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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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The expectation of the total number of pairs of keys in a hash table that collide using universal hashing

I am reading CLRS relating to perfect hashing. When computing the $$ \mathbb{E}[\sum_{j=0}^{m-1}{n_j\choose{2}}] $$ where $m$ is the number of slots in the hash table, and $n_j$ is the number of keys ...
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Limitations of DFA [duplicate]

In this link it is mentioned: A DFA is not powerful enough to recognize many context-free languages because a DFA can't count. But counting is not enough -- consider a language of ...
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38 views

Finding combinations of variables that can take value of -1/0/1 that produce sum of 0 with added constraint

I have 64 variables that can either take a value of -1, 0, or 1 and I am interested in finding all possible combinations of variables such that I have n variables in each the positive and negative ...
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List maximum number of three-sets

Suppose i'm given with pairs component-quantity. I need to list maximum number of possible three-sets. {AA}, {BB}, {C}, {D} => {ABC},{ABD}. To list two-sets ...
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Can someone give me the definition of #Monotone-2SAT?

In the decision problem, I set all variables to true and see if the formula is satisfiable. My question is because I do not understand how there can be multiple solutions, though all variables are ...
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66 views

Find total count of all paths starting from a fixed vertex to all other vertexes of the graph

Given an directed graph (may contain cycles) we have to find total number of simple paths from a fixed source vertex to all other vertices of the graph, i.e. $$ \text{#(paths from 1 to 2)}+\text{#(...
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Min no.of operations required to convert an array to which it should contain elements of equal frequency

I have come across this tricky problem. An array of N elements should be converted to another array within k operations such ...
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1answer
19 views

Is there an FPRAS for the number of min st cuts in general graphs?

Provan and Ball [1] showed that the problem of counting the number of minimum st cuts is #P-Complete. What is known about the problem of approximating the number of min st cuts? Is it possible to get ...
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Count paths in matrix that visit each number exactly once [duplicate]

Let's say we are given matrix of size $N \leq 21 \text{ by } M \leq 21$ each element of the matrix is either $-1$ or number in the interval $[0, 20]$. We want to count the number of paths that start ...
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182 views

Count-min sketch

I don't understand the use case of count min sketch. Based on Count–min sketch, it says "serves as a frequency table of events in a stream of data.". If I know there are N types of events, why can't ...
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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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1answer
27 views

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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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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1answer
60 views

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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1answer
44 views

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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1answer
113 views

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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1answer
45 views

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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1answer
181 views

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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1answer
26 views

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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226 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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1answer
56 views

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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1answer
593 views

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
156 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
57 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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1answer
693 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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1answer
237 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
71 views

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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1answer
201 views

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
317 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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1answer
166 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-...