Questions tagged [lower-bounds]

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Lower bound for merging $m$ sorted arrays (decision tree leaves count - permutations)

I need some help understanding how to calculate the lower bound on the time complexity of merging $m$ sorted arrays of length $n$. The bound should be $nm \lg(m)$. I need to prove this using a ...
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20 views

Useful conditions for proving super polynomial lower bound for some kind of recurrences

Given a recurrence of the form $\forall n,m.\ \ T(n,m)=\begin{cases}1,&,m=1\\\sum_i{T(n_i,m_i)}&,\text{else}\end{cases}$ Note: both $n_i$ and $m_i$ are dependent on $n,m$ so they should have ...
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49 views

Number of comparisons in array where each element appears n/k times [duplicate]

Given an array of $n$ elements with $k$ distinct elements, each appearing $n/k$ times, how can I show that the number of comparisons to the sort the array in the worst case is in $\Omega(n \log k)$?
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1answer
38 views

Lower Bound for Time Complexity of Pairing Problem

Given an array X and array Y both of length n, the pairing algorithm will return the elements of the arrays matched so that the smallest element in X will be matched with the smallest element of Y, ...
3
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1answer
42 views

Pebble game lower bound?

This paper says pebble games have super linear lower bound for every fixed $k$ https://dl.acm.org/citation.cfm?doid=62.322433. Why is it not considered proof of constructive example for a function in ...
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1answer
40 views

Dealing with test condition '=' for a while loop when determining a bound function/loop variant

The following is the definition of what a bound function for a while loop must satisfy: The bound function is an integer-valued, total function of some of the inputs, variables and global data that ...
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1answer
35 views

Asymptotics of a sinusoid

Consider the function $$ f(n) = 2n^2 |\sin(\pi \cdot n/2)|. $$ Which of the following classes does $f(n)$ belong to? $$ O(n^2), \Omega(n^2), \Theta(n^2), \omega(n^2), o(n^2). $$ I'm working in this ...
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1answer
32 views

Comparison-based lower-bound for finding duplicates in an array of $n$ numbers

Decision Problem: Given $n$ real numbers, give an algorithm that outputs "1" iff there are at least two numbers that are identical and outputs "0" otherwise. (Assume that comparison between any two ...
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2answers
71 views

Lower bound of disjointness by discrepancy?

I need to show that $Disc_\mu(Disj) \geq \frac{1}{2n+1}$ for any distribution $\mu: \{0,1\}^n \times \{0,1\}^n \to [0,1]$. Disjointness is defined as $Disj(X,Y)=\left\{ \begin{array}[ll]+1 & \...
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1answer
71 views

Why is finding minimum number of comparisons to sort $n$ elements so difficult?

In The Art of Computer Programming 2nd Ed, Vol 3, Section 5.3.1 then discuss a function $S(n)$ which is define as: $S(n)$ : The minimum number of comparisons that suffice to sort $n$ elements. ...
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1answer
31 views

Finding maximum takes at least $\lceil n/2 \rceil$ comparisons

We are given an array $A$ with $n$ elements, $n \in \mathbb{N}$ and all elements are in the set $\{1,2,3, \cdots, n \}$. I want to prove that finding the maximum in $A$ (that is, outputting the index ...
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1answer
325 views

What is an optimal algorithm?

I'm a computer science newbie and I thought I understood cases and bounds when I first studied them. I would take worst case as upper bound and best case as lower bound, but now I know that they are ...
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1answer
60 views

What is Ironic complexity? What are some good resources to learn about it?

The term "Ironic complexity" was coined by Scott Aaronson for the stuff Ryan Williams does in the area of complexity theory. Could anyone tell me what kind of problems and approaches does Ryan ...
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27 views

Dynamic Perfect Hashing and Lower Bound

I am writing a Seminar about dynamic perfect hashing and its lower bound by the FKS schema using the the adversary method mentioned here by using a Tree data structure. But somehow i don t get how ...
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64 views

Postive interval problem lower bound

I was trying to solve the question given below. Algorithm : Using divide and conquer technique, divide the input till we get a very small size array's ( let us say of size 2 ). Solve these small ...
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2answers
87 views

Why do you need at least ln(n!) many comparison to sort a list?

"If every element comparison (testing whether $a_i \le a_j$ ) provides at most one bit of information, argue that you need at least on the order of $\ln(n!)$ many tests/comparisons to sort the list." ...
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1answer
67 views

Do you >have< to define the upper and lower bound? (context: traveling salesman)

Do one have to define the upper and lower bound to be able to solve the tsp, or is that just an unnecessary intermediate step? And if so, why would one define those bounds? (context: the traveling ...
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2answers
120 views

Why are greedy algorithms used to find upper/lower bounds? (when they doesn't guarantee an optimal solution)

Take the nearest neighbor algorithm for the traveling salesman problem as an example. Why is it used to find the upper bound? When can't it guarantee an optimal solution? (Thanks to many comments ...
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1answer
213 views

Does finding a cycle with $\log n$ length in $\text{P}$?

Let $G$ be an arbitrary graph with $n$ vertices and we want to find a simple cycle with $\log n$ length. Is there exists a known polynomial algorithm for this problem?
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21 views

Complexity class without fixed-poly size circuit

$PP$ is shown to have no fixed-poly size circuit by Vinodchandran. Bounded inside the polynomial hierarchy, $\Sigma^2_p$ is also shown to possess no fixed-poly size circuit by Kannan. In notation, ...
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1answer
105 views

Using Yao's principle to find a lower bound

This is a HW question, so I'm not expecting any answers, just a general guidance/help. Definition. Given $\underset{\neq0}{\underbrace{s}}\in\left\{ 0,1\right\} ^{n}$, a function $f:\left\{ 0,1\right\...
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1answer
27 views

Showing $2^x$ is a lower bound

How do I show that $2^x - x^2 \in \Omega(2^x)$? Basically, I know that this means that $\exists a, x_0 \in \mathbb{R^+}, \forall x \in \mathbb{N}, a.2^x \leq 2^x - x^2$. I worked around a bit with ...
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2answers
49 views

findMax reduce to sort

Can I reduce the find max (or find min) problem to the sort problem? Because if so, knowing the lower bound for find max is Ω(n) I can also infer that the lower bound for sorting is Ω(n) too? I'm ...
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1answer
162 views

lower bound proof with adversary argument

We have to run a song on a Walkman, for that we need 2 full batteries. Let's say we have a mixed set of 30 batteries (15 are empty and and 15 are full) and then only way to test if the battery is full ...
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29 views

Query complexity of exact learning and combinatorial parameter

When defining the query complexity of exact learning for a concept $c$ (considered as a function from $\{0,1\}^n \mapsto \{0,1\}$) in a concept class $\mathcal{C}$, we often come across the following ...
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1answer
59 views

Minimum and maximum of sum of inverse degree of a graph

Suppose we have a simple undirected graph $G(V,E)$, where $V$ and $E$ are the set of vertices and edges respectively. we denote $d(v)$ as the degree of a vertex $v \in V$. I am interested to find ...
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57 views

Highest lower bound on an NP complete problem

What is the highest time complexity lower bound that has been proven on any (non-contrived) NP complete problem?
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1answer
223 views

Correctness of lower bound proof

I am working on this exercise with the purpose of learning how to provide proper proofs and I would like to know if my proof for the following problem is correct. Given a sorted array $A$ (of $n$ ...
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1answer
70 views

Proof of Lower Bound for Deterministic Distinct Elements Algorithm

There is a proof in this document (page 8, Section 4, Lemma 3: https://inst.eecs.berkeley.edu/~cs170/fa16/lecture-11-29.pdf) that mirrors a proof my professor gave in my algorithms class. The lemma ...
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1answer
23 views

Use substitute algorithm to prove T(n) =4T(n/2) + n^2 compact lower bound [duplicate]

how to use substitute algorithm to prove T(n) =4T(n/2) + n^2 compact lower bound. this is a algorithm of my class home work,but i dont know how to solve it.
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84 views

How hard is APPROXIMATE-#SAT? [closed]

It is well known that the problem of counting the satisfying assignments of SAT, namely the problem #SAT, is #P-complete. It is also suspected (somewhat less widely) that even deciding SAT should ...
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1answer
121 views

Is this lower bound proof for the comparison-based sorting problem correct?

Here is a lower bound proof for the comparison-based sorting problem: Any comparison-based sorting algorithm can be considered to work by putting elements into their final positions one by one (Take ...
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1answer
76 views

How to prove that a problem can't be solved in time $\mathcal O(n^{1/2 - \epsilon})$?

I have seen many problems in graph theory and in other related fields which admit a sublinear (in input size) running time $\mathcal O(n^{1/2})$ algorithm, where $n$ is the input size. I am not sure ...
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168 views

How to find lower bound of f(n) for master theorem

I'm studying the master method to solving a recurrence. It describes three cases, the last one of which depends on what lower bound a function f(n) has. I usually ...
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0answers
38 views

From SETH to circuit lowerbounds

Are there reductions from SETH (Strong Exponential Time Hypothesis) to lowerbounds against threshold circuits? (maybe for computing Boolean functions of the form OR-of-AND-of-OR) In threshold ...
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2answers
777 views

Is it possible to solve 3SUM in $O(n^2)$ time?

Problem: 3SUM Input: Three lists A, B and C of integers and an integer k. Each list contain $n$ numbers. Task: Decide whether there exists a tuple (a, b, c) ∈ A × B × C such that a + b + c = k. ...
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29 views

Circuit Lower bound for $EXP^{NP}$

By Burhman, Fortnow and Thierauf result Paper Link, we know that $MA_{EXP} \not\subset P/poly$. Also, we know that $MA \subseteq P^{NP}$ (or $\Delta_{2}^{P}$ in some literatures). By using the ...
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1answer
77 views

Tight bound on the number of intersections between a line and a triangulation

I'm interested in the maximum number of intersections that a line and a triangulation on $n$ points could have. More specifically, given $n$, we are interested in the worst-case (maximum) number of ...
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2answers
3k views

How to prove that matrix multiplication of two 2x2 matrices can't be done in less than 7 multiplications?

In Strassen's matrix multiplication, we state one strange ( at least to me) fact that matrix multiplication of two 2 x 2 takes 7 multiplication. Question : How to prove that it is impossible to ...
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4answers
567 views

Is $Ω(n\log ⁡n)$ the lower-bound for *all* sorting algorithms or *just comparison-based* sorting algorithms?

Is $Ω(n\log n)$ the lower-bound for all sorting algorithms or just comparison-based sorting algorithms? If the latter, is it possible for there to be general-purpose sorting algorithms which perform ...
2
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1answer
107 views

“Natural” reductions vs “Polynomial-time many-one” reductions (Karp Reductions)

For two problems $A$ and $B$ and a Karp Reduction $R$ from $A$ to $B$, we call the reduction $R$ natural if, for any instance $I$ of problem $A$, the size of $R(I)$ (as well as the possible numerical ...
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1answer
518 views

How to prove that matrix inversion is at least as hard as matrix multiplication?

Suppose we are given a matrix $A$ over real numbers and we want to computer the inverse of matrix $A$. There are various algorithms to do so and it also turn out that we can use matrix multiplication ...
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1answer
94 views

Find a subset in constant many queries

Black box of $f(x)$ means I can evaluate the polynomial $f(x)$ at any point. Input: A black box of monic polynomial $f(x) \in\mathbb{S}[x]$ of degree $d$. Question : $\mathbb{S} \subseteq \mathbb{Z}$,...
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1answer
4k views

how to find upper bound and lower bound of quadratic equation

I am relatively new to algorithms, I wrote one pattern matching algorithm and its running time is $O(n^2)$, I tried it by step count method, direct method and also the constant method which all yields ...
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1answer
329 views

Best Algorithm for searching for an index in an array such that A[i] = i

Recently i got a question in one of my exams about asking for an algorithm which searches an element in a sorted array such that $A[i] = i$. My algorithm was based on binary search and did a $O (\log ...
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0answers
200 views

Reductions: Lower Bound and Upper Bound

The question is from my complexity-theory course. Explain the concept of polynomial reduction between problems and explain how, and under what circumstances, lower bound and upper bound problems can ...
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1answer
82 views

Lower bounds: Detecting a length 2 path

Prove that determining if a non-directed graph of $n$ vertices has or doesn't have a length $2$ path requires time $\Omega(n^2)$, assuming that the graph is represented as an adjacency matrix.
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1answer
104 views

Proving that converting min-heaps to max-heaps requires time Ω(n)

Suppose I have a min-heap SH stored inside an array. I can perform the operations: view-min(SH) in $O(1)$ extract-min(SH) in $O(\log n)$ insert(SH) in $O(\log n)$ is-empty(SH) in $O(1)$ If I want to ...
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1answer
298 views

Doubt regarding address calculation in two-dimensional arrays

I am reading about the address calculation formulas for one and two-dimensional arrays. I have two related doubts concerning it. In one of the problems, we are asked to calculate the ...
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1answer
153 views

Lower bound on space of DFS keeping the running time linear

$\mathsf{DFS(G, u) \text{}}$, $G = (V,E)$ Input : A Directed graph $G$ and a source vertex $u$. Find : Is $v$ reachable from vertex $u$ for all $v \in V$ ? Model of computation : Word RAM , one ...