Questions tagged [lower-bounds]

questions about lowerbounds on functions, usually the complexity of an algorithm or a problem

6
votes
0answers
242 views

Data structures lower bounds on Turing machines

Have there been any results on lower bounds for implementing data structures on Turing machines, e.g. stacks, queues, etc ? I guess that people are mostly interested in models with random access, but ...
45
votes
0answers
1k views

Monotone complexity of s-t connectivity

In the problem CONN, we obtain a directed $n$-vertex graph (encoded as a boolean string of $n^2$ bits, one for each potential edge), and want to decide whether there is a path between all $n^2$ pairs $...
15
votes
1answer
394 views

Characterization of read-once formulae over the full binary basis

Background A read-once formula over a set of gates (also called a basis) is a formula in which each input variable appears once. Read-once formulas are commonly studied over the De Morgan basis (...
21
votes
2answers
987 views

Can addition be carried out in less than depth 5?

Using carry look ahead algorithm we can compute addition using a polynomial size depth 5 (or 4?) $AC^0$ circuit family. Is it possible to reduce the depth? Can we compute the addition of two binary ...
4
votes
1answer
259 views

Exponential blowup in Simple Proof of a theorem of Statman by Mairson

I'm studying "A simple proof of a theorem of Statman" by H.G. Mairson. At page 4, he encodes set/type theory in lambda calculus. In particular, note che "op" trick in the definition of $eq_{k+1}$. ...
13
votes
2answers
394 views

Is it possible to use random restrictions to obtain a lower-bound for $\mathsf{TC^0}$?

There are several well-known $\mathsf{AC^0}$ circuit size lower-bound results based on random restrictions and the Switching Lemma. Can we develop a Switching Lemma result to prove a size lower-bound ...
-3
votes
1answer
162 views

Lower bounds on $Q_{\epsilon}(IP)$

I want to show that $Q_{\epsilon}(IP) \geq (1-O(\epsilon))n$, where $IP:\{0,1\}^n \times \{0,1\}^n \rightarrow \{0,1\}$ is the usual mod 2 inner product. I have Nayak's lower bound, but I am not sure ...
7
votes
1answer
986 views

How to generate a permutation uniformly by repeating using an one-bit uniform random generator?

If I have an one-bit uniform random generator, how can I use it to generate a permutation uniformly for the sequence {1, 2, ..., n}. I have a solution: run the one-bit random generator n*n times to ...
8
votes
1answer
357 views

Lower Bounds on Running time of Graph Algorithms

Are there any non-trivial lower bounds on the running time of graph algorithms in RAM/PRAM/ models of computation ? I am not looking for the NP-Hardness results here. Following is a result that I ...
8
votes
0answers
256 views

Existence of “colouring matrices” — a generalisation

This is a generalisation of the following post: Existence of "colouring matrices". As the base case turned out to be fairly straightforward (in essence, precisely equal to the existence of Sperner ...
9
votes
2answers
190 views

Existence of “colouring matrices”

Edit: there is now a follow-up question related to this post. Definitions Let $c$ and $k$ be integers. We use the notation $[i] = \{1,2,...,i\}$. A $c \times c$ matrix $M = (m_{i,j})$ is said to be ...
7
votes
1answer
500 views

Razborov's Approximation methods

The approximation mothods is used for deriving lower bounds on the monotone circuit size of k-cliue and perfect matching problem. People in parameterized complexity theory strongly believe that k-...
0
votes
1answer
153 views

Functions and Counting Problems in Streaming Computation

I have read a stream computation paper in STOC07(Paul Beame, T. S. Jayram, and Atri Rudra. Lower bounds for randomized read/write stream algorithms.) and FOCS08(Paul Beame and Trinh Huynh. On the ...
11
votes
1answer
493 views

Lower bounds for learning in the membership query and counterexample model

Dana Angluin (1987; pdf) defines a learning model with membership queries and theory queries (counterexamples to a proposed function). She shows that a regular language that is represented by a ...
2
votes
1answer
84 views

Lower bounds on batched query search

I am not much in the field of databases. But the problem I m facing is the following: given a database $D$, we receive a batch of distinct queries $Q = \{q_1, ..., q _k\}$, where each $q_i$ is a ...
1
vote
1answer
324 views

Questions about computing matrix rigidity

Matrix rigidity was introduced by Valiant in 1977: The rigidity $Rig_M(r)$ of boolean matrix $M$ over GF(2) is the smallest number of entries of $M$ that must be changed in order to reduce its ...
8
votes
2answers
1k views

(0,1)-vector XOR problem

this is a rewrite of another recent question of mine [1] that wasnt stated well (it had a semi obvious simplification, mea culpa) but I think theres still a nontrivial question at the heart of it. ...
12
votes
1answer
425 views

Monotone circuit complexity of computing functions on sparse inputs

The weight $|x|$ of a binary string $x\in\{0,1\}^n$ is the number of ones in the string. What happens if we are interested in computing a monotone function on inputs with few ones? We know that ...
0
votes
0answers
151 views

covering an NxN grid using overlapping vs. non-overlapping windows residing k points in each

Let the problem, $P_{overlapping}$, be the following. We have an $N_1 \times N_2$ grid. Each cell of the grid can have the value either 0 or 1. Assume that we have $a \times b$ overlapping windows as ...
16
votes
2answers
526 views

Number of binary gates needed to compute AND and OR of n input bits simultaneously

What is the minimal number of binary gates needed to compute AND and OR of $n$ input bits simultaneously? The trivial upper bound is $2n-2$. I believe that this is optimal, but how to prove this? The ...
8
votes
1answer
498 views

Conditional results implying difficulty of improving upper/lower bounds for permanent

Let $A$ be a given square matrix. Is there any evidence that beating quadratic lower bounds for $B$ such that $\text{det}(B) = \text{per}(A)$ could be hard? Is there any plausible conjecture which ...
12
votes
1answer
469 views

Problems to reduce from to prove an $\Omega(n\log n)$ lower bound

What are the standard problems we can reduce from to prove $\Omega(n\log n)$ lower bounds? Of course, state problems other than sorting and element distinctness.
20
votes
5answers
420 views

Reducing space usage of st-connectivity with multiple passes?

Suppose a graph $G$ with $n$ vertices is presented as a stream of $m$ edges, but multiple passes are allowed over the stream. Monika Rauch Henzinger, Prabhakar Raghavan, and Sridar Rajagopalan ...
14
votes
3answers
670 views

How can I show a Gap-P problem is outside #P

There are a number of problems in combinatorial representation theory and algebraic geometry for which no positive formula is known. There are several examples I am thinking of, but let me take ...
39
votes
2answers
2k views

Are the problems PRIMES, FACTORING known to be P-hard?

Let PRIMES (a.k.a. primality testing) be the problem: Given a natural number $n$, is $n$ a prime number? Let FACTORING be the problem: Given natural numbers $n$, $m$ with $1 \leq m \leq n$, ...
18
votes
1answer
845 views

Most efficient way to convert an $\text{AC}^0$ circuit to a circuit (of any depth) with gate fanout 1

EDIT (Aug 22, 2011): I am further simplifying the question and putting a bounty on the question. Perhaps this simpler question will have an easy answer. I'm also going to strikethrough all the parts ...
25
votes
2answers
1k views

Formula size lower bounds for AC0 functions

Question: What is the best known formula size lower bound for an explicit function in AC0? Is there an explicit function with an $\Omega(n^2)$ lower bound? Background: Like most lower bounds, ...
6
votes
0answers
202 views

Lowerbounds for in-situ permutation

What is the best known lowerbound for the worst case complexity of in-situ permutation (also called in-place rearrangement)? Has there been any reported progress after the 1970's article by Knuth (...
1
vote
0answers
192 views

Circuit lower bound for NAND based arbitrary deterministic computation [closed]

What is the lowest known lower-bound on the number of NAND gates needed in order to perform an arbitrary deterministic computation with a fixed length input and output?
13
votes
2answers
620 views

Progress on generalized star-height problem?

The (generalized) star height of a language is the minimum nesting of Kleene stars required to represent the language by an extended regular expression. Recall that an extended regular expression over ...
33
votes
2answers
1k views

Cohomological approach to boolean complexity

A few years ago, there was some work by Joel Friedman relating lower circuit bounds to Grothendieck cohomology (see papers: http://arxiv.org/abs/cs/0512008, http://arxiv.org/abs/cs/0604024). Has this ...
10
votes
2answers
539 views

What happens if we improve the time hierarchy theorems?

In a nutshell, the time hierarchy theorems say that a Turing machine can solve more problems if it has more time for computation. In detail for deterministic TM and time-constructable functions $f,g$ ...
14
votes
3answers
1k views

Lower Bounds for Data Structures

Are results known which rule out the existence of "too-good-to-be-true" data structures? For example: can one add $Split$ and $Join$ functionality to an order maintenance data structure (see Dietz ...
4
votes
1answer
208 views

Why does deterministic recognition of DYCK(2) languages in the streaming model take linear space?

I was reading the paper "Recognizing Well-Paranthesized Expressions in the Streaming Model" by Magniez, Mathieu and Nayak where they give upper and lower bounds on the space required to recognize DYCK(...
10
votes
0answers
272 views

Lower bound method for ordered binary decision diagrams

This is an idea/question inspired by the question and answer of Boolean functions with exponential size OBDD representation in all orders except one order?: If you want to prove some exponential ...
8
votes
0answers
248 views

For median is it optimal to compare in pairs first?

Median can be done in linear time and is now down to (I think) $2.97n$. The lower bounds is (I think) $(2+\epsilon)n$ where $\epsilon$ is very small. The following theorem, if true, may help improve ...
18
votes
2answers
2k views

Hierarchy theorem for circuit size

I think that a size hierarchy theorem for circuit complexity can be a major breakthrough in the area. Is it an interesting approach to class separation? The motivation for the question is that we ...
9
votes
1answer
611 views

Lower bounds on the Threshold function

In decision tree complexity of a boolean function, a very well know lower bound method is to find a (approximate) polynomial that represents the function. Paturi gave a characterization for symmetric ...
14
votes
1answer
353 views

Lower bounds on the size of CFGs for specific finite languages

Consider the following natural question: Given a finite language $L$, what is the smallest context-free grammar generating $L$? We can make the question more interesting by specifying a sequence of ...
17
votes
3answers
758 views

Succinct data structures survey?

Fischer's paper this month reminded me how little I know about the art of succinct data structures, and algorithms to use them. For those that don't know about succinct data structures: Given a ...
15
votes
2answers
502 views

Complexity lower bound: the gap between decision trees and RAMs

I recently discovered a quadratic lower bound on the complexity of a problem in the decision tree model, and I wonder whether this result could be partially generalized to the random access machine ...
12
votes
2answers
516 views

Which problems in computational geometry or graph theory are believed to be $\Omega(n^3)$?

This is intended as a follow up question to Robin Kothari's previous post on polynomial time hardness results. Specifically, I'm interested in seeing some hardness proofs for problems that are ...
8
votes
3answers
715 views

One way randomised communication complexity of disjointness

I am looking for a reference for the (classical) one way randomised communication complexity of disjointness when the universe can be large. Say Alice and Bob both have sets of size $m$ chosen from a ...
6
votes
1answer
1k views

Highest lower bound on NP problems (TSP)

I'll try another question that I haven't been able to find almost any kind of information about, thanks a lot for any kind of pointers or explanations. Is there a list of the proven lower bounds of ...
12
votes
2answers
2k views

Reversing a list using two queues

This question is inspired by an existing question about whether a stack can be simulated using two queues in amortized $O(1)$ time per stack operation. The answer seems to be unknown. Here is a more ...
22
votes
2answers
1k views

Protocol partition number and deterministic communication complexity

Besides (deterministic) communication complexity $cc(R)$ of a relation $R$, another basic measure for the amount of communication needed is the protocol partition number $pp(R)$. The relation between ...
5
votes
1answer
342 views

Stronger Lower Bounds on Nondeterministic Multiparty Communication

This is a continuation of my previous question on Lower bounds for Nondeterministic Multiparty Communication. From the answer, the $\mu^\infty$ norm lower bounds nondeterministic multiparty ...
11
votes
1answer
432 views

Lower bounds for Nondeterministic Multiparty Communication

This is a continuation of my previous question on communication lower bounds for partial boolean functions. Can someone suggest any reference on lower bounds for nondeterministic multiparty ...
7
votes
2answers
548 views

Communication lower bounds for partial boolean functions

There are well known techniques for proving lower bounds on the communication complexity of boolean functions, like fooling sets, the rank of the communication matrix, and discepancy. 1) How do we ...
29
votes
1answer
2k views

Fourier coefficients Boolean Functions described by Bounded Depth Circuits with AND OR and XOR gates

Let $f$ be a Boolean function and let's think about f as a function from $\{-1,1\}^n$ to $\{ -1,1 \}$. In this language the Fourier expansion of f is simply the expansion of f in terms of square free ...