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3
votes
0answers
56 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 - ...
3
votes
1answer
70 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
81 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
81 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
23 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
68 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 ...
2
votes
1answer
55 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
35 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
36 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
138 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
37 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
159 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
38 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
28 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
121 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
31 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
336 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 ...
6
votes
0answers
69 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
344 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
480 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
416 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
1k 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
477 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 ...
3
votes
1answer
192 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 ...
13
votes
1answer
241 views

Is detecting “doubly” arithmetic progressions 3SUM-hard?

This is inspired by an interview question. We are given an array of integers $a_1, \dots, a_n$ and have to determine if there are distinct $i \lt j \lt k$ such that $a_k - a_j = a_j - a_i$ $k - j = ...
3
votes
1answer
91 views

Complexity of transposing matrices represented as list of row or column vectors

Given [[1,4,7],[2,5,8],[3,6,9]] which is a list of the column vectors of matrix |1, 2, 3| |4, 5, 6| |7, 8, 9| is $ \Omega(n^2) $ a lower bound for transposing? ...
4
votes
2answers
113 views

Constraint violation and efficiency in search

It seems that (in a broad sense) two approaches can be utilized to produce an algorithm for solving various optimization problems: Start with a feasible solution and expand search until constraints ...
3
votes
2answers
315 views

Methods for Finding Asymptotic Lower Bounds

I've found in many exercises where I'm asked to show that $f(n)=\Theta(g(n))$ where the two functions are of the same order of magnitude I have difficulty finding a constant $c$ and a value $n_0$ for ...
10
votes
0answers
138 views

Is there a data-structure which is more efficient than both arrays and linked lists? [duplicate]

Background: In this question we care only about worst-case running-time. Array and (doubly) linked lists can be used to keep a list of items and implement the vector abstract data type. Consider the ...
5
votes
1answer
118 views

Simple lower bounds against AC0

It is known that $Parity \notin AC^0$ (nonuniform), but the proof is rather involved and combinatorial. Are there simpler, but weaker lower bounds, say for $NP \not \subseteq AC^0$ or $NEXP \not ...
1
vote
3answers
118 views

Search spaces and computation time

This question follows on previous questions (1), (2), where we define an initial space of possibilities and reason about how a solution is chosen from that. Consider a problem P where we are given: ...
1
vote
1answer
274 views

Input that causes an operation on a binomial heap to run in $\Omega(\log n)$ time?

I was studying binomial heaps and its time analysis. Are there any inputs that cause DELETE-MIN, DECREASE-KEY, and DELETE to run in $\Omega(\log n)$ time for a binomial heap rather than $O(\log n)$?
1
vote
1answer
58 views

Progress of algorithms in problem spaces

Continuing in the vein of two prior questions (1) and (2), we started with sorting, where we had a set of $n!$ input possibilities a goal space of only one element consisting of the one correct ...
3
votes
2answers
85 views

Lower bound on size of proof that a list of integers is sorted

Suppose we have a list of unbounded integers, written in binary, and we want to write a (formal) proof that the list is sorted in ascending order. Such a proof might look (informally) like: "2 < ...
5
votes
2answers
319 views

Space complexity below $\log\log$

Show that for $l(n) = \log \log n$, it holds that $\text{DSPACE}(o(l)) = \text{DSPACE}(O(1))$. It's well known fact in Space Complexity, but how to show it explicitly?
4
votes
1answer
73 views

Proofs based on narrowing down sets of possibilities

Consider the argument made in this question based on the comparison sorting lower-bounds proof, which runs as follows. First, the comparison sorting lower-bounds proof was recited: For $n$ ...
6
votes
2answers
178 views

Is there any nontrivial problem in the theory of serial algorithms with a nontrivial polynomial lower bound of $\Omega(n^2)$?

In the theory of distributed algorithms, there are problems with lower bounds, as $\Omega(n^2)$, that are "big" (I mean, bigger than $\Omega(n\log n)$), and nontrivial. I wonder if are there problems ...
6
votes
1answer
313 views

Generalizing the Comparison Sorting Lower Bound Proof

Let's start with the comparison sorting lower bound proof, which I'll summarize as follows: For $n$ distinct numbers, there are $n!$ possible orderings. There is only one correct sorted sequence of ...
2
votes
1answer
354 views

Lower bound for Convex hull

By making use of the fact that sorting $n$ numbers requires $\Omega(n \log n)$ steps for any optimal algorithm (which uses 'comparison' for sorting), how can I prove that finding the convex-hull of ...
4
votes
2answers
471 views

Lower bounds: queues that return their min elements in $O(1)$ time

First, consider this simple problem --- design a data structure of comparable elements that behaves just like a stack (in particular, push(), pop() and top() take constant time), but can also return ...
12
votes
4answers
493 views

Is every linear-time algorithm a streaming algorithm?

Over at this question about inversion counting, I found a paper that proves a lower bound on space complexity for all (exact) streaming algorithms. I have claimed that this bound extends to all linear ...
6
votes
1answer
199 views

Bound on space for selection algorithm?

There is a well known worst case $O(n)$ selection algorithm to find the $k$'th largest element in an array of integers. It uses a median-of-medians approach to find a good enough pivot, partitions ...
5
votes
1answer
818 views

How to use adversary arguments for selection and insertion sort?

I was asked to find the adversary arguments necessary for finding the lower bounds for selection and insertion sort. I could not find a reference to it anywhere. I have some doubts regarding this. I ...