Questions tagged [integers]

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

111 questions
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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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Complexity of Integer Division

My favourite algorithm textbook: The Algorithm Design Manual S. Skiena.pdf had this interesting problem: 1-28.  Write a function to perform integer division without using either the / or * ...
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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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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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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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Merge Sort meaning of a bit part of code

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