scholarly journals The Complexity of Malign Ensembles

1990 ◽  
Vol 19 (335) ◽  
Author(s):  
Peter Bro Miltersen
Keyword(s):  
Polynomial Time ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Worst Case ◽  
Average Case ◽  
Exponential Time ◽  
Running Time ◽  
Exponential Time Algorithms

We analyze the concept of <em> malignness</em>, which is the property of probability ensembles of making the average case running time equal to the worst case running time for a class of algorithms. We derive lower and upper bounds on the complexity of malign ensembles, which are tight for exponential time algorithms, and which show that no polynomial time computable malign ensemble exists for the class of superlinear algorithms. Furthermore, we show that for no class of superlinear algorithms a polynomial time computable malign ensemble exists, unless every language in P has an expected polynomial time constructor.

Chapter 16. Worst-Case Upper Bounds

2021 ◽  
Author(s):  
Evgeny Dantsin ◽  
Edward A. Hirsch
Keyword(s):  
Structural Complexity ◽  
Upper Bounds ◽  
Worst Case ◽  
Exponential Time ◽  
Running Time ◽  
Dichotomy Theorem ◽  
Exponential Time Hypothesis ◽  
Satisfiability Algorithms

The chapter is a survey of ideas and techniques behind satisfiability algorithms with the currently best asymptotic upper bounds on the worst-case running time. The survey also includes related structural-complexity topics such as Schaefer’s dichotomy theorem, reductions between various restricted cases of SAT, the exponential time hypothesis, etc.

REDUCING SIMPLE GRAMMARS: EXPONENTIAL AGAINST HIGHLY-POLYNOMIAL TIME IN PRACTICE

2007 ◽  
Vol 18 (04) ◽  
pp. 715-725
Author(s):  
CÉDRIC BASTIEN ◽  
JUREK CZYZOWICZ ◽  
WOJCIECH FRACZAK ◽  
WOJCIECH RYTTER
Keyword(s):  
Polynomial Time ◽  
Experimental Analysis ◽  
Time Complexity ◽  
Packet Classification ◽  
State Machines ◽  
Worst Case ◽  
Exponential Time ◽  
Push Down

Simple grammar reduction is an important component in the implementation of Concatenation State Machines (a hardware version of stateless push-down automata designed for wire-speed network packet classification). We present a comparison and experimental analysis of the best-known algorithms for grammar reduction. There are two approaches to this problem: one processing compressed strings without decompression and another one which processes strings explicitly. It turns out that the second approach is more efficient in the considered practical scenario despite having worst-case exponential time complexity (while the first one is polynomial). The study has been conducted in the context of network packet classification, where simple grammars are used for representing the classification policies.

scholarly journals An Extended Quadratic Frobenius Primality Test with Average Case Error Estimates

2001 ◽  
Vol 8 (45) ◽  
Author(s):  
Ivan B. Damgård ◽  
Gudmund Skovbjerg Frandsen
Keyword(s):  
Error Probability ◽  
Prime Numbers ◽  
Upper Bounds ◽  
Worst Case ◽  
Average Case ◽  
Closed Expression ◽  
Incremental Search ◽  
Starting Point ◽  
Primality Test ◽  
Small Set

We present an Extended Quadratic Frobenius Primality Test (EQFT), which is related to the Miller-Rabin test and the Quadratic Frobenius test (QFT) by Grantham. EQFT is well-suited for generating large, random prime numbers since on a random input number, it takes time about equivalent to 2 Miller-Rabin tests, but has much smaller error probability. EQFT extends QFT by verifying additional algebraic properties related to the existence of elements of order 3 and 4. We obtain a simple closed expression that upper bounds the probability of acceptance for any input number. This in turn allows us to give strong bounds on the average-case behaviour of the test: consider the algorithm that repeatedly chooses random odd k bit numbers, subjects them to t iterations of our test and outputs the first one found that passes all tests. We obtain numeric upper bounds for the error probability of this algorithm as well as a general closed expression bounding the error. For instance, it is at most 2^{-143} for k=500, t=2 . Compared to earlier similar results for the Miller-Rabin test, the results indicates that our test in the average case has the effect of 9 Miller-Rabin tests, while only taking time equivalent to about 2 such tests. We also give bounds for the error in case a prime is sought by incremental search from a random starting point. While EQFT is slower than the average case on a small set of inputs, we present a variant that is always fast, i.e. takes time about 2 Miller-Rabin tests. The variant has slightly larger worst case error probability than EQFT, but still improves on previous proposed tests.

scholarly journals On the Comparison Complexity of the String Prefix-Matching Problem

1995 ◽  
Vol 2 (46) ◽  
Author(s):  
Dany Breslauer ◽  
Livio Colussi ◽  
Laura Toniolo
Keyword(s):  
Linear Time ◽  
String Matching ◽  
Random Access ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Matching Problem ◽  
Machine Model ◽  
Worst Case ◽  
On Line ◽  
Sequential Comparison

In this paper we study the exact comparison complexity of the string<br />prefix-matching problem in the deterministic sequential comparison model<br />with equality tests. We derive almost tight lower and upper bounds on<br />the number of symbol comparisons required in the worst case by on-line<br />prefix-matching algorithms for any fixed pattern and variable text. Unlike<br />previous results on the comparison complexity of string-matching and<br />prefix-matching algorithms, our bounds are almost tight for any particular pattern.<br />We also consider the special case where the pattern and the text are the<br />same string. This problem, which we call the string self-prefix problem, is<br />similar to the pattern preprocessing step of the Knuth-Morris-Pratt string-matching<br />algorithm that is used in several comparison efficient string-matching<br />and prefix-matching algorithms, including in our new algorithm.<br />We obtain roughly tight lower and upper bounds on the number of symbol<br />comparisons required in the worst case by on-line self-prefix algorithms.<br />Our algorithms can be implemented in linear time and space in the<br />standard uniform-cost random-access-machine model.

scholarly journals Fair Pairwise Exchange among Groups

2021 ◽  
Author(s):  
Zhaohong Sun ◽  
Taiki Todo ◽  
Toby Walsh
Keyword(s):  
Polynomial Time ◽  
Real World ◽  
Research Question ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Upper Bounds ◽  
Individual Rationality ◽  
Running Time ◽  
Graph Structures ◽  
Real World Applications

We study the pairwise organ exchange problem among groups motivated by real-world applications and consider two types of group formulations. Each group represents either a certain type of patient-donor pairs who are compatible with the same set of organs, or a set of patient-donor pairs who reside in the same region. We address a natural research question, which asks how to match a maximum number of pairwise compatible patient-donor pairs in a fair and individually rational way. We first propose a natural fairness concept that is applicable to both types of group formulations and design a polynomial-time algorithm that checks whether a matching exists that satisfies optimality, individual rationality, and fairness. We also present several running time upper bounds for computing such matchings for different graph structures.

Lower and upper bounds for the linear arrangement problem on interval graphs

2018 ◽  
Vol 52 (4-5) ◽  
pp. 1123-1145
Author(s):  
Alain Quilliot ◽  
Djamal Rebaine ◽  
Hélène Toussaint
Keyword(s):  
Lower Bound ◽  
Polynomial Time ◽  
Feasible Solution ◽  
Interval Graph ◽  
Upper Bounds ◽  
Unit Interval ◽  
Lower And Upper Bounds ◽  
Linear Arrangement ◽  
Arrangement Problem ◽  
Linear Arrangement Problem

We deal here with theLinear Arrangement Problem(LAP) onintervalgraphs, any interval graph being given here together with its representation as theintersectiongraph of some collection of intervals, and so with relatedprecedenceandinclusionrelations. We first propose a lower boundLB, which happens to be tight in the case ofunit intervalgraphs. Next, we introduce the restriction PCLAP of LAP which is obtained by requiring any feasible solution of LAP to be consistent with theprecedencerelation, and prove that PCLAP can be solved in polynomial time. Finally, we show both theoretically and experimentally that PCLAP solutions are a good approximation for LAP onintervalgraphs.

scholarly journals The multidimensional truncated moment problem: Carathéodory numbers from Hilbert functions

2021 ◽  
Author(s):  
Philipp J. di Dio ◽  
Mario Kummer
Keyword(s):  
Moment Problem ◽  
Hankel Matrix ◽  
Upper Bounds ◽  
Moment Problems ◽  
Algebraic Varieties ◽  
Lower And Upper Bounds ◽  
Worst Case ◽  
Flat Extension ◽  
Truncated Moment Problem ◽  
Carathéodory Number

AbstractIn this paper we improve the bounds for the Carathéodory number, especially on algebraic varieties and with small gaps (not all monomials are present). We provide explicit lower and upper bounds on algebraic varieties, $$\mathbb {R}^n$$ R n , and $$[0,1]^n$$ [ 0 , 1 ] n . We also treat moment problems with small gaps. We find that for every $$\varepsilon >0$$ ε > 0 and $$d\in \mathbb {N}$$ d ∈ N there is a $$n\in \mathbb {N}$$ n ∈ N such that we can construct a moment functional $$L:\mathbb {R}[x_1,\cdots ,x_n]_{\le d}\rightarrow \mathbb {R}$$ L : R [ x 1 , ⋯ , x n ] ≤ d → R which needs at least $$(1-\varepsilon )\cdot \left( {\begin{matrix} n+d\\ n\end{matrix}}\right) $$ ( 1 - ε ) · n + d n atoms $$l_{x_i}$$ l x i . Consequences and results for the Hankel matrix and flat extension are gained. We find that there are moment functionals $$L:\mathbb {R}[x_1,\cdots ,x_n]_{\le 2d}\rightarrow \mathbb {R}$$ L : R [ x 1 , ⋯ , x n ] ≤ 2 d → R which need to be extended to the worst case degree 4d, $$\tilde{L}:\mathbb {R}[x_1,\cdots ,x_n]_{\le 4d}\rightarrow \mathbb {R}$$ L ~ : R [ x 1 , ⋯ , x n ] ≤ 4 d → R , in order to have a flat extension.

scholarly journals Obtaining a Proportional Allocation by Deleting Items

Algorithmica ◽  
2021 ◽  
Author(s):  
Britta Dorn ◽  
Ronald de Haan ◽  
Ildikó Schlotter
Keyword(s):  
Control Problem ◽  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Upper Bounds ◽  
Fair Allocation ◽  
Indivisible Goods ◽  
Lower And Upper Bounds ◽  
Proportional Allocation ◽  
Minimum Number

AbstractWe consider the following control problem on fair allocation of indivisible goods. Given a set I of items and a set of agents, each having strict linear preferences over the items, we ask for a minimum subset of the items whose deletion guarantees the existence of a proportional allocation in the remaining instance; we call this problem Proportionality by Item Deletion (PID). Our main result is a polynomial-time algorithm that solves PID for three agents. By contrast, we prove that PID is computationally intractable when the number of agents is unbounded, even if the number k of item deletions allowed is small—we show that the problem is $${\mathsf {W}}[3]$$ W [ 3 ] -hard with respect to the parameter k. Additionally, we provide some tight lower and upper bounds on the complexity of PID when regarded as a function of |I| and k. Considering the possibilities for approximation, we prove a strong inapproximability result for PID. Finally, we also study a variant of the problem where we are given an allocation $$\pi $$ π in advance as part of the input, and our aim is to delete a minimum number of items such that $$\pi $$ π is proportional in the remainder; this variant turns out to be $${{\mathsf {N}}}{{\mathsf {P}}}$$ N P -hard for six agents, but polynomial-time solvable for two agents, and we show that it is $$\mathsf {W[2]}$$ W [ 2 ] -hard when parameterized by the number k of

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