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38
votes
5answers
117k 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 ...
23
votes
2answers
1k views

Is there a "natural" undecidable language?

Is there any "natural" language which is undecidable? by "natural" I mean a language defined directly by properties of strings, and not via machines and their equivalent. In other words, if the ...
20
votes
5answers
575 views

Efficient compression of unlabeled trees

Consider unlabeled, rooted binary trees. We can compress such trees: whenever there are pointers to subtrees $T$ and $T'$ with $T = T'$ (interpreting $=$ as structural equality), we store (w.l.o.g.) $...
16
votes
3answers
1k views

Efficient encoding of sudoku puzzles

Specifying any arbitrary 9x9 grid requires giving the position and value of each square. A naïve encoding for this might give 81 (x, y, value) triplets, requiring 4 bits for each x, y, and value (1-9 =...
16
votes
3answers
3k views

What are the possible sets of word lengths in a regular language?

Given a language $L$, define the length set of $L$ as the set of lengths of words in $L$: $$\mathrm{LS}(L) = \{|u| \mid u \in L \}$$ Which sets of integers can be the length set of a regular language?...
10
votes
3answers
6k views

Minimum number of clues to fully specify any sudoku?

We know from this paper that there does not exist a puzzle that can be solved starting with 16 or fewer clues, but it implies that there does exist a puzzle that can be solved from 17 clues. Can all ...
11
votes
5answers
3k views

A sufficient and necessary condition about regularity of a language

Which of the following statements is correct? sufficient and necessary conditions about regularity of a language exist but not discovered yet. There's no sufficient and necessary ...
20
votes
1answer
524 views

Ratio of decidable problems

Consider decision problems stated in some “reasonable” formal language. Let's say formulae in higher-order Peano arithmetic with one free variable as a frame of reference, but I'm equally interested ...
23
votes
1answer
1k views

Is there an equivalent of van Emde Boas trees for ropes?

Someone I know is planning on implementing a text editor in the near future, which prompted me to think about what kind of data structures are fast for a text editor. The most used structures are ...
4
votes
1answer
571 views

How to determine if a database schema violates one of the less known normal forms?

In database normalization, 1NF (no multivalued attributes), 2NF (all non-PK attributes depending only on PK attributes) and 3NF (all non-PK attributes depending on all of the PK attributes) are widely ...
10
votes
1answer
279 views

Analysis of and references for Koch-snowflake-like (and other exotic) network topologies

In computer networking and high-performance cluster computer design, network topology refers to the design of the way in which nodes are connected by links to form a communication network. Common ...
20
votes
2answers
4k views

Type-checking algorithms

I am starting a personal bibliographic research on type-checking algorithms and want some tips. What are the most commonly used type-checking algorithms, strategies and general techniques? I am ...
20
votes
1answer
428 views

Equivalence of Kolmogorov-Complexity definitions

There are many ways to define the Kolmogorov-Complexity, and usually, all these definitions they are equivalent up to an additive constant. That is if $K_1$ and $K_2$ are kolmogorov complexity ...
17
votes
4answers
1k views

Is this finite graph problem decidable? What factors make a problem decidable?

I want to know if the following problem is decidable and how to find out. Every problem I see I can say "yes" or "no" to it, so are most problems and algorithms decidable except a few (which is ...
18
votes
1answer
251 views

Natural occurrences of monads that make use of the category-theoretical framework

Today, a talk by Henning Kerstan ("Trace Semantics for Probabilistic Transition Systems") confronted me with category theory for the first time. He has built a theoretical framework for describing ...
20
votes
3answers
327 views

Extension of SQL capturing $\mathsf{P}$

According to Immerman, the complexity class associated with SQL queries is exactly the class of safe queries in $\mathsf{Q(FO(COUNT))}$ (first-order queries plus counting operator): SQL captures safe ...
7
votes
1answer
440 views

Analyzing load balancing schemes to minimize overall execution time

Suppose that a certain parallel application uses a master-slave design to process a large number of workloads. Each workload takes some number of cycles to complete; the number of cycles any given ...
26
votes
3answers
2k views

What are the conditions for a NFA for its equivalent DFA to be maximal in size?

We know that DFAs are equivalent to NFAs in expressiveness power; there is also a known algorithm for converting NFAs to DFAs (unfortunately I do now know the inventor of that algorithm), which in ...
29
votes
3answers
2k views

Do subqueries add expressive power to SQL queries?

Does SQL need subqueries? Imagine a sufficiently generalized implementation of the structured query language for relation databases. Since the structure of the canonical SQL ...
52
votes
2answers
7k views

How to define quantum Turing machines?

In quantum computation, what is the equivalent model of a Turing machine? It is quite clear to me how quantum circuits can be constructed out of quantum gates, but how can we define a quantum Turing ...
8
votes
1answer
916 views

What Is The Complexity of Implementing a Particle Filter?

In a video discussing the merits of particle filters for localization, it was implied that there is some ambiguity about the complexity cost of particle filter implementations. Is this correct? ...
23
votes
1answer
1k views

How fundamental are matroids and greedoids in algorithm design?

Initially, matroids were introduced to generalize the notions of linear independence of a collection of subsets $E$ over some ground set $I$. Certain problems that contain this structure permit greedy ...
34
votes
2answers
3k views

Are there improvements on Dana Angluin's algorithm for learning regular sets

In her 1987 seminal paper Dana Angluin presents a polynomial time algorithm for learning a DFA from membership queries and theory queries (counterexamples to a proposed DFA). She shows that if you are ...
44
votes
2answers
2k 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 ...
39
votes
2answers
6k views

Are there inherently ambiguous and deterministic context-free languages?

Let us call a context-free language deterministic if and only if it can be accepted by a deterministic push-down automaton, and nondeterministic otherwise. Let us call a context-free language ...
32
votes
2answers
1k views

Equivalence of Büchi automata and linear $\mu$-calculus

It's a known fact that every LTL formula can be expressed by a Büchi $\omega$-automaton. But, apparently, Büchi automata are a more powerful, expressive model. I've heard somewhere that Büchi automata ...
10
votes
3answers
4k views

Algorithm to test whether a binary tree is a search tree and count complete branches

I need to create a recursive algorithm to see if a binary tree is a binary search tree as well as count how many complete branches are there (a parent node with both left and right children nodes) ...
22
votes
2answers
3k views

Recursive definitions over an inductive type with nested components

Consider an inductive type which has some recursive occurrences in a nested, but strictly positive location. For example, trees with finite branching with nodes using a generic list data structure to ...
39
votes
6answers
9k views

Clock synchronization in a network with asymmetric delays

Assume a computer has a precise clock which is not initialized. That is, the time on the computer's clock is the real time plus some constant offset. The computer has a network connection and we want ...
23
votes
1answer
368 views

Is there any nongeneral CFG parsing algorithm that recognises EPAL?

EPAL, the language of even palindromes, is defined as the language generated by the following unambiguous context-free grammar: $S \rightarrow a a$ $S \rightarrow b b$ $S \rightarrow a S a$ $S \...
10
votes
0answers
128 views

Applying the graph mining algorithm Leap Search in an unlabeled setting

I am reading Mining Significant Graph Patterns by Leap Search (Yan et al., 2008), and I am unclear on how their technique translates to the unlabeled setting, since $p$ and $q$ (the frequency ...
22
votes
1answer
1k views

Natural candidates for the hierarchy inside NPI

Let's assume that $\mathsf{P} \neq \mathsf{NP}$. $\mathsf{NPI}$ is the class of problems in $\mathsf{NP}$ which are neither in $\mathsf{P}$ nor in $\mathsf{NP}$-hard. You can find a list of problems ...
15
votes
2answers
3k views

Time spent on requirement and its effect on project success and development time

Is there any evidence suggesting that time spent on writing up, or thinking about the requirements will have any effect on the development time? Study done by Standish (1995) suggests that incomplete ...
24
votes
2answers
325 views

Is Smoothed Analysis used outside academia?

Did the smoothed analysis find its way into main stream analysis of algorithms? Is it common for algorithm designers to apply smoothed analysis to their algorithms?
11
votes
2answers
621 views

Which kind of branch prediction is more important?

I have observed that there are two different types of states in branch prediction. In superscalar execution, where the branch prediction is very important, and it is mainly in execution delay rather ...
13
votes
2answers
5k views

Random Sudoku generator

I want to generate a completely random Sudoku. Define a Sudoku grid as a $9\times9$ grid of integers between $1$ and $9$ where some elements can be omitted. A grid is a valid puzzle if there is a ...
29
votes
2answers
719 views

Characterization of lambda-terms that have union types

Many textbooks cover intersection types in the lambda-calculus. The typing rules for intersection can be defined as follows (on top of the simply typed lambda-calculus with subtyping): $$ \dfrac{\...
97
votes
3answers
31k 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 ...
70
votes
1answer
13k 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 ...
31
votes
1answer
2k views

Rice's theorem for non-semantic properties

Rice's theorem tell us that the only semantic properties of Turing Machines (i.e. the properties of the function computed by the machine) that we can decide are the two trivial properties (i.e. always ...
18
votes
3answers
2k views

Clever memory management with constant time operations?

Let's consider a memory segment (whose size can grow or shrink, like a file, when needed) on which you can perform two basic memory allocation operations involving fixed size blocks: allocation of ...
11
votes
4answers
26k views

Evaluating the average time complexity of a given bubblesort algorithm.

Considering this pseudo-code of a bubblesort: ...
34
votes
9answers
4k views

What is the significance of context-sensitive (Type 1) languages?

Seeing that in the Chomsky Hierarchy Type 3 languages can be recognised by a state machine with no external memory (i.e., a finite automaton), Type 2 by a state machine with a single stack (i.e. a ...
29
votes
7answers
13k views

Generating Combinations from a set of pairs without repetition of elements

I have a set of pairs. Each pair is of the form (x,y) such that x,y belong to integers from the range [0,n). So, if the n is 4, then I have the following pairs: <...
15
votes
2answers
307 views

Which method is preferred for storing large geometric objects in a quadtree?

When placing geometric objects in a quadtree (or octree), you can place objects that are larger than a single node in a few ways: Placing the object's reference in every leaf for which it is ...
19
votes
4answers
2k views

Does cooperative scheduling suspend processes when they perform an I/O operation?

Many operating systems references say that with cooperative (as opposed to preemptive) multitasking, a process keeps the CPU until it explicitly voluntarily suspends itself. If a running process ...
365
votes
12answers
334k 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 $O(n \log n)$ ...
19
votes
2answers
1k views

Does the 'difference' operation add expressiveness to a query language that already includes 'join'?

The set difference operator (e.g., EXCEPT in some SQL variants) is one of the many fundamental operators of relational algebra. However, there are some databases ...
10
votes
1answer
829 views

Algorithm to translate a deterministic Büchi automaton to LTL (when possible)

Linear temporal logic and deterministic Büchi automata are incomparable: DBA cannot express $FGa$, and LTL cannot express "at least each odd letter is 'a'". But sometimes it is interesting to know ...
12
votes
3answers
10k views

All NP problems reduce to NP-complete problems: so how can NP problems not be NP-complete?

My book states this If a decision problem B is in P and A reduces to B, then decision problem A is in P. A decision problem B is NP-complete if B is in NP and for every problem in A in ...

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