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text:
Cramond ist ein Dorf am Rand von Edinburgh in Schottland, das heute zu den Vororten von Edinburgh gehört. Es liegt am nordwestlichen Ende der Stadt an der Mündung des Almond in den Firth of Forth.

In der Gegend von Cramond wurden Hinweise auf frühzeitliche Besiedlung aus dem Mesolithikum, aus der Bronzezeit und der Römerzeit gefunden. Außerdem ist das Dorf der Geburtsort des schottischen Ökonomen John Law.

Etymologie
Lange wurde angenommen, dass die Römer ihre Befestigung Alaterva nannten. Ein Steinaltar, der den "alatervanischen Müttern und den Müttern des Paradeplatzes" gewidmet war, wurde unter dem Fundament des Cramond House gefunden. Archäologen interpretierten die Inschrift als Bezugnahme auf den Fundort, aber diese Theorie gilt unter Wissenschaftlern als widerlegt. "Alatervae" ist vermutlich ein alter Name für die Matronae Alaterviae.

Geschichte

Steinzeit
Bei archäologischen Ausgrabungen in der Gegend von Cramond wurden Hinweise auf eine Besiedlung aus der Zeit um 8500 v. Chr. gefunden. Damit ist Cramond die älteste bekannte frühgeschichtliche Siedlung in Schottland. Die Bewohner des mesolithischen Lagers waren nomadische Jäger und Sammler. Alle Knochen wurden vom säurehaltigen Boden zersetzt, aber es wurden Müll- und Vorratsgruben sowie zahlreiche Haselnussschalen gefunden. Es wird angenommen, dass der Platz wegen seiner Nähe zum River Almond und zum Firth of Firth gewählt wurde, da die dortigen Austern- und Muschelbestände eine verlässliche Nahrungsquelle boten. Außerdem wurden einige mikrolithische Steinwerkzeuge, die am Lagerplatz hergestellt wurden, gefunden.

Römerzeit

Um das Jahr 142 kamen römische Streitkräfte nach Cramond. Sie wurden von Kaiser Antoninus Pius befehligt, der ihnen den Auftrag gegeben hatte, ein Fort an der Mündung des River Almond zu errichten. Dieses Lager sollte die östliche Flanke der Grenze zu Schottland schützen. Fast fünfhundert Menschen arbeiteten daran und bauten ein Fort, das beinahe 24.281 m² abdeckte sowie einen Hafen. Jedoch wurde das Fort nur für ungefähr fünfzehn Jahre bewohnt, bevor die Truppen es wieder verließen, da sie im Süden gebraucht wurden, um den Hadrianswall zu beschützen. Tonwaren und Münzen aus späterer Zeit zeigen, dass das Fort und der Hafen ungefähr zwischen 208 und 211 erneut besiedelt und als Basis für die Armee und die Flotte des Kaisers Septimius Severus benutzt wurden.

1997 wurde im Hafenschlamm die Cramond Lioness entdeckt und als Sandsteinstatue einer Löwin identifiziert, die eine männliche Figur verschlingt, wahrscheinlich eine von einem Paar am Grab eines Militärkommandanten. Es ist eines der ehrgeizigsten Stücke römischer Skulptur, die in Schottland überlebt haben.

Quellen

Weblinks

Ort in Edinburgh (Council Area)

redpajama_set_name: RedPajamaWikipedia


text:
"However, DPI's response includes certain implications that are misleading or inaccurate."
by donnamartinez
From today's release of the investigative report into the Department of Public Instruction comes these key findings and recommendations, as well as a fascinating state auditor's note. First:
• An analyst for DPI submitted false travel reimbursement claims totaling about $3,400 even though the job required no travel.
• The Division Director acknowledged that no one reviewed the details of the reimbursement requests and that the payments would not have been approved if they had been properly scrutinized.
• A former Division Director for DPI engaged in activity and discussions regarding a return to service within six months of her effective retirement date in violation of state law.
• DPI should take action to recover funds inappropriately paid to the analyst in accordance with state law and take disciplinary action against the analyst for knowingly submitting false statements.
• DPI should thoroughly review details of all travel reimbursement claims before authorizing payment.
• DPI should ensure all retirees are made aware of and provided guidance on the North Carolina Retirement Systems' Return-to-Work laws.
• The State Retirement Systems Division should determine whether the former Division Director should forfeit retirement pay and state health plan coverage already received.
And now, the state auditor's note:
In general, DPI management agreed with the findings and recommendations in this report and stated that it initiated both disciplinary and corrective action. However, DPI's response includes certain implications that are misleading or inaccurate.
Durham wants me to pay rent on something I already own
Tax-funded rail spur, costing $24 million, is unused at Global Transpark

redpajama_set_name: RedPajamaCommonCrawl


text:
pysal.viz.mapclassify.gadf
==========================

.. currentmodule:: pysal.viz.mapclassify

.. autofunction:: gadf

redpajama_set_name: RedPajamaGithub


Every year the WMC and IFMA host an event in honour of his Majesty Maha Vajiralongkorn Bodindradebayavarangkun the King of Thailand's birthday. This year is no different and will feature a Super 8 competition in the 72kg weight category.
The Super 8 competition is considered the toughest in the muaythai world as finalists will have to compete a grueling total of 3 times across 2 hours.
This will be the main event of the competition and the first and second athletes will be awarded trophies by WMC President General Chetta Thanajaro and the Minister of Sports Mr. Weerasak Kowsurat.
The bouts will begin on Monday the 30th July from 6 pm onwards across 2 rings next to the Sports Authority of Thailand in Ramkhamhaeng.
This entry was posted on Monday, July 30th, 2018 at 8:30 AM	and is filed under News.


text:
Meet the 2019 SENT Seminar Presenters
The NBA Incubate SENT Seminar equips Social ENTrepreneurs for leadership and change! This four-day seminar is designed for Disciples and their leadership teams who are starting new health and social services ministries. The SENT Seminar will cover the basics of nonprofit ministry startups, as well as skills for leadership and change in our global and faith communities.
This year, the SENT Seminar will be in Scottsdale, Arizonia, and run from September 11-14, 2019. We are grateful for the expert leadership of the following SENT Seminar presenters.
Dietra Wise Baker, NBA Program Coordinator for Advocacy and Activism
Rev. Dr. Dietra Wise Baker first began serving with NBA as an Organizing Specialist with the Ferguson Justice Initiative. In 2018, she transitioned to Program Coordinator for Advocacy and Activism. This work flows from NBA's mission to "advocate for the well-being of humanity" and is done in collaboration and consistent communication with fellow NBA staff to equip NBA and our ministry partners to deepen engagement in advocacy and activism. Following the trajectory of the work begun with the Ferguson Justice Initiative, these efforts utilize relationships, wisdom, and processes built and tested through that work.
Based in St. Louis, Missouri, Dietra nurtures engagement strategies with NBA and our partners to transform root causes of social injustice. This includes equipping and training partners to address the critical social justice issues most relevant to their work, and collaborating with staff to cultivate partnerships across the life of the church to coordinate efforts related to social justice theology, action, and advocacy.
Dietra has a Bachelor's degree from Howard University in Washington, DC, a Master of Divinity degree from Eden Theological Seminary in St. Louis and she recieved her Doctorate of Ministry in Preaching from Aquinas Institute of Theology in St. Louis. Pastor Dietra loves to dance, sing, and teach. She is married to Cornell R. Baker, and they have a beautiful teen daughter, Alexis Baker.
Peter Drury, Chief Strategy Officer at Well Spring Family Services
Peter Drury serves as Chief Strategy Officer, where he is charged with keeping Wellspring future-focused, aligned, growing, and increasingly wise. A social worker with a business education – plus years of innovative leadership and consulting for philanthropists, executives, and boards – Peter is uniquely positioned to facilitate strategy, planning, performance measures, and optimization of the whole. Peter has supported many high-performing organizations during critical stages of growth, such as Splash, Odessa Brown Children's Clinic, Seattle Children's, Sightline Institute, and Make-A-Wish. He is a member of the faculty in Seattle University's acclaimed Master of Nonprofit Leadership Program, and holds MSW, MBA, and M.Div. degrees from the University of Illinois, Seattle University, and Yale University.
Victoria Sun Esparza, CEO and Lead Designer for In the Water Design
Victoria Sun Esparza is the CEO and Lead Designer for In the Water Design, a consulting firm that specializes in design-thinking strategy for religious organizations and non-profits. Victoria's non-traditional background in theological education, vocal performance, and design-thinking gives her a unique perspective on the world and its problems. Her work aims to connect religious organizations and nonprofits to better tools for solving problems and meeting the needs of the people around them. Victoria holds degrees from Abilene Christian University and Southern Methodist University. When she's not traveling, Victoria lives her life in Dallas, Texas with her husband and fox-like dog, Leila.
Maria Francesca French, Executive Director of Hatchery LA
Maria currently serves as the Executive Director at Hatchery LA. Hatchery LA works towards innovation and spiritual entrepreneurship informed by asking new questions of God, community, faith and church. Maria holds a Master of Arts in Religion and Theology as well a Master of Arts in Christian Thought. She is currently engaged in Doctoral work with Wesley Theological Seminary in the Cambridge track studying the intersection of faith, theology and culture, in a world where the philosophical landscape is ever changing. As a teacher of things Christian and Post Christian, a purveyor of story and faith futurist, Maria is passionate about moving into the future with a contextual faith that matters. She is a native New Yorker and now lives in Los Angeles, California.
Hannah Hasan, Owner and Spoken Word Storyteller at Epoch Tribe
Hannah Hasan was gifted with a love for words and the power to express those words with heart and soul so that she can inspire, motivate and change the world around her. She is a spoken word storyteller. She uses the art of the spoken word through poetry and storytelling to create change, build bridges of connection, and shed light on some of the most pressing social issues of our time. In addition to writing commissioned poetry, leading story sharing master classes, and public speaking through her business, Epoch Tribe, Hannah plans and executes events, master classes and workshops that are centered around using stories for social impact.
Hannah obtained her Bachelor of Arts degree from North Carolina A&T State University in 2006 and her Master of Fine Arts degree from Full Sail University in 2012. She has extensive professional experience working in programming for nonprofit organizations, social ventures, and arts-based programming. She currently lives in Charlotte, North Carolina, with her husband Jamaal.
Héctor J. Hernández, NBA Connect Coordinator
Rev. Héctor J. Hernández serves as a full-time Connect Coordinator. In his role, Héctor is responsible for both mental health outreach with particular focus on Disciples Hispanic communities, and the coordination of NBA peer learning and wellness groups.
Héctor has been serving the Latino/Latina immigrant community of Indianapolis, Indiana, since 2009. He is an ordained minister of the Christian Church (Disciples of Christ) of the United States and Canada and served as interim solo pastor of Iglesia Nueva Creación (New Creation Hispanic Church) in the west side of Indianapolis. Héctor has a Master of Divinity and a Master of Arts in Marriage and Family Therapy from the Christian Theological Seminary. He recieved his undergraduate degree in Communications with a minor in Visual Arts. As an artist, clergy and immigrant, Héctor has been reflecting artistically and theologically on topics like immigration, the humanitarian crisis in our border, the unaccompanied Central America children, hope, and social justice.
Belinda King, Vice President of Disciples Church Extension Fund
Belinda R. King has served at Disciples Church Extension Fund since 2012 where she began as the Assistant Treasurer for Loan Services. This year, she was elevated to Vice President of Building and Capital Services.
Belinda has over 20 years of banking experience in business banking, commercial lending, investment management, and client service. Currently, she is responsible for creating, maintaining, and enhancing relationships with Disciples clergy, regional ministers and other adjudicatory leaders, and ecumenical church partners. She provides congregations with financial management, building planning, capital fundraising helping to raise over $1 million, and providing $167million loans. She has been recognized and acknowledged for her exceptional work in helping churches to live into the mission that they have been called to serve.
Belinda has a Bachelor of Science in Public Administration & Labor Relations from Indiana University and an MBA from Indiana Wesleyan. She also holds the Executive Certificate in Religious Fundraising (ECRF) and FINRA Series 6 and 63 licenses. Belinda is a member of the board of the Indiana Christian Church region, now in her sixth year, and formerly served as Interim Treasurer. The accomplishments of which Belinda is most proud, however, are her roles as a devoted mother to three children and a dedicated life partner to her husband of 29 years, Rod. She resides in Fishers, Indiana.
Carla Leon, Lead for Innovation at the United Church of Canada
Carla is the lead for Innovation at the United Church of Canada. She has a background in for-profit having worked at Coca-Cola Canada, McCain Foods, and Canadian Pacific Railway. She completed her undergrad at Wilfrid Laurier Business School and recieved an MBA at the University of British Columbia with a specialization in organizational behaviour and supply chain management. Ten years ago, she started her own consulting business and quickly pivoted it into Small Business Solver, a leading pay-what-you-can small business resource used worldwide including hundreds of communities in Canada. This is not the typical entry point to working with the United Church!
Over the last 7 years, Carla has worked with the United Church of Canada where she launched Buying United, Embracing the Spirit, the Social Innovation Challenge and most recently the coaching program. She works directly in supporting their 350 new initiatives that have been funded through Embracing the Spirit, the 500 social innovations that have launched through the Social Innovation Challenge, and working directly with communities of faith to help them reimagine church outside of Sunday.
LaShaundra McCray, Owner of 3C: Creative Communications and Consulting
Rev. LaShaundra McCarty is the owner of 3C: Creative Communications & Consulting, which has been in business since 2009. She began her post-graduate journey as a graphic designer and social media manager/trainer providing support to nonprofits, denominations and entrepreneurs. Within Disciples, LaShaundra has worked as a consultant to Reconciliation Ministry, National Convocation, Disciples Home Missions, North Carolina Region and Disciples Women. Over the years, that has transformed into providing workshops on leadership, communication, social media, storytelling, and brand development. She has expanded from developing the image of an organization to fostering the development of its people.
The NBA incubates new ministries, supporting social entrepreneurs of faith who are serving their communities in a variety of innovative ways and empowering these Disciples-led health and social service projects to focus on growth, impact, and sustainability. Learn more at nbacares.org/incubate


text:
Q: Spring-Data-Cassandra Sample Web App not Starting (Spring 4.1.2.Release,Spring-data-cassandra 1.1.1-Release,JBoss 8 Wildfly,Java 7) I am building a web application with spring 4.1.2 and Cassandra 2.1.2 as database. During the integration of spring-data-cassandra , i am encountering bean creation exception . Could anyone guide on where i am going wrong?
Error trace is

org.springframework.beans.factory.BeanCreationException: Could not autowire field: private org.springframework.data.cassandra.core.CassandraOperations someapp.dao.role.JdbcSomeDao.cassandraOperations; nested exception is org.springframework.beans.factory.NoSuchBeanDefinitionException. No qualifying bean of type [org.springframework.data.cassandra.core.CassandraOperations] found for dependency: expected at least 1 bean which qualifies as autowire candidate for this dependency.
Dependency annotations: {@org.springframework.beans.factory.annotation.Autowired(required=true)}

My Source code is as following:
  The web initializer file where web application gets initialized
public class WebInitConfig implements WebApplicationInitializer {

    @Override
    public void onStartup(ServletContext servletContext) throws ServletException {
        AnnotationConfigWebApplicationContext rootContext = new           AnnotationConfigWebApplicationContext();
        rootContext.register(DispatcherConfig.class);
        servletContext.addListener(new ContextLoaderListener(rootContext));
        servletContext.setInitParameter("defaultHtmlEscape", "true");

        AnnotationConfigWebApplicationContext mvcContext = new AnnotationConfigWebApplicationContext();
           mvcContext.register(CassandraConfig.class);
        // The main Spring MVC servlet.
        ServletRegistration.Dynamic appServlet = servletContext.addServlet("appServlet", new DispatcherServlet(mvcContext));
                appServlet.setLoadOnStartup(1);
        Set<String> mappingConflicts = appServlet.addMapping("/");
        if (!mappingConflicts.isEmpty()) {
            for (String s : mappingConflicts) {
               System.out.println("Mapping conflict: " + s);
            }
            throw new IllegalStateException(
                 "'appServlet' cannot be mapped to '/' under Tomcat versions <= 7.0.14");
           }
        }

     }

The DispatcherConfig file that contains spring dispatcher code and view resolvers.
@Configuration
@EnableWebMvc
@PropertySource("classpath:cassandra.properties")
@ComponentScan("randompackage.*")
public class DispatcherConfig extends WebMvcConfigurerAdapter {

     @Override
     public void addResourceHandlers(ResourceHandlerRegistry registry) {
        registry.addResourceHandler("/css/**").addResourceLocations("/css/");
     }

     @Bean
     public ViewResolver configureViewResolver() {
         InternalResourceViewResolver viewResolve = new InternalResourceViewResolver();
         viewResolve.setPrefix("/WEB-INF/jsp/");
         viewResolve.setSuffix(".jsp");

         return viewResolve;
     }

     @Override
     public void configureDefaultServletHandling(DefaultServletHandlerConfigurer configurer){
       configurer.enable();
     }
}

The spring-data-cassandra configuration file "CassandraConfig.java, which is loaded on application start-up in webinitconfig.java
@Configuration
@PropertySource(value = { "classpath:cassandra.properties" })
@EnableCassandraRepositories(basePackages = { "someapp.dao" })
public class CassandraConfig {
/*
      @Autowired
      private Environment env;
*/
      @Bean
      public CassandraClusterFactoryBean cluster() {

        CassandraClusterFactoryBean cluster = new CassandraClusterFactoryBean();
        cluster.setContactPoints("10.20.13.157");
        cluster.setPort(Integer.parseInt("9042"));

        return cluster;
      }

      @Bean
      public CassandraMappingContext mappingContext() {
        return new BasicCassandraMappingContext();
      }

      @Bean
      public CassandraConverter converter() {
        return new MappingCassandraConverter(mappingContext());
      }

      @Bean
      public CassandraSessionFactoryBean session() throws Exception {

        CassandraSessionFactoryBean session = new CassandraSessionFactoryBean();
        session.setCluster(cluster().getObject());
        session.setKeyspaceName("spring_data");
        session.setConverter(converter());
        session.setSchemaAction(SchemaAction.NONE);

        return session;
      }

      @Bean
      public CassandraOperations cassandraTemplate() throws Exception {
        return new CassandraTemplate(session().getObject());
      }

      @Bean(name="someManager")
      public SomeManager getSomeManager() throws Exception {
          return new someClassManager();
      }

      @Bean(name="someDao")
      public SomeDao getSomeDao() throws Exception {
          return new someDao();
      }

}

And I have the respective Controllers,Service and Daos.
@Controller
public class SomeRegistrationController {

    @Autowired
    SomeManager someManager;

    @RequestMapping(value="registration.htm")
    public String registration() {
        someManager.Registration(new SomeDom());

        return "registration";

    }

}

Service Class for some mappings
@Service("someManager")
public class SomeClassManager implements SomeManager {

    @Autowired
    SomeDao someDao;

    @Override
    public void registration(SomeDom details) {
        someDao.registration(details);
    }

}

Dao Class for jdbc interaction
@Repository("someDao")
//@EnableAutoConfiguration
public class JDBCSomeDao implements SomeDao{

    @Autowired
    private CassandraOperations cassandraOperations;

    @Override
    public void Registration(SomeDom details) {
        System.out.println("JDBC");
        System.out.println(cassandraOperations.toString());
    }
}

redpajama_set_name: RedPajamaStackExchange


The Church of St. John on Hamilton's Charlton Street has the same cordial atmosphere and acoustics as does St. John's in Oakville. Such ambiance somehow lends itself to an intimate environment for enjoying classical music. VOX METROPOLIS' trio of flautist Sara Traficante; cellist Kirk Starkey and the piano of David Jones chose music from the mid-18th century right up to 2015. All of the works presented were – while some not melodic…certainly interesting.
The Mississauga Players' in-house-authored bit of fluff plays strictly for laughs; and in that context – succeeds. As theatre, both the playwright and director need a smidgen of dramaturgical tweaking. Frank De Francesco's pastiche 'GRAND HOTEL' presents six situations occurring on New Year's Eve, as a long-suffering bellboy retires and a young replacement is thrust into the position of night manager. His involvements with the diverse hotel guests range from improbable to downright absurd. All are giggle-gleaning situations.
The Hamilton Philharmonic closed its 2014-2015 season at Hamilton Place with a concert was directed by former conductor James Sommerville, who also appeared as the soloist for the evening. The program included major orchestral works from the 18th and19th centuries. The concert opened with Mozart's Horn Concerto No. 4 in E-flat Major, K.495. This work was completed in 1786 and was composed for Ignaz Leutgeb. Many soloists conduct while performing, but the French horn soloist plus conductor present unique challenges.
He was the only American soldier executed for desertion since the Civil War and some still consider it a blot on Military justice. Based on W.B. Huie's 1954 book; Robert Knuckle has crafted a balanced and almost objective iteration of this sorry chapter in American history. Under the aegis and delicate touch of Willard Boudreau; who is neither maudlin nor melodramatic, Kayson Productions of Dundas has staged a perfect and dramatic gem. We were fascinated and touched.
Shakespeare's Richard III is a convoluted yet still fascinating play 6 centuries after his death at Bosworth – the battle that ended the Wars of the Roses and the House of York. The interpretive envelope-push had to be the Dreyfuss reading in 'The Goodbye Girl"; but Tim Welham's one-man definition is an incredible thespian piece of artistry. For 1¼ hours, Welham captures not only the physically flawed but morally challenged British King. By amazingly diverse facial expression; body language; voices and stance, the audience meets each of his friends, associates, victims and relatives.


In 2004 Elgin Rotary and MacAllan's Distillery celebrated 21 years of 'The Annual Festival of Bothy Ballads', known to most as 'The Champion of Champions' contest. In effect of course it is quite simply, the annual bothy world championship! A commemorative CD was issued capturing performances from most of those songsters who have starred in the event throughout the years. So, here it is - over an hour of couthy songs from the best in the business. Jist grand !


AhmedabadSale.com was the first Real Estate Portal (Launch Date: Oct 15, 2003) of India based on Java/J2EE Standards. Since it is java based platform the useability and consistency in software can be felt in every day use. We place all kinds of Real Estate Advertisements.
AhmedabadSale.com also hosts Motorcycle and Car Advertisements. Advertisements can be placed by filling on-line form and they can also upload photos through on-line form. Website has advanced search facility by which one can search properties by location and property type. Real Estate Property Developers can place Panoramic Street View of their Project which currently no website in India does that.
Portal has a photographic album of all the major attractions in Ahmedabad. It has total of 600 photographs of major streets, museums, lakes, river, parks etc. Perhaps AhmedabadSale.com is the only portal in the world that has this comprehensive photographs of one city. Another unique feature of photo album is that one can search landmark buildings photographs through advanced search facility and you can browse photo albums which are 10 years or older.
AhmedabadSale.com in 15 years had 30 million hits from all around the world.
In 2006 our website won Jboss (Red Hat, Inc., USA) Honorable Mention Award.
Online Yellow Pages & Place Your Ad.
Looking to Buy Residence then Submit your Ad.
How do I see java applications through Browser?
JDK and JRE are developed by Sun and Oracle since 1996 to Present. We have (jdk1.0 (1996), jdk2.0...... to jdk11.0 (Present- 2018)).
with javafx 8.0. No configuration is required for javafx with jre 8.0. So if you want ot watch videos and audio files than use jre 8.0.
This you can download from Oracle website.
This is another example of javafx audio application which shows photographs, text data and audio file. Enjoy!!!!!
exception site list. Select Ok and you are done for browsing the Audio Application.
- Swing Components and much more.
It can also connect to database.
- For OS Windows XP, Vista and 7 having 32 bit OS; JRE (Java Runtime Environment) should have JRE version 8.0 (32bit).
For 64 bit OS Win 8 or 10 same JRE Version 8.0 (64 bit) should be used.
exception site list. Select Ok and you are done for browsing the Video Application.
keeps the profile of firefox browser.
system or damage network files. Hence it is very safe application.
Photo Album in the form of Applet - 2014 - 2017, Atlast after 3 years Applet is without error!!
front end we can have millions of such apps on the internet.
you will see Java Icon.
there is Exception Site List. Input http://www.ahmedabadsale.com/photos_1/photo_Applet.html url in the exception site list.
Select Ok and you are done for browsing the applet.


text:
\section{Introduction}
Geometric routing is an appealing routing technique that uses the position of the nodes for routing data in communication networks. In particular, {\it greedy routing} consists in using routing paths such that at each hop the distance to the destination decreases. Greedy routing is local. Unfortunately, there are simple examples where all neighboring nodes of a node are at larger distance to the destination and greedy cannot be applied. It is then relevant to determine in which situations greedy routing guarantees the delivery of data. Besides the relevance to routing data in communication networks, greedy routing has lead to interesting works in graph theory and computational geometry.

The paper \cite{DBLP:conf/algosensors/PapadimitriouR04} is usually considered as a first landmark. In particular,  it is shown how a $3$-connected graph can embedded in ${\mathbb R}^3$ in such a way that greedy routing guarantees delivery. Moreover, it is {\it conjectured}  that any $3$-connected planar graph can be embedded in $\mathbb R^2$ such that greedy routing guarantees delivery, what is now called a {\it greedy embedding} (in $\mathbb R^2$). This conjecture leads to intensive research and an exhaustive survey of all the results is not presented here. We mention key contributions in the main directions around this conjecture and position the contribution of this article among these works.

The conjecture was proved to be true. In \cite{DBLP:conf/focs/MoitraL08,eppstein2011succinct} it is proved for $3$-connected graphs, in \cite{bose1999online} in Delaunay triangulations, in \cite{chen2007distributed} for graphs that satisfy conditions with respect to the power diagram. In \cite{DBLP:conf/infocom/Kleinberg07}, a greedy embedding in the hyperbolic plane of a connected finite graphs is constructed. More related to our approach, in \cite{Dhandapani08} the conjecture is proved for planar triangulations (maximal planar graphs), see also \cite{he2010schnyder,he2011succinct}. All these approaches rely on a distance metric that is used by greedy routing. The papers \cite{eppstein2011succinct,he2010schnyder,he2011succinct} limit the memory requirement to ${\cal O}\bigl(log(n)\bigr)$ bits to represent the coordinates. Such coordinate systems are called {\it succint} and this property is important for the design of scalable routing scheme. Our coordinate system, see definition \ref{def:coordinates}, is {\it succint}. In \cite{Dhandapani08,he2010schnyder,he2011succinct}, Schnyder's caracterization of (maximal) planar graphs \cite{schnyder} is used. This characterization is discussed in section \ref{vrac} and is also used in this work. However, our approach is to avoid the definition of a metric and the computation of the planar embedding, see for instance \cite{angelini2009algorithm} for an algorithm to compute the greedy embedding of planar triangulations. We rely on the metric-free definition of greedy paths in \cite{DBLP:journals/tmc/LiYL10} - without embedding the graph. Moreover, the coordinate system used in \cite{Dhandapani08,he2010schnyder,he2011succinct} is more complex to compute than the one we use. Briefly, in Schnyder work, planar graphs are characterized by the existence of three total order relations on the vertex set of the graph (and extra conditions). Using these order relations Schnyder builds three spanning\footnote{ Actually, spanning internal nodes of the triangulation.} directed trees called the {\it realizer} and the coordinates are computed using these trees. This coordinate system has relevant properties, for graph drawing, see for instance \cite{nishizeki2004planar}, that we do not need for routing\footnote{Although it is relevant to ask if these properties are necessary for constructing a greedy embedding.}. The particularities of greedy routing with respect to greedy drawing are already pointed out in \cite{eppstein2009succinct}.

In this article, we use directly the order ranks for coordinates, see Definition \ref{def:coordinates} without (non-local) extra-computations. This approach is coherent with previous works. In particular \cite{DBLP:conf/HucJLR12,DBLP:conf/algosensors/HucJLR10} where the three order relations are obtained by the nodes composing a communication network by measuring the distances to three distinguished nodes, see Figure \ref{fig:exvrac}. The motivation for this approach to avoid the computation of the coordinates, i.e. the usual scheme for localizing the nodes is $1)$ measure distances and, 2) compute the coordinates. In the subsequent works \cite{DBLP:conf/icpads/SamarasingheL12,DBLP:conf/sensornets/SamarasingheL14}, we design the classical {\it greedy-face} routing paradigm in this simple coordinate system.

We emphasize that the underlying assumptions in these works (or generally in geographic routing) is that the communication graph is a Unit Disk Graph (UDG) and the nodes are located in a $2D$ region, in particular we can use the Jordan curve theorem to prove that face routing guarantees the delivery. In this work, we do not make such assumption. We assume that the graph is given by the sets of nodes and edges in an abstract way.

\noindent{\bf Our contribution} In this paper we show that given a maximal planar graph we can
\begin{itemize}
\item Provide a metric-free definition of greedy paths.
\item Design a local greedy routing algorithm that guarantees delivery.
\item Decouple the problems of greedy routing and greedy embedding.
\end{itemize}
Moreover, all our computations are constructive and the coordinate system requires ${\cal O}\bigl(log(n)\bigr)$ bits and can be qualified of {\it succint} \cite{eppstein2011succinct}.

In section \ref{vrac} we present Schnyder's characterization of planar maximal graphs, fix the notations and define greedy paths. In section \ref{greedy} we prove the properties that we need to build a greedy path between any two nodes.  Finally, in section \ref{routingmaxplanargraph} we state the main result of the paper about the existence of greedy paths.

\section{Schnyder three-dimensional representation and coordinates}
\label{vrac}

Given a planar graph $G=(V,E)$, it is proved in \cite{schnyder} that there exists three total order relations on $V\times V$, denoted $<_1, <_2, <_3$ such that
\begin{equation}
\begin{array}{l l}\label{eq:schnydercond}
&a)~~\bigcap_{i=1,2,3} <_i =\emptyset,\text{, and}\\
&b)~~\forall (x,y)\in E,  \forall z\not\in\{x,y\}~ \exists i\in\{1,2,3\} \text{ s.t. } (x,z)\in <_i \text{ and } (y,z)\in <_i.
\end{array}
\end{equation}
This is called a {\it (3-dimensional) representation} of the planar graph.
We also use the notation $x<_iz$ for $(x,z)\in <_i$ and we say $v$ is a neighboring node of $u$ to say that $(u,v)\in E$.

\begin{definition}(standard representation, internal and external nodes)\label{def:internal} The representation is {\it standard} if there are three distinguished elements $A_1, A_2, A_3$ such that $A_i$ is the maximal element for $<_i$ and the remaining $A_{i-1}, A_{i+1}$\footnote{We use the notation $ i+1\equiv i \text{ mod } 3 +1$.} are the two smallest elements of $<_i$ (in any order, $A_{i-1}<_1 A_{i+1}$ or $A_{i+1}<_1 A_{i-1}$). Any representation can be turned to a standard one \cite{schnyder}.

 The distinguished elements $A_i, i=1,2,3$ of a standard representation are called {\it external}, the others nodes are called {\it internal}.
\end{definition}

\begin{definition}\label{def:coordinates}(Coordinates) Given a node $u\in V$ the rank of $u$ in the order $<_i$ is denoted by $rank_i(u)$. If there are $n$ elements in $V$,  the smallest element has rank $1$ and the largest rank $n$. The coordinates of the node $u$ are $(u_1,u_2,u_3)=(rank_1(u), rank_2(u), rank_3(u))$. This coordinate system requires 3 integers per nodes and hence, it scales like ${\cal O}\bigl(log(n)\bigr)$ bits and is {\it succint} \cite{eppstein2011succinct,he2010schnyder,he2011succinct}.
\end{definition}

\begin{figure}[!]\hskip -2cm
\includegraphics[scale=0.3]{coordonnees}\hskip -1cm\includegraphics[scale=0.3]{coordonnees2}\vskip -2cm
\caption{Two realistic way of getting the order relations $<_i$. Coordinate assignment with with raw distances from anchors on the left and  perpendicular distances from heights of the triangles on the right}
\label{fig:VRAC2}\label{fig:VRAC}\label{fig:exvrac}
 \end{figure}

In ~\cite{DBLP:conf/sensornets/SamarasingheL14} the three order relations are obtained by measuring the distances from the node $u$ to three anchors $d(u,A_i)$ or,  in  \cite{DBLP:conf/icpads/SamarasingheL12} by the three heights of the triangles (needs more computation and information) $uA_iA_j$, see Figure \ref{fig:exvrac}. The order relations are defined by $u<_i v \Leftrightarrow d(u,A_i)>d(v,A_i)$. The motivation for using this VRAC coordinate system \cite{DBLP:conf/algosensors/HucJLR10}, in ad-hoc communication networks, is to avoid the usual computations for the  localization of the nodes. Indeed, most localization schemes start to measure distances and then proceed to computations. In the VRAC coordinate system the computations are avoided. Although in this article we do not assume that the order relations are given in this way, it is helpful to keep these representations in  mind. To help visualize the definitions and results we show how they can be represented if such a geometrical model is assumed. We emphasize that the goal of the article is the design of greedy routing algorithm without geometry.

The three order relations are total\footnote{A total order is a binary relation which is valid for \textit{all the pairs} in a set} and it makes sense to associate the minimum of a set with respect to one of the three order. We will denote this by $\text{min}_i$ for $i=1,2,3$.

These three orders permit the definition of sectors associated with a node $u$.\\
\begin{definition}(Sectors) \label{def:sectors} We define the following sectors associated to a node $u\in V$, see Figure \ref{fig:VRAC}. Note that the reference node $u$ does not belong to the sectors.
\begin{align*}
s_1^u = \{v\mid ~u~<_ 1 ~v, ~u~>_ 2 ~v, ~u~>_ 3 ~v\}.\\
s_2^u = \{v\mid ~u~<_ 1 ~v, ~u~<_ 2 ~v, ~u~>_ 3 ~v\}.\\
s_3^u = \{v\mid ~u~>_ 1 ~v, ~u~<_ 2 ~v, ~u~>_ 3 ~v\}.\\
s_4^u = \{v\mid ~u~>_ 1 ~v, ~u~<_ 2 ~v, ~u~<_ 3 ~v\}.\\
s_5^u = \{v\mid ~u~>_ 1 ~v, ~u~>_ 2 ~v, ~u~<_ 3 ~v\}.\\
s_6^u = \{v\mid ~u~<_ 1 ~v, ~u~>_ 2 ~v, ~u~<_ 3 ~v\}.\\
\end{align*}

\noindent Notice that the coordinates of the nodes in Definition \ref{def:coordinates} make possible to determine in which sector a node belongs relatively to another one. Sometimes, the sector $s^u_i$ is also referred to as {\it the sector $i$ of $u$}\footnote{We use the notation $ i+1\equiv i \text{ mod } 6 +1$ if $i$ is the index of a sector, i.e. $s^u_i$.}.
\end{definition}

\begin{definition}\label{def:sectornotation}
Given a node $D$, we also use the convenient notation $s^u_D$ to denote the sector $j$ of $u$ such that $D\in s^u_j$, i.e. $D\in s^u_D$.
\end{definition}

There is a useful way to distinguish the edges that uses the definition of partial orders $<_1^*, <_2^*, <_3^*$, see Lemma 3.1 in \cite{schnyder}
\begin{definition} For each $i\in\{1,2,3\}$ we define
\begin{equation*}
(u,v)\in <_i^*~ \Longleftrightarrow~ (u,v)\in<_i \text{ and } (v,u)\in<_{i+1} \text{ and } (v,u)\in<_{i-1}.
\end{equation*}
Or equivalently
\begin{equation*}
(u,v)\in <_i^*~ \Longleftrightarrow~  v\in s^u_{2i-1}.
\end{equation*}
\end{definition}
\begin{property}\label{prop:order*}
The empty intersection property $a)$  in $(\ref{eq:schnydercond})$ implies that for each $u,v\in V$ there exists exactly one $i\in{1,2,3}$ such that $(u,v)\in <_i^*$ or $(v,u)\in <_i^*$ (equivalently $v\in s^u_{2i-1}$ or $u\in s^v_{2i-1}$). It is convenient to rememember that if $(u,v)<_i^*$ then $v\in s^u_{2i-1}$, i.e. $s^u_1$ or $s^u_3$ or $s^u_5$, the indexes are odd and even otherwise.
\end{property}

\begin{property}\label{prop:sectors}
A node $u$ has at most one edge $(u,v)\in E$ such that $v\in s_{2i-1}^u$. Moreover,  such a node $v$ satisfies that $v<_i z ~\forall z\in s_{2i-1}^u, i=1,2,3$, i.e. $v=min_i\{z\mid z\in s^u_{2i-1}\}$,  see Lemma 3.1 of \cite{schnyder},  this follow from condition $b)$ in $(\ref{eq:schnydercond})$. These properties can be written
\begin{align}
\left.
{\bf\text{If } v\in s^u_1,~z\not=v\text{  we have }}~~\begin{array}{l l}
z<_2 u\\
z<_3 u
\end{array}\right\} \Rightarrow z>_1 v.\label{eq:propertesu1}\\
\left.
{\bf\text{If } v\in s^u_3,~z\not=v\text{  we have }}~~\begin{array}{l l}
z<_1 u\\
z<_3 u
\end{array}\right\} \Rightarrow z>_2 v.\label{eq:propertesu3}\\
\left.
{\bf\text{If } v\in s^u_5,~z\not=v\text{  we have }}~~\begin{array}{l l}
z<_1 u\\
z<_2 u
\end{array}\right\} \Rightarrow z>_3 v.\label{eq:propertesu5}
\end{align}
\end{property}
Property \ref{prop:sectors} is from \cite{schnyder} and the proof uses part $ b)$ of the graph representation $(\ref{eq:schnydercond})$. There is a nice geometric {\it void} condition associated to this property. Indeed, if we assume that the edge $(u,v)$ belongs to $s^u_1$ then the existence of a node $ u<_1 w <_1 v$ violates the second condition of $(\ref{eq:schnydercond})$, see Figure \ref{fig:voidregion} and Figure 3c of \cite{Dhandapani08}. It is interesting to compare this void region with the corresponding ones of the planar Relative Neighborhood Graphs (RNG) or Gabriel Graphs (GG) \cite{DBLP:conf/mobicom/KarpK00}.

\begin{figure}\hskip-1cm
\includegraphics[scale=0.3]{voidregion}\hskip -1cm\includegraphics[scale=0.3]{localview}\caption{{\bf Left: }The geometric interpretation of Property \ref{prop:sectors}. The hatched region is void. {\bf Right:}  Locally a node $u$ of a maximal planar graph as exactly three edges in the sectors $s_1^u, s_3^u, s^5_u$ if the representation is standard. The undetermined (0,1,2,...) number of edges in the sectors $s^u_2, s^u_4, s^u_6$ are not represented. }\label{fig:localview}\label{fig:voidregion}
\end{figure}
\subsection*{Informal presentation of the routing strategy}

\noindent If the graph is planar maximal  and the representation standard then each internal node $u$, see Definition \ref{def:internal},  has exactly one edge in each sector $s^u_1, s^u_3, s^u_5$ and an indeterminate ($0,1,2,\ldots$) number in the remaining sectors $s^u_2, s^u_4, s^u_6$. Indeed, because the representation is standard the sectors $s^u_1, s^u_3, s^u_5$ contain the nodes $A_1, A_2, A_3$ respectively and are not empty. The maximality assumption implies that if there is an option of adding an edge and keeping the planarity property then the edge is present \cite{schnyder}. It is helpful to look at the geometric visualization in the right of Figure \ref{fig:localview}.

For routing from a node $u$ to a destination $D\in s^u_1\cup s^u_3\cup s^u_5$ the natural option is to follow the edge $(u,v)$ such that $v\in s^u_D$ ($=s^u_1$ or $s^u_3$ or $s^u_5$). Next, from $v$, if $D\in s^v_1\cup s^v_3\cup s^v_5$ we repeat the same strategy. However, it may happen that $D\not\in s^v_1\cup s^v_3\cup s^v_5$, see Proposition \ref{lemmagreedy}.  In this case $D\in  s^v_2\cup s^v_4\cup s^v_6$ and the existence of an edge in the sector $s^u_D$ is not provided by the Schnyder's characterization $(\ref{eq:schnydercond})$. Nevertheless, in Proposition \ref{prop:greedyeven} we show how we can route the data in this case.

\section{Greedy Routing}
\label{greedy}

Our greedy routing technique differs from the classical ones as we do not assume that a metric is given. Instead, we use  the metric-free axioms characterizing greedy paths provided in \cite{DBLP:journals/tmc/LiYL10}, i.e. given a destination $D$ we have 1. ({\bf transitivity})  if node $v$ is greedy for $u$ and $w$ is greedy for $v$ then $w$ is greedy for $u$ as well and  2. ({\bf odd symmetry}) if $v$ is greedy for $u$ then $u$ is not greedy for $v$. Moreover, the coordinate system that we use is different than the one used in the others works that are based on Schnyder's characterization of planar graph, for example  in \cite{DBLP:dblp_journals/tcs/HeZ13,Dhandapani08}. Indeed, they use a coordinate system used in \cite{schnyder} that is more complex to compute than our in Definition \ref{def:coordinates}. This difference is possible here because initially the coordinate system was designed to draw the planar graph, while we limit our purpose to route the data.

\begin{definition} For destination node $D$, a path $\{u^k\}$ is a greedy path if there exists $i\in\{1,2,3\}$ such that
\begin{equation}\label{eq:shrinkingregion}
\forall k~~ u^{k+1}<_i{u^k},\text{ or } \forall k~~ u^{k+1}>_i{u^k}.
\end{equation}
For a greedy path there is a coordinate that changes monotonically.
\end{definition}
Because the coordinates change by at least one unit along a path, a  greedy path must stop. In the following we build greedy paths from $u$ to $D$ such that $u<_i u^k<_iD$ the fact that $D$ is an upper bound and the construction continues while $u^k<_i D$ implies the convergence of the sequence to $D$.
\remark In the proofs of Propositions \ref{prop:greedyeven} and \ref{lemmagreedy} we use the assumption that the graph is maximal to say that given a node $u$ there exists neighboring nodes in the sectors $s^u_1, s^u_3, s^u_5$. Unfortunately we must proceed with caution if the node $u$ is one of the distinguished nodes $A_1, A_2, A_3$ since these nodes may not have any neighboring nodes in these sectors. Actually, these nodes do not cause any trouble because there is a path from any internal nodes to them with increasing coordinate $rank_1, rank_2, rank_3$ respectively. They are also all connected to each others. For these reasons and in order to make our best to simplify the exposition we no longer make any reference to these particular nodes in the proofs.

In the proof of Proposition \ref{prop:greedyeven} we need the following Proposition
\begin{proposition}\label{lemmatransitivity} $D'\in s^D_i$ and $D''\in s^{D'}_i$ then $D''\in s^D_i$
\end{proposition}
\begin{proof}
This property follows directly from the transitivity of the inequalities in the definition of the sectors $(\ref{def:sectors})$.
\end{proof}

\begin{proposition}\label{prop:greedyeven} We assume that the graph $G$ is triangular (or equivalently planar maximal, this implies the existence of a unique edge in each sectors $s^u_1, s^u_3, s^u_5$ of $u$ for all $u$ internal nodes). Then provided that the destination $D$ belongs to $s^u_2$  (or $s^u_4$, or $s^u_6$) then there is a path  $\{u^i\}$ in $G$ with $u^0=u$ such that $u^{i+1} \in s_2^{u^i}$ ($u^{i+1} \in s_4^{u^i}$ or $u^{i+1} \in s_6^{u^i}$ respectively), and the path converges to $D$.

Along the path the coordinate $rank_3$ ($rank_1$, $rank_2$) decreases monotonically if $D\in s^u_2$ ($D\in s^u_4$, $D\in s^u_6$ respectively).
\end{proposition}
\begin{proof}
For concreteness we consider $D\in s^u_4$. If $u$ is connected to $D$ we define $u^1=D$ and the proposition is true. Else, we prove below that there exists a neighboring node of $u$, $u^1$ such that $D\in s^{u^1}_4$ and $D<_1 u^1<_1 u$. Hence, by applying the construction iteratively we construct the sequence of points that satisfy $u^{i+1}\in s^{u^i}_4$, lower bounded by $D$ and decreases with respect to $<_1$, i.e. $D<_1 u^{i+1}<_1 u^i$. Such a sequence converges to $D$.

{\bf Let us prove that given $u$ such that $D\in s^u_4$ there exists $v$ such that $(u,v)\in E$, $D\in s^v_4$ and $D<_1 v<_1 u$. } $u$ is internal, by the maximality (triangulation) assumption there exists two neighboring nodes of $u$ such that $v\in s^u_3$ and $w\in s^u_5$. we then have
\begin{align}
&D <_1 u, ~D >_2 u,~D>_3 u~\Leftrightarrow D\in s^u_4\label{eqq1}\\
&v<_1 u,~v>_2 u,~v<_3 u~\Leftrightarrow v\in s^u_3\label{eqq2}\\
&w<_1 u,~w<_2 u,~w>_3 u~\Leftrightarrow w\in s^u_5\label{eqq3}
\end{align}
If $v$ (or $w$) is such that $D\in s^v_4$ (or $D\in s^w_4$) the next point on the path is $u^1=v$ (or $u^1=w$) and $(\ref{eqq2})$ shows that $v=u^1<_1 u$, and $D\in s^v_4\Rightarrow v>_1 D$ (or $(\ref{eqq3})$ shows that $w=u^1<_1 u$ , and $D\in s^w_4\Rightarrow w>_1 D$).

\noindent Else, we have to prove that there exists a neighboring node of $u$ in the sector $s^u_4$ that satisfies the conditions. We have that $D>_2u>_2w$, and $D>_3u>_3v$ (using $(\ref{eqq1},\ref{eqq2},\ref{eqq3})$) and  $D\not\in s^v_4$ and $D\not\in s^w_4$ imply
\begin{align}
&D\not\in s^v_4\Rightarrow\left.\begin{array}{l l} D>_1v ~D<_2 v~D>_3v~~\text{ or}\\ D<_1v~D<_2v~D>_3v\end{array}\right\}\Rightarrow D<_2v\label{eq:notinsv4}\\
&D\not\in s^w_4\Rightarrow\left.\begin{array}{l l} D>_1w ~D>_2w~D<_3w~~\text{ or}\\ D<_1w~D>_2w~D<_3w\end{array}\right\}\Rightarrow D<_3w\label{eq:notinsw4}
\end{align}
Next, because $D\in s^u_4\Rightarrow u\in s^D_1$ and the maximality assumption, there exists an edge $(D,D')$ with $D'\in s^D_1$. If $D'=u$ we are done.

\noindent Else we have by the property $(\ref{eq:propertesu1})$ and $u\in s^D_1$ that $\boxed{D'<_1 u}$.

By gathering the inequalities corresponding to $u\in s^D_1$ with the ones deduced from $(\ref{eq:notinsv4})$,$(\ref{eq:notinsw4})$ we obtain  $D<_1D', v>_2D>_2 D', w>_3D>_3D'$. Using $D' <_1u$, $D'<_2v$ with property $(\ref{eq:propertesu3})$ we obtain $\boxed{D' >_3 u}$.

Last from $D'<_1 u$, $D' <_3 w$ and property $(\ref{eq:propertesu5})$ (with edge $(u,w)$ instead of $(u,v)$) we obtain $\boxed{D' >_2 u}$.
Finally, we have proved that $D'\in s^u_4$ with the boxes equations and $D<_1D' <_1 u$. The node $D'$ plays the same role as $D$ in the statement of the proposition but with an increasing $<_1$ order position. Because of the bound $D' <_1 u$ we see that by applying iteratively the construction we obtain a sequence $D', D'', \ldots$ that converges to $u$ and such that all the points belong to $s^u_4$. Moreover, along the sequence we have $D'\in s^{D}_1$,  $D''\in s^{D'}_1$,... and Lemma \ref{lemmatransitivity} implies that all the points in the sequence belong to $s^D_1$. In particular, for the point $x$ that is connected to $u$ $x\in s^D_1\Leftrightarrow D\in s^x_4$. We have then proved the existence of a point $x\in s^u_4$ that satisfies $D\in s^x_4$ and such that $D<_1x<_1u$.
\end{proof}

\begin{remark}{\bf Construction of the greedy path if $D\in s^u_{2i}$}\label{remarkeven}

\noindent In order to route from $u$ to $D\in s^u_{2i}$ the node $u$ must first check whether for $v\in s^u_{2i+1}$ and $w\in s^u_{2i-1}$ one of the condition $D\in s^v_{2i}$ or $D\in s^w_{2i}$ is satisfied and if yes sends the message accordingly. Otherwise, the message is forwarded to (the existing) neighboring node in $x\in s^u_{2i}$ such that $D\in s^x_{2i}$. This routing scheme converges because the coordinate $i$ decreases along the path and the path don't step over $D$ because all the points in the path are $>_1 D$.

\end{remark}

\begin{proposition}\label{lemmagreedy} Let us assume that $(u,v)\in E$ and $D,v\in s^u_1$ (or $s^u_3$ or $ s^u_5$). Then, $D\not\in s^v_3\cup s^v_4\cup s^v_5$ (or $s^v_1\cup s^v_5\cup s^v_6$ or $s^v_1\cup s^v_2\cup s^v_3$).
\end{proposition}
\begin{proof} Let us consider $v,D\in ^u_1$ the other cases are proved similarly by a permutation of the indices. We have
\begin{equation*}
\begin{array}{l l}
v\in s^u_1\Leftrightarrow u<_1 v~~u>_2 v~~u>_3 v\\
D\in s^u_1\Leftrightarrow u<_1 D~~u>_2 D~~u>_3 D.
\end{array}
\end{equation*}
Part b) of the Schnyder's conditions $(\ref{eq:schnydercond})$ implies that $D$ must be larger than $u$ and $v$ for one order and we see on the two inequalities above that it can  only be $<_1$. The condition $D\in s^v_3\cup s^v_4\cup s^v_5$ implies that $v>_1 D$ and hence there is no $i\in\{1,2,3\}$ such that $u,v<_i D$ and the result in proved.
\end{proof}

\begin{remark}{\bf Construction of the greedy path if $D\in s^u_{2i-1}$}\label{remarkodd}

\noindent The practical implication of Proposition \ref{lemmagreedy} for routing is to prove the existence of a greedy path from $u$ to $D\in s_{2i-1}^u$. We decompose the construction in two parts and for concreteness we consider $D\in s^u_1$.

\noindent {\bf Part 1.}  The maximality assumption implies the existence of a node  $v\in s^u_1$ such that $(u,v)\in E$. If $v=D$ we are done. Else, $u$ sends the message to $v$ and the first coordinate $rank_1$  increases, the second one $rank_2$ and the third one $rank_3$  decrease. If $D\in s^v_1$ then $v$ repeats the same procedure and the coordinates continue to be updated monotonically and $D>_1 v$ because $D\in s^v_1$ and this implies that the first part of the construction converges to $D$ or switches to the second part.

\noindent {\bf Part 2.} If the path reaches a node $v$ such that $D\not\in s^v_1$ the construction of the path continue with this second part. In this case $D\in s^v_2$ or $D\in s^v_6$ must be satisfied because of Proposition \ref{lemmagreedy}. In both cases we have $D>_1 v$ and we can apply Proposition \ref{prop:greedyeven} that shows the existence of a sequence of nodes $v'$ with $D\in s^{v'}_2$ or  $D\in s^{v'}_6$ respectively and this sequence eventually reaches $D$. If  $D\in s^{v'}_2$ then by Proposition \ref{prop:greedyeven} the coordinate $rank_3$ continues to decrease along the second part of the construction. If $D\in s^{v'}_6$ the coordinate $rank_2$ continues to decrease. In both cases we have shown that along the two parts of the construction one coordinate ($rank_2$ or $rank_3$) decreases monotonically and the resulting path is then {\it greedy}.
\end{remark}

\section{Routing in maximal planar graph}\label{routingmaxplanargraph}
In \cite{Dhandapani08}, it is proved using Schnyder's characterization of planar graphs $(\ref{eq:schnydercond})$ that there exists an embedding of the graph in the plane\footnote{Actually in the plane in $\mathbb R^3$ such that $x+y+z=1$.} such that greedy routing is successful (using the natural metric). In \cite{DBLP:dblp_journals/tcs/HeZ13} the authors use a similar coordinate system  and design a (simple) routing algorithm. Both papers use a {\it realizer} as defined in \cite{schnyder}. In the setting of unit Disk Graph (UDG) it is shown in \cite{DBLP:conf/algosensors/HucJLR10} that Schnyder's characterization prove to be useful for planarizing and routing on the communication graph. In \cite{DBLP:conf/icpads/SamarasingheL12,DBLP:conf/sensornets/SamarasingheL14}\footnote{Notice that our definition of greedy routing is different than the one used in \cite{DBLP:conf/icpads/SamarasingheL12,DBLP:conf/sensornets/SamarasingheL14}} the classical greedy-face routing paradigm is designed using coordinate similar to the ones used in the present article, see Figure \ref{fig:exvrac}, in the setting of UDG.

\noindent Our construction of the greedy routing algorithm on planar triangulations is summarized in the next Theorem. The pseudo-code of the algorithm is provided in Algorithm \ref{fig:speudocode} and the correctness of the algorithm is proved in the remarks \ref{remarkeven} and \ref{remarkodd} of the construction of the path if $D\in s^u_{2i}$ or $D\in s^u_{2i+1}$ that follow the Propositions \ref{prop:greedyeven} and \ref{lemmagreedy}.
\begin{theorem} Let us consider two nodes $u$ and $D$ of a planar triangulations. Using the coordinate system in Definition \ref{def:coordinates} there exists a greedy path between the two nodes. Moreover, the routing algorithm is local, only the coordinate of the destination and the neighboring nodes are necessary.
\end{theorem}

\begin{algorithm}[t]
\caption{Pseudo-code of the greedy routing}\label{fig:speudocode}
\begin{algorithmic}[1]
\State \textsc{input} Source $u$, Destination $D$
\Repeat
\If {$D\in {\cal N}_u$} $u=D$\Comment ${\cal N}_u$ is the set of neighbors of $u$
\Else
\If  { $D\in s^u_{2i-1}$}
	\State $u=v\in s^u_{2i-1}$ s.t. $(u,v)\in E$ \Comment $v$ is unique \vspace{0.1cm}
\Else \Comment  $D\in s^u_{2i}$ consider $v\in s^u_{2i-1}$ and $w\in s^u_{2i+1}$ s.t. $(u,v), (u,w)\in E$
	\If { $D\in s^v_{2i}$}
		\State $u=v$
	\Else
		\If{$D\in s^w_{2i}$}
			\State $u=w$
		\Else \State $u=x\in s^u_{2i}$ s.t. $D\in s^x_{2i}$ \Comment must exist by Proposition \ref{prop:greedyeven}
		\EndIf
	\EndIf
\EndIf
\EndIf
\Until{u=D}
\end{algorithmic}
\end{algorithm}
\vspace{-0.2cm}

\section{Conclusion}
It this article we provide a definition of greedy routing that is independent of any graph embedding in metric space. Moreover, we use a new coordinate system such that greedy routing guarantees delivery and again without reference to any embedding of the graph. Besides the theoretical relevance, an important motivation for this article is to make geographic routing a real practical solution. This is the reason for avoiding the computation in the localization phase of the nodes. Moreover, the coordinate system that we use in the present paper uses only 3 integers and requires  ${\cal O}\bigl(log(n)\bigr)$ bits and can be qualified of {\it succint} \cite{eppstein2011succinct}. We emphasize that our algorithm does not require to embed the graph in a metric space. The next step towards the development of a general practical routing algorithm is to provide a distributed algorithm that extract  the standard representation of the communication graph (or a subgraph). Such an algorithm would generalize the approach in  \cite{DBLP:conf/icpads/SamarasingheL12,DBLP:conf/sensornets/SamarasingheL14} where the coordinates are measured and the techniques could be merged to overcome the situation where the graph is not maximal.

From a theoretical point of view, the result presented in the present article asks the question whether there exists graphs that admit greedy routing but no greedy embeddings.

We have implemented our algorithm and simulated the routing process for some random planar triangulations. We obtained the random networks by placing the three distinguished nodes $A_1, A_2, A_3$ on the top of an equilateral triangle on the plane and generated nodes at random inside the triangle. The three order relations are defined by measuring the distances from the nodes to the distinguished nodes, i.e. $u<_i v\Leftrightarrow d(A_i,u)>d(A_i,v)$, $i=1,2,3$ where $d$ is the Euclidean distance on the plane, see the left of Figure \ref{fig:VRAC}. The equilateral triangle property implies that the representation is standard and constraining the nodes inside the triangle implies the Schnyder's empty property, see Property $a)$ of $(\ref{eq:schnydercond})$, \cite{DBLP:conf/sensornets/SamarasingheL14}. Then, the edges are generated in order to satisfy the condition $b)$ of $(\ref{eq:schnydercond})$. This is easily done by connecting each node $v$ to the three nodes  $v_1\in s^v_1, v_2\in  s^v_3,$ and $v_3\in  s^v_5$ such that $v_{i}=min_i\{ z\mid z\in s^v_{2i-1}\}$, $i=1,2,3$, accordingly to Property \ref{prop:sectors}. An instance of the random graphs is presented on Figure \ref{fig:experiment}.

 Our experiments validated the algorithm positively. The code and more complete explanations can be obtained upon request.

To conclude, we ask the question whether all planar triangulations can be drawn in the plane by distinguishing three nodes  $A_1, A_2, A_3$ and placing the other nodes on the plane inside the triangular region delimited by  $A_1, A_2, A_3$ \footnote{Similarly to how we produced random planar triangulations.} in such a way that the Schnyder orders are similar to the orders we obtain by measuring the distances from the nodes to the distinguished nodes. Answering positively would lead to a description (or even an algorithm) of a greedy embedding for planar triangulations.

\begin{figure}[!]\hskip -2cm
\hspace{2cm}
\centering
\includegraphics[scale=0.2]{experiment-figure}
\caption{Triangular Graph Generated by Experiments}
\label{fig:experiment}
 \end{figure}

\bibliographystyle{elsarticle-num}

redpajama_set_name: RedPajamaArXiv


Most likely, you already see where this is going. Of course the ducks did well in swimming and flying, the rabbits in running, the eagles in flying and the squirrels in running and climbing…and occasionally their own version of flying. So who were the "good" students and who were the "bad" ones? None of them? All of them? How are we supposed to answer that question? They weren't necessarily good or bad. They were just different. And different is not the same.
Well, guess what? These students eventually struggled through their school years and they now work for your company.
That's right. Look around you. There are ducks, rabbits, squirrels and eagles, and they all want to contribute to the mission of your company. But chances are you currently have some ducks doing the running and some rabbits doing the swimming. How's that working for you? Are you frustrated? Are they?
Wouldn't it be so much easier if they looked like ducks, rabbits, squirrels and eagles? Then you could see their differences right there in front of your face, and you would know how to structure your team. This is what Talent Insights and the Talent Insights Blog is designed to do—to show you how each of your team members is uniquely wired. It starts with Myers-Briggs.
Which issue is their hill to die on, or take you out over, and a whole lot more!
This Myers-Briggs Type Indicator cheat sheet is great for helping a manager keep track of the types he or she might have on the team. Keep this in front of you as you are delegating projects or restructuring the team for maximum impact. Share it with your whole team to improve communication and teamwork across the board. Help your team members understand each other!
Who's on your team? Share your team's dynamics with us in the comment section below.


text:
By Amy Cassaniti November 20, 1973 was the date for the network TV premiere of A Charlie Brown Thanksgiving. In the special, after attempting to prepare a non-traditional meal for his friends, Charlie Brown is finally able to invite his friends to come with him for Thanksgiving at his grandmother's condo. The children end the episode singing the classic "Over the River and Through the Wood." This classic song has been a staple of elementary Thanksgiving pageants since its
By Amy Cassaniti Many modern aficionados of classical music think that the musician, Yo-Yo Ma, is the GOAT (greatest of all time) cellist. Having played the cello since he was four years old, Ma has a repertoire that includes classical, traditional Chinese melodies, Argentinian tangos, Brazilian, contemporary, even American bluegrass. Throughout his career, Ma has demonstrated time and again the versatility of the cello and the versatility of his playing. Despite being the mo
By Amy Cassaniti November marks the birth month of The March King, John Philip Sousa. Sousa was a prolific composer. In his lifetime he composed 136 Marches and 11 Operettas. Sousa's marches continue to be some of the most performed here in the U.S. and globally. But just what makes a great march? First and foremost is Meter. Marches are written in an even meter with a strongly accented first beat to assist in military marching. For example, each measure has two beats p
By Amy Cassaniti The March King. As anyone who has spent even one season in a marching band knows, the great crowd pleasers are any marches written by John Philip Sousa. Born November 6, 1854, John Philip Sousa began his music career studying the violin. As a young musician, Sousa thought he might fulfill the dream of many a young boy and run away to join the circus. Fortunately, in 1868, at the age of 14, Sousa's father enlisted him in the United States Marine Band as an


From BioWare and EA comes Anthem, one of 2019's most anticipated games. Offering intense PvE action combined with RPG-style skill trees, Anthem promises to be one of the most heavily played games of the year. With jaw-dropping graphics, an incredible movement engine (that includes flying) and immersive, involved third person gameplay, Anthem could well prove to be one of those games that maintains a thriving community for years to come.
Coming to both PS4 and Xbox One, Anthem offers an immersive online experience that combines play with others and single player elements, all taking place within one world. You play as a Freelancer, tasked with leaving your home community in order to explore the wider reaches of the universe. With Fort Tarsis as your headquarters, you can take assignments and form squads with up to three other players in order to complete missions. A veritable Mos Eisley, Fort Tarsis plays host to a variety of different factions, each with their own characteristics.
An entirely new IP, Anthem is brimming with mysteries and new lore that is begging to be discovered. However, one of the major draws to the game will undoubtedly be the technology it unleashes. As with any futuristic shooter, you can expect some daring, inventive weaponry that will allow you to approach tasks in a variety of ways. One of the most eye-catching technological features of the Freelancers is, however, the Javelin suit that every player wears. These suits grant you superhuman abilities, enabling enhanced fighting skills and, most notably, flight.
Your Javelin suit is completely customiseable, and you can save different load-outs so you can refer back to set ups that have worked particularly well. No matter if it's speed, precision, or brute force that you choose to favour, you can make your Javelin respond accordingly.
Anthem is available on Xbox One S, Xbox One X, PS4 and PS4 Pro, as well as PC. It also requires an internet connection at all times, even if you're not planning on teaming up with other players in a squad, so bear that in mind! We'll be tracking all the best prices and deals for all console platforms, including digital keys and physical copies, right here.


Excursion duration - 3 hours, price is valid for a minimum group 3 people.
ALZHIR - Akmola camp for the wives of traitors of Motherland - one of the largest camps of Soviet Union and one of the three "islands" of the "Gulag Archipelago". The camp was opened in the beginning of 1938, when it received the first stages of the prisoners - mostly members of the families of traitors. According to incomplete data, the number of repressed exceeded 18,000 convicts.In 2007 at the former site of the correctional camp was opened the Museum and Memorial Complex of Victims of Political Repression and Totalitarianism, which you will visit during the tour. At the entrance to the museum there is "Arch of Grief", and further are two compositions "Despair and Powerlessness" and "Fight and Hope" – in memory to the families forever separated by cruel time. The tunnel, where on the walls represented the tragic moments of parting with children, cruel and unfair court of the Three, a prison cell and a punishment cell, helps to plunge into the atmosphere of repressions. The exposition of the museum is located on hall perimeter, that involves the visitor in a vicious circle and immerses in the historical past of Kazakhstan. At the end of excursion you will have an opportunity to watch 20-minute documentary film.
Keep your fitness routline and improve your shape in Kazzhol Hotel spa-center!


There are several dealers selling repro pickelhaube in the US and Europe, 90% are made in India.
There are two manufactures in Europe that produce Excellent, Museum quality Helmets and they cost from $400-$1500.
The majority of what is being offered by others rates from Poor to Fair. Dealers are charging anywhere from $65-150 for these, they are not well made, have cheap, flimsy brass pieces and they are usually incorrect in the details.
Our Spiked Helmets are Very Good Reproductions. They are the most accurate and well made for the money. We've seen what everyone else has to offer and we've tested all the manufactures. You will not find a better made, historically accurate Pickelhaube anywhere at our prices.


Espresso Dark Balsamic Vinegar Condimento - Palmetto Olive Oil Co.
This aged Balsamic Vinegar is amazingly complex, with layers of authentic espresso and dark roasted coffee. It makes a fantastic glaze or barbecue base for pork, beef or chicken. Add a shot to an "affogato float", which calls for a tall mug of vanilla gelato scoops covered with coffee.


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Amor Towles '83 Talks Passion, Competence, Honesty
Uncategorized Oct 3, 2012
English teacher and former Headmaster Dick Baker took to the stage on Wed., Oct. 3, to introduce best-selling author Amor Towles, Class of '83. He briefly described Towles' time at Nobles, his successful career in finance, and the road to his highly-acclaimed, first published novel, Rules of Civility. Although it's been more than 30 years since Towles attended Nobles, Baker said it "felt like yesterday," and vividly described his former student as a consummate actor, outstanding public speaker and determined scholar. "Amor wasn't a rebel, but he wasn't conventional…it's one of the lessons I think he lives by. You don't have to commit yourself to convention."
As soon as Towles took the stage, it was clear Baker's words would ring true. The Assembly audience would have been easily captivated with stories from his world book tour, life as a published writer, or details about the probable movie adaptation. It's what one might expect from a best-selling author. But, Towles set the topic of his book aside and spoke to students about a valued life lesson, as illustrated by '60s rock group Led Zeppelin. Unconventional? Yes. And everyone gathered in Lawrence Auditorium was hooked.
Towles detailed the English rock band's creative process while recording the 1971 album, Led Zeppelin IV, and the history behind their remake of Kansas Joe McCoy's "When the Levee Breaks." Towles shared that years later, when lead guitarist Jimmy Page was asked about the album's success and components of the recording process, he responded simply, "Passion, honesty and competence." Towles has adapted those three words as a personal motto, and he urged Nobles students to take a closer look into their meaning. He talked about their impact and said, "If you develop competence, have passion for what you do and approach everything with honesty, the outcome will always be fine."
Long Assembly ended with a Q&A session, Towles took the opportunity to explain how life's unexpected curve balls often lead to bigger and better things, advising students to take things as they come. Prodded by audience questions, he also used the time to circle back to Rules of Civility. Conventional? Maybe. But with a best-selling author of such talent among them, Nobles students, faculty and staff couldn't help themselves.