Questions tagged [time-complexity]

The amount of time resources (number of atomic operations or machine steps) required to solve a problem expressed in terms of input size. If your question concerns algorithm analysis, use [tag:runtime-analysis] instead. If your question concerns whether or not a computation will *ever* finish, use [tag:computability] instead. Time-complexity is perhaps the most important sub-topic of [tag:complexity-theory].

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18 views

What's the decoding time complexity of LT codes?

LT codes are practical fountain codes that are near-optimal erasure correcting codes. Simply stated, for encoding a $n$-block message, each packet first chooses a degree $d\in\{1,\ldots,n\}$ ...
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1answer
42 views

Time complexity of algorithm inversely proportional to size of sub problem?

Let's say I have an algorithm with time complexity $T_n = T_\frac{n-1}2 + 1$, $T_0 = 0, T_1 = 1$. Assume (Induction hypothesis) $T_n = C\log_2(n+1)$ for some $C$. $T_1$ imposes $C \geq 1$. Therefore ...
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Quick Clarification Question about Time Complexity in CLRS

I'm reading about the Hiring Problem in "Introduction to Algorithms" and read Interviewing has a low cost, say $c_i$, whereas hiring is expensive, costing $c_h$. Letting $m$ be the number of ...
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Complexity of Towers of Hanoi

I ran into the following doubts on the complexity of Towers of Hanoi, on which I would like your comments. Is it in NP? Attempted answer: Suppose Peggy (prover) solves the problem & submits it ...
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1answer
51 views

The padding argument in the proof of NTIME(n) ⊆ DTIME(n^1.2) implies Σ2-TIME(n^8) ⊆ NTIME(n^9.6)

In "Computational Complexity, A modern approach", Arora & Barak proof the following claim (Claim 5.11.2): Suppose that $\mathsf{NTIME}(n) \subseteq \mathsf{DTIME}(n^{1.2})$. Then $\Sigma_2$-$\...
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1answer
45 views

Time complexity of combinations of n pairs of parentheses

I have the following code snippet for combinations of n pairs of parentheses. ...
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2answers
290 views

Algorithm for scheduling unit time tasks with arrival times and deadlines

Suppose we have $n$ tasks to order over $n$ days. Each tasks takes 1 day to be completed. Each task has a start date when the task becomes available and a deadline when the task must be delivered. ...
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1answer
99 views

Improving time complexity from O(log n/loglog n) to O((log ((nloglog n)/log n))/loglog ((nloglog n)/log n))

Suppose I have an algorithm whose running time is $O(f(n))$ where $f(n) = O\left(\frac{\log n}{\log\log n}\right)$ And suppose I can change this running time in $O(1)$ steps into $O\left(f\left(\...
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What is the run time of this algorithm, written in pseudocode?

count = 0 for i = 1 to n: for j = 1 to i: count += 1 So from my understanding, we can break this up into 2 summations, by nesting the $j$ loop as a ...
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Fastest algorithm for transforming points into graph

Given a set of $n$ two-dimensional points in the plane $$\{ (x_1, y_1), (x_2, y_2), \dots, (x_n, y_n)\}$$ and a real number $M$, I want to transform this set of points into a graph with the points as ...
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Comparing asymptotic running time of two algorithms $\sqrt n$ and $2^{\sqrt{\log _{2}n}}$

Given two algorithms with their time-complexity $t_a(n)=\sqrt{n}$ and $t_b(n) = 2^{\sqrt{\log _{2}n}}$ and i have to show $t_b(n) = O(t_a(n)) $. I´ve made a program to check this statement and it ...
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Compare two complexity functions having the same asymptotic complexity

For a certain problem two solution algorithms (A1 and A2) with the following execution times have been found: $A1: T_{A1}(n)=4n^2 +7log(n^2)$ $A2: T_{A2}(n) = 4T(n/2) + log(n)$ Say, technically ...
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1answer
441 views

What is the time complexity in big-O notation of Algorithm X?

Knuth's Algorithm X is a recursive, nondeterministic, depth-first, backtracking algorithm that solves the NP-complete problem exact cover (EC). (It actually finds all solutions to EC.) I do not ...
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1answer
49 views

Time complexity of a 2-heap question

The problem statement is pretty straight forward: given an array of integers and a window size, return an array of doubles of the median of each window. arr = 1, 3, 5, 10, 6, 9, 2 k = 3 would yield ...
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1answer
34 views

What is the time complexity of FC_MRV algorithm?

I am studying CSP and read the papers on it.I wanted to know the time complexity of Forward checking with Minium Remaining Value algorithm.
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Is this computational complexity of the k-NN (custom distance) correct?

I read on a book that in general k-NN (no optimizations), given $d$ dimensions $n$ examples every computation of distance is $O(d)$. Since every example has to be compared with all the other ones, ...
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1answer
143 views

“Fuzzy” Chinese Remainder Theorem

I have some "fuzzy" congruences like these: \begin{align} \\ x&\equiv a_1 \mod 3 \text{ with } a_1 \in \{0,1\},\\ x&\equiv a_2\mod 5 \text{ with } a_2 \in \{2,3,4\},\\x&\equiv a_3 \mod 7 \...
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1answer
55 views

Are problems in NP $\cap$ coNP less difficult than those in NP-complete?

I am taking a complexity class now, and I struggle to understand the concept of "hardness": Assume that $L \in \textsf{NP } \cap \textsf{coNP}$. In means that under the assumption $\mathsf{NP} \neq \...
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Examples of simple/common problems with quadratic running time

Are there some examples of simple or common problems which have (in a practical sense) quadratic complexity? I would have thought there were a lot of them, but upon further investigation, I can't seem ...
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Which function grows faster: N Log N or N^(1+ε/√(log N)) [duplicate]

How would you go about solving this problem? I thought about using a limit infinity approach, but got confused and Wolfram Alpha didn't provide any explanation.
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Units of time in time analysis (frequency count method)

In time analysis, how many units of time will the piece of code z=2x+3y; take? will it take 1 unit of rime or 4 units of time ?
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What is the most efficient algorithm to compute polynomial coefficients from its roots?

Given $n$ roots, $x_1, x_2, \dotsc, x_n$, the corresponding monic polynomial is $y = (x-x_1)(x-x_2)\dotsm(x-x_n) = \prod_{i}^n (x - x_i)$. To get the coefficients (i.e. $y = \sum_{i}^n a_i x^i$), a ...
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How to get Algorithm complexty based on another 2 algorithms?

I had quiz last week and it says: suppose algorithms $A_1$ and $A_2$ have worst-case time bound $p$ and $q$, respectively. Suppose algorithm $A_3$ consists of applying $A_2$ to the output of $A_1$. (...
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1answer
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Optimizing the problem

I have a recurrence relation: $$f(a,b) = \begin{cases} 1 & (a,b) = (0, 0)\\ 1 & (a,b) = (a, 0)\\ 0 & (a,b) = (0, b)\\ 2a & (a,b) = (a,1)\\ f(a-1,b) + f(a-2, b-1) + f(a-1,b-1)...
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1answer
52 views

Understanding $O(2^n)$ time complexity due to recursive functions

Consider the following binary recursive fibonassi program: ...
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1answer
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how to proof ${ NPC \bigcap CO-NPC \ne \varnothing then NP = P ? }$

how proof ${\ \ NPC \ \ \bigcap \ \ CO-NPC \ne \varnothing }$ then ${NP = P ? }$
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Does the master theorem apply to T(n) = 3T(n/3) + nlogn?

I am given an example of a case where the master theorem does not apply, but it seems like it should apply. This was the reasoning: $T(n) = 3T(n/3) + n \log n$ with $ a = 3, b=3, f(n) = n\log n$ ...
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1answer
22 views

Partitioning a set based on binary predicate

Given a collection of objects $X = (x_0,x_1,...,x_{N-1})$ and a binary predicate $F$ which takes as parameters elements of the collection, find a better than $\mathcal{O}(N^2)$ algorithm which ...
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If NP is a subset of DTIME[n^O(log n)] then what happens?

If $\mathsf{NP}\subseteq \mathsf{DTIME}[n^{O(\log n)}]$ then what happens? Does it imply $\mathsf{NP}\neq \mathsf{EXP}$? Is there any other consequences such as $\mathsf{BPP}\neq \mathsf{EXP}$? Does ...
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1answer
46 views

Assume that NP = DTIME(2^sqrt(n)), prove that DTIME(2^sqrt(n)) = DTIME(2^n)

I tried using the padding argument to prove such a thing (as it appeared in Arora's book), but I am not sure how this technique will help me here. I am trying to get to a contradiction to the Time ...
2
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1answer
63 views

Interpretation of an asymptotic notation

Assume that we measure the complexity of an algorithm (for some problem) by two parameters $n$ and $m$ (where $m \le n$). What is the formal interpretation of the following claim: there is no ...
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1answer
30 views

How does $n^c \lg n, 0<c<1$ compare to other common time complexities

Between what two common time complexities would you place $n^c lg n, 0<c<1$? The following table illustrates the common time complexities. Source: wikipedia
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1answer
257 views

Modular exponentiation running time

I read on Wikipedia that modular exponentiation can be done in polynomial time. I've a few questions regarding it (sorry if they seem a bit easy – I'm not a comp sci student). Is it poly ...
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1answer
41 views

Description logics with decision problems within NP

Is there any description logic where important decision problems (e.g. abox consistency or concept satisfiability) lie within NP with respect to their time complexity? The well-researched family of $\...
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1answer
31 views

Analysis of Pan-cake sorting

i was implementing pan-cake sorting. We can implement it by taking largest element to start and flipping it recursively (Like selection sort). However it is mentioned that the A[i] has to be a ...
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1answer
111 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
257 views

Complexity class for polynomial algorithm for 3SAT, but with exponentially large values?

I have a hypothetical question: suppose there exists an algorithm that solves an NP-Complete problem polynomial time, but requires the computation of values that grow exponentially big (or small). ...
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3answers
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Measuring Complexity of Boolean Satisfiability Problem

How exactly is the complexity of a SAT solver measured? My main concern is that, for $N$ variables, you can have, e.g., an OR of $O(2^N)$ AND terms, which would take at least $O(2^N)$ time to process. ...
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What is the difference between super-polynomial time and exponential time?

What is the difference between super-polynomial time and exponential time? Any differences?
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How can I solve the recurrence $f(n) = 3f(\frac{n}{4}) + \log(n)$? [duplicate]

The master theorem didn't work here. I tried to do the substitution method but I ended up with an additional term: $2Σ(i \cdot 3^i)$. Also I should find the solution $g(n)$ such as $f=\Theta(g)$.
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1answer
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please tell Time complexity of following program [closed]

please tell the time complexitiy of the following code
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1answer
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How to find time complexity of this pseudocode

Recently, I came across a question about finding sum of all values in range $[low, high]$ in BST $T$. Then I formulated following algorithm to carry out that task: We do inorder traversal of ...
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1answer
25 views

Non-deterministic Turing machine for $L_1 = \{w\#0^n|w \text{ is a suffix of some $x$ in $L$ with } |x|=n\}$

Show if L is in NP, then also L1 is in NP $$L_1 = \{w\#0^n|w \text{ is a suffix of some $x$ in $L$ with } |x|=n\}$$ I know that if L is in NP, then there exists a NTM $M_L$ than accepts $x$, ...
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1answer
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Is SAT a single language or a union of languages?

I know that a language is in NP if a Turing machine can decide the language of its checking relation $\{\text{boolean formula }\#\text{ truth assignment | truth assignment is correct}\}$ in polynomial ...
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1answer
22 views

Time complexity to find out the number of ways to parenthesize N matrices

I am trying to figure out the $time$ $complexity$ to find out the number of ways we can parenthesize $N$ $matrices$. I have approached this problem as, say if we have $N+1$ matrices then we can ...
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2answers
162 views

Complexity of cyclic sort

I have this algorithm ("cyclic sort") to sort an array which contains unique numbers from 1 to $n$: ...
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1answer
101 views

Survival algorithm for Network deterministic failures

Consider an undirected network $G = (V,E)$ in which edge $e$ $\in$ $E$ fails after (deterministic) time $t(e) > 0$. Network failure occurs at the first instant in which $G$ is no longer connected. ...
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0answers
32 views

Find sub-matrix containing the maximum number of elements consisting only of 1's [closed]

I am trying to get help on it here, originally posted first at: https://stackoverflow.com/questions/59446920/find-sub-matrix-containing-the-maximum-number-of-elements-consisting-only-of-1s Basically ...
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1answer
30 views

What is $f(n)$ in $NTIME(n)\subseteq DTIME(f(n))$ if $CIRCUITSAT$ is in $P$?

If $CIRCUITSAT$ in $n$ variables and $m$ gates has an $O((nm)^c)$ algorithm for a fixed $c>0$ then $NTIME(n)\subseteq DTIME(O(f(n)))$ for large enough $f(n)$. What is the smallest $f(n)$ in $NTIME(...
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1answer
22 views

Arbitrary Turing machine run time analysis on the empty word

Consider $L = \{ \langle M,n \rangle : M $ accpets $\epsilon $ in less than $T(n)$ steps$\}$ This language is decidable because a decider can simulate $M$ on $\epsilon$ and accept if it accepts and ...