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1answer
36 views

How often can a linear speed sort succeed?

Let's say you have sorting function. It is allowed to exit with failure (but if it does not it must return a correctly sorted sequence). It is also $\mathcal O (n)$. What kind of bounds can we place ...
3
votes
1answer
30 views

Replacing n with 2n in asymptotic bounds

I am going through Normal Subgroup Reconstruction and Quantum Computation Using Group Representations by Hallgren et al. In the proof of the theorem $6$ of the paper on page 632, the authors go on ...
1
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1answer
20 views

How to give an upper bound on this bin packing problem?

In the bin packing with fragile objects (BPFO) problem one is given a set of objects $\{1,\ldots,n\}$ where each object $i$ has a weight $w_i$ and a fragility $f_i$ for all $i$ in the set ...
0
votes
1answer
54 views

Estimating the number of distinct elements

Need to understand "intuition" part. It does not make sense to me why $log(d)$ is a good approximation. We have a stream $\sigma = \{a_1, ..., a_n\}$, with each $n \in [n]$, and this implicitly ...
6
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2answers
71 views

Lower Bounds for Size of Independent Set in a Graph?

I recently learnt that for any instance of a k-SAT problem with $m$ clauses and $n$ literals , we have an assignment of literals such that at least $m(1 - 2^{-k})$ clauses are satisfied. I was ...
1
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2answers
61 views

Time complexity of comparing two $N \times N$ Matrices?

So each matrix has $N^{2}$ elements, and so just by comparing each element we would be doing $O(N^{2})$ operations. Is there any other way to compare these two matrices such that the number of ...
3
votes
1answer
32 views

How is the space hierarchy theorem different for non space constructible functions?

Sipser first introduces space constructible functions. Then uses the definition to prove the space hierarchy theorem: if f(n) is a space constructible function then there are languages that can ...
2
votes
1answer
40 views

Regex to NFA to complement

So I've found out that a regular expression $n$ symbols long converts to an NFA with $O(n)$ states, it is linear. Now to go from that NFA to the complement of the NFA, since I can't just flip accept ...
0
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1answer
59 views

Prove by induction that the running time of recursive Fibonacci is exponential

This example followed from a Fibonacci algorithm in class. The professor showed us how to compute $T(n) = T(n-1) + T(n-2) + 3$, but left this step for us to prove, so I decided to attempt to prove it! ...
5
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1answer
70 views

Finding a small element in a changing array

Consider having an integer array $A$ with $n$ elements, in addition to any data structure you like. The array is initialized to zeros. The goal to to support two operations: ...
1
vote
1answer
88 views

Does Heapsort work in time o(n log n) in the best case?

Is it possible for Heapsort to work in time $o(n\log n)$ on certain inputs? For example in case of Insertion sort it is possible, however when it comes to Quickssort it is not possible. What about ...
2
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1answer
46 views

Using Context free language to simulate regular expression in finite automata

Is there a minimum number of non terminal we need to use in order to simulate a finite automata with n states? When we try to convert a language accepted by NFA to context free language, do we need n ...
3
votes
1answer
47 views

Lower bound for number of nonterminals in a CFG

Let's say we have a context-free grammar for the language $a\mbox{*}b\mbox{*}c\mbox{*}$. Is there a way to determine a lower bound for the number of nonterminals in this grammar? I'm pretty sure you ...
0
votes
0answers
18 views

How to prove log^b n = o(n^a) [duplicate]

I'm trying to prove $$log^b n = o(n^a)$$ Method of Induction Base Case: Holds n =1 ...
0
votes
1answer
55 views

Lower Bound for Comparison-based sorting algorithms

We know that the lower bound for comparison-based sorting algorithms is Ω(nlogn), where logn being the binary logarithm of n. But what about for the best-case scenario of the bubble sort, which takes ...
7
votes
1answer
187 views

$O(\frac{\log n}{\log \log n})$ algorithm for the prefix parity problem

The prefix parity problem can be defined as follows. You are given a string $S$ of length $n$ and initially every character is $0$. Then you want to build a data structure that can support updates ...
3
votes
2answers
66 views

Can one increment an $n$ bit integer using fewer than $2 - 2^{1-n}$ bit inspections on average?

Given an $n$-bit integer, I am interested in performing an increment operation using as few bit reads as possible. The standard binary code (standard binary representation of numbers), requires $n$ ...
1
vote
1answer
43 views

$2$-sorted array. How to sort it in minimal number of comparisons ?

It is given array $2$-sorted array $a[1..n]$. $2$-sorted denotes that $a[1]\le a[3]\le...\le$ and $a[2]\le a[4]\le ..\le$ Obviously we may split array into two sorted arrays and then merge two ...
4
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1answer
65 views

space complexity of DFA intersection problem

the DFA-intersection computation problem, given two DFAs specified on the input, compute the intersection DFA, or the decision problem to determine its emptiness, turns out to have wider/ deeper ...
1
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1answer
63 views

Sorting array with two elements - in place and minimal number of comparisons, lower bound

Algorithm must be in place. I would like to find lower bound for comparison algorithm. Algorithm will sort array with only two elements - without loss of generality let assume that there are only $1s$ ...
1
vote
1answer
109 views

Partition array - elements non-negative and negative. Minimal number of replacements of elements

I consider following problem: It is given array with numbers, for example $[-2,0,1,-324,213,321,-2]$. The problem is: replace elements such that negative numbers precede the non-negative. For our ...
3
votes
2answers
386 views

Why doesn't decision tree work in case find minimum

When we would like to prove lower bound comparison algorirthm, we often use decision tree, for example sorting by comparisons. So let's consider find minimum in array $a[1..n]$ by comparison. Lower ...
1
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0answers
38 views

Minimum exchanges for heap sort

I'm studying heap sort and was presented with the following question. What is the minimum number of items that must be exchanged during a remove the maximum operation in a heap of size N? Give a ...
0
votes
1answer
46 views

Comparing two graphs [closed]

I want to compare the vertices of two graphs. Given two graphs, $G_1 = (V_1, E_1)$ and $G_2 = (V_2, E_2)$, I want to compare $u_n$ and $v_m$ for $u \in V_1$ and $v \in V_2$. I came up with double ...
0
votes
1answer
30 views

lower bound of checking if in array are two different elements [closed]

Considering the lower limit to the problem of checking whether the array are only the same number via comparisons. And thnik about $ n-1 $. Consider the diagram hasse. This diagram must be consistent ...
3
votes
1answer
65 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 ? ...
4
votes
1answer
41 views

Corner cases in the Interleave Lower Bound for BSTs

The Interleave lower bound is a lower bound for the amount of operations any Binary Search Tree needs to make for a sequence of accesses. It is used in the construction of Tango Trees, and is based on ...
1
vote
2answers
234 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
28 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
130 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
2k 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
372 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
83 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
193 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
102 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
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0answers
27 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
51 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
151 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
38 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.
8
votes
2answers
186 views

Is integer sorting possible in O(n) in the transdichotomous model?

To my knowledge there doesn't exist a $O(n)$ worst-case algorithm that solves the following problem: Given a sequence of length $n$ consisting of finite integers, find the permutation where every ...
4
votes
1answer
45 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
185 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
325 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 ...
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2answers
73 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 ...
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3answers
2k 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 ...
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4answers
492 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 ...
2
votes
1answer
33 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 ...
6
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1answer
97 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
51 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 ...