two quicksort partition methods mentioned in Cormen: ...
26
votes
7answers
5k views

What are the simplest examples of programs that we do not know whether they terminate?

The halting problem states there is no algorithm that will determine if a given program halts. As a consequence, there should be programs about which we can not tell whether they terminate or not. ...
17
votes
1answer
1k views

What classes of data structures can be made persistent?

Persistent data structures are immutable data structures. Operations on them return a new "copy" of the data structure, but altered by the operation; the old data structure remains unchanged though. ...
43
votes
2answers
1k views

Determining capabilities of a min-heap (or other exotic) state machines

See the end of this post for some clarification on the definition(s) of min-heap automata. One can imagine using a variety of data structures for storing information for use by state machines. For ...
19
votes
2answers
7k views

Changing variables in recurrence relations

Currently, I am self-studying Intro to Algorithms (CLRS) and there is one particular method they outline in the book to solve recurrence relations. The following method can be illustrated with this ...
19
votes
1answer
539 views

Rigorous proof for validity of assumption $n=b^k$ when using the Master theorem

The Master theorem is a beautiful tool for solving certain kinds of recurrences. However, we often gloss over an integral part when applying it. For example, during the analysis of Mergesort we ...

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