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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Fastest algorithm for finding the number of primes in a range

Is there an algorithm for finding the number of primes in a given range $[N, M)$ that works in time linear to $M-N$? For context, $N$ and $M$ can go up to $10^{10}$, but the distance between N and M ...
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1answer
42 views

Can all types of computational problems be modeled as decision problems?

Can all types of computational problems (search, counting, optimization...) be modeled as (sets of) decision problems? Rephrased: For every type of computational problem is there a set of decision ...
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1answer
34 views

Number of permutations of set {1, 2, …, n} for which insertion sort will perform exactly n permutations

I have had the following problem at my last exam: For how many permutations of set {1, 2, ..., n} where n > 2 will insertion sort (without guard element) perform exactly n comparisons. My thinking ...
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2answers
58 views

To calculate how many times a certain year repeats itself in the calendar within a given year range

Let's say I have a year $Y$. I wanna know how can I calculate the number of times the calendar configuration of the year $Y$ repeats itself in the year range $[A, B]$. Is there any method to it ...
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58 views

Counting triplets from three arrays satisfying the equation x^2 = yz

Let's say I have three arrays of positive integers X, Y and Z. You can assume that each of ...
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1answer
157 views

Count total number of k length paths in a tree

This is a question from a competitive programming competition. Given a tree with n nodes and a number k, find the total number of paths of length k in that tree. I know for a fact that a solution can ...
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15 views

Count to infinity problem (routing) between unsynchronized stations(tricky)

i was wondering, will count to infinity can occur in the following cases? if so, will it necessarily occur or can the routing tables stabilize themselves? Distances: From A to B - 3 from B to C - 4 ...
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94 views

How is the set of functions from ${\{a,b\}}$ to $N$ countable?

Assume a set of functions from ${\{a,b\}}$ to $N$ Where $N$ is the set of Natural numbers. Let us assume that the size of $N$ is $n$. i.e $|N|=n$ The first element $a$ have $n$ choices for mapping....
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1answer
66 views

Counting big enough elements

Let $v$ be a vector of positive integers of length $n$. I want to find the highest $k$ such as there are at least $k$ elements of $v$ that are greater or equal to $k$. What would be the best ...
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1k views

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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33 views

Estimating number of points in 1D space

There are some arbitrary-chosen points in 1D space. What needs to be found is the approximate number of them without counting all of them. It is possible to choose some coordinates (numbers) and for ...
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1answer
59 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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131 views

How to find the Big-O for finding combinations of balanced parentheses?

Given n pairs of parentheses, a function which returns the total number of all combinations well-formed parentheses could be: ...
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3answers
2k views

Finding all solutions to an integer linear programming (ILP) problem

My problem is to find all integer solutions to an ILP. As an example, I'm using an ILP with two variables, but I may have more than two variables. I describe the method I currently use to solve this ...
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1answer
62 views

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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1answer
27 views

Build a histogram from a very large data sample

I want to calculate a histogram from an array of size N. N is very large. I know 2 ways to do so: The naive approach is to ...
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1answer
5k views

How to calculate an accurate estimated reading time of text?

I suppose the calculation should not be done by only two factors (average reading speed/words per minute, and word count). But at least by a third parameter, that in my opinion should measure the ...
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2answers
110 views

Fastest algorithm to find all the possible paths of length $n$ from a give node in a directed graph?

I am trying to find the fastest algorithm to find all the possible paths of length $N$ from a given node in a directed graph. My solution is to do a modification of breadth first search from the ...
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1answer
27 views

Given a set of intervals $(I_n)_n$ contained in $[0, L]$, compute the longest interval in $[0, L]$ which has empty intersection with all $(I_n)_n$

Let be $(I_n)_n$ a set of $p$ intervals each contained in $[0, L]$ for $L \geq 1$. I define $(J_n = [a_n, b_n])_n$ the set of intervals which have empty intersection with $I_n$ for all $n \in [[1, p]]...
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Can approximation help find the exact answer?

Lets assume we have an array with 100 numbers and we want to find how many '1's there are. Best solution will be reading every numbers and counting. Now we get a hint that there are 50,51 or 52 '1' in ...
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49 views

Counting number of apples in an apple tree with given number of layers

This problem comes from a competitive programming question, and it seems to require dynamic programming. There are several layers of apples arranged in a formation with each apple having a value ...
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15 views

Calculating Time Complexity of Algorithm Using Incrementor Variable [duplicate]

I am trying to calculate the time complexity of an algorithm using n in the code below. I have a working solution to a coding challenge to sort a stack using only ...
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1answer
100 views

Number of ways painting graph in two colors, such that two nodes of same color are linked by edge

We are given undirected graph of $N$ nodes and $M$ edges, we want to count the number of possible ways to paint this graph in $2$ colors such that for each two nodes having the same color, there must ...
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38 views

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 $VP=VNP$? Is there a way to show $\#P$ is in $FP^...
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Are $\mathsf{\#P}$ problems harder than $\mathsf{NP}$ problems

I have a method to solve the $\mathsf{\#P}$ version of 3SAT in a way that seemingly reduces it to an $\mathsf{NP}$ problem. - I don't have a formal understanding of these terms so I will just show an ...
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1answer
42 views

Estimation of the number of solutions by Counting

This is a question from a quantum computation textbook. Consider a classical algorithm for counting the number of solutions to a problem. The algorithm samples uniformly and independently $k$ times ...
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2answers
71 views

Prove that this language is NP-Hard

Given $$\mathrm{\#3SAT} = \{ (w, y) \mid w\text{ is a $\mathrm{3SAT}$ instance with at least $y$ satisfying assignments}\}\,,$$ prove that $\mathrm{\#3SAT}$ is NP-Hard. I am currently stuck with ...
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1answer
357 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
48 views

Why does not valiant's reduction show NP=RP?

Valiant converts $SAT$ formula to a $0/1$ matrix such that $Permanent$ of the matrix is $4^m\#SAT$. We know $Permanent$ can be approximated to $1+\epsilon$ factor with probability $1-\frac1\delta$ in ...
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1answer
55 views

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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1answer
886 views

How many ways to express N as sum of 2, 3 and 5?

I've learnt about problems about express N as sum of 2, 3, 5. For examples, if N = 7: N = 5 + 2 N = 2 + 5 N = 2 + 2 + 3 N = 2 + 3 + 2 N = 3 + 2 + 2 But most of I found on the Internet that the ...
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1answer
554 views

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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1answer
98 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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11 views

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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1answer
42 views

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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1answer
102 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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135 views

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
26 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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35 views

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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2answers
291 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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1answer
414 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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69 views

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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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
66 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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1answer
527 views

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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85 views

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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3answers
175 views

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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2answers
76 views

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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1answer
37 views

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 ...