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1answer
22 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. ...
1
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1answer
66 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
33 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
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0answers
72 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
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1answer
34 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 deļ¬ning a hyperplane in $R^n$. Thus, tracing a path through the tree computes ...
1
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1answer
17 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
109 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?
2
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0answers
26 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 ...
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3answers
138 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 ...
5
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0answers
58 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
205 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 ...
3
votes
1answer
315 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
963 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
396 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
130 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 ...
12
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1answer
216 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
87 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
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2answers
104 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
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2answers
212 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
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0answers
136 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
108 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
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3answers
109 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
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1answer
234 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
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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
82 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
281 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
71 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
172 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
271 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 ...
1
vote
1answer
238 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
374 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
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4answers
469 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
175 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
638 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 ...