On Identifying and Locating-Dominating Codes in the Infinite King Grid

Siirretty Doriast

Automated Discharging Arguments for Density Problems in Grids

Discharging arguments demonstrate a connection between local structure and global averages. This makes it an effective tool for proving lower bounds on the density of special sets in infinite grids. However, the minimum density of an identifying code in the hexagonal grid remains open, with an upper bound of $\frac{3}{7} \approx 0.428571$ and a lower bound of $\frac{5}{12}\approx 0.416666$. We present a new, experimental framework for producing discharging arguments using an algorithm. This algorithm replaces the lengthy case analysis of human-written discharging arguments with a linear program that produces the best possible lower bound using the specified set of discharging rules. We use this framework to present a lower bound of $\frac{23}{55} \approx 0.418181$ on the density of an identifying code in the hexagonal grid, and also find several sharp lower bounds for variations on identifying codes in the hexagonal, square, and triangular grids.Comment: This is an extended abstract, with 10 pages, 2 appendices, 5 tables, and 2 figure

Optimal local identifying and local locating-dominating codes

We introduce two new classes of covering codes in graphs for every positive integer $r$. These new codes are called local $r$-identifying and local $r$-locating-dominating codes and they are derived from $r$-identifying and $r$-locating-dominating codes, respectively. We study the sizes of optimal local 1-identifying codes in binary hypercubes. We obtain lower and upper bounds that are asymptotically tight. Together the bounds show that the cost of changing covering codes into local 1-identifying codes is negligible. For some small $n$ optimal constructions are obtained. Moreover, the upper bound is obtained by a linear code construction. Also, we study the densities of optimal local 1-identifying codes and local 1-locating-dominating codes in the infinite square grid, the hexagonal grid, the triangular grid, and the king grid. We prove that seven out of eight of our constructions have optimal densities

Fault-Tolerant Detection Systems on the King's Grid

A detection system, modeled in a graph, uses "detectors" on a subset of vertices to uniquely identify an "intruder" at any vertex. We consider two types of detection systems: open-locating-dominating (OLD) sets and identifying codes (ICs). An OLD set gives each vertex a unique, non-empty open neighborhood of detectors, while an IC provides a unique, non-empty closed neighborhood of detectors. We explore their fault-tolerant variants: redundant OLD (RED:OLD) sets and redundant ICs (RED:ICs), which ensure that removing/disabling at most one detector guarantees the properties of OLD sets and ICs, respectively. This paper focuses on constructing optimal RED:OLD sets and RED:ICs on the infinite king's grid, and presents the proof for the bounds on their minimum densities; [3/10, 1/3] for RED:OLD sets and [3/11, 1/3] for RED:ICs

Finding codes on infinite grids automatically

We apply automata theory and Karp's minimum mean weight cycle algorithm to minimum density problems in coding theory. Using this method, we find the new upper bound $53/126 \approx 0.4206$ for the minimum density of an identifying code on the infinite hexagonal grid, down from the previous record of $3/7 \approx 0.4286$.Comment: 18 pages, 5 figure

On Error-detecting Open-locating-dominating sets

An open-dominating set S for a graph G is a subset of vertices where every vertex has a neighbor in S. An open-locating-dominating set S for a graph G is an open-dominating set such that each pair of distinct vertices in G have distinct set of open-neighbors in S. We consider a type of a fault-tolerant open-locating dominating set called error-detecting open-locating-dominating sets. We present more results on the topic including its NP-completeness proof, extremal graphs, and a characterization of cubic graphs that permit an error-detecting open-locating-dominating set

On location, domination and information retrieval

The thesis is divided into two main branches: identifying and locatingdominating codes, and information retrieval. The former topics are motivated by the aim to locate objects in sensor networks (or other similar applications) and the latter one by the need to retrieve information in memories such as DNA data storage systems. Albeit the underlying applications, the study on these topics mainly belongs to discrete mathematics; more specically, to the elds of coding and graph theory. The sensor networks are usually represented by graphs where vertices represent the monitored locations and edges the connections between the locations. Moreover, the locations of the sensors are determined by a code. Furthermore, the desired properties of the sensor network are deeply linked with the properties of the underlying code. The number of errors in reading the data is abundant in the DNA data storage systems. In particular, there can occur more errors than a reasonable error-correcting code can handle. However, this problem is somewhat oset by the possibility to obtain multiple approximations of the same information from the data storage. Hence, the information retrieval process can be modelled by the Levenshtein's channel model, where a message is sent through multiple noisy channels and multiple outputs are received. In the rst two papers of the thesis, we introduce and study the new concepts of self- and solid-locating-dominating codes as a natural analogy to self-identifying codes with respect to locating-dominating codes. The rst paper introduces these new codes and considers them in some graphs such as the Hamming graphs. Then, in the second paper, we broaden our view on the topic by considering graph theoretical questions. We give optimal codes in multiple dierent graph classes and some more general results using concepts such as the Dilworth number and graph complements. The third paper focuses on the q-ary Hamming spaces. In particular, we disprove a conjecture proposed by Goddard and Wash related to identifying codes. In the fourth paper, we return to self- and solid-locating-dominating codes and give optimal codes in some graph classes and consider their densities in innite graphs. In the fth paper, we consider information retrieval in memories; in particular, the Levenshtein's channel model. In the channel model, we transmit some codeword belonging to the binary Hamming space through multiple identical channels. With the help of multiple dierent outputs, we give a list of codewords which may have been sent. In the paper, we study the number of channels required to have a rather small (constant) list size when the properties of the channels, the code and the dimension of the Hamming space are xed. In particular, we give an exact relation between the number of channels and the asymptotic value of the maximum list size.Väitöskirja käsittelee kahta aihetta: identioivia ja paikantavia peittokoodeja sekä tiedon noutamista muistista. Ensimmäisen aiheen motivaationa on objektien paikantaminen sensoriverkoista (sekä muut samankaltaiset sovellukset) ja jälkimmäisen tiedonnouto DNA-muisteista. Näiden aiheiden tutkimus kuuluu diskreettiin matematiikkaan, täsmällisemmin koodaus- ja graa-teoriaan. Sensoriverkkoja kuvataan yleensä graafeilla, joissa solmut esittävät tarkkailtuja kohteita ja viivat yhteyksiä näiden kohteiden välillä. Edelleen sensorien paikat määräytyvät annetun koodin perusteella. Tästä johtuen sensoriverkon halutut ominaisuudet pohjautuvat vahvasti alla olevaan koodiin. Luettaessa tietoa DNA-muisteista tapahtuvien virheiden määrä saattaa olla erittäin suuri; erityisesti suurempi kuin kiinnitetyn virheitä korjaavan koodin korjauskyky. Toisaalta tilanne ei ole aivan näin ongelmallinen, sillä DNA-muisteista voidaan saada useita eri arvioita muistiin tallennetusta tiedosta. Näistä syistä johtuen tietojen noutamista DNA-muisteista voidaan mallintaa käyttäen Levenshteinin kanavamallia. Kanavamallissa yksi viesti lähetetään useiden häiriöisten kanavien kautta ja näin vastaanotetaan useita viestejä (yksi jokaisesta kanavasta). Väitöskirjan kahdessa ensimmäisessä julkaisussa esitellään ja tutkitaan uusia paikantavien peittokoodien luokkia, jotka pohjautuvat aiemmin tutkittuihin itse-identioiviin koodeihin. Ensimmäisessä julkaisussa on esitelty nämä koodiluokat sekä tutkittu niitä joissain graafeissa kuten Hammingin graafeissa. Tämän jälkeen toisessa julkaisussa käsitellään yleisiä graa-teoreettisia kysymyksiä. Julkaisussa esitetään optimaaliset koodit useille graaperheille sekä joitain yleisempiä tuloksia käyttäen mm. Dilworthin lukua sekä graakomplementteja. Kolmas julkaisu keskittyy q-arisiin Hammingin avaruuksiin. Erityisesti julkaisussa todistetaan vääräksi Goddardin ja Washin aiemmin esittämä identioivia koodeja koskeva otaksuma. Neljäs artikkeli käsittelee jo kahdessa ensimmäisessä artikkelissa esiteltyjä paikantavien peittokoodien luokkia. Artikkeli esittää optimaalisia koodeja useille graaperheille sekä käsittelee äärettömiä graafeja. Viides artikkeli käsittelee tiedonnoutoa ja erityisesti Levenshteinin kanavamallia. Kanavamallissa binääriseen Hammingin avaruuteen kuuluva koodisana lähetetään useiden identtisten kanavien läpi. Näistä kanavista vastaanotetaan useita eri arvioita lähetetystä koodisanasta ja rakennetaan lista mahdollisesti lähetetyistä sanoista. Artikkelissa tutkitaan kuinka monta kanavaa tarvitaan, jotta tämän listan koko on pieni (vakio), kun kanavien ominaisuudet, koodi ja Hammingin avaruuden dimensio on kiinnitetty. Erityisesti löydetään täsmällinen suhde kanavien lukumäärän ja asymptoottisesti maksimaalisen listan koon välille

On using distances to locate vertices: resolving sets and metric bases of graphs, two generalisations and their forced vertices

A graph consists of vertices that are connected by edges. A resolving set of a graph is a subset of its vertices that gives a unique combination of distances to every vertex of the graph. We can use the distances we are given to locate a vertex within the graph we are considering. Resolving sets were introduced by Slater in 1975 and independently by Harary and Melter in 1976. Robot navigation and network discovery and verification are examples of applications that have been suggested for resolving sets. In this dissertation, we consider resolving sets and two of their generalisations that can be used to locate subsets of vertices instead of individual vertices. We consider how these generalisations are connected to other concepts such as locatingdominating sets and the boundary of a graph. We place special emphasis on studying the minimum cardinalities of resolving sets and the two generalisations. In addition to proving general bounds to these minimum cardinalities, we consider their exact values in some graph families. Natural decision problems arise from some of the concepts that we consider and we study their algorithmic complexities. We also investigate which vertices of a graph must be included in an optimal resolving set or one of the two generalisations. For the resolving sets that can be used to locate subsets of vertices, there exist vertices that are in all such resolving sets. We call these vertices forced vertices of the graph. Such vertices do not exist for regular resolving sets. However, for minimum resolving sets they can exist, and we call them basis forced vertices of the graph. In this dissertation, we characterise the forced vertices of a graph, and consider some extremal properties of graphs that contain basis forced vertices