5 research outputs found
Byzantine Approximate Agreement on Graphs
Consider a distributed system with n processors out of which f can be Byzantine faulty. In the approximate agreement task, each processor i receives an input value x_i and has to decide on an output value y_i such that
1) the output values are in the convex hull of the non-faulty processors\u27 input values,
2) the output values are within distance d of each other.
Classically, the values are assumed to be from an m-dimensional Euclidean space, where m >= 1.
In this work, we study the task in a discrete setting, where input values with some structure expressible as a graph. Namely, the input values are vertices of a finite graph G and the goal is to output vertices that are within distance d of each other in G, but still remain in the graph-induced convex hull of the input values. For d=0, the task reduces to consensus and cannot be solved with a deterministic algorithm in an asynchronous system even with a single crash fault. For any d >= 1, we show that the task is solvable in asynchronous systems when G is chordal and n > (omega+1)f, where omega is the clique number of G. In addition, we give the first Byzantine-tolerant algorithm for a variant of lattice agreement. For synchronous systems, we show tight resilience bounds for the exact variants of these and related tasks over a large class of combinatorial structures
Enumerating the Digitally Convex Sets of Powers of Cycles and Cartesian Products of Paths and Complete Graphs
Given a finite set , a convexity , is a collection of subsets
of that contains both the empty set and the set and is closed under
intersections. The elements of are called convex sets. The
digital convexity, originally proposed as a tool for processing digital images,
is defined as follows: a subset is digitally convex if, for
every , we have implies . The number of
cyclic binary strings with blocks of length at least is expressed as a
linear recurrence relation for . A bijection is established between
these cyclic binary strings and the digitally convex sets of the
power of a cycle. A closed formula for the number of digitally convex sets of
the Cartesian product of two complete graphs is derived. A bijection is
established between the digitally convex sets of the Cartesian product of two
paths, , and certain types of binary arrays.Comment: 16 pages, 3 figures, 1 tabl
A Steiner general position problem in graph theory
Let be a graph. The Steiner distance of is the minimum size of a connected subgraph of containing . Such a subgraph is necessarily a tree called a Steiner -tree. The set is a -Steiner general position set if holds for every set of cardinality , and for every Steiner -tree . The -Steiner general position number \sgp_k(G) of is the cardinality of a largest -Steiner general position set in . Steiner cliques are introduced and used to bound \sgp_k(G) from below. The -Steiner general position number is determined for trees, cycles and joins of graphs. Lower bounds are presented for split graphs, infinite grids and lexicographic products. The lower bound for the latter product leads to an exact formula for the general position number of an arbitrary lexicographic product