scholarly journals On Communication Complexity of Fixed Point Computation

2021 ◽  
Vol 9 (4) ◽  
pp. 1-27
Author(s):  
Anat Ganor ◽  
Karthik C. S. ◽  
Dömötör Pálvölgyi
Keyword(s):  
Fixed Point ◽  
Lower Bound ◽  
Communication Complexity ◽  
Search Problem ◽  
Communication Model ◽  
List Type ◽  
Content Type ◽  
Approximation Factor ◽  
Approximate Fixed Point ◽  
Fixed Point Computation

Brouwer’s fixed point theorem states that any continuous function from a compact convex space to itself has a fixed point. Roughgarden and Weinstein (FOCS 2016) initiated the study of fixed point computation in the two-player communication model, where each player gets a function from [0,1]^n to [0,1]^n , and their goal is to find an approximate fixed point of the composition of the two functions. They left it as an open question to show a lower bound of 2^{\Omega (n)} for the (randomized) communication complexity of this problem, in the range of parameters which make it a total search problem. We answer this question affirmatively. Additionally, we introduce two natural fixed point problems in the two-player communication model. Each player is given a function from [0,1]^n to [0,1]^{n/2} , and their goal is to find an approximate fixed point of the concatenation of the functions. Each player is given a function from [0,1]^n to [0,1]^{n} , and their goal is to find an approximate fixed point of the mean of the functions. We show a randomized communication complexity lower bound of 2^{\Omega (n)} for these problems (for some constant approximation factor). Finally, we initiate the study of finding a panchromatic simplex in a Sperner-coloring of a triangulation (guaranteed by Sperner’s lemma) in the two-player communication model: A triangulation T of the d -simplex is publicly known and one player is given a set S_A\subset T and a coloring function from S_A to \lbrace 0,\ldots ,d/2\rbrace , and the other player is given a set S_B\subset T and a coloring function from S_B to \lbrace d/2+1,\ldots ,d\rbrace , such that S_A\dot{\cup }S_B=T , and their goal is to find a panchromatic simplex. We show a randomized communication complexity lower bound of |T|^{\Omega (1)} for the aforementioned problem as well (when d is large). On the positive side, we show that if d\le 4 then there is a deterministic protocol for the Sperner problem with O((\log |T|)^2) bits of communication.

Using Digitised X-Ray Powder Diffraction Scans as Input for a New Pc-At Search/Match Program

1987 ◽  
Vol 31 ◽  
pp. 423-430 ◽  
Author(s):  
P. Caussin ◽  
J. Nusinovici ◽  
D.W. Beard
Keyword(s):  
Fixed Point ◽  
Data Base ◽  
Powder Diffraction ◽  
Phase Identification ◽  
Interactive Graphics ◽  
List Type ◽  
Fast Method ◽  
X Ray ◽  
Fixed Point Computation ◽  
Peak Location

AbstractA Search/Matcti program lias 'beea written for the IBM PC AT computer that is capable of -using "background - subtracted, digitized 2-ray powder diffraction scans as inputs in addition to the d/I data traditionally used. This novel procedure has proved especially effective when numerous unresolved lines are present in the pattern. The method is also less demanding of data quality thaii the peak location programs. The program may he extended to searching & data "base of digitized standard patterns.The program, has several parameters that can- "be adjusted, including chemistry. The results from the Johnson/Vand list type of output are directly accessible to the interactive graphics program. This gives the diffraction!st a fast method for verifying the phase identification. Because of the speed of fixed point computation techniques, the 52,791 pattern file can be scanned in about 90 seconds.This paper will illustrate the utility of the program.

scholarly journals On the Nielsen root theory of n-valued maps

2021 ◽  
Vol 23 (4) ◽  
Author(s):  
Robert F. Brown
Keyword(s):  
Fixed Point ◽  
Lower Bound ◽  
Lie Group ◽  
Hausdorff Distance ◽  
Root Number ◽  
Root Numbers ◽  
Fixed Point Result

AbstractLet $$\phi :X \multimap Y$$ ϕ : X ⊸ Y be an n-valued map of connected finite polyhedra and let $$a \in Y$$ a ∈ Y . Then, $$x \in X$$ x ∈ X is a root of $$\phi $$ ϕ at a if $$a \in \phi (x)$$ a ∈ ϕ ( x ) . The Nielsen root number $$N(\phi : a)$$ N ( ϕ : a ) is a lower bound for the number of roots at a of any n-valued map homotopic to $$\phi $$ ϕ . We prove that if X and Y are compact, connected triangulated manifolds without boundary, of the same dimension, then given $$\epsilon > 0$$ ϵ > 0 , there is an n-valued map $$\psi $$ ψ homotopic to $$\phi $$ ϕ within Hausdorff distance $$\epsilon $$ ϵ of $$\phi $$ ϕ such that $$\psi $$ ψ has finitely many roots at a. We conjecture that if X and Y are q-manifolds without boundary, $$q \ne 2$$ q ≠ 2 , then there is an n-valued map homotopic to $$\phi $$ ϕ that has $$N(\phi : a)$$ N ( ϕ : a ) roots at a. We verify the conjecture when $$X = Y$$ X = Y is a Lie group by employing a fixed point result of Schirmer. As an application, we calculate the Nielsen root numbers of linear n-valued maps of tori.

scholarly journals Sample Examination Questions

1958 ◽  
Vol 1 (2) ◽  
pp. 134-138
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
List Type ◽  
Color Problem ◽  
Euler Path

1. Explain briefly what is meant by each of the following: (i) Euler path(ii) fixed point theorem(iii) four color problem(iv) unit braid of order n(v) MBbius band.

Rising from the ashes: Satyam's story

2014 ◽  
Vol 4 (1) ◽  
pp. 1-8
Author(s):  
Tanvi Gautam
Keyword(s):  
Change Management ◽  
Subject Area ◽  
Executive Education ◽  
List Type ◽  
Corporate Communications ◽  
It Industry ◽  
Market Reactions ◽  
Content Type ◽  
Employee Trust ◽  
Starting Point

Subject area Leadership, human resource management, crisis management, change management and communication. Study level/applicability Executive education; postgraduate; undergraduate. Case overview This case study describes the collapse of Satyam, a leading IT industry service provider from India. Satyam went into a crisis mode after revelation of financial fraud by its Chairman. This resulted in a crisis not just for the company, its clients and employees – but it also had the potential to shake up the entire Indian IT industry the world over, by shattering investor and client confidence in the Indian IT sector. The case provides the students with an inside view of the unfolding of events at Satyam and the people challenges that emerge in a crisis scenario. The case outlines reactions from the industry, government, clients and employees as they tried to make sense of a very chaotic situation, and its multi-level ramifications both within India and outside. The case ends with Thallapalli Hari, the Global Head of Marketing and Communication and ex-head of HR, trying to visualise and prioritise a course of action to propose to other members of the leadership team. Expected learning outcomes The key aim of this case is to provide a backdrop to the crisis, and also help students put themselves in the role of an HR crisis manager as well as portray the decision making and communication challenges that emerge in chaotic situations. The importance of an immediate and yet strategic response is emphasised and the case is a great starting point to have a discussion on the competencies and skills required in HR to lead under unusual circumstances. This case allows participants to get an in-depth understanding of the collapse of Satyam. The case also illustrates principles of leadership, change management and communication, in particular: Leadership: The Satyam story is an HR and leadership crisis nightmare come true. What should an HR leader do when you wake up to find your company with a ruined reputation, minimal financial capital, 53,000 employees on the payroll and more than 500 clients with pending deliverables worldwide. Where do you begin? The case illustrates a situation where immediate action is required to stop the tailspin into which the company was heading. Change management: The situation demanded that change be managed from a chaotic system to a stable system. The big issue though remains as to how one can get a system into a state of stability when everything is changing at the same time. Most change management plans have some stable variables, however in the case of Satyam there were multiple changes taking place simultaneously. A combination of change in leadership, client relationships, employee trust and confidence, market reactions together make for a perfect storm. Dealing with even one of these changes is a challenge for a company. In the case of Satyam, its entire existence was at stake. Communication: The demands for communicating effectively in a crisis situation are different than communicating under stable systems. The choice of medium, the speed of response, the content all need careful monitoring. Whereas most companies have teams that separately deal with internal and external communication, Satyam provides a unique situation where managing both effectively at the same time was critical to the future of the firm. The stakes for effective communication are much higher under the circumstances. This case can be used in organizational behaviour, human resources and corporate communications modules being taught to under-graduates, post-graduates and for executive education. Supplementary materials Teaching notes are available for educators only. Please contact your library to gain login details or email [email protected] to request teaching notes.

scholarly journals Generalizations of the Poincaré Birkhoff fixed point theorem

1977 ◽  
Vol 17 (3) ◽  
pp. 375-389 ◽  
Author(s):  
Walter D. Neumann
Keyword(s):  
Fixed Point ◽  
Lower Bound ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Periodic Points ◽  
Open Problems ◽  
Number Of Components ◽  
Birkhoff Theorem

It is shown how George D. Birkhoff's proof of the Poincaré Birkhoff theorem can be modified using ideas of H. Poincaré to give a rather precise lower bound on the number of components of the set of periodic points of the annulus. Some open problems related to this theorem are discussed.

scholarly journals On the Ramsey Number $R(4,6)$

10.37236/2102 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Geoffrey Exoo
Keyword(s):  
Fixed Point ◽  
Automorphism Group ◽  
Lower Bound ◽  
Heuristic Search ◽  
Ramsey Number ◽  
Complete Graphs ◽  
Search Procedure ◽  
Edge Colorings

The lower bound for the classical Ramsey number $R(4,6)$ is improved from 35 to 36. The author has found 37 new edge colorings of $K_{35}$ that have no complete graphs of order 4 in the first color, and no complete graphs of order 6 in the second color. The most symmetric of the colorings has an automorphism group of order 4, with one fixed point, and is presented in detail. The colorings were found using a heuristic search procedure.

Lower bounds for cutting planes proofs with small coefficients

1997 ◽  
Vol 62 (3) ◽  
pp. 708-728 ◽  
Author(s):  
Maria Bonet ◽  
Toniann Pitassi ◽  
Ran Raz
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Cutting Planes ◽  
Communication Complexity ◽  
Proof System ◽  
Small Weight ◽  
Deduction Rule ◽  
Special Case ◽  
Exponential Lower Bound

AbstractWe consider small-weight Cutting Planes (CP*) proofs; that is, Cutting Planes (CP) proofs with coefficients up to Poly(n). We use the well known lower bounds for monotone complexity to prove an exponential lower bound for the length of CP* proofs, for a family of tautologies based on the clique function. Because Resolution is a special case of small-weight CP, our method also gives a new and simpler exponential lower bound for Resolution.We also prove the following two theorems: (1) Tree-like CP* proofs cannot polynomially simulate non-tree-like CP* proofs. (2) Tree-like CP* proofs and Bounded-depth-Frege proofs cannot polynomially simulate each other.Our proofs also work for some generalizations of the CP* proof system. In particular, they work for CP* with a deduction rule, and also for any proof system that allows any formula with small communication complexity, and any set of sound rules of inference.

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