We present two constant-time dynamic data-structures that support insertions, deletions, and queries with one-sided errors: a space-efficient dynamic (key-only) filter and a compact dynamic data-structure that combines retrieval and filtering (called a key-value filter). A one-sided error occurs when a query for a key not in the dataset is issued and the outcome is wrong, i.e., a “yes” in a filter or a non-null in the key-value filter. The response to a query with a key in the dataset always returns the correct answer, i.e., a “yes” in a filter and the correct value in a key-value filter. The probability of the one-sided error in our data-structures is (1/(log n)) , where n is the maximum cardinality of the dataset, and the probability space is over the random bits of the data-structure (i.e., random choice of hash function). The computational framework is the Word RAM model. We differentiate between accesses to non-modifiable memory (i.e., read-only memory that stores the program instructions, hash function seed or tables, etc.) and accesses to modifiable memory (i.e., read-write memory that stores the representation of the dataset). We are not aware of previous works that make this distinction in the context of data-structures. Our dynamic filter design requires only a single access to the modifiable memory per operation in the worst-case. We also present a dynamic key-value filter for values of O(loglog n) bits that requires 1+o(1) accesses to the modifiable memory per operation in expectation. Previous dynamic filter designs require, in the worst case, at least two accesses to modifiable memory for queries with keys not in the dataset. Previous dynamic retrieval data-structure designs always require two dictionary accesses for queries with keys not in the dataset even for single bit values. We prove bounds on the number of balls that overflow in a dynamic balls-into-bins random process for a range of bin capacities that extends the Iceberg Lemma of [Bender et al., JACM 2023]. The correctness of the key-value filter is based on the previously unstudied natural case of unit-capacity bins with more bins than balls. Finally, we observe that the splitting technique for achieving succinct representation of hash functions is not necessary for our data-structures.
The k -CombDMR problem is that of determining whether an n × n distance matrix can be realised as a sub-matrix by n vertices in some unweighted undirected graph with n + k vertices. This problem has a simple solution in the case k=0 . In this paper we show that this problem is polynomial-time solvable for k=1 and k=2 , and we provide algorithms to construct such graph realisations by solving appropriate 2-SAT instances. For the case where k ≥ 3 , we prove that the problem becomes NP-complete. We show this by a reduction from the k-colourability problem, where k≥ 3 . Finally, we present how the simpler problem of tree realisability can be solved in polynomial time for all k ≥ 0 .
In this paper we study geodetic convex hulls in graphs and prove new bounds on the size of hull sets and related parameters. A set of vertices in a graph is geodetic convex if no shortest path between vertices in the set leaves the set. The geodetic hull of a vertex set is its smallest convex superset. A hull set is a set of vertices whose geodetic hull is the entire graph. Computing a hull set of minimum size poses an NP-hard problem. We present an exact kernelization algorithm which implies that the problem is FPT in the vertex cover number. Furthermore we provide new bounds on the size of a minimum hull set in relation to other graph parameters. For practical application, we design an improved heuristic as well as an algorithm to obtain strong instance-based lower bounds. We evaluate the efficiency and quality of the heuristic and the lower bound on diverse graphs. Our experiments demonstrate that our methods outperform previous ones by a large margin.
In this paper, we study the All-Colors problem: given a graph G each of whose vertices is equipped with a button and assigned a color value from the set {0,1, … , m-1} and an integer k, can we reach color value 1 (mod m) on every vertex of G by pressing the button at most k times. The rule we follow is the following: if a button of a corresponding vertex is pressed one time, then the color values of the vertex and its neighbors are incremented by 1. This problem is known to be NP-hard on bipartite graphs even when m =2 [Theor. Comput. Sci., 2007], although linear time solvable on trees [SIAM J. Comput., 2004]. In this work, we study this problem in the realm of parameterized complexity with respect to several parameters. In particular, we show the following for All-Colors.
We consider the two-watchtower problem in the plane in which, given a 1.5-dimensional terrain T, we want to compute the minimum common height of two watchtowers that are erected at points of T and together guard the entire T. The problem is encountered in three versions, the discrete, the semi-continuous, and the continuous [1, 3, 5]. In this paper, we focus on the discrete and semi-continuous versions of the problem. We prove important geometric properties of the problem which enable us to describe a simpler and faster decision process for each of these two versions; this leads to a faster O(n(2) log(2) n)-time algorithm for the discrete two-watchtower problem and an algorithm matching the complexity of the algorithm in [1] for the semi-continuous version. Additionally, for the discrete version, we present a simple algorithm that does not use parametric search; it runs in O(n(3) log n) time, which improves upon the currently best algorithm for this problem.
