the amount of time resources (number of atomic operations or machine steps) required to solve a problem or run an algorithm with respect to the input size.

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41 views

Linear Search complexity ϴ (1)? [on hold]

Suppose that we have an array $A[1...n]$ and this array has $m$ different keys. Is it possible for $n \rightarrow\infty$ the complexity to become $\Theta (m)$? Which means that if $m$ is constant then ...
-1
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0answers
10 views

PSPACE subset of EXPTIME [duplicate]

In computational complexity theory, PSPACE is the set of all decision problems that can be solved by a Turing machine using a polynomial amount of space. In computational complexity theory, the ...
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1answer
33 views

Time-complexity of recursive defined code [duplicate]

How would I set up a recursive formula for time-complexity for this code: ...
2
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1answer
23 views

Can we separate P and E?

Let $\mathsf E$ be deterministic exponential time with linear exponent. Do we know that the inclusion $\mathsf P\subseteq\mathsf E$ is strict? If so, what's the proof? The time hierarchy ...
10
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3answers
614 views

Is it really possible to prove lower bounds?

Given any computational problem, is the task of finding lower bounds for such computation really possible? I suppose it boils down to how a single computational step is defined and what model we use ...
0
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0answers
13 views

Creating an algorithm with a certain worse case runtime [duplicate]

The inputs are x sorted lists (in increasing order) and in each list there are j/x elements (we are assured the numbers will work out to be a natural number. eg: j = 9, x = 3 L1 = [1, 2, 5], L2 = ...
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1answer
30 views

Whats is the meaning of polynomial run-time in input size ? [duplicate]

If an algorithm runs in exponential time with exponential input then we say it runs in polynomial time ? Why ? Doesn't the algorithm run in exponential time anyway ? How the input size affects ? ...
-1
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1answer
16 views

Scalar by N component vector multiplication faster than O(N)?

Is there a way to multiply scalar by vector faster than just multiplying each element of the vector by that scalar? It feels to me that there should be some exploit to do that. After all we will ...
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0answers
35 views

Running time of partial algorithms

What is the correct term for the maximal running time of a given algorithm on all inputs of length bounded by given $n$, on which the algorithm halts? Assume, if necessary, that the halting problem ...
2
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1answer
41 views

What is the implication of NP-completeness if P=NP?

If a certain problem $X$ is NP-complete and $P\neq NP$, then $X$ is not polynomial. But we still don't know that $P\neq NP$, so in theory $X$ may be polynomial. Does the fact that $X$ is NP-complete ...
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1answer
65 views

Is this problem P or NP?

Given a set of whole numbers $M=\{z_0, ..., z_n\}$ Are there $z_i$ and $z_j$ with $i \neq j$ but $z_i = z_j$? Is this Problem (surely or only probably) in $P$ or in $NP$? Is it $NP-hard$?
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0answers
9 views

Creating knapsack instance from 2 instances with specific number of solutions

Let 2 instances of knapsack. (items 1...n, with weights $w_1$,$w_2$,.. and some $W_1$, and items 1',2',..m' with $w'_1,w'_2$,.. and $W_2$ size of the container). Suppose now that the first instance ...
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1answer
50 views

All optimisation problems have equivalent decision problems

How can we prove the theorem that every optimization problem has an equivalent decision problem, and the optimisation problem is at least as hard as that decision problem? And secondly, I'm not sure ...
1
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2answers
26 views

Time complexity for searching $k-th$ element from starting and ending of a linked list

What are the time complexities of finding $8th$ element from beginning and $8th$ element from end in a singly linked list? Let $n$ be the number of nodes in linked list, you may assume that $n > ...
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3answers
40 views

Complexity of BST [duplicate]

I have the following pseudo-code for printing all nodes of a BST : ...
3
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1answer
69 views

Why is the complexity of this nested for loop not $O(n^2)$?

I have the following pseudo-code: mystery(n): if n <= 50 : for i = 1 ... n : for j = 1 ... n : print i*j else : mystery(n-1) For ...
9
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2answers
128 views

Efficient algorithm to compute the $n$th Fibonacci number

The $n$th Fibonacci number can be computed in linear time using the following recurrence: ...
2
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2answers
37 views

Contradiction between best-case running time of insertion sort and $n\log n$ lower bound?

If the best case for Insertion sort & bubble sort is $O(n)$ then how is lower bound for any comparison sort is $\Omega(n\log n)$? I mean, $O(n)$ is obviously smaller than $\Omega(nlogn)$. What am ...
1
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1answer
35 views

IS and matching

I have 2 different but similar problems, one belongs to NP and one to L and I don't understand why. First problem: Input: an undirected graph G with n^2 vertices. Question: Is there exist in G a ...
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2answers
67 views

What is complexity of checking whether a natural number is a perfect square? [closed]

As the title says, what is complexity of checking whether a natural number is a perfect square?
2
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1answer
28 views

Solving recurrence relation

I am trying to find a $\Theta$ bound for the following recurrence equation: $$ T(n,p,k)=T(n,p,k/2)+T(n,p/4,k)+T(n/8,p,k)+npk $$ ...
1
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1answer
32 views

Big Omega of 3-Sum Algorithm [duplicate]

An optimized algorithm for the 3-sum problem with an input array N has O(N^2logN) however I read that the Big Omega for this ...
4
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3answers
100 views

Termination in infinite-time

Does it make sense to speak of algorithms that take an infinite amount of time to terminate? In particular, suppose we have a loop with a bound function that is initially positive and is decreased ...
1
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1answer
43 views

What is the correct representation of Master Theorem?

What I'm taught in my class - $T(n)=aT(\frac{n}{b})+\theta(n^k\log^pn)$ where $a\geq1$, $b>1$, $k\geq1$ and $p$ is a real number. if $a>b^k$ then, $T(n)=\theta(n^{\log_ab})$ if ...
3
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0answers
80 views

Are there any algorithms where the recovery of a witness changes the time complexity?

In many algorithms, such as the solution to the longest-subsequence problem using dynamic programming, finding the length of an answer (or signaling the nonexistence of an answer) is easy, but ...
0
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2answers
87 views

What is the complexity of recurrence $T(n) = T(n-1) + 1/n$ [duplicate]

What is the complexity of the follwoing recurrence? $$T(n) = T(n-1) + 1/n$$ I highly suspect the answer to be $O(1)$, because your work reduces by $1$ each time, so by the $n$th time it would be ...
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3answers
967 views

Has the graph isomorphism problem been solved?

Wikipedia's graph isomorphism problem page would seem to indicate that, no, it has not been solved. However, a friend of mine pointed out A Polynomial Time Algorithm for Graph Isomorphism . I am not ...
1
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1answer
31 views

Asymptotic analysis for quicksort on special case

I have the following problem for homework: Given an array of the form $[m+1, m+2,..., n, 1, 2,..., m]$ as an input, analyze quicksort's run time complexity. TIP: check for $m > \frac{n}{2}$ and ...
3
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1answer
141 views

Is it axiomatic that the Time Hierarchy Theorem holds true in all relativized worlds?

I learned from this post that ${\sf DTIME}^{\text{EXP}}(n^k) \neq \text{EXP}$ for a fixed $k$ for otherwise the Time Hierarchy Theorem would fail in that relativized world. However, is it possible to ...
2
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1answer
170 views

Why does the Time Hierarchy Theorem not relativize?

Is it true that $DTIME^A(n^k) = EXP$ for any fixed $k$ and EXPTIME-complete oracle $A$? If not, what do these complexity classes equal and why (because I know that $P^A = EXP$ for any ...
1
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3answers
63 views

If A is in P and B is non-trivial, then A ≤p B [duplicate]

On wikipedia's article on Polynomial-time reduction it states: Every nontrivial decision problem in P (the class of polynomial-time decision problems, where nontrivial means that not every input ...
6
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2answers
203 views

algorithm time analysis “input size” vs “input elements”

I'm still a bit confused with the terms "input length" and "input size" when used to analyze and describe the asymptomatic upper bound for an algorithm Seems that input length for the algorithm ...
2
votes
1answer
57 views

$T(n) = \sqrt{n}\,T(\sqrt{n}) + n\log n$ [duplicate]

I tried to solve the recurrence $T(n) = \sqrt{n}\,T(\sqrt{n}) + n\log n$ with the master theorem but I can't get it to work. How many arrays exist in each step in the recursion tree? Or can I solve ...
1
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0answers
32 views

Connection between formula size and time complexity

Supposing we have a problem $P$ with input size $n$ encoded as a boolean formula $f$ in $n$ variabes which is a multilinear polynomial. Let $f$ have the smallest degree. Is there a connection between ...
3
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1answer
64 views

Hierarchy of complexity classes $\bigcup_{c > 0} \mathrm{Time}(2^{c \log^k n})$, w.r.t. $k$

This is a true/false question: For each integer $k > 1$, define the complexity class $\sf QP_k := \bigcup_{c > 0} Time(2^{c \log^k n})$. Then for all integers $k > 1$, $\sf QP_k ...
2
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0answers
60 views

#SAT Complexity

I have been looking at algorithms for solving #SAT and calculated that simply extending a SAT algorithm like DPLL by adding the negation of a solution to the original formula and solving again takes ...
1
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1answer
24 views

Implement opposite() method to tell if there are two opposite numbers, (x,-x)

Let a dictionary with the operations insert(), delete() and search(). Each one of them ...
0
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0answers
62 views

Time Complexity of Queue Problem

What is the time complexity of the following problem? Definitions Given a discrete time axis, define a FIFO as a queue unit with the commands: PUSH (data to back of queue), POP (the head of the ...
2
votes
1answer
32 views

Complexity of Independent Set on Triangle-Free Planar Cubic Graphs

I know that IS (is there independent set of size at least $k$?) on planar cubic graphs is NP-Complete, and IS on triangle-free graphs is also NP-Complete. But how about IS on triangle-free planar ...
9
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1answer
342 views

Is there an efficient algorithm for expression equivalence?

e.g. $xy+x+y=x+y(x+1)$ ? The expressions are from ordinary high-school algebra, but restricted to arithmetic addition and multiplication (e.g. $2+2=4; 2.3=6$), with no inverses, subtraction or ...
2
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1answer
86 views

Are there any coP problems

Is there a notion of coP problem? Also is there a notion of every problem being reducible to one problem in P (like 3SAT in NP completeness)?
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1answer
37 views

Time complexity for two multiplications modulo $p$

The time complexity of computing $MK\bmod P$ is $O((\log n)^2)$. What is the time complexity of computing $MK^2\bmod P$? Is it $O(2(\log n)^2)$ or $O((\log n)^2)$?
3
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3answers
157 views

Shouldn't complexity theory consider the time taken for different operations?

I have read the answer found here which considers the size of integers when doing comparisons and how that affects on the basic cost of comparison. I am trying to understand why each basic operation ...
2
votes
1answer
122 views

Average time to solve maze through backtracking

Given a set A consisting of all possible solvable mazes on an n by n square grid, what is the average running time to solve the mazes in A using a standard backtrack algorithm with no optimizations? ...
1
vote
2answers
44 views

What is the complexity of finding the two prime numbers a composite number (used in RSA Protocol) is made of?

I am aware that as the number increases in Digits the process of locating the two prime numbers that when multiplied produce the given number is increased as well. I also know that is it somewhat ...
2
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1answer
58 views

What is the lower bound for finding the third largest in a set of $n$ elements?

The problem is easy to describe: What is the lower bound for finding the third largest in a set of $n$ elements? Particularly, do we have to know both the largest and the second largest for ...
1
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1answer
32 views

Big-O Notation for Menezes-Vanstone Elliptic Curve Cryptography?

I need someone help me about . how can compute time complexity for this algorithm (Menezes-Vanstone Elliptic Curve Cryptography). I have spent much time reading journals and papers but as yet have ...
1
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1answer
79 views

How many comparisons in the worst case, does it take to merge 3 sorted lists of size n/3?

How many comparisons in the worst case, does it take to merge 3 sorted lists of size n/3? (where n is a power of 3) I was told it takes: $$2(n-2) + 1 = 2n-3$$ However, I can't seem to figure out ...
1
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1answer
40 views

Sorting with a recursive oracle

It is known that the runtime complexity of sorting is $\Theta (n \log n)$. But what if we have, for every input array of size $n$, an oracle that can sort any array of $k<n$ numbers in constant ...
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1answer
55 views

Master Theorem Questions?

NOTE: I asked this on mathstackexchange, but didn't get the responses I wanted, thought I should post in CS. Sorry if i did something wrong but i am a newbie. State the asymptotic (worstcase) ...