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Questions tagged [integers]

Questions about properties of, working with and algorithms on integers.

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What is the equivalent of the integers symbol Z for n bit only integers?

We refer to the set of all integers as $\mathbb{Z}$. Now suppose we have a set of integers that can be held within a computer variable of $n$ bits width. Clearly they can only be of $2^{n}$ range, ...
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1k views

Complexity of Integer Division

My favourite algorithm textbook: The Algorithm Design Manual S. Skiena.pdf had this interesting problem: 1-28. [5] Write a function to perform integer division without using either the / or * ...
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1answer
45 views

Reversible Merge of Integer Hash Values

Context: I am working with a tree-like data structure. I would like every node in the tree to have an integer hash value that is the result of combining the integer hash values for the node's ...
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87 views

Efficient data structure for storing integers in a range?

Say I'm constantly given integers from the range $[1,2^{32}]$ in a random order and have to store these so that when a duplicate arrives I can deal with it. By the end of this algorithm all $2^{32}$ ...
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2answers
3k views

Partition of a set of integer into 3 subsets of approximately equal sum

I'm having a very hard time trying to figure out how to solve this problem efficiently. Let me describe how it goes: "A hard working mom bought several fruits with different nutritional values for ...
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1answer
19 views

Merge Sort meaning of a bit part of code

I am studying this part for a merge sort implementation: ...
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11 views

Structure of numbers in descriptive complexity

Descriptive complexity is a useful way to free yourself from computational considerations when studying complexity. One of those considerations is the encoding of the structures you are working on. ...
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19 views

Partitioning the columns of a matrix

I thought about this problem for a while now and am not able to find a solution for it, be it a direct algorithm or a reduction to a known problem, so I'm asking here: Suppose you have a matrix $A\in\...
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81 views

How does prefix summing the count histogram in counting sort result in an array of output indices?

My implementation of counting sort is based on the description I found here. I've also included my code in this REPL, but here it is: ...
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20 views

Is finding a sum of two squares representation for a number computationally assymptotically equivalent to integer factorization?

I have been wondering this question because given a number $n$ with prime factors of the form $4k+3$ when we square $n$ and find a sum of two squares which should reveal these types of factors (as ...
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71 views

Algorithm for implementing the modulus “%” operator?

How can an efficient modulus operator be implemented? Here's a naive way of defining A % B: given $(a,b) \in \mathbb{Z}$ (represented as ...
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35 views

Using half float to represent scaled short (int16), do I lose precision comparing to using double?

A device is generating 14-bit integer which is stored as int16 (short), a scaling process will then scale the data to value of order 10E-3. Does it then matter if I store these number with half float ...
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645 views

Find 8 numbers whose sum is closest to a defined value

I have a file that has a number (a positive integer) on each row. Given a number $q$, I want to find a value that's a sum of some 8 numbers in the file, and is as close to $q$ as possible. So, ...
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60 views

Given a RxC grid, how to generate N 2D points randomly such that no 3 points are collinear?

Context, I have a geometric algorithm that is sensitive to collinear points and receives as input a list of points in 2D generated randomly. Suppose that I have a Boolean function nonCollinear(x,y,z) ...
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3answers
1k views

Computing a histogram with the number of extant values not known in advance

(This may be more fitting for CSTheory, I'm not sure.) I'm looking for an practical or theoretical work (that is, academic papers, online jots, pseudocode or code) regarding efficient algorithms for ...
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What is the most efficient way to compute factorials modulo a prime?

Do you know any algorithm that calculates the factorial after modulus efficiently? For example, I want to program: ...
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1answer
805 views

Calculate storage requirements for a data set

I have a simple problem. I can't seem to even find the right search terms to get me pointed in the direction I need to be heading. I'm writing a bunch of integers to disk. Lot's of them. Starting ...
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2answers
758 views

How can I find minimum number required to add to sequence such that their xor becomes zero

Given a sequence of natural numbers, you can add any natural number to any number in the sequence such that their xor becomes zero. My goal is to minimize the sum of added numbers. Consider the ...
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1answer
2k views

Algorithm to determine if a number is perfect on a Turing Machine

I've been trying for a while now to find a solution for the problem in the title: determining if a number is perfect using a Turing Machine. I only had one class on the TM and while I did "get" how it ...
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42 views

Sets whose decimal expansions form a regular language

Write $\bar n$ for the decimal expansion of $n$ (with no leading 0). For a set $S$ of natural numbers, let its set of expansions (in base 10) be $\bar S = \{\bar n \...
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Improving an integer factoring algorithm

I've written an algorithm for integer factorization (specifically RSA-like coprimes - products of two large primes, roughly of the same number of decimal digits) which is not based on QS, GNFS or any ...
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75 views

Given two arrays of length n and n - 1, order the first array such that no partial sum is in the second array

Two arrays of natural numbers are given of length $n$ and $n - 1$: e.g. $A: [a_0, a_1,..., a_{n-1}, a_n]$ $B: [b_0, b_1, ..., b_{n-2}, b_{n-1}]$ All elements of $A$ are unique (can be in $B$), all ...
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Find all pairs (i, j), such that i + (i+1) + (i+2) + … + j = n

We have give positive integer $n$, and we want to find all the pairs $(i, j), i\leq j$ such that: $$i + (i+1) +(i+2)+(i+3)+ \dots + j = n$$ Clearly we can try all possible pairs in $O(N^2)$, but that ...
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1answer
67 views

Mistake in the Algorithm Design Manual?

Look at this excerpt from the Algorithm Design Manual by Skiena, 2nd Edition The sum of the first $n$ even integers? Surely, the two sums given do not include only even numbers, right? Is this a ...
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3answers
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Determine missing number in data stream

We receive a stream of $n-1$ pairwise different numbers from the set $\left\{1,\dots,n\right\}$. How can I determine the missing number with an algorithm that reads the stream once and uses a memory ...
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1answer
148 views

Linear time multiplication on RAM machine?

This page says following: Integer Multiplication has an O-optimal linear-time algorithm on a RAM or SMM Is this page fooling me or how can we multiply 2 numbers in linear time (bitwise complexity) ...
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1answer
49 views

Compress 32 bit integer into a normalized float

and sorry if this is a duplicate of some sort, my poor technical vocabulary doesn't get me very far with google. I am trying to implement some well-known PRNG algorithms. For the time, the algorithm ...
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2answers
82 views

How to find the closest N to the power of X to the given number?

Let's say we have number 4920 and we want to find the closest $n^x$ to 4920 2 ^ 12 = 4096 but it's not the closest possible $n^x$, for example 17 ^ 3 = 4913 is closer to 4920 The question is, how do ...
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1answer
793 views

How can repeated addition/multiplication be done in polynomial time?

I can see how adding 2 unsigned $n$-bit values is $O(n)$. We just go from the rightmost digits to the leftmost digits and add the digits up sequentially. We can also perform multiplication in ...
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1answer
63 views

Swap elements using integer addition and multiplication gates

I need to swap two integers using only integer addition and multiplication gates. I can't subtract them. I'm dealing with a sorting network, so I need to compare and swap. The compare and swap ...
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1answer
27 views

Gray-like code with maximum value <= maximum value of original symbol

I want to iterate through the numbers $0,1,2,\dots,n-1$ in some order, where each number in the sequence differs by only one bit from the previous bit. I'm going to be using each number as an index ...
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64 views

Low-level division operator implementation

A few weeks ago, I asked this question on the implementation of the shift operator for an architecture that implements boxing. Reviewing my implementation I found out that the division operator wasn't ...
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1answer
58 views

Low-level shift operator implementation

I want a shift operator for an architecture that implements boxing. Basically, the high-level constructs represent integers as 31 bits followed by 1 bit set to one as a tag (so to represent binary int ...
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289 views

4-partition elements summation NP completeness

How can we prove that the following problem $A$ is NP complete? Given a set of integers $S={a_1, ..., a_n}$ and a number $D$, is it possible to find disjoint sets $S_1, S_2, S_3, S_4$ such that $S_1 \...
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491 views

Best complexity of parity/comparison in the Residue Number System

Let: $\left\{m_1, ~...~, m_k\right\}$ be a set of coprime natural numbers, $M=\prod_{i=1}^{k} m_i$ $X$ be a natural integer, such that $X < M$ Then $X$ can be expressed in the Residue Number ...
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1answer
134 views

How to uniquely encode a vector of non-increasing positive integers

Given a vector of positive integers $A=[a_1, \cdots, a_n]$, where $\sum_{1 \leq i \leq n}a_i = n(n-1)$ and $a_i \geq a_j$ iff $i \geq j$, I am interested in encoding the vector, maintaining the ...
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57 views

How to determine the carry vector of an integer addition

An integer addition can be represented as following: $A + B = A \oplus B \oplus carry_{vector}$ My question is how to determine the carry vector from $A$ and $B$ (not from the result)?
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What are applications to sort plain integer arrays?

A lot of research and engineering effort is put into finding fast methods to sort an array of integers; e.g., Java's runtime library has highly-tuned methods to sort arrays of each primitive type (see ...
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1answer
245 views

There are n numbers. Find the maximal set of pairwise NON coprime numbers

I have to take input in an array(no. of elements in array <=10^5). For ex:- Let the array be {2,3,4,16,9,45,81,27} Now I need to find the order of the maximal set such that any pair of elements in ...
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2answers
6k views

Algorithm to find optimal currency denominations

Mark lives in a tiny country populated by people who tend to over-think things. One day, the king of the country decides to redesign the country's currency to make giving change more efficient. The ...
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4answers
482 views

Counting an integer's divisors without just enumerating them (or estimating if not possible)?

I'm trying to count the number of divisors an integer $n$ has. The simple way to do this is to just enumerate all integers from 1 to $\sqrt{n}$, and count how many integers evenly divide $n$. For ...
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1answer
136 views

Complexity class of integer factorization

Is integer factorization confirmed to be an NP-complete problem? If not, then if one could transform IF into an equivalent problem which is already proved to be NP-complete, would it mean that IF is ...
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1answer
140 views

Practical Implementation for Refinement Order on Integer Partitions

The refinement order on partitions of an integer $n$ can be defined as follows: $\lambda=(\lambda_1,\dots,\lambda_k)\leq\mu=(\mu_1,\dots,\mu_\ell)$ if there is a partition of the parts of $\lambda$ ...
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2answers
348 views

Find the duplicates in a list of floating point numbers

I receive a list of real numbers ( float ) between $0$ and $1$. The list has length $N+1$ and I need to find two numbers on the list which are $\le \frac{1}{N}$ ...
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361 views

What algorithms exist for solving natural number linear systems?

I'm looking at the following problem: Given $n$-dimensional vectors of natural numbers $v_1, \ldots, v_m$ and some input vector $u$, is $u$ a linear combination of the $v_i$'s with natural number ...
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Problems that become far easier when restricted to only integer values

I know that there are some problems that are very hard to solve in general, but become much easier and asymptotically faster if restricted to only integer values. One such example would be sorting ...
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Sum of 3 integers with full adder

1)Is it possible for a full adder to add three e.g 4 bit numbers? I mean I know the full adder has 3 inputs and two outputs but the second bit of C comes from the previous block (as shown in the image ...
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2answers
173 views

Efficient algorithm for getting from 1 to n with 3 specific operations

The question: Given those 3 valid operations over numbers and an integer $n$: add $1$ to the number multiply the number by $2$ multiply the number by $3$ describe an efficient ...
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1answer
597 views

Convert integer of mixed radix to standard positional numeral system and vice versa

I have multiple numbers (e.g. [1, 4, 2]) where each number can be one of a specified range of numbers (e.g. [0-1, 0-5, 0-3]). I ...