Questions tagged [runtime-analysis]

Questions about methods for estimating the increase in runtime of an algorithm as the input size increases.

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How to apply AC-3(Arc-consistency 3) algorithm in N-Queen problem?

I am building N-Queen Solver with java. I confused with AC-3 algorithm. I heard that AC-3 can be applied with backtracking algorithm before processing and during the search.The latter is called MAC-3 ...
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Choosing algorithms and/or data structures at runtime based on input characteristics

I've been reading about Adaptive Computing, i.e. the idea of computer programs taking feedback from the environment at runtime to improve the output in some way. More precisely, my current focus is in ...
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BigO time complexity of 3 nested for loops

I'm debating with a friend whether a particular function I wrote is $O(N^3)$ or $O(N \times M \times X)$ I believe it is the latter since all 3 variables differ in size. $N = 100, M = 50, X = 10000$ ...
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Shortcuts/Patterns for being able to calculate the running time of a loop/algorithm? [duplicate]

This is my first question here. I, like many people, suffer from the lack of the ability to be able to determine the running time of algorithms just by looking at them. I've picked up on a few ...
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26 views

Big oh notation run time [duplicate]

I have this Question , I want the answer and show me how to solve it Please : Analyze the running time of the following algorithm using Big-Oh notation ...
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1answer
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Understanding proof of upper bound on complexity of recursive computation of graph chromatic polynomial

This question is about section 2.3 of Wilf's ``Algorithms and Complexity'' https://www.math.upenn.edu/~wilf/AlgoComp.pdf in which he analyses the complexity of a recursive computation of the ...
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Growth rate and runtime [duplicate]

Sorry if this maybe a dumb question, just a little confused But with Big-Oh notation, does it measure the runtime or growth rate of an algo? or both?
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26 views

Karp-Rabin - what is the input for the worst case time complexity?

I'm trying to determine the input for the worst case time complexity of Karb-Rabin regardless of the used hash function. However, I'm seeing both of these answers on the Internet: String "AAAAAAAA" ...
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1answer
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Running time for Breadth-First-Search vs Depth-First-Search

Can someone explain why BFS is $O(V + E)$ whereas DFS is $\Theta(V + E)$. I understand the definitions of both notations, but I just don't see why the bound for DFS should be tighter than that of BFS. ...
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Convex hull partition of a set of points

Given a set $S$ of $n$ points in $\mathbb R^2$, denote by $\mathrm{convb}(S)$ the boundary of the convex hull of $S$. Let \begin{align*} S_1 &= \mathrm{convb}(S)\\ S_{i+1} &= \mathrm{convb}\...
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2answers
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Running time correct with Omega?

I have the following statement. I would say it's correct as it's either equal or higher than $\Omega(\log^{10}(n))$. Because: I know $\log(2^n) = n$. By that I would guess the same goes for $\log(n^{...
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1answer
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Big O of runtime operation n-m+1 [duplicate]

I have a loop that runs n-m+1 times where m = len(list_m) n = len(list_n) for i in range(n-m+1): Is this time complexity O(n-m+1)?
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Run time of pseudo code in big theta notation [duplicate]

I am looking for the run time of the following pseudo code. ...
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1answer
38 views

Worst Case Scenario for Quicksort algorithm with pivot element n/2

What would the worst case array look like if I decide to always take the element on the position $\frac{n}{2}$ as the pivot element? I know that if I choose the left or rightmost element as pivot ,the ...
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3answers
82 views

What should I call algorithms with non-linear non-constant time?

I am writing a paper in which I want to refer to a group of algorithms. Some of these algorithms are of complexity O(NlogN), and some of the are more complex (e.g ...
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2answers
379 views

All superlinear runtime algorithms are asymptotically equivalent to convex function?

Is it true that every algorithm with runtime complexity of $T(n)=\Omega(n)$ satisfies that $T(n)=\Theta(f(n))$ for some convex function $f$? All the examples that I could think of satisfy the above ...
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2answers
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Is purely functional programming in some situations less efficient than imperative programming?

I am used to implementing algorithms in imperative languages. Many of the algorithms I have implemented use hash maps, hash sets, mutable arrays, heaps, doubly linked lists, etc. I understand that ...
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Why does the knapsack dynamic programming solution has runtime of O(nW)? [duplicate]

Can you please help me analyse the runtime of the knapsack dynamic programming (which i have seen somewhere is O(nW))? This is the algorithm i an using: Define M[i,s] to be the maximum value that ...
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1answer
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Big Oh Time Complexity Involving 'for i in range(n)' [closed]

Given the code below and the comment analysis: ...
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1answer
490 views

Complexity of many constant time steps with occasional logarithmic steps

I have a data structure that can perform a task $T$ in constant time, $O(1)$. However, every $k$th invocation requires $O(\log{n})$, where $k$ is constant. Is it possible for this task to ever take ...
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2answers
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Does my solution converge to O(N) for worst-case time complexity?

Forgive me if this should be in StackOverflow or Mathematics instead! I was given the following question at an interview: ...
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28 views

Time and space complexity of a recursive problem (code included)

I am having trouble finding out the time and space complexity for this recursive solution. I have to create a list of words in order of word length. Each word, must be one character insertion off from ...
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Worst Case Analysis of a Multivariate Recurrence of a Graph Algorithm

I have a graph algorithm that runs in: $$ T(n, m) = \begin{cases} c_1 & n \leq 2 \lor m = 1\\ T(n - i,\ m - j - k) + T(i, k) + c_2 m + c_3 n & m \leq (n-i)i\\ T(n - i,\ m) + T(i, m) + ...
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1answer
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Are all hypothetical machine models for calculating runing time of an alogrithm same?

Im learning about time complexity analysis, and cant seem to figure out why do we consider a hypothetical machine that takes 1 unit of time for arithemitic and logical instructions and 1 unit of time ...
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How to prove that the time complexity of this algorithm is O($\sqrt{N}$)?

int n; cin >> n; int sum = 0; for (int i = 1; sum <= n; i++) { sum += i; } If I assumed that $N = 100$, the loop will run $13$ steps, ...
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1answer
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Mergable heap with no key knowledge cannot EXTRACT-MIN in $o(\log n)$ amortized time

We are looking into Fibonacci heaps in class at the moment, but I am stuck with this problem. Let $H$ be a mergable heap structure, by which is meant a data structure, where each element has a key, ...
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How to find big-O for an in-place perfect shuffle algorithm

I've found a simple algorithm to interleave two halves of an array in place. It involves swapping the first 1/2 of the items into the correct place, then unscrambling the permutation of the 1/4 of ...
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2answers
51 views

How to find the big o running time if the recursion function have different cases of recursion with different fraction of n?

How to find the big o running time if the recursion function have different cases of recursion with different fraction of n? If I have a recursive function like this for example (This is just an ...
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2answers
28 views

Running time complexity of finding maximal power of divisor that divides natural number

Given $n \in \mathbb{N}$, a divisor $p\vert n$, I would like to efficiently find $e\in\mathbb{N}$ with $p^e \vert n$, and $e$ maximal with this property. I will assume that multiplication/division of ...
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1answer
34 views

algorithm analysis - complex dependant nested loop

First of all, I know there are many questions like this on the site. But I think this case is a bit different. Consider the following code: ...
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1answer
23 views

Optimal scalability of a distributed algorithm

What's the optimal scalability of some algorithm when I implement it in a distributed manner? Intuitively, it seems to me that any algorithm can scale at most linearly with number of computing nodes....
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Highest stack of rectangles

Suppose we have a set of $n$ dimensional rectangles $R = \{(x_{i,1}, \ldots, x_{i,n}), i \in 1 \ldots k\}$. We want to create the highest stack in say the first dimension such that each side of the ...
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Reccurent runtime

If $T(n) = T\left(\frac{n}{2}\right) + 5$ (binary search), then the runtime in Big-O notation is $O(\log(n))$. If $T(n) = T\left(\frac{n}{2}\right) + 0$, then is it correct to say that the the ...
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1answer
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Euclidean algorithm and well define ness on the underlying set

Euclidean algorithm is given below: gcd($a$,$b$):   if $a=0$, return $b$   otherwise, return gcd($b \bmod a$, $a$) Let us first argue that the algorithm terminates. The reason ...
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Finding runtime of a recurrence relation with a fractional power

Consider the following algorithm and find the tightest Big-$O$: Assume $\texttt{multiplyKS}$($A,B$) is $O(n^{1.58})$ and $\texttt{Add}($A,B$)$ is $O(n)$. If my runtime is $T(n)$, I have: Lines 1 ...
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3answers
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What is the asymptotic complexity of the following code snippet?

for (i = 2; i < n; i = i * i) { for (j = 1; j < i / 2; j = j + 1) { sum = sum + 1; } } I know that the outer loop can run for a maximum of $n^2$ ...
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1answer
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Heap operating in time $\Gamma^{-1}(n)^2$

I have a priority queue implementation which I claim has the following worst case asymptotic run-times for the given operations: PEEK_MIN …………………………… O(1) POP_MIN…………………………… O( (INVERSE_Γ(n)) ^2). ...
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2answers
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Run-time of Hungarian algorithm - matrix formulation

There are many different explanations of the Hungarian algorithm. My favorite explanation is the one based on matrices, for example here, since it is very intuitive and easy to carry out in a ...
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1answer
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Djikstra algorithm analysis

My textbook says that the Dijkstra algorithm's runtime is $O(n) + O(m \log(n)) = O((n+m) \log(n))$. How did they come up with that? Dijkstra algorithm pseudocode: ...
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What would be the big-o time complexity of this scenario? [duplicate]

I am wondering what the time complexity of a for loop that increments the control variable, but also multiplies it inside the loop. For example ...
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2answers
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Running Time of Sorting Algorithm

Determine the asymptotic running time of the sorting algorithm maxSort. Algorithm maxSort(A) Input: An integer array A Output: Array A sorted in non-decreasing order ...
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1answer
69 views

Running Time for Finding Maximum

Consider the algorithm findMax that finds the maximum entry in an integer array. Algorithm findMax($A$) Input: An integer array $A$ Output: The maximum entry of $A$ ...
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1answer
74 views

Complexity of brute force primality test in the number of digits

I'm wondering how to express the complexity of a brute force primality testing algorithm in the number of digits the number under test has. The brute force algorithm just checks whether $n$ is prime ...
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1answer
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Why $\Theta(n^2)$ multiplication of coefficient required for canonical form of polynomial?

I was working through a textbook (Probability & Computing by Michael Mitzenmacher & Eli Upfal) and am not able to understand the following: Let $F(x)$ be given as a product $F(x) = \prod_{...
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2answers
374 views

Time complexity of Dijkstra's algorithm for sparse graph

I'm not sure I understand the answer to this question: Question 9. What is the running time of Dijkstra's algorithm in a graph that is sufficently sparse - in particular, $E=o(V^2/\log V)$, ...
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1answer
138 views

Why is a heap better than a linked list for implementation of a priority queue?

Using a heap, you have O(log(n)) insertion and O(log(n)) removal. Using a linked list, you have O(n) insertion and O(1) removal. Why is it better to have log-n for both than n for one and constant ...
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Runtime explanation of this function [duplicate]

I am trying to understand the runtime complexity of the below code in terms of n. I know that it is $Θ(n^{4/3})$, but I don't get why. I thought the outer loop runs $log(n)$ times, the second one ...
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2answers
99 views

Is there a useful algorithm with a decreasing asymptotic time?

Algorithmic complexity is usually increasing and almost always strictly increasing based on input size. This is logical since algorithms take time to execute steps, and for almost all problems, the ...
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1answer
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Why $xx^TM$ requires $O(dk)$ operations?

Suppose $x \in \mathbb{R}^d$ and $M \in \mathbb{R}^{d \times k}$. Why $xx^TM$ requires $O(dk)$ operations?
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4answers
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Why aren't primality tests easily linear in time complexity?

Why don't we consider them as linear? I don't understand. You just have to check for factorization up to sqrt of n. So it's even faster than linear. I assume it's not linear only if we compare the ...