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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 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
31 views

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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0answers
52 views

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
20 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
25 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: ...
0
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1answer
14 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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30 views

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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0answers
35 views

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 ...
2
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1answer
40 views

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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34 views

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
59 views

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
33 views

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
62 views

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 ...
2
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1answer
49 views

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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1answer
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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
52 views

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 ...
2
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1answer
61 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 $A$ ...
2
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1answer
39 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 ...
3
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1answer
50 views

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
86 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
42 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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0answers
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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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1answer
62 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
22 views

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
390 views

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 ...
0
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1answer
38 views

How do you find set of keywords present in set of words in linear time or log time?

I am trying to optimize a program, where I need to know whether a given set of keywords present in the set of words. I believe using the dictionary is the only way to optimize it. Any other technique ...
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3answers
80 views

Show that the following algorithm takes $O(n)$ time

You are given a linked list of size $n$. An element can be accessed from the start of the list or the end of the list. The cost to access any location is $\min(i,n-i)$, if the location being accessed ...
5
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1answer
57 views

Least number of guesses needed to determine all unknown subsets of a set

Say I have a set $\mathbb{S}=\{1,2,...,n\}$. I have an adversary who breaks up $\mathbb{S}$ into $k$ unknown and disjoint subsets. Denote this new set $\mathbb{A}$. I can guess any combination $s$ and ...
0
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1answer
92 views

Create a binary search tree from a sorted array: Space complexity reasoning

Here is the Python code. The solution is fairly common and is seen in most textbooks like 'Cracking the Coding Interview' and 'Element of Programming Interviews'. ...
0
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1answer
36 views

What will be big O complexity for this loop? [duplicate]

I am not able to understand time complexity of this for loop. While outer loop is O(n) the inner loop jumps certain calculation. How to find the complexity? ...
0
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1answer
34 views

Big O analysis for problem where number of items searched is unknown

Consider this problem: you are searching an array of elements and are comparing the square of the current element to some number K. Essentially, you are looking to see if the square root of K is in ...
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0answers
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Merge Sort Recurrence analysis for list of strings [duplicate]

I saw this question and tried to find out what the time complexity was: Using the recurrence relation for Merge Sort: $$T(n)\; =\; merge\_time\; +\; 2T(n/2)$$ Here, since we have a list of strings, ...
2
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1answer
84 views

How to find efficiently the minimum modification to avoid close consecutive numbers?

I have an array of sorted numbers: arr = [-0.1, 0.0, 0.5, 0.8, 1.2] I want the difference (dist below) between consecutive ...
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0answers
28 views

Worst case lower bound of binary search

For the question below, it is asking to prove the lower bound on the worst case is log(n). I have no problem proving this and the solution makes 100% sense to me. However, there is a comment at the ...
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1answer
24 views

Time complexity dependent on magnitude of input [duplicate]

I'm trying to analyze an algorithm that looks like this: def foo(L): for n in L: for x in range(n): ... What would be its time complexity?...
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1answer
28 views

Find the efficiency class

I have to find the efficiency class of this algorithm b = 3 a = 4 for i = 4 to n^2 if (i mod 2 == 0) a = a+2 else b = b*3 end for I ...
0
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1answer
117 views

Big-O notation for nested loops that might skip iterations

When you have an algorithm that may skip a lot of iterations due to a hash table lookup, do you still count the iterations that are exited immediately? Hypothetical example: ...
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1answer
121 views

Finding time complexity when while loop included [duplicate]

There are two sorted arrays nums1 and nums2 of size m and n respectively. Find the median of the two sorted arrays. Example 1: nums1 = [1, 3] nums2 = [2] The median is 2.0 Example 2: nums1 = [1, ...
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1answer
24 views

The time complexity of the Wikipedia version of Pollards $(p-1)$ algorithm

I am trying to understand the runtime of Pollard's $(p-1)$-algorithm as presented on Wikipedia. There the author writes that it takes $\mathcal{O}(B\log B\log^2n)$ time, but I do not see why. Here ...
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0answers
17 views

Power of 2 assumption in Divide and conquer [duplicate]

Currently doing an Algorithms course in my 2nd year of university (I am a maths student, but thankfully at Warwick University, we have quite a flexible degree). One of the topics we cover is the ...
0
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1answer
511 views

Complexity of finding a majority element

I was given a question that is stated that; Suppose you’re consulting for a bank that’s concerned about fraud detection, and they come to you with the following problem. They have a collection ...
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1answer
27 views

My code works, But how do I make this code run in a deterministic time?

The Problem: Given 3 inputs Bounce, Ball drop height, and ball view height. How do I calculate the number of times the observer can see the ball pass. So my code gives correct output, but it takes ...
2
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0answers
50 views

Tail recursion can't work with dynamic programming programs

I am doing some exercises on dynamic programming in order to get familar with this concept. I've noticed that most of the time it's not difficult to calculate the complexity of a program using dynamic ...
2
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1answer
43 views

minimum number of nodes that traverse all the graph

In the following graph, we can traverse entire graph if we select the nodes 0 and 2. I am looking for an efficient algorithm which returns this two nodes. Note that this is neither vertex-cover ...
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1answer
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0answers
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Upper bound on the average-case runtime of shell sort

I found that shell sort with the gaps of Fibonacci sequence has the lower bound complexity $\Omega(N \log N)$ in average cases. I want to know the upper bound complexity in average cases, so I write ...
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0answers
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Complexity of bisection method for finding an interval

Let $f$ be a continuous function and $[a,b]$ be an interval where $f(c)=0$ for some unique number $c \in [a,b]$ and where $f(a) f(b) \leq 0$. Suppose there exists a sub-interval $[a_0,b_0]\subset [a,b]...
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1answer
24 views

Running time based on smallest subtree

I've constructed an algorithm on (rooted) binary trees, where the running time at a node depends on the size of its smaller subtree, where we compute from each leaf upwards towards the root. More ...
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1answer
148 views

Finding the Kth largest element can be optimized to O(n) only if k is a constant?

There's a famous question posted on this site which asks about finding the $k$th largest element. Many answers are written there which optimized it and found algorithms with expectation of $O(n)$. ...