Questions tagged [counting]
The term Counting in Computer Science is usually used to refer to counting objects in certain arrangements or with certain properties.
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Finding a 2SAT instance that has a specific solution set
Is there a 2SAT instance of variables $(a,b,c,d,e,f,g)$ that has exactly the solution set $S=\{ (1,0,0,0,0,0,0),(0,1,0,0,0,0,0),(0,0,1,0,0,0,0),(1,1,0,1,0,0,0),(1,0,1,0,1,0,0),(0,1,1,0,0,1,0),(1,1,1,1,...
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Counting independent sets
I know the Independent set problem is NP-complete. But could there be a more efficient way to count the exact number of different independent sets in an arbitrary, given graph?
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BSTs with repeating keys
The problem is to count number of unique binary search trees with keys $a_1,a_2,...,a_n$, given that some of the keys are not unique. For example, $a$ could be 2, 1, 1, 4, 3, 4.
We could try an ...
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Count number of non-contiguous occurrences in string
Given strings $S,T$ such that $n=|T|>|S|$ , I'd like an algorithm to count number of occurrences of $S$ in $T$ (as a subsequence), not necessarily contiguous.
Example:
if $T=aababc, S=abc$, the ...
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how to count all pairs such that w^x=y^z, where 1<=w,x,y,x<=n and 1<=n<=1000000
how to count all pairs such that w^x=y^z, where 1<=w,x,y,x<=n and 1<=n<=1000000
for example for n=3, there is 15 solutions
1^1=1^1
1^1=1^2
1^1=1^3
1^2=1^1
1^2=1^2
1^2=1^3
1^3=1^1
1^3=1^2
1^...
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Let F be a function defined for all nonnegative integers by the following recursive definition
Let F be a function defined for all nonnegative integers by the following recursive
definition.
F(0) = 0, F(1)= 1
F(n + 2) = 2F(n) + F(n +1), n>0
Compute the first six values of F; that is, write ...
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Why solving #2SAT in polynomial time implies P = NP?
The wikipedia article for #P states that if we have a polynomial-time algorithm for a #P-complete problem, P = NP is true.
As #2SAT is #P-complete, this would mean that providing a polynomial-time ...
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What are the definitions of countable and measurable colourings of a graph?
In this paper, the author discusses colourings of the plane, or in other words, of the underlying graph. I suppose a finite colouring is a colouring using at most $k$ colours for some natural number $...
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What is the (intuitive) relation of NP-hard and #P-complete problems?
From Wikipedia on $\mathrm{NP}$-completenes: "a [decision] problem is NP-complete if it is both in NP and NP-hard." [link] I think we can paraphrase this as the first statement:
An $\mathrm{...
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Efficiently count distinct in large range
I have a pubsub channel where an event is fired every time a user logs in, and I want to be able to query the unique users in a date range.
Solutions I thought:
Put the data in bigquery, and then use ...
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Algorithm to find number of occurrences in mutually exclusive sets
Given multiple sets of three items, how can I find the most-commonly occurring item among the sets using only one item from each set. The sets don't have duplicates, but if I'm thinking about this ...
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DJNZ command in Universal Register Machine
How do I represent DJNZ command of counting machine via commands of Universal Register Machine, those commands are CLR JNE INC and TR, via this commands i have to represent DJNZ command, any help ...
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Sorted list of counters in constant time
Summary.
A data structure maintains in constant time a sorted list of counter values, for a dynamic set of counters. I am interested in references using this structure, and in possible improvements.
...
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Universal quantification and the number of solutions
How would one count the amount of solutions to quantified formulas that have universal quantifiers? For example, for a boolean formula $\Phi(X)$ with a number of solutions $\#\Phi(X)$ let's construct ...
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Can we show that #3CNF is in FPTAS
If we have a deterministic algorithm $A$ such that $\#3CNF \in APX$, how can we show that there is a fully polynomial deterministic approximation scheme for $\#3CNF$? How can we show that $\#3CNF \in ...
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Does the existence of an $\alpha$-approximation scheme for a problem $f$ imply there exists a fully polynomial (deterministic) approximation scheme?
If you have an $\alpha$-approximation algorithm $A$ for some problem $f \in \#P$, such that (for $0 < \alpha \leq 1$)
$$
\alpha f(x) \leq A(x) \leq \frac{f(x)}{\alpha},
$$
does that automatically ...
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How to count this operation for (int interval = n/2; interval > 0; interval /= 2) using counting primitive operation?
I was confused how to label this for (int interval = n/2; interval > 0; interval /= 2) with counting operation and estimating this operation
so that I can get ...
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Applications of the DGIM algorithm
In the field of mining of data streams the algorithm of Datar-Gionis-Indyk-Motwani (DGIM, M. Datar, A. Gionis, P. Indyk, and R. Motwani, “Maintaining stream statistics over sliding windows,” SIAM J. ...
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Why can’t we use FPRAS for #DNF to estimate #CNF?
Why cant we approximate the number of satisfying assignments of a CNF formula $g$ by first counting the solutions to $\neg g$ (which is in DNF) using the FPRAS for $\#DNF$ and then estimating the $\#g$...
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Counting all subsequences constrained to a condition
I was trying to find all subsequences constrained to the following conditions:
remove element from the end.
remove element from the beginning.
remove element from both sides.
For example, given the ...
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Determining the number of reachable vertices from every vertex in a directed acyclic graph
Let $G = (V, E)$ be a directed acyclic graph, which is quite sparse (in the examples I have in mind, $|E| \approx 10|V|$ or so). For each vertex $v \in V$, let $f(v)$ be the number of vertices ...
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How many functions require precisely $n^2$ gates?
I'm trying to determine an asymptotic bound on the cardinality of the following set of functions. It is the functions with $n$-bit inputs, $\{0,1\}$ output, and requires precisely $n^2$ NAND gates. I'...
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Proving that deterministic approximate counting uses log(n) space
We just saw the Morris algorithm in class and we were asked the following:
In class, we saw a constant factor approximate randomized counting
algorithm with space complexity $O(\log \log n)$, where $...
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Is there a reduction from 2sat to bpm?
Given a 2SAT instance can we convert into bipartite perfect matching in parsimonious reduction?
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Sorting elements into k subarrays
Given are $n$ integer numbers in the range $0$ to $5n$.
A SubSort algorithm organizes the numbers into $k=n/100$ sets,
$s_{1}$, …, $s_{k}$ , each containing $100$ numbers, such that the
following ...
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Determine if for given some $L$, $S_L={L(M) : <M>\in L}$ then for any $L$, if $S_L=RE$ then $L\in R$ is True or False and explain
Determine if for given some $L$, $S_L=\{\ L(M) | <M>\in L \}$ then for any
$L$, if $S_L=RE$ then $L\in R$. Correct or Incorrect and explain why.
I think the claim is incorrect, and I'm trying ...
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Find the total no. of strings ( len n ) possible given a set of sets of letters such that no two letter from a single set should be in that string
This was an algorithm problem but I am having problems in formulating it.
I have a certain approach but I do not know how to fully execute:
Given
26 letters in total
All possible strings of length n
...
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How can we count the number of pairs of coprime integers in an array of integers? (CSES)
For reference, I am trying to solve this CSES Problem.
The problem basically states that given up to $10^5$ positive integers in the range $[1, 10^6]$, find the number of pairs of those positive ...
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Count of different ways to express N as the sum of given numbers
I'm working on a case and I need some help :)
I need to find number of ways and solutions itself to express N as the sum of given numbers.
So, Sum (N) = 600 and the numbers from which I need to get ...
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Model Counting for Sum of Conjunctive Formulas
Problem:
Let $X=\{x_1, ..., x_N \}$ be a set of binary variables. Each variable can be assigned to either $0$ or $1$ so there are $2^N$ possible assignments.
Input: We are given a positive integer $C$ ...
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Why does my code work: bijecting binary trees to Dyck paths
The number of Dyck paths (paths on a 2-d discrete grid where we can go up and down in discrete steps that don't cross the y=0 line) where we take $n$ steps up and $n$ steps down follows the Catalan ...
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Counting substrings that belong to a regular language
Given a regular language $L$ and a string $x$ give an efficient algorithm to count the occurrences of substrings $x[i,j] \in L$.
More in particular, I am looking for a linear time algorithm in the ...
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Number of substrings possible with even characters
Consider a string 'ABBAA'
Possible substrings with even number of characters are $4$
'ABBA' : Count of 'A' is even and 'B' is even
'AA' : Count of 'A' is even and 'B' is even - ($0$)
Similarly 'BB' ...
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Is there a #$P$-complete counting problem such that every (valid) instance of its decision version is a Yes-instance?
I want to know whether there is a decision problem, written EasyProblem, satisfying the follow property:
For every valid instance $x$, $x$ is a Yes-instance for EasyProblem (if we construct ...
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#perfectMatchings is self-reducible
How can one show that the counting problem:
Given a graph, output the number of perfect matchings
Is self reducible?
I found a hint in Moore's Chapter on Counting, Sampling and Statistical Physics:
...
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What is the solve of F(n,n) = F(n-1,n) + F(n, n-1) + 1 Where F(0,a) = 1 and F(a, 0) = 1 for every a
I'm given the following python function:
...
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Counting paths by their type
An edge-labelled directed graph is the data of $G = (V, E, l)$ where $(V, E)$ is a directed graph, and $l \colon E \to \mathbb{P}$ is some function. (For the graph I am considering, labels take values ...
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Iterate unique sets of integers
I'm trying to figure out of if there's a way to generate all unique sets of integers of length K, where each member has an upper bound of N, and a lower bound of M, without tracking which sets have ...
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Efficient Algorithm to Find the n-th Odious Number
An odious number is defined as an integer that has odd binary Hamming weight. I need an implementation of algorithm that finds the nth odious number, preferably recursive. Any ideas? A python script ...
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Polynomial time counter of solutions of 2SAT expression with pure literals
As per the title, is there any polynomial time algorithm to count the number of satisfying arguments for a 2SAT expression with pure literals? An even shallower case: Is there any such counter when ...
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Count number of ways in which atomic operation(s) of n different processes can be interleaved
PROBLEM: Count the number of ways in which atomic operation(s) of n different processes can be interleaved. A process may crash mid way before completion.
Suppose there are a total of n different ...
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Enumerating every "partnering" without repeating partners
I'm taking a class. In this class every week we have a partner. There are an even number of people in the class. We'd like avoid having repeat partners if possible so that everyone gets to work with ...
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Unambiguous context-free grammar for strings with at least as many a's as b's
I have designed this Grammar but it is ambiguous:
$$S\to aSbS \mid bSaS \mid aS \mid\epsilon$$
Would anyone help me make it unambiguous? Assume the alphabet is $\{a,b\}$.
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Prove that $\#k-colouring$ graph problem is $\#P-complete$
I need to prove, that the $\#k-colouring$ graph problem is $\#P-complete$. I want to construct the reduction from $\#3SAT$ problem, so $\#3SAT \leq \#k-colouring$. The reduction between the counting ...
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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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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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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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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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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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