Questions tagged [lower-bounds]

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Lower Bound on Parity of Boolean Functions

Let's say we have boolean functions $f_1, \cdots, f_n$, each of which operates on pairwise disjoint variables (i.e. the variables for each function are unique to that function). Then, how can we show ...
3 votes
0 answers
56 views

Lower bound of continuous random walk

Let $X_1,...X_T$ be i.i.d. random variables supported on the $[-1,1]$ segment, with an expected value of 0 and positive variance. Let $H_t = | \sum_{i=1}^t X_i|$, and let $H = \max_{1\leq t\leq T} H_t$...
4 votes
1 answer
91 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 ...
4 votes
2 answers
2k views

Minimal number of comparisons - sorting $6$ elements

I've been thinking about sorting $6$ elements with the minimal possible number of comparisons. I can do it in $10$ comparisons but I've no idea if this is optimal. Or is there a better algorithm ? ...
0 votes
1 answer
35 views

Universal lower bound of the multi message problem

The multi message problem is: Let there be an undirected graph $G = (V,E)$ with $n$ vertices, and let $r \in G$. The algorithm sends a message $M_i$ of size $\Omega(\log(n))$ to each vertex $v_i$ ...
0 votes
1 answer
62 views

Given a boolean circuit that computes a boolean function, can we always find an equivalent circuit with optimal size?

Let's say that we have a decision problem $P$. Let's also say that $I_n$ is the set of all instances of size $n$ that exist for this problem, and that its cardinality is finite. There is a sequence of ...
0 votes
0 answers
28 views

Definition of an algebraic decision tree

I am trying to understand what an algebraic decision tree is but wikipedia lacks a formal definition, just an intuition. So I need to check if my understanding is correct. From what I have read it ...
0 votes
0 answers
33 views

What is an "almost tight bound"

I am doing some research for a paper that I am writing and one of the papers that I have come across that has some interesting results talks about an "almost tight bound". I am not a ...
1 vote
0 answers
50 views

Bounding function for Travelling Salesman Problem

I have been studying the Branch and Bound paradigm. I came across an approach to solve the Travelling Salesman Problem using branch and bound where a specific kind of bounding function was used. I've ...
2 votes
1 answer
83 views

Lower bound for ϵ-tester with one-sided error for the "2-injective" property of functions

An $\epsilon$-tester given an input and a property, is defined as follows: If the input holds the property then the tester should accept with probability at least $\frac 2 3$. Otherwise if the input ...
1 vote
1 answer
78 views

How to evaluate the tightness of a bound on a function?

I recently submitted a paper where in part of the paper I derived a bound on a function (note it is an upper bound). The benefit of the bound is that it is much less complex to compute in contrast to ...
1 vote
2 answers
118 views

Lower bound union of a unsorted array with sorted array

I read this link and I have similar question. Suppose given two Arrays $A$ that is sorted array with length $n$ and $B$ unsorted array with length $n$. We want to find union of two arrays (i.e. we try ...
0 votes
0 answers
45 views

Lower bound union of a sorted array and unsorted array [duplicate]

Suppose given two arrays $A$ and $B$ with length $n$. Array $A$ is sorted and $B$ is unsorted. Is there any lower bound for computing $A\cup B$?
0 votes
0 answers
28 views

Tight bounds for expected maximum of k binomial(n,p) IIDs

What is the tightest lower and upper bound for the expected maximum value of k IID Binomial(n, p) random variables I tried to derive it : $$Pr[max \leq C] = (\sum_{i = 0}^C {n \choose i}p^i(1 - p)^i)^...
2 votes
2 answers
77 views

What is the lower bound on retrieving an item in a collection if no arrays(Random access memory) are allowed?

I know that retrieving an item in a collection can be done in $O(1)$ time(on average) using hash tables. I would like to know if there is an algorithm that could be as performance without using arrays....
0 votes
0 answers
49 views

Lower bound of solving a optimization problem [duplicate]

Suppose given $k$-sorted arrays of numbers that contains total of $n$ elements. we try to choose $k$ elements in $k$ arrays (each arrays exactly one element) such that minimize difference between ...
0 votes
0 answers
14 views

non-linear lower bounds for polynomial time decision problems [duplicate]

Are there any decision problems that have deterministic polynomial time algorithms and proven non-linear lower bounds?
1 vote
4 answers
2k views

Worst run-time for 3 nested loop

Suppose we need to find a tight asymptotic bound on the worst case run time of the following program ...
1 vote
1 answer
299 views

Why does IP = PSPACE

Can anyone give an intuitive explanation to why IP = PSPACE, or at least one direction of it? I looked at many research papers but its very hard to understand the formalism unless you have a solid ...
6 votes
1 answer
202 views

Counting number of swaps to make two strings equal in linear time

The input to our problem is a pair of strings, say $x$ and $y$. We treat our alphabet size as a constant, i.e., our input is effectively a pair of arrays with the values therein bounded by a constant. ...
1 vote
1 answer
119 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}...
1 vote
3 answers
384 views

Best-case time: comparison-based sorting on a list of size n must make n-1 comparisons (reference to proof)

I am looking for a reference to a proof that for every list of size $n$ comparison-based sorting cannot make less than $n-1$ comparisons. Do you have a reference of a book that covers it (with page ...
0 votes
2 answers
236 views

Sorting algorithm which sorts half the possible inputs in linear time

Prove that there isn't any comparison sort algorithm which for an input of size $n$ can sort at least half of the permutations of the input in linear time. (For the other half the algorithm can ...
0 votes
1 answer
42 views

Lower bound on computing $x^n$

I know that we can compute $x^n$ in $\log n$. Are there any lower bound for computing $x^n$?
1 vote
0 answers
217 views

Adversary argument and proving a lower bound of an algorithm. How does it work?

I need to understand how adversary argument works to prove the lower bound of an algorithm. For now, I am looking to prove that a "certain" algorithm that takes in input array requires omega(...
0 votes
0 answers
154 views

Problems/properties of dynamic graphs with strong lower bounds

I know from [1] that the lower bound for the maximum hitting time of simple random walk on a dynamic graph is $\Omega(2^n)$. Smoothed analysis has been applied to the maximum hitting time [3] and ...
3 votes
1 answer
1k views

CLRS Question 8-6 Lower bound on merging sorted lists

I'm doing the CLRS Problems and there's a part I'm having trouble following. The question is: Part a) Given 2n numbers, compute the number of possible ways to divide them into two sorted lists, each ...
2 votes
1 answer
66 views

Coming up with an adversary strategy for a clique of maximum size

I’m having trouble coming up with a good adversary strategy for this problem: Input: a graph G Output: the maximum size of any clique in G Where the algorithm asks each time, “are vertices x and y ...
0 votes
1 answer
159 views

Lower bound on worst-case time complexity of all sorting algorithms neglecting reading input and accessing elements time

We know that the worst-case time complexity of any comparison sorting algorithm is $\Omega(n\log n)$. Is there a lower bound on the worst-case running time of sorting algorithms of any type? Not just ...
5 votes
5 answers
965 views

Prove a lower bound

Prove: $n^{5}-3n^{4}+\log\left(n^{10}\right)∈\ Ω\left(n^{5}\right)$. I always get stuck in these types of questions, where there is a $"-(xy^{z})"$ in the expression. Whenever I see the solutions for ...
0 votes
3 answers
179 views

Is it possible to prove that this algorithm is big Omega $n^2logn$ time complexity?

Considering the following recursive algorithm: $ T(n)= T(\frac{n}{2})+c_1(\frac {n}{2})^2+c_2n$. I was able to prove that this algorithm is $O(n^2 logn)$ I was trying to understand whether it is a ...
2 votes
2 answers
71 views

Showing asymptotic lower bound on log of recurrence

I'm trying to prove a lower bound on some computational problem, but in order to do it, I need an $\Omega(n\log(n))$ lower bound on $\log(T(n))$, where $T(n)$ is a recurrence defined as follows: $T(1) ...
0 votes
1 answer
61 views

Prove that the following algorithm is $\Theta(n^3)$ by induction

I have the following algorithm runtime: $T(1) = b $ for some positive constant. Otherwise, $T(n)=8T(\frac n 2) + 100n^2$ I am trying to prove that it is $\Theta(n^3)$ by induction. I proved that it is ...
2 votes
1 answer
101 views

$\log n$ lower bound for space complexity

I am currently reading Arora and Barak's Computational complexity. In Chapter 4 (Space complexity), they say the following: Since the TM's work tapes are separated from its input tape, it makes sense ...
1 vote
1 answer
56 views

Communication complexity of equality gap problem

I'm interested to know what is the biggest known $0\le \epsilon\le 1$ such that the $gap-EQUALITY$ problem that is defined by: $$f_\text{GEQ}(x,y)=\cases{1&$x=y$\\0 & $x$ and $y$ differ in at ...
0 votes
1 answer
308 views

Decision tree and information-theoretic lower bound

Consider the following problem : ...
0 votes
0 answers
202 views

How to prove that the lower bound of the Huffman coding problem is $\Omega(n \log n)$?

how to prove that the lower bound of the Huffman coding problem is $\Omega(n \log n)$? Here Huffman coding problem is Huffman encoding. For example, ...
2 votes
0 answers
80 views

Prove lower bound on boolean circuit

Given matrix $A \in \{0,1\}^{n \times m}$ with $n$ rows and $m = 2^n - 1$ columns. Where $j$-th column is binary decomposition of $j$ ($j = 1 \dots 2^n - 1$). For example, if $n = 3$: $ A = \begin{...
2 votes
0 answers
82 views

Number of planar graphs with linear edges, given a fixed embedding

Suppose we are given a set of $n$ points on the plane. How many different planar graphs can we form on those $n$ vertices, assuming that each edge must form a straight line between the two vertices ...
3 votes
1 answer
60 views

Number of planar graphs, given an embedding

I want to find an upper bound on the number of planar graphs with $n$ vertices, assuming that we are given some embedding for those vertices beforehand. In particular, Im interested in either showing ...
2 votes
1 answer
113 views

Information-theoretic lower bound for succinct string dictionary of the Unicode Name property

Background The literature on succinct data structures refers often to the “information-theoretic lower bound” of encoding data, i.e., the minimum number of bits needed to store the data – a concept ...
4 votes
1 answer
242 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 ...
7 votes
1 answer
10k 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 ...
6 votes
1 answer
2k 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 ...
0 votes
0 answers
109 views

Techniques to prove lower bounds on randomized algorithms

I am interested in proving lower bounds for AM-like languages. The usual techniques for lower bounds in non-probabilistic machines don't work for probabilistic machines. Intuitively, when I think ...
1 vote
1 answer
43 views

"Equality" problem in distributed computation

I recently started learning about distributed computation on graphs (not to be confused with parallel computation with threads). I have seen as a side note in a few lower bound proofs, a reference ...
0 votes
0 answers
30 views

How many strings for CLOSEST STRING lower bound to apply

In the CLOSEST STRING problem, one is given (bit-)strings $s_1, \dots, s_t$, each of length $L$ and an integer $d$. The question to be answered is whether there exists a string $s$ that has hamming ...
1 vote
1 answer
61 views

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 ...
1 vote
1 answer
121 views

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 $...
2 votes
0 answers
55 views

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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