Questions tagged [lower-bounds]
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206
questions
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Hardness of boolean functions
For a boolean function $f:\{0,1\}^n\longrightarrow\{0,1\}$, $H_{avg}(f)$ is a function from $\mathbb{N}\longrightarrow \mathbb{N}$, termed as the average case hardness, if $\forall$ circuit $C_n$ of ...
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1answer
47 views
Lower bound on algorithm solving certain recurrence
I have to find the lower bound of the following recursion:
$A_1 = C_1 = p_1$, $B_1 = D_1 = 1-p_1$, $F_k = A_k + B_k$. Evaluate $F_n$.
\begin{align}
A_{k+1} &= (A_k + 2C_k) p_{k+1} + (1-p_k) p_{k+1}...
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How to Show that an Algorithm Depends on all N inputs (and thus has a lower bound of Omega(n)) [duplicate]
In the algorithm detailed on this page: Fastest Algorithm for Computing Expected Value, I was wondering how to prove that the algorithm depends on all $n$ of the inputs it is fed (and thus has $\Omega(...
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Is there a super-linear lower bound on the time complexity of all solutions of NP complete problems?
$P \ne NP$ would imply that any polynomial is a lower bound on the time complexity of any NP complete problem.
Is some non-trivial lower bound known at all?
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1answer
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Connection between Pseudo random generators and hardness
For a boolean function $f:\{0,1\}^n\longrightarrow\{0,1\}$ $H_{avg}(f)$ is defined as the largest $S(n)$ s.t. for all circuit $C_n$ of size $S(n)$, $\Pr_{x\in U_n}[C_n(x)=f(x)]<1/2+1/S(n)$. Here $...
1
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1answer
62 views
Existence of boolean function with exponential average case hardness
Show that for every large enough $n$, there is a boolean function $f\colon \{0,1\}^n\longrightarrow\{0,1\}$, whose average case hardness is exponential. The question is taken from Arora Barak ...
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76 views
What is the difference between saying there is no ϵ>0 such that a problem can be solved in $O(n^{2-\epsilon})$ time and $n^{2-o(1)}$ or $\Omega(n^2)$?
I have seen the formulations
there is no ϵ>0 such that a problem can be solved in $O(n^{2-\epsilon})$ time
a problem requires time $n^{2-o(1)}$
a problem requires time $\Omega(n^2)$
being used ...
3
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2answers
110 views
Information-theoretic limits for a weighing puzzle
Consider the following problem:
You are given $n$ coins with labels $1, \ldots, n$. You know that coins have weights $1, \ldots, n$, but you don't know whether the labels are correct (i.e. they can ...
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2answers
70 views
Why decision tree method for lower bound on finding a minimum doesn't work
(Motivated by this question. Also I suspect that my question is a bit too broad)
We know $\Omega(n \log n)$ lower bound for sorting: we can build a decision tree where each inner node is a comparison ...
2
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1answer
63 views
Counting circuits with constraints
Please forgive me if this question is trivial, I couldn’t come up with an answer (nor finding one).
In order to show that there are boolean functions $f : \{0,1\}^n \rightarrow \{0,1\}$ which can be ...
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1answer
26 views
Lower bound on comparison-based sorting
I have a question from one of the exercises in CLRS.
Show that there is no comparison sort whose running time is linear for at least half
of the $n!$ inputs of length $n$. What about a fraction of $1/...
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0answers
7 views
Lower bounds for orthogonal matrix multiplication
Is it possible, according to the current state of knowledge, that orthogonal matrices can be multiplied faster than arbitrary matrices?
More precisely, let $T(N)$ denote the worst-case time of the ...
1
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1answer
23 views
Communication Complexity for Product Distributions
In general for the (two-party) set disjointness problem for inputs of length n, we know that the parties need to communicate $\Omega(n)$. Surprisingly, today I discovered (if I understood correctly) ...
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2answers
29 views
Algebra for min/max bounds
I am trying to model some set operations which are only well-defined if one is a subset of the other. The way the sets are constructed, I'll have a series of constraints of the form $x \subseteq y$, ...
1
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1answer
25 views
Lower bound time complexity for obtaining an arbitrary entry in a hashtable
I just answered this question on StackOverflow, which asks for an efficient algorithm such that given a nonempty hashtable,
the algorithm should return a pointer to an arbitrary nonempty entry in the ...
1
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1answer
40 views
Decision tree lower bound for finding two array elements summing to zero
I have to solve this exercise:
Given an unordered array $A[1], \ldots, A[n]$ of positive and negative integers,
determine if there are two indices $i \neq j$ such that
$A[i] + A[j] = 0$. ...
1
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1answer
12 views
Resolution exponential lower bound… alternative proofs?
I am reading the Resolution proof system exponential lower bound via Haken's bottleneck method for the Pigeonhole Principle as presented in Arora and Barak's Computational Complexity: A Modern ...
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Finding the lower bound through decision trees
One way to find the lower bound of a comparison based algorithm is to use the decision tree. U have two questions regarding this method :
1) We know that the height of the tree is path that connects ...
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1answer
25 views
A Question related to the method of find lower bound : Trivial lower bounds
In Trivial lower bounds we just need to count the number of items in the input that needs to be processed and the number of items that need to be generated and the trivial lower bound time is then the ...
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2answers
85 views
Lower bound and worst case scenario
We know that the lower bound is the minimum amount of work needed to solve a problem. So for a given problem say x it has the best algorithm ( the most efficient algorithm to solve this problem ) say ...
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1answer
28 views
Why there is no polynomially large sequence of polynomial large weights that derandomize the isolation lemma?
I was studying the paper Derandomizing the Isolation Lemma and Lower Bounds for Circuit Size by Arvind and Mukhopadhyay and came across the following claim (Observation 1.2 on page 3):
"More ...
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22 views
What is an $O(n \log(n))$ binary sorting algorithm with a guaranteed low scaling constant on the run-time?
Let $O_c(f(n))$ denote that $c$ is the scaling constant for the run-time (e.g. $\text{run time} \leq c\cdot f(n) + B$ if $n$ is large enough)
The absolute lower limit on the run-time for a binary ...
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2answers
61 views
Efficiently computing minimal elements over partially ordered sets
I have a list of sets that I would like to sort into a partial order based on the subset relation.
In fact, I do not require the complete ordering, only the minimal elements.
If I am not mistaken, ...
3
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1answer
52 views
Standard information-theoretic lower bound?
There should be a simple argument, but I'm struggling to see it.
Suppose Alice has a string $x \in \{0, 1\}^n$ and sends a message $s = s(x)$ to Bob. And suppose that given $s$, Bob can reconstruct ...
4
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1answer
105 views
An example where the algorithm of Hopcroft and Karp performs poorly?
I have been trying to construct an example, where Hopcroft and Karp's algorithm for the maximum matching problem performs poorly (say at least $\Omega(\log n)$ rounds). However, all the examples I ...
3
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2answers
139 views
Why is the lower bound for sorting strings Ω(d + nlogn)?
I'm taking an Advanced Algorithms course. I'm currently studying efficient algorithms for sorting strings. In this chapter, it is provided a lower bound for the time complexity of $\Omega(d + n\log{n})...
4
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1answer
68 views
Conditional lower bounds on the running time of the single source shortest path problem
Just out of curiosity, I was wondering whether there is a conditional lower-bound on the running time of an algorithm for the Single Source Shortest Path Problem (on directed or undirected graphs). I ...
2
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2answers
329 views
Find both lower and upper asymptotic bounds for $T(n) = 2T(\frac{n}{2})+n^4$
So far we have learned Recursion Tree, Substitution Method, and Master's Theorem.
I'm not sure how we can find lower AND upper bounds.
I know that using Master's Theorem, we get $T(n) = \Theta(n^4)$, ...
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0answers
88 views
Decision Tree for searching an element in an n*n matrix
I just learnt decision tree concept in class. I have a question for homework. It says to prove that for searching an element in n*n matrix the lower bound is logn and prove it using decision tree.
My ...
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26 views
Minimum amount of rectangles to create a 2-dimensional matrix
From this codegolf question.
Consider an $r$ by $c$ matrix of nonnegative integers, called $M$. You also have a zero matrix of the same dimensions, called $N$. A "move" consists of replacing a ...
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1answer
40 views
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 ...
2
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0answers
25 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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51 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 $\Omega(n \log k)$?
3
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1answer
93 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
53 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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185 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 ...
1
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1answer
38 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 ...
1
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1answer
197 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
101 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 & \...
3
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1answer
93 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.
...
3
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1answer
35 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
4k 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
126 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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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
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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
111 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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195 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
263 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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1answer
147 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\...
