Mathematical questions about Algorithms, including the analysis of algorithms, combinatorial algorithms, proofs of correctness, invariants, and semantic analyses. See also (computational-mathematics) and (computational-complexity).

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7
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1answer
134 views

Put a mouse to the last cell

We have (n=12) cells $C_1, C_2 ,\dots, C_{12}$ which are initially empty. At each step, we can do one of two operations: $\mathbf{P}$: Put only in the first cell $C_1$ 2 mice. $\mathbf{M}$: Move ...
7
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2answers
9k views

Fastest prime generating algorithm

What is the fastest known algorithm that generates all distinct prime numbers less than n? Is it faster than Sieve of Atkin?
7
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1answer
1k views

Why are Hornsat, 3sat and 2sat not equivalent?

I have been reading a little bit about complexity theory recently, and I'm having a bit of a stumbling block. The horn satisfiability problem is solvable in linear time, but the boolean satisfiability ...
7
votes
1answer
5k views

Split a set of numbers into 2 sets, where the sum of each set is as close to one another as possible

Given a set of numbers, I'd like to split this set into 2 sets, where the sum of each set is as close to equal as possible. How would I go about doing this in a programmatic way? Thanks in advance ...
7
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2answers
517 views

Accelerating Convergence of a Sequence

Suppose I had a monotonically increasing sequence $\{d_{n}\}$ which is also bounded above. The $d_{n}$'s satisfy a given recurrence, however computationally they tend very slowly to the limit. What ...
7
votes
1answer
461 views

Importance of Constructible functions

A function $f$ is called fully time-constructible if there exists a Turing machine $M$ which, given a string $1^n$ consisting of $n$ ones, stops after exactly $f(n)$ steps. Analogously, we can call a ...
7
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1answer
599 views

Fast algorithm for approximating Eigenvalue distribution of large sparse matrix

I am interested in the eigenvalue distribution of a huge $2^{16}$x$2^{16}$ Hermitian sparse matrix with spectrum contained in $[-1,1]$. That is I don't need to know all eigenvalues exactly, but rather ...
7
votes
3answers
286 views

Probability that a vertex in the spanning tree of an $N$ x $N$ grid graph is a leaf

Suppose we have an $N$ x $N$ grid graph $G(V,E)$ and we construct a spanning tree of this graph in the following way. Start with a set $S$ which contains only the vertex at the top left corner of the ...
7
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1answer
241 views

Tuning the birthday paradox

I have limited access to a collection $X_1,\ldots,X_m$ of sets of positive integers. Each $X_i$ is "moderately large" (a brief survey has found them to contain about $10^6$ elements in each set), but ...
7
votes
1answer
147 views

Bounds on the gaps in a variant of polylog-smooth numbers.

Sorry for the long intro. I think the explanation motivates the question and puts it in context. But if you want to skip the story, then just move on to the grey boxes; they should contain enough ...
7
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2answers
623 views

Buckets of Balls, Will one fill if I add another Ball?

I was refereed here by stackoverflow.com. With some searching I found this: another balls and bins question, but its not quite what I am looking for. Rather the inverse. IE the expected number of ...
7
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1answer
169 views

Accelerating approximations for arccos

I have recently built a method to accelerate drastically the accuracy of the following approximation of $\arccos(x)$ : $f_n(x)=2^n\sqrt{2-2g^{n-1}(x)}$ where $g(x)=\frac{1}2\sqrt{2+2x}$ and ...
7
votes
3answers
2k views

Calculating Non-Integer Exponent

I just wanted to directly calculate the value of the number $2^{3.1}$ as I was wondering how a computer would do it. I've done some higher mathematics, but I'm very unsure of what I would do to solve ...
7
votes
2answers
224 views

Who is the oldest? (in a non-revealing way)

How can a group of people figure out who is the oldest, without revealing any other information? Revealing all the ages to a trusted third party is not allowed. Preferably I'm looking for solutions ...
7
votes
2answers
210 views

Density of black cells in rule 110

Is there a way to compute the limit of the ratio (number of black cells)/(number of white cells), in the rule 110 or rule 30 automaton? With initial state = 1 black cell. Simulation of first 120000 ...
7
votes
1answer
88 views

Find the diameter of a set faster than $O(|M|^2)$

$M$ is a finite set of points in a metric space. I want to calculate the diameter of the set, i.e. the greatest distance between two points. Is there a smarter way to do this than to calculate the ...
7
votes
1answer
617 views

The Average Running Time Of Euclid Algorithm?

What is the average running time of Euclid Algorithm with respect to all possible input pairs $(m,n)$ such that $\gcd(m,n) = d$? It seems very hard to deduce from the recurrence $T(m,n) = T(n, m ...
7
votes
1answer
201 views

How to study results of Diophantine equation?

I finally managed to learn a bit of number theory and Diophantine equation(with the help of Arturo Magidin's great answer in what type of math is this?). But I'm wondering what's the next step after? ...
7
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0answers
467 views

Generating a stochastic matrix with a given second dominant eigenvalue

I need a procedure (iterative or otherwise) that, given a positive integer $N$ and a (possibly complex) number $\lambda$ such that $0 < \vert \lambda \vert < 1$, will be able to generate an $N ...
7
votes
1answer
110 views

Quickest way to solve a matrix one step at a time.

I have a $14\times14$ matrix with a possibility of six states in each position The matrix is random each time. An example matrix would be: $$ \begin{pmatrix} ...
7
votes
0answers
392 views

Problem with an algorithm to $3$-colour the edges of cubic graphs

I'm currently trying to implement an algorithm to $3$-colour the edges of cubic graphs. (I want to do this with Matlab's Symbolic toolbox). After restricting to planar graphs to ensure the existence ...
7
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0answers
234 views

What’s the best way to cut an apple?

Take the apple in one hand, and the knife in the other. In the first cut, the apple is divided in two pieces: a small one that drops into the plate and a big one that is still hold with the hand. This ...
7
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0answers
318 views

Does this calculation have a name, or a generic formulation?

Background I would appreciate help in identifying / explaining this operation: To calculate each of the $n$ values of $f(\Phi)$: sample from the distribution of each of $i$ parameters, $\phi_i$ ...
7
votes
1answer
242 views

Factoring some integer in the given interval

Let N be a positive integer. Is there an efficient (i.e. probabilistic polynomial time) algorithm which, on input a sufficiently large N, outputs the full factorization of some integer in the interval ...
6
votes
6answers
4k views

Calculate $\pi$ to an accuracy of 5 decimal places?

In this message at point 18 I saw following programming question: Given that $\pi$ can be estimated using the function $4(1 – 1/3 + 1/5 – 1/7 + \ldots)$ with more terms giving greater accuracy, ...
6
votes
5answers
217 views

How can I find the value of $a^n+b^n$, given the value of $a+b$, $ab$, and $n$?

I have been given the value of $a+b$ , $ab$ and $n$. I've to calculate the value of $a^n+b^n$. How can I do it? I would like to find out a general solution. Because the value of $n$ , $a+b$ and $ab$ ...
6
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5answers
330 views

pandigital rational approximations to the golden ratio and the base of the natural logarithm

Steven Stadnicki suggested in a comment that I post the following as a question. The golden ration $\phi$ is given by $$\phi = \frac{1+\sqrt{5}}{2} \approx 1.618033988.$$ A rational approximation is ...
6
votes
2answers
3k views

Minimum distance of a binary linear code

I need to find parameters $n$, $k$ and $d$ of a binary linear code from its Generator Matrix. How can I find parameter $d$ efficiently? I know the method that compute all the codewords and take ...
6
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4answers
2k views

calculating n choose k mod one million

I am working on a programming problem where I need to calculate 'n choose k'. I am using the relation formula $$ {n\choose k} = {n\choose k-1} \frac{n-k+1}{k} $$ so I don't have to calculate huge ...
6
votes
3answers
3k views

What is the most efficient algorithm to find the closest prime less than a given number $n$

Problem Given a number n, $2 \leq n \leq 2^{63}$. $n$ could be prime itself. Find the a prime $p$ that is closet to $n$. Using the fact that for all prime $p$, $p > 2$, $p$ is odd and $p$ is of ...
6
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10answers
11k views

Is there an algorithm to find all subsets of a set?

I'm trying to find a way to find all subsets of a set. Is there an algorithm to calculate this?
6
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2answers
2k views

Lights Out Variant: Flipping the whole row and column.

So I found this puzzle similar to Lights Out, if any of you have ever played that. Basically the puzzle works in a grid of lights like so: 1 0 0 00 0 0 00 1 0 0 0 0 1 0 When you selected a ...
6
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2answers
744 views

Algorithm fot planarity test in graphs

I am implementing a graph library and I want to include some basic graph algorithms in it. I have read about planar graphs and I decided to include in my library a function that checks if a graph is ...
6
votes
2answers
5k views

Which is asymptotically larger: $\lg(\lg^* n)$ or $ \lg^*(\lg n)$?

This definition is extracted from "Introduction to Algorithm, 2nd Edition". The iterated logarithm function We use the notation $\lg^* n$ (read "log star of $n$") to denote the iterated ...
6
votes
5answers
303 views

Fibonacci-like sequence

Today I have to deal with something which reminds Fibonacci sequence. Let's say I have a certain number k, which is n-th number of certain sequence. This sequence however is created by recursive ...
6
votes
4answers
227 views

Recursive Sequence Tree Problem (Original Research in the Field of Comp. Sci)

This question appears also in http://cstheory.stackexchange.com/questions/17953/recursive-sequence-tree-problem-original-research-in-the-field-of-comp-sci. I was told that cross-posting in this ...
6
votes
2answers
739 views

Computing nth term of fibonacci-like sequence for large n

Sum up to nth term of fibonacci sequence for very large n can be calculated in O($\log n$) time using the following approach: $$A = \begin{bmatrix} 1&1 \\\\1&0\end{bmatrix}^n$$ ...
6
votes
3answers
3k views

“Closest pair of points” algorithm

I'm having a problem understanding why I just have to consider the next 7 points in the Closest pair of points - algorithm. Can someone explain it in greater detail?
6
votes
4answers
20k views

Algorithms for Finding the Prime Factorization of an Integer

As practice, I am currently writing a program that takes a given integer $n$ as input, and then finds the (unique) prime factorization of $n$, provided $n$ is composite. My question is about ...
6
votes
4answers
542 views

Algorithm for constructing primes

Are there any good algorithms which can be used to construct a prime greater than $n$, for arbitrary $n$? There are some brute force approaches: for example, factoring $n!+1$. However, I'm looking ...
6
votes
1answer
2k views

Why is sorting pancakes NP hard?

An article posted yesterday (http://www.i-programmer.info/news/112-theory/3280-pancake-flipping-is-hard-np-hard.html) references a new study released on Arxiv (http://arxiv.org/abs/1111.0434v1) with ...
6
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2answers
1k views

Subset sum problem is NP-complete?

If I know correctly, subset sum problem is NP-complete. Here you have an array of n integers and you are given a target sum t, you have to return the numbers from the array which can sum up to the ...
6
votes
2answers
8k views

Continuous coloring of a Mandelbrot fractal

I've recently started making a small fractal app in Javascript using the famous Mandelbrot bulb $(z = z^2 + c)$. I've been trying to find the best method of coloring the points on the complex plane, ...
6
votes
4answers
265 views

What is the number of full binary trees of height less than $h$

Given a integer $h$ What is $N(h)$ the number of full binary trees of height less than $h$? For example $N(0)=1,N(1)=2,N(2)=5, N(3)=21$(As pointed by TravisJ in his partial answer) I can't ...
6
votes
3answers
737 views

FFT with powers of 3

Classic Fast Fourier Transfrom (FFT) works fine, when $n$ is power of 2. How to generalize FFT procedure when $n$ is power of 3? Is it possible to easily modify the algorithm and preserve its ...
6
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1answer
4k views

XOR properties in set of numbers

Say I have n positive numbers A1,A2...An and Ak is minimum among them. And d is a number ...
6
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4answers
2k views

The longest sum of consecutive primes that add to a prime less than 1,000,000

In Project Euler problem $50,$ the goal is to find the longest sum of consecutive primes that add to a prime less than $1,000,000. $ I have an efficient algorithm to generate a set of primes ...
6
votes
3answers
792 views

A balanced latin rectangle (more rows than columns)

In psychology we sometimes use balanced latin squares for the order of our tests to counterbalance first-order carry-over effects (fatigue, learning, etc.) . For our current study we want to pretest ...
6
votes
3answers
1k views

Inverse of symmetric matrix $M = A A^\top$

I have a matrix, generated by the product of a non-square matrix with its own transpose: $$M = A A^\top.$$ I need the inverse of $M$, assuming $\det(M) \neq 0$. Given the nature of the matrix $M$, ...
6
votes
1answer
234 views

Efficient algorithm for finding how many times a point is inside the triangles formed by given points

Given n 2D points and a special point p, what would be the best way to find how many times p is inside among those $^nC_3$ triangles formed by the n points.