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

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0
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2answers
30 views

Traveling salesman problem (TSP): what is the Relation with number of vertices and length of the found route?

I know that there are many algorithms (exact or approximate) which implement the traveling salesman problem. I would like to know the relation between the number of the vertices (i.e., the places to ...
0
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0answers
22 views

Node potentials of minimum cost flow successive shortest path algorithm

I have a simple directed graph $G(V,E)$ that has a source $s$ and sink $t$. Each edge $e$ of $G$ has positive integer capacity $c(e)$ and positive integer cost $a(e)$. I am trying to find the minimum ...
3
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2answers
33 views

Decidability of predicate calculus with equality only

I read in some books that propositional calculus is decidable (e.g. with truth tables), and predicate calculus is not decidable (as proved by Church and Turing). Unfortunately, I do not exactly ...
0
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1answer
32 views

Sequence who converges to $\sqrt{a}$ for every $a\geq0$

If we have: $$x_n=\left\lbrace \begin{matrix} b\in\mathbb{R}\setminus \{0\} & ,n=1 \\ \dfrac{a+x_{n-1}^2}{2x_{n-1}} & , n>1\end{matrix} \right.$$ Then, is easy to prove that $(x_n)\to ...
1
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0answers
16 views

Prove that your algorithm is optimal by showing that no other algorithm can solve the problem faster in the worst case.

You are given 5 balls that look identical but one of them has a different weight from the others. You are given a balance scale to find the odd ball. Give an algorithm to solve this task that ...
5
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3answers
901 views

One number divisible by all prime factors of another?

Given two numbers $x$ and $y$, how to check whether $x$ is divisible by all prime factors of $y$ or not?, is there a way to do this without factoring $y$?.
2
votes
1answer
130 views

Aliens to the Moon

$N$ Aliens want to reach their Moon ($D$ meters away), but they can only put on each other, making a vertical chain. Every $Alien(i)$ has an height $X(i)$ and a lenght of their arms $Y(i)$. ...
2
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1answer
59 views

Solving $T(n) = 3T(n-1)$

How is the constant before the $T$ important to the result from $T(n)$ I know that \begin{equation*} T(n) = T(n-1) + 3 \Rightarrow \theta(n)~\text{and}~T(n) = T(n-1) + n \Rightarrow \theta(n^2) ...
0
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1answer
34 views

Multi-level bin-packing problem?

Let us consider we have B Bags and each Bag has some number of Items. The problem is to distribute Bags of Items uniformly across as minimum number of Bins as possible. However, there are two ...
3
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3answers
36 views

Prove that Euclid's algorithm computes the GCD of any pair of nonnegative integers

I've been struggling with a basic exercise involving Euclid's algorithm and mathematical induction. Given the following definition of the Euclid's algorithm (in Java): ...
1
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1answer
60 views

Struggling with difference between greedy and naive but optimal algorithms? (Graph theory)

I've been thinking about the following problem for quite a while and tried multiple solutions, but I'm having difficulty telling the difference between a greedy algorithm and an inefficient naive ...
0
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1answer
30 views

Real logarithm of a real matrix?

What is the real logarithm of \begin{equation} \begin{pmatrix} -1 & 1 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 1 \\ 0 & 0 & 0 & -1 \end{pmatrix}? ...
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0answers
26 views

Is the following algorithm for finding the minimum diameter spanning tree correct?

You are given an undirected and unweighted connected graph $G(V, E)$ for which you've been asked to find a spanning tree that has minimum diameter. I have an idea but I'm not sure if it's a ...
0
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0answers
79 views

completely stuck on this question about decidability

A decision problem X is as follows: $X = f\langle M \rangle\mid M$ is a program that accepts the empty string. Now, we define another decision problem Y as follows: $Y = f\langle P \rangle ...
4
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4answers
240 views

Show the probability that the sum of these numbers is odd is 1/2

Setting Let $S$ be a set of integers where at least one of the integers is odd. Suppose we pick a random subset $T$ of $S$ by including each element of $S$ independently with probability $1/2$, Show ...
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0answers
19 views

Looking for examples of path finding algorithms that use “metagraphs”

I don't think this is the right term (metagraphs), but it seems appropriate. I'm looking for any examples of path finding algorithms that take 2 graphs into account, G1 and G2, where a copy of G2 is ...
7
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1answer
157 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 ...
1
vote
1answer
25 views

Switching lights in a matrix

I'm interested in papers and articles on the following problems (not necessarily solutions). At least is there a name to these that I can lookup ? Say that $a_{ij} \in \{-1, +1\}$ for $1 \leq i, j ...
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2answers
22 views

Analysis of Algorithms - Big O Notation Equivalences - Limits

Please see below block question from review for test. True Or False? Justify Your answers A) is 2^(n+1) = O($2^n$) B) is 2^2n = O($2^n$) C) is log($n^2$) = O(logn) D) is ...
0
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1answer
28 views

Where does the root of this tree come from?

I am doing a practice question from Midterm Dynamic Programming The Problem : Consider a row of n numbers a1, ..., an. The numbers are all positive, and n is even. We play a game against an ...
1
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1answer
56 views

How to find worst case in chain matrix multiplication

The question we got was Determine a worst-case parenthesization of the matrix-chain product whose sequence of dimensions is (5, 2, 3, 10, 4, 6, 7, 8). what i dont understand is how do we determine ...
2
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1answer
39 views

Wouldn't this Greedy Algorithm achieve the highest possible of money in this situation?

I am doing a practice question from Midterm Dynamic Programming The Problem : Consider a row of n numbers a1, ..., an. The numbers are all positive, and n is even. We play a game against an ...
1
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0answers
28 views

people passing a bridge (a proof for a greedy algorithm)

the problem some people are passing a bridge . each one takes a different time to pass . assume the people are sorted by their passing time increasingly . these are the conditions of passing the ...
0
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1answer
33 views

Looking for algorithms capable of modifying graph structure

I realize this is a quite a general request. I'm just looking for examples of path searching algorithms for directed graphs which are capable of utilizing simple modifications (adding vertices, adding ...
2
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1answer
17 views

Finding regular expressions for which a given Turing machine halts and accepts, halts and rejects, and diverges.

Consider the Turing machine M = (Q,Σ,Γ,δ,q,F) F = {t} Q = {q,r,s,t,v,x} Σ = {a,b,c} Γ = {B,a,b,c} δ = [q,a,q,a,R] [q,b,q,b,R] [q,c,v,b,R] [q,B,r,B,L] [r,a,s,B,L] [r,b,s,B,L] [r,c,s,B,L] ...
4
votes
1answer
22 views

Find the “surface vertices” of a collection of points.

I am currently doing some experiments in order to simulate liquids. I have a collection of 3D points that interact with each other to form a body of water. I would like to form a mesh from these ...
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0answers
9 views

recurrence tree final step - binary search

Starting with the base case and recursive case run times as follows: 􏰀 t(N) = 1 , if N = 1 t(N)= 1+t(N/2) ,ifN > 1 At the end of my tree I have ...
0
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0answers
12 views

Examples of Algorithms capable of modifying graph structure?

I'm currently working on a problem where one is presented with 2 connected digraphs (call them G1 and G2), each with an associated set of logical constraints. Each vertex of each graph represents a ...
2
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0answers
53 views

Efficient computation of matrix determinant in finite ring

I am trying to implement generalization of Hill cipher. My idea is very simple: the size of key matrix should be arbitrary number not only three. All steps of this cipher is trivial except computation ...
1
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0answers
17 views

Learning Booth's Algorithm, I Can't Find the Issue on Final Result

I am practicing using Booth's Algorithm to multiply a positive number and a negative number (specifically the problem is $-12 \times 4$). I have included my attempt, but I can't find the issue. If ...
1
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2answers
29 views

Simple Adding and Subtracting algorithm to get a current amount

So this is my first post and its probably a really dumb question but I cannot get my head around a proper algorithm to handle the following situation. I have a dataset that contains employees who ...
1
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1answer
24 views

0/1 knapsack upper bound

I'm new to the 0/1 knapsack problem and I've ordered my nodes into profit/weight as: Knapsack max weight: 12 ...
0
votes
1answer
31 views

Find all nearest points

I have two sets: $$P = \{p_1, p_2, ..., p_n\}$$ $$Q = \{q_1, q_2, ..., q_m\}$$ For each $p_i$ point I need to find all nearest points in $Q$. I.e., $$p_i \rightarrow \{ q_{i_1}, q_{i_2}, ..., ...
0
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2answers
80 views

Minimum number of operations to make all of the strings (objects) the same

Let $A$ be an alphabet, $K$ and $N$ be natural numbers and $X$ be a list of $N$ strings over $A$, each one consisting of $K$ letters. You have one operation ($@f$): convert a string from $X$ to ...
3
votes
1answer
69 views

Algorithm to solve gridlock (traffic jam)

Is there an algorithm for solving a gridlock (traffic jam)? Given a gridlock situation, this algorithm should tell which moves each car should make, and in which order the cars should move, so that ...
0
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0answers
24 views

How to approximate a trigonometric to make less computation complexity

I having a trigonometric function such as $$ p_2(s) = \begin{cases} \frac {1}{(2 \pi)^2}(1-\cos (2 \pi s)), & \text{if $s \le1$ } \\ \frac {1}{2 }(s-1)^2, & \text{if $s >1$ } ...
1
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1answer
24 views

Origin of divergence in a divergent field (2D)

I have a field of measured vectors, see example of four vectors in image below. If there was no noise they would all point outward exactly from one "central point". i.e. there would be a circle whose ...
0
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3answers
34 views

Efficient algorithm for taking powers of binary numbers?

Is there any efficient algorithm for taking powers of binary numbers? Or even just squaring one? Thanks ahead of time.
1
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2answers
37 views

Problem implementing a QR factorization

I'm trying to write a Fortran subroutine to compute a QR factorization using the Householder method. To test my routine, I compute the factorization of the following matrix: $$ A = \begin{pmatrix} ...
0
votes
2answers
41 views

Writing a GCD of three numbers as a linear combination

i know how to find the $\gcd(5,11,2^{2015}-1)$. but i can't seem to find the linear combination. do you find it the same way we find the linear combination of two integers.
1
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1answer
30 views

How to set up this algorithm?

I'm doing a simple budget app, to make some calculations to improve private economy. I've made an option where you can enter the value of the money that are in the bank right now, and how many days ...
0
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0answers
25 views

How to use mobius-inversion to solve this problem?

Currently, I'm trying to solve this problem using mobius-inversion. the function f(d) means the number of (i, j, k) equals d, and function g(d) means the numbers that satisfying: d | (i, j, k). Then ...
1
vote
1answer
39 views

Given that Minimum Spanning Tree is NP-complete show that Hamiltonian Cycle is NP complete

So first of all I know finding MST is in P and is not NP complete. But I checked last year exams from my University and there is a problem: Given that Minimum Spanning Tree is NP-complete show that ...
0
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0answers
17 views

Implementing specific SVD algorithms

My goal is to learn to implement the two-sided Jacobi SVD, a method of SVD for bidiagonal matrices, and a method of SVD for tridiagonal matrices. Can anyone recommend a place to learn about these, or ...
1
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1answer
23 views

How to efficiently list prime with a very specific property

I noticed that my phone number 06 xx xx xx xx is a prime number. Ok that cool ... But if you had the country code (+33 for france), ...
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0answers
8 views

Is the mapping achieved by the discrete Frechet Distance the best achievable pair decomposition?

I have the following proble: There are 2 sequences $P$ and $Q$ composed by $M$ and $N$ points respectively. I want to calculate the discrete Frechet distance between them. This operation outputs also ...
0
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0answers
17 views

A suitable algorithm to change strings to zero strings

I found a programming exercise I couldn't solve. Namely, I have been given a long binary string and a fixed set of operations. Here, an operation means that it takes a substring and transforms its ...
0
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0answers
19 views

Regular Subresultant PRS from Euclidean PRS?

Is there any way to compute regular subresultant polynomial remainder sequence if we know the Euclidean polynomial remainder sequence of two univariate polynomials and vice versa?
0
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0answers
37 views

How many counts the square number?

For example I have a number is $55$. Can I check this for how many counts number's square create this number. For example I have $55$ So, I know this number total from 5 number, which are $$55 = ...
0
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1answer
22 views

Why is $X_m$ and $Y_m$ not included in the shaded region(where median can lie)?

This problem is from Algorithms, problem 2 The Problem Given two sorted list of numbers $X$[1..$n$] and $Y$[1..n]. we need to come up with a O($\log n$) time algorithm to find the median of the 2$n$ ...