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Questions tagged [counting]

The term Counting in Computer Science is usually used to refer counting objects in certain arrangements or with certain properties.

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Find xor sum of all pairs raised to power of 3

We are given array $A$ of $N$ integers each in the range $1 \leq A_i \leq 2^{30}$, that is we can write each integer with at most 30 bits. The target is to compute $\sum_{1\leq i \leq N,1\leq j<i} (...
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Does $P=PP$ or $P=PSPACE$ have consequences for algebraic class problem $VP=VNP$?

Deciding majority of counting problem is $PP$ class. Is there any relation between $PP$ and $VNP$ and is there consequence of $P=PP$ or $P=PSPACE$ to $VP=VNP$?
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Counting on a matrix

I have an $n \times m$ matrix, and fill it with numbers of $1 \dots k$. If a matrix can be turned into another matrix by exchanging its lines and exchanging its columns, the two matrices are ...
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Given tree with 0 or 1 assigned to each node, count paths with odd number of ones in it

Let's say we have given tree of $N$ nodes and $N-1$ edges, each of the $N$ nodes is assigned one integer, either $0$ or $1$. We want to count all paths between two nodes $u$ and $v$ such that on the ...
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Counting rectangles in one big histogram

Let's say we have given array of $N$ integers which actually represent the height of a histogram, now for each rectangle with lower side at the base of the histogram we need to count how many elements ...
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Given graph with N nodes colored in K colors, count paths visiting each color exactly once

Let's say we have given graph of at most $N = 400$ nodes, and each node is colored in one of $K = 21$ colors. We need to count paths of length $K$, such that each node on the path is in one of the $K$ ...
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Count submatrices with only zeros for each element of the matrix

Let's say we have given matrix of size $N \cdot M$, only with zeros and ones. For each element in the matrix, we want to count subrectangles that are covering this element and are made only of zeros. ...
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Sum of unique elements in all sub-arrays of an array

Given an array $A$, sum the number of unique elements for each sub-array of $A$. If $A = \{1, 2, 1, 3\}$ the desired sum is $18$. Subarrays: ...
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Count numbers less than $x$ co-prime to $p$

We have given two numbers $x$ and $p$. We want to count how many numbers are less than $x$ and are co-prime with $p$. I know that we can solve the problem in $O(x\log x)$ with iterating over all ...
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How to find all topological sortings of a special DAG in O(N^2)

I came across the following question in a hackerrank competition, which is based on topological sorting of a DAG. https://www.hackerrank.com/contests/hourrank-29/challenges/birthday-assignment/forum ...
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Count number of the ways to fill a N-lengthed binary string

From the problem, count the number of ways to fill a binary string of length $N$ with at least one $1$'s consecutive sequence of length $K$ and other $1$'s consecutive sequences have length no more ...
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Counting subarrays where each number either doesn't occur or occurs odd number of times

We have given array $V$ of $N$ integers, we want to count the number of subarrays of the array such that each elements in the subarray either doesn't occur at all, or it occurs odd number of times. ...
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Why it is not $O(m)$ but $O(\log m)$?

I am reading the lecture notes and have a question. I am trying to understand the beginning of Section 3 on page 2. Problem: Given an input stream $\sigma$, compute (or approximate) its length $m$. ...
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Defining decision-problem complexity classes by counting branches of a polynomial-time NTM

This answer on another SE community discusses the concept of a "counting complexity class". As far as I can tell, the author is using that term in a slightly nonstandard way: most sources (PS format) ...
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Can I make last part of counting sort become to be starting from lowest index?

Counting sort is originally like below code. ...
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How to count number of 1s of the first k positions in a bitset (bitmap)?

Input: Given a bitset $b$, which is is an array of 0s and 1s, of length $L$, and a positive integer $k$. Output: the total number of 1s in the first $k$ position of $b$. Obviously it can be done in $...
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Counting (enumerating) minimal solutions of a dual horn formula

Is there an efficient algorithm ("does not necessarily have to be a polynomial time algorithm") to compute all "minimal" solutions for a Dual Horn formula (conjunction of clauses where each clause ...
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On FPTAS and many one parsimonious reductions

We have two $NP$ complete problems $\Pi_1$ and $\Pi_2$. Suppose $\Pi_1\rightarrow\Pi_2$ be a many one parsimonious reduction. If $\Pi_1$ has an FPTAS then does $\Pi_2$ also have? If $\Pi_2$ has an ...
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Counting solutions of a particular type in HORN SAT

I am interested in counting the number of solutions of a particular type (say #) in HORN SAT. I have 2 questions concerning the same. Suppose we have a HORN SAT -: $(x_1) \land (x_2 \implies x_1)$, ...
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Modified Counting Inversions problem using divide and conquer

Given an array $A$, find the number of pairs $(i, j)$, such that $ i > j$ and $A[i] \ge A[j]$. This is a modified version of the famous problem of Counting Inversions, only in this version it ...
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How hard is APPROXIMATE-#SAT? [closed]

It is well known that the problem of counting the satisfying assignments of SAT, namely the problem #SAT, is #P-complete. It is also suspected (somewhat less widely) that even deciding SAT should ...
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Find smallest enclosing circle

On a 2d plane, there is a large circle centered at $(0, 0)$ with a radius of $R_{{o}}$. It encloses $\sim 100$ or so smaller circles distributed randomly across the parent circle otherwise with known ...
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Number of distinct single-assignment forms with $j$ binary function calls?

Given $n$ inputs and $k$ outputs and $j$ identical binary function calls to $g$, how many possible distinct single-assignment forms are there? The only assumption made about $g$ is that if $a = c \...
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Is there a simple way to construct a Boolean formula that is true if and only if at most $k$ of the input variables are true? [duplicate]

I could of course construct a truth table for the function $$f(x) = \left(\sum_i x_i\right) \leq k$$ Where $x$ is an assignment and I'm slightly abusing notation to count Booleans. And then I could ...
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Count points on same distance from set of points

Let's consider finite grid of points with size of $N$ by $M$ and set of $x$ points ($x$ is small number, up to 10, $N$ and $M$ are big numbers, up to 30000 )). Each of the $x$ points is described with ...
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Applications of counting queries

Following the paper on AD-Trees (a data structure to answer counting queries efficiently). The paper discussed the applications of such a data structure to useful for machine learning methods such as ...
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Counting number of permutations respecting partial order

Suppose that we have an array $A$ of $n$ elements with some partial order known, e.g. for example as a $n\times n$ matrix containing $c_{ij} \in \{-1, 0, 1\}$ where $0$ represents unknown and $-1, 1$ ...
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Is there any simpler way to have for each sequence element the amount of succeeding larger elements than to implement an AVL tree?

I have a sequence. And now for each element in this sequence I would like to know how many subsequent elements are larger. Or, in other words, I have a sequence $a_1, \ldots, a_n$, and for each $1\leq ...
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183 views

How many 3-SAT expressions with up to N variables are satisfiable?

TL;DR There are exactly 255 possible 3-sat expressions with exactly 3 variables (more meticulously defined below). Of those, exactly 254 are satisfiable. There are exactly 4,294,967,295 possible 3-...
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Count numbers less than $K$ in array

Let's say we have given array $A$ consisting of $n$ integers, and integer $K$. Now we want to count number of indexes $i$ such that $A_i<K$. What is the easiest way to pre-process the array and ...
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Proving or disproving a set of total functions is countable

Let S be the set of total functions from $N \rightarrow M$, such that for each $f \in S$, there is $i > 1$ such that for all $j < i$, $f(i)$ and $f(j)$ are not equivalent Turing machines. ...
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Radix, merge, counting sort and when to use

Okay can't figure this out. I want to make sure I understand it. There are n random keys each being float numbers with p decimal places. So, for example, 123.456, 343.645, 234.543, 863.238, 956....
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Algorithm. Input: pointers to k unsorted arrays of different lengths. Needed output: k sorted arrays

$k = \Theta(n)$ The arrays consist only natural numbers $1$ to $n$ The sum of the length of all arrays = $\Theta(n)$ It should return the $k$ original arrays, each sorted on its own. The running ...
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Approximate count per element in list/stream via Counting Bloom+Morris

I've a large list A of elements. Given another list B of elements, I need B to be sorted by ...
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1answer
122 views

Sorting an array in linear time

I need to find a method to sort an array in $O(n)$ time complexity. I saw this link, however I'm not sure how to apply it to the elements I need. Input: an array $A$ of length $n$, containing ...
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1answer
54 views

On lowness of $\oplus P$

$\oplus P$ is low for itself ($\oplus P^{\oplus P}=\oplus P$). Are there other complexity classes $\mathcal D$ that satisfy $\mathcal D^{\oplus P}=\oplus P$? Are there complexity classes $\mathcal C$ ...
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585 views

How to count all contiguous subsequences with positive sum?

I have array $t$ with size $n \leq 10^6$. It has only two kinds of elements inside: $1$ or $-1$. I need to count how many contiguous subsequences have positive sum. This pseudocode demonstrates ...
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Count Wildcard Parenthesizations of a String

Let $\Sigma = \{ (, ), ? \}$ be an alphabet. For a given string $s \in \Sigma^*$, we denote by $f(s)$ the number of ways to replace each symbol $?$ either with $($ or with $)$ such that $s$ is ...
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185 views

Using Pascal's Triangle to implement queues and stacks using heaps

I have the following question as homework in an algorithms, analysis and data structures class: And here's an answer I wrote up: A queue is a first-in-first-out data structure. A heap is a data ...
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1answer
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Number of minimal unsatisfiable partial assignments in 2-SAT/3-SAT

A minimal unsatisfiable partial assignment for 3-CNF is a partial assignment that: There exist a clause where all variables are unsatisfied. Unfixing any variable make every clause contain at least ...
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How to use Bitmasking to solve this problem?

http://codeforces.com/problemset/problem/535/B The problem is: You are given a lucky number n. Lucky numbers are the positive integers whose decimal representations contain only the lucky digits 4 ...
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1answer
280 views

Can counting problems have optimal substructure?

I understand that for a problem to be solvable using dynamic programming, it needs to have the following properties: optimal substructure overlapping subproblems I stumbled upon an article which ...
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147 views

How many different trees can we form from given graph?

I'm trying to practice some combinatorics and I faced this problem, let's say we have given graph with N nodes and M edges. $$N\leq500, M \leq N\cdot(N - 1)/2$$ In this graph I want to count the sub-...
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Are counting problems the same as problems involving listing all possible combinations?

I recently tried coming up with an algorithm that uses dynamic programming for the counting variant of the change problem. Given a set of target and a set of denominations, print the number of ...
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What's wrong with this [closed]

Here's the question: Do more baby names start with "A" or "B"? Write code to count and print those two counts ("A" count, then "B" count). I then write this code down: table = new SimpleTable("baby-...
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Count arrays with size n, sum k and largest element m

I'm trying to solve pretty complex problem with combinatorics. Namely, we have given three numbers N, K, M. Now we want to count how many different arrays of integers are there with length N, sum K ...
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Count of distinct substrings in string inside range

Having string $S$ of length $n$, finding the count of distinct substrings can be done in linear time using LCP array. Instead of asking for unique substrings count in whole string $S$, query $q$ ...
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How to encode a sequence of non-decreasing integers with an integer without redundancy, loops, and recursions

How to encode a sequence of n non-decreasing integer of [0, ..., m] fulfilling the following conditions: no or minimal redundancy only use 1 integer variable or k independent integer variables with a ...
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1answer
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Proving $\#CYCLE \in \#P$

I'm trying to find a way to count distinct simple cycles in a graph in order to prove that $\#CYCLE \in \#P$, if I could represent a distinct cycle, then I'll have a witness. I saw this question: ...
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Display Counting Algorithm

I am writing some firmware for a display that will take measurements and present them in real time on an LCD screen. I would like for the measurements to display as smoothly as possible... What I mean ...