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

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4
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1answer
77 views

Complexity of Linear Diophantine equations

My question is simply, can linear Diophantine equations be solved in polynomial time? Specifically, I am looking at equations of the form $a_1 x_1+a_2 x_2 + ... + a_n x_n = k$, where $a_i,x_i,k$ are ...
5
votes
1answer
76 views

Difference in Sorting 32- and 64-bit Integers

In 2007, Barrack Obama was interviewed at Google. The question was, "What is the best way to sort a million 32-bit integers?" Does the fact that the size range of the integer was specified elude to a ...
0
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1answer
74 views

How to store factorials? [closed]

Can someone help me to store the factorial of large numbers such as 100! efficiently? UPDATE: obviously, storing the argument rather than the factorial digits themselves achieves a significant ...
2
votes
1answer
37 views

Checking whether a number is a square or higher power modulo n

Is there an algorithm to check whether an integer $x$ is a square modulo $n$, where $n$ is an integer whose factorization we do not know? Is the Jacobi symbol helpful? What about higher powers, ...
5
votes
1answer
99 views

Checking whether an integer is a square or higher power

Is there an algorithm to check whether an integer $n$ is a square? What about higher powers, e.g., testing whether $n$ is a $k$th power? I understand that the Jacobi symbol ...
5
votes
1answer
174 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 ...
6
votes
2answers
93 views

Understanding Intel's algorithm for reducing a polynomial modulo an irreducible polynomial

I'm reading this Intel white paper on carry-less multiplication. It describes multiplication of polynomials in $\text{GF}(2^n)$. On a high level, this is performed in two steps: (1) multiplication of ...
1
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1answer
118 views

Algorithm to decide if $n \le m!$

This is an assignment of an introductory course of complexity theory and I need to find a way to do the following: Given $n,m \in \Bbb N$, is $n \le m!$ ? The idea is to provide a Post Machine that ...
5
votes
2answers
68 views

Quick calculation for $(x^y) \bmod z$

What are the possible ways to calculate $(x^y) \bmod z$ quickly for very large integers? Integers $x,y \lt 10^{10000}$ and $z \lt 10^6$.
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2answers
82 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 ...
3
votes
1answer
449 views

Shift-and-or multiplication operation

Continuing in the same vein as Carry-free multiplication operation, a followup question is as follows (differences in bold): Let $r = p \oplus q$ be an operation similar to multiplication, but ...
3
votes
2answers
145 views

Carry-free multiplication operation

In long-multiplication, you shift and add, once for each $1$ bit in the lower number. Let $r = p \otimes q$ be an operation similar to multiplication, but slightly simpler: when expressed via ...
3
votes
0answers
219 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 ...
5
votes
1answer
111 views

Division by a constant

After skimming Multiplication by a Constant is Sublinear (PDF), (slides (PDF), slides with notes (PDF)) I was wondering if this could be extended to division by a constant in sublinear time? ...
6
votes
2answers
84 views

Isn't std::bernoulli_distribution inefficient? Designing a bit-parallel Bernoulli generator

C++11 has a convenient Bernoulli RNG, illustrated at http://en.cppreference.com/w/cpp/numeric/random/bernoulli_distribution . However, distilling an entire random integer into a single random bit ...
6
votes
3answers
526 views

What data structure would efficiently store integer ranges?

I need to keep a collection on integers in the range 0 to 65535 so that I can quickly do the following: Insert a new integer Insert a range of contiguous integers Remove an integer Remove all ...
1
vote
2answers
226 views

Filling Rows of a Matrix Subject to Conditions

I'm seeking to write an algorithm which, given a value of N, will fill a matrix consisting of (N+1)(N+2)(N+3)/6 rows and 4 columns with the integers from 0, ... , N, subject to the conditions that: ...
10
votes
3answers
138 views

Number of multisets such that each number from 1 to $n$ can be uniquely expressed as a sum of some of the elements of the multiset

My problem. Given $n$, I want to count the number of valid multisets $S$. A multiset $S$ is valid if The sum of the elements of $S$ is $n$, and Every number from $1$ to $n$ can be expressed ...
3
votes
3answers
272 views

How should I store and sort a large number of 64-bit integers?

I have about 500,000,000 64-bit integers, so these numbers could be very large. I want to sort them as quickly as possible. I have a couple of questions: What data structure do you suggest for ...
3
votes
2answers
66 views

Remove divisors from a set of integers

I have a set $S$ of integers. I want to remove all elements of $S$ that are divisors of another element of $S$. In other words, I want to compute $T = \{y \in S : \forall d \in S . d \nmid y \}$. ...
0
votes
1answer
419 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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3answers
1k views

What are two's complement integers?

Can someone explain in plain English what "two's complement integer" means? I read this: in Java long is a 64-bit signed two's complement integer
0
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1answer
21 views

VAR autoincrement with constant space consumption for super large tables

Assume there was a database system that had a data type called VARINT or some variant that allowed instead of fixed-length INTs ...
4
votes
1answer
583 views

Algorithm to return largest subset of non-intersecting intervals

I need an efficient algorithm that takes input a collection of intervals and outputs the largest subset of non-intersecting intervals. i.e. Given a set of intervals $I = \{I_1, I_2, \ldots, I_n\}$ ...
2
votes
2answers
124 views

Solve Integer Factoring in randomized polynomial time with an oracle for square root modulo $n$

I'm trying to solve exercise 6.5 on page 309 from Richard Crandall's "Prime numbers - A computational perspective". It basically asks for an algorithm to factor integers in randomized polynomial time ...
0
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3answers
125 views

How to guess the value of $j$ at the end of the loop?

for ( i = n , j = 0 ; i > 0 ; i = i / 2 , j = j + i ) ; All variables are integers.(i.e. if decimal values occur, consider their floor value) Let ...
1
vote
1answer
262 views

Shifting subset sum solution by constant positive integer

While reading the Wikipedia article about the subset sum problem I came across this example: "is there a non-empty subset whose sum is zero? For example, given the set $\{ −7, −3, −2, 5, 8 \}$, the ...
6
votes
3answers
494 views

Comparing rational numbers

Given $a,b,c,d \in \mathbb N$ and $b,d \notin \{0\}$, $$ \begin{eqnarray*} \frac a b < \frac c d &\iff& ad < cb \end{eqnarray*} $$ My questions are: Given $a,b,c,d$ Assuming we can ...
2
votes
1answer
187 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 ...
5
votes
2answers
113 views

Numeral systems other than unary used in nature or in animal and human behaviours

Representing numeric values using positional notation is one of the milestones in the history of arithmetic. Babylons used a base 60 system, Maya a base 20 system; base 10 system became "the standard" ...
4
votes
1answer
656 views

Fastest square root method with exact integer result?

I am dealing with the problem of computing $ s = \lfloor sqrt(x)\rfloor$ with $x \in [0,30000^2]$. The common sqrtf(x) on C language is too slow for this case, ...
3
votes
4answers
432 views

Get specified bit in addition of two large binary numbers

I am performing an addition operation on two large binary numbers that have an equal number of bits. Both numbers are stored in an array of length $N$, which is rather large. At first I tried running ...
5
votes
3answers
471 views

Converting function to bitwise only?

I have a function to count upper bits of a 32 bit value. So if a number is 11100011111..., the result is 3 as there are 3 ones in the most significant place before a 0 is hit. I need to convert the ...
3
votes
1answer
207 views

How to compute linear recurrence using matrix with fraction coefficients?

What I'm trying to do is generate Motzkin numbers mod a large number $10^{14} + 7$ (not prime), and it needs to compute the $n$th Motzkin number as fast as possible. From Wikipedia, the formula for ...
2
votes
1answer
75 views

Run time of product of polynomially bounded numbers

Let $M$ denote a set of $n$ positive integers, each less than $n^c$. What is the runtime of computing $\prod_{m \in M} m$ with a deterministic Turing machine?
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2answers
209 views

Complexity of computing $n^{n^2}$

What is the complexity of computing $n^{n^2},\;n \in \mathbb{N}$?
7
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2answers
1k 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 ...
2
votes
2answers
890 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 ...
10
votes
4answers
1k views

Most efficient algorithm to print 1-100 using a given random number generator

We are given a random number generator RandNum50 which generates a random integer uniformly in the range 1–50. We may use only this random number generator to ...
5
votes
3answers
401 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 ...
6
votes
3answers
435 views

What is the bitwise xor of an interval?

Let $\oplus$ be bitwise xor. Let $k,a,b$ be non-negative integers. $[a..b]=\{x\mid a\leq x, x\leq b\}$, it is called a integer interval. What is a fast algorithm to find $\{ k\oplus x\mid x\in ...
7
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2answers
1k views

Representing Negative and Complex Numbers Using Lambda Calculus

Most tutorials on Lambda Calculus provide example where Positive Integers and Booleans can be represented by Functions. What about -1 and i?
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3answers
669 views

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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2answers
154 views

From FACTOR To KNAPSACK

If there were an algorithm that factored in polynomial time by means of examining each possible factor of a complex number efficiently, could one not also use this algorithm to solve unbounded ...
11
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1answer
2k views

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: ...
7
votes
2answers
225 views

Detecting overflow in summation

Suppose I am given an array of $n$ fixed width integers (i.e. they fit in a register of width $w$), $a_1, a_2, \dots a_n$. I want to compute the sum $S = a_1 + \ldots + a_n$ on a machine with 2's ...
10
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1answer
178 views

Overflow safe summation

Suppose I am given $n$ fixed width integers (i.e. they fit in a register of width $w$), $a_1, a_2, \dots a_n$ such that their sum $a_1 + a_2 + \dots + a_n = S$ also fits in a register of width $w$. ...
3
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3answers
159 views

Language of the graph of an affine function

Write $\bar n$ for the decimal expansion of $n$ (with no leading 0). Let : be a symbol distinct from any digit. Let $a$ and $b$ ...
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5answers
538 views

Language of the values of an affine function

Write $\bar n$ for the decimal expansion of $n$ (with no leading 0). Let $a$ and $b$ be integers, with $a > 0$. Consider the language of multiples of $a$ plus a ...