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4
votes
0answers
32 views

One way communication complexity lower bounds techniques

I have been teaching myself communication complexity. I am starting to understand the general methods for proving randomized lower bounds. However, I don't yet understand what techniques are ...
1
vote
2answers
93 views

If an NP-complete problem is shown to have a non-polynomial lower bound, would that prove that P != NP?

I understand that the Cook-Levin theorem proved that any NP problem is reducible to an NP-complete problem, which signifies that if a polynomial-time algorithm for an NP-complete problem is found, it ...
1
vote
1answer
22 views

number of comparison in sort algorith with special operation

Let's define: $ a_i:a_j \Longleftrightarrow a_i < a_j;\ a_i=a_j;\ a_i > a_j $ So it is similiar to normal operation $<$, but $:$ give information when elements are equal. I want show that ...
7
votes
1answer
113 views

Find the central point in a metric-space point set, in less than $O(n^2)$?

I have a set of $n$ points which are defined in a metric space – so I can measure a 'distance' between points but nothing else. I want to find the most central point within this set, which I ...
6
votes
1answer
440 views

Least number of comparisons needed to sort (order) 5 elements

Find the least number of comparisons needed to sort (order) five elements and devise an algorithm that sorts these elements using this number of comparisons. Solution: There are 5! = 120 possible ...
2
votes
3answers
157 views

In complexity, why do we find upper bounds, not lower bounds?

In algorithms we use to find Big-O (upper bound), Big-omega (lower bound) and Big-Theta but why we are always interested in finding upper bounds instead of lower bounds?
2
votes
2answers
70 views

Can you get O(n) with a word frequency algorithm?

By a word frequency algorithm: An algorithm gets a document as an input, and returns each unique word along with the number of times it has appeared in the document. For example: in:"Hello my name ...
4
votes
2answers
85 views

Algorithm to find sequence of minimum moves to sort 13 card hand

Just for fun I am trying to write a program to sort the 13 cards (from a standard pack of 52) in a Bridge hand by performing human-like moves on the hand. A sorted bridge hand is arranged by suit, ...
6
votes
1answer
81 views

TM recognizing $0^n1^n$ requires Ω(log n) space

I am trying to prove that any deterministic 1-tape Turing Machine which recognizes the language $L = \lbrace{0^n1^n | n \geq 0 \rbrace}$ requires $\Omega(\text{log }n)$ space. I believe this can be ...
0
votes
0answers
21 views

Skip lists with probability p [closed]

Let's say we have a skip list that in which probability $p$ is the chance that an element will get promoted. What boundary for height can we get with probability $1-1/n$? What about expected search? ...
3
votes
1answer
39 views

Lower bound for maxima on 2D plane

Given $n$ points $(x_1, y_1), \ldots, (x_n, y_n)$ on a 2-dimensional plane. A point $(x_1, y_1)$ dominates $(x_2, y_2)$ if $x_1 > x_2 \land y_1 > y_2$. A point is called a maxima if no ...
6
votes
1answer
138 views

Is there an intuitive proof for the existence of hard functions?

I am referring to the theorem on page 115 of the book by Arora and Barak, which states that, ``For every $n>1$, there exists a function $f:\{0,1\}^n \rightarrow \{0,1\}$ that cannot be computed by ...
-2
votes
1answer
24 views

Quadratic lower bound for deciding the set of palindromes

How to prove a single tape Turing machine needs at least n squared time to decide palindrome? This is an exercise from the "computational complexity - a modern approach" book.
4
votes
1answer
38 views

Average case lower bound for sorting

The $\Omega(n\lg{n})$ lower bound for sorting in the comparison model is well known. Is there a similar average case lower bound for sorting in the comparison model and if so, which random ...
7
votes
1answer
148 views

Searching the space of permutations

I'm given n objects, and a set of n permutations of these n objects (out of n! total permutations). There is a true underlying permutation, which I know is one among the set of n permutations, but I ...
5
votes
1answer
209 views

Lower bound for finding majority element in a sorted array

Suppose $A$ is a sorted array with $n$ elements. I want to know whether we can determine if there are majority elements in $A$ with time complexity $O(1)$. Recall that a majority element of $A$ is ...
1
vote
1answer
30 views

Particular function communication complexity computation

Consider a boolean function $f:\{0,1\}^n\rightarrow\{0,1\}$. If $f$ satisfies $f(\bar{0})=0$ where $\bar{0}$ is vector of $0$, $f(x)=1$ with every $0/1$ vector of hamming weight $1$, then ...
1
vote
2answers
32 views

Morgenstern proof for FFT lower bound

I looked at my notes from a class about fast forier transform , and the professor proved in class theorem thanks to Morgenstern , first he defined linear algorithm as a algorithm that inly uses ...
10
votes
3answers
845 views

Is it really possible to prove lower bounds?

Given any computational problem, is the task of finding lower bounds for such computation really possible? I suppose it boils down to how a single computational step is defined and what model we use ...
6
votes
2answers
181 views

Can element uniqueness be solved in deterministic linear time?

Consider the following problem: Input: lists $X,Y$ of integers Goal: determine whether there exists an integer $x$ that is in both lists. Suppose both lists $X,Y$ are of size $n$. Is there a ...
3
votes
1answer
30 views

Lower bounds for space with some probability of error

There is an information theoretic lower bound of $\log_2 {U \choose x}$ for the number of bits to represent a subset of $x$ elements chosen from a universe of size $U$. We can in principle use this ...
7
votes
1answer
92 views

Determining if $G$ contains $K_4$ as a minor in polynomial time

I am trying to devise an algorithm for determining if an undirected graph $G$ contains $K_4$ as a minor. I was able to show in a previous problem how to test for $K_{2,3}$ by looking at all pairs of ...
1
vote
1answer
48 views

Lower-bounds of a given problem

I have the following problem: You have n objects that have identical weight except for one that is a bit heavier than the others. You have a balance scale. You can place objects on each side ...
0
votes
1answer
36 views

simple lower bound for constructing a Spanning tree

i have to demonstrate that under the assumptions{Bidirectional Links, Total Reliability (no error during the execution), Connectivity, Distincts ids values, Multiple inititators (entities that starts ...
2
votes
1answer
74 views

What is the lower bound for finding the third largest in a set of $n$ elements?

The problem is easy to describe: What is the lower bound for finding the third largest in a set of $n$ elements? Particularly, do we have to know both the largest and the second largest for ...
3
votes
1answer
40 views

Understanding the flaw in a proof attempt of the Communication Complexity of Equality

I'm new to communication theory and I've been wondering where the following simple argument fails: Equality Problem We have two players, player 1 Alice who gets an $n$-bit vector $X$ and player 2 Bob ...
8
votes
1answer
121 views

Problems that provably require quadratic time

I'm looking for examples of problem which has a lower bound of $\Omega(|x|^2$) for input $x$. The problem needs to have the following properties: $\Omega(n^2)$ runtime proof for any algorithm - ...
4
votes
1answer
146 views

Lower space bound on a turing machine accepting palindromes

Let $$ PAL = \lbrace x \in \lbrace 0, 1, \# \rbrace^* | x = rev(x) \rbrace $$ How do I show that a turing machine deciding $PAL$ must use space $\Omega(\log n)$? I have a feeling that I need to use ...
1
vote
1answer
132 views

What is the min # of moves to sort an array from 1 to n?

Problem: You are required to sort an array with numbers from 1 to n. You can do a "move", which means choosing one element and moving it to any place you want (insert to any place, not swap). Prove ...
6
votes
0answers
86 views

Problems with Θ(n³) complexity on TMs with lower bounds by communication complexity arguments

One of the most used simple examples of application of Communication Complexity is the $\Omega(n^2)$ lower bound for recognizing palindromes of length $2n$ on a single tape Turing machine. Is ...
0
votes
1answer
28 views

How to bound a running time equation? [duplicate]

I simply need a standard way to find the upper and lower bound of a running time equation (please no shortcuts that only work for this specific problem).... Example: $T(n)=\frac{c}{5}(4^{\left ...
-3
votes
1answer
84 views

exponential lower bound on boolean formula conjunctions, what complexity class? [closed]

This new paper A Lower Bound for Boolean Satisfiability on Turing Machines by Hsieh asserts an exponential lower bound for a TM time complexity on a problem of finding whether a solution exists to a ...
3
votes
1answer
86 views

Lower-bounding the Membership Problem in the Bitprobe Model

I am working through the following paper "Data Structures for Storing Small Sets in the Bitprobe Model" by Radhakrishnan et al. and am confused regarding one of their arguments about a lower bound. ...
5
votes
1answer
79 views

Linearithmic lower bound for 1-D “distinct” closest pair of points problem

The 1-D distinct closest pair of points problem is as follows: Given a set of n distinct integer points on real line, find a pair of points with the smallest distance between them, here the distance ...
1
vote
1answer
42 views

Lower bound on number of comparisons needed to search for a number in a sorted 3-d array

Suppose we have an $N \times N \times N$ 3-d sorted array meaning that every row,column, and file is in sorted order. Searching for an element in this structure can be done using $O(N^2)$ comparisons. ...
2
votes
1answer
231 views

Traveling Salesman: how to use a lower bound?

Let me preface this question by giving some helpful background material. I'm trying to solve the traveling salesman problem using branch and bound. Concretely, for a partial solution, I'm using the ...
6
votes
1answer
45 views

Implications of the $\Omega(\frac{2^n}{n})$ circuit lower bound being tight

There is a basic result in circuit complexity that says: There exists a language that cannot be solved with circuits of size $o(\frac{2^n}{n})$. The argument is a simple counting argument on the ...
2
votes
0answers
197 views

Trying to understand the Gilmore-Lawler lower bound

For a class project we're developing a software that solves a common optimisation problem. After some research we've found out that our problem is called QAP (Quadratic Asssignment Problem) and the ...
1
vote
1answer
39 views

Help in geometrically understanding “Linear Decision Trees”

In the words of (http://www.cs.utah.edu/~suresh/5962/lectures/17.pdf, section 17.2), "Each $f(x)$ can be interpreted as defining a hyperplane in $R^n$. Thus, tracing a path through the tree computes ...
1
vote
1answer
43 views

Doubt in the correctness of decision tree models for constructing a lower bound

If we were to intuitively construct a lower bound for searching an element in a list $A$ containing $n$ integers, it would be in $\Omega(n)$. But with the decision tree model, the number of leafs is ...
4
votes
1answer
135 views

Lower bound on running time for solving 3-SAT if P = NP

Is there a lower bound on the running time for solving 3-SAT if P = NP. For instance, is it known that 3-SAT can't be solved in linear time? What about quadratic?
3
votes
0answers
35 views

Adversarial bin packing

An adversary gives you a set of items whose total size is $x$ (he gets to choose how $x$ is distributed. e.g. there may be $k-1$ items of size $\frac{x}{k}$ and 2 items of size $\frac{x}{2k}$). The ...
1
vote
3answers
533 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 ...
7
votes
0answers
79 views

guillotine cuts versus general cuts

Cutting problems are problems where a certain large object should be cut to several small objects. For example, imagine you have a factory that works with large boards of raw glass, of width W and ...
2
votes
1answer
549 views

How to analyze the lower bound of the horse racing problem using adversary argument?

Consider the following problem: Give an algorithm to find the $1^{st}, 2^{nd}, 3^{th}$ fastest horses from 25 horses. In each round, at most 5 horses can race and you can get the exact position of ...
5
votes
2answers
862 views

Can you do an in-place reversal of a string on a vanilla turing machine in time $o(n^2)$?

By a vanilla Turing machine, I mean a Turing machine with one tape (no special input or output tapes). The problem is as follows: the tape is initially empty, other than a string of $n$ $1$s and $0$s ...
3
votes
1answer
583 views

Lower-bound complexities for finding common elements between two unsorted arrays

I'm facing some problems that deal with finding common elements between unsorted arrays and I'd like to know whether there are well-known lower-bounds for the worst-case and, eventually, what are ...
2
votes
1answer
2k views

Time complexity in Big O notation for Harmonic series with first k terms missing

Firstly, let's suppose there exists an algorithm where $i$ iterates from $1$ to $n$, spending $\frac{n^2}{i}$ time in each iteration. Thanks to the well known $O(\log n)$ upper bound for the Harmonic ...
2
votes
2answers
658 views

A Problem on Time Complexity of Algorithms

For every integer $t$, is there a problem whose solutions can be verified in $O(n^{s})$ time but cannot be found in $O(n^{st})$ time? By verifying, I mean that given a candidate solution $y$, we can ...
4
votes
1answer
254 views

Lower bound for sorting n arrays of size k each

Given $n$ arrays of size $k$ each, we want to show that at least $\Omega(nk \log k)$ comparisons are needed to sort all arrays (indepentent of each other). My proof is a simple modification of the ...