arXiv subjects

Daniel S. Hirschberg

Publications and source records attributed to Daniel S. Hirschberg.

9 recordsLinked to original sources

arXiv2025

The Rectilinear Marco Polo Problem

We study the rectilinear Marco Polo problem, which generalizes the Euclidean version of the Marco Polo problem for performing geometric localization to rectilinear search environments, such as in geometries motivated from urban settings, and to higher dimensions. In the rectilinear Marco Polo problem, there is at least one point of interest (POI) within distance $n$, in either the $L_1$ or $L_\infty$ metric, from the origin. Motivated from a search-and-rescue application, our goal is to move a search point, $Δ$, from the origin to a location within distance $1$ of a POI. We periodically issue probes from $Δ$ out a given distance (in either the $L_1$ or $L_\infty$ metric) and if a POI is within the specified distance of $Δ$, then we learn this (but no other location information). Optimization goals are to minimize the number of probes and the distance traveled by $Δ$. We describe a number of efficient search strategies for rectilinear Marco Polo problems and we analyze each one in terms of the size, $n$, of the search domain, as defined by the maximum distance to a POI.

cs.CG↗

arXiv2025

The Marco Polo Problem: A Combinatorial Approach to Geometric Localization

We introduce and study the Marco Polo problem, which is a combinatorial approach to geometric localization. In this problem, we are told there are one or more points of interest (POIs) within distance $n$ of the origin that we wish to localize. Given a mobile search point, $Δ$, that is initially at the origin, a localization algorithm is a strategy to move $Δ$ to be within a distance of $1$ of a POI. In the combinatorial localization problem we study, the only tool we can use is reminiscent of the children's game, "Marco Polo," in that $Δ$ can issue a probe signal out a specified distance, $d$, and the search algorithm learns whether or not there is a POI within distance $d$ of $Δ$. For example, we could imagine that POIs are one or more hikers lost in a forest and we need to design a search-and-rescue (SAR) strategy to find them using radio signal probes to a response device that hikers carry. Unlike other known localization algorithms, probe responses do not inform our search algorithm of the direction or distance to a POI. The optimization problem is to minimize the number of probes and/or POI responses, as well as possibly minimizing the distance traveled by $Δ$. We describe a number of efficient combinatorial Marco Polo localization strategies and we analyze each one in terms of the size, $n$, of the search domain. Moreover, we derive strong bounds for the constant factors for the search costs for our algorithms, which in some cases involve computer-assisted proofs. We also show how to extend these strategies to find all POIs using a simple, memoryless search algorithm, traveling a distance that is $\mathcal{O}(\log{k})$-competitive with the optimal traveling salesperson (TSP) tour for $k$ POIs.

cs.CG↗

arXiv2015

From Discrepancy to Majority

We show how to select an item with the majority color from $n$ two-colored items, given access to the items only through an oracle that returns the discrepancy of subsets of $k$ items. We use $n/\lfloor\tfrac{k}{2}\rfloor+O(k)$ queries, improving a previous method by De Marco and Kranakis that used $n-k+k^2/2$ queries. We also prove a lower bound of $n/(k-1)-O(n^{1/3})$ on the number of queries needed, improving a lower bound of $\lfloor n/k\rfloor$ by De Marco and Kranakis.

cs.DS↗

arXiv2013

Combinatorial Pair Testing: Distinguishing Workers from Slackers

We formalize a problem we call combinatorial pair testing (CPT), which has applications to the identification of uncooperative or unproductive participants in pair programming, massively distributed computing, and crowdsourcing environments. We give efficient adaptive and nonadaptive CPT algorithms and we show that our methods use an optimal number of testing rounds to within constant factors. We also provide an empirical evaluation of some of our methods.

cs.DS↗

arXiv2011

Fully De-Amortized Cuckoo Hashing for Cache-Oblivious Dictionaries and Multimaps

A dictionary (or map) is a key-value store that requires all keys be unique, and a multimap is a key-value store that allows for multiple values to be associated with the same key. We design hashing-based indexing schemes for dictionaries and multimaps that achieve worst-case optimal performance for lookups and updates, with a small or negligible probability the data structure will require a rehash operation, depending on whether we are working in the the external-memory (I/O) model or one of the well-known versions of the Random Access Machine (RAM) model. One of the main features of our constructions is that they are \emph{fully de-amortized}, meaning that their performance bounds hold without one having to tune their constructions with certain performance parameters, such as the constant factors in the exponents of failure probabilities or, in the case of the external-memory model, the size of blocks or cache lines and the size of internal memory (i.e., our external-memory algorithms are cache oblivious). Our solutions are based on a fully de-amortized implementation of cuckoo hashing, which may be of independent interest. This hashing scheme uses two cuckoo hash tables, one "nested" inside the other, with one serving as a primary structure and the other serving as an auxiliary supporting queue/stash structure that is super-sized with respect to traditional auxiliary structures but nevertheless adds negligible storage to our scheme. This auxiliary structure allows the success probability for cuckoo hashing to be very high, which is useful in cryptographic or data-intensive applications.

cs.DS↗

arXiv2009

Improved Adaptive Group Testing Algorithms with Applications to Multiple Access Channels and Dead Sensor Diagnosis

We study group-testing algorithms for resolving broadcast conflicts on a multiple access channel (MAC) and for identifying the dead sensors in a mobile ad hoc wireless network. In group-testing algorithms, we are asked to identify all the defective items in a set of items when we can test arbitrary subsets of items. In the standard group-testing problem, the result of a test is binary--the tested subset either contains defective items or not. In the more generalized versions we study in this paper, the result of each test is non-binary. For example, it may indicate whether the number of defective items contained in the tested subset is zero, one, or at least two. We give adaptive algorithms that are provably more efficient than previous group testing algorithms. We also show how our algorithms can be applied to solve conflict resolution on a MAC and dead sensor diagnosis. Dead sensor diagnosis poses an interesting challenge compared to MAC resolution, because dead sensors are not locally detectable, nor are they themselves active participants.

cs.DS↗

arXiv2007

The minimum size required of a solitaire army

The solitaire army is a one-person peg jumping game where a player attempts to advance an "army" of pegs as far as possible into empty territory. The game was introduced by John Conway and is also known as "Conway's Soldiers". We consider various generalizations of this game in different 2D geometries, unify them under a common mathematical framework, and find the minimum size army capable of advancing a given number of steps.

math.CO↗

arXiv2005

Improved Combinatorial Group Testing Algorithms for Real-World Problem Sizes

We study practically efficient methods for performing combinatorial group testing. We present efficient non-adaptive and two-stage combinatorial group testing algorithms, which identify the at most d items out of a given set of n items that are defective, using fewer tests for all practical set sizes. For example, our two-stage algorithm matches the information theoretic lower bound for the number of tests in a combinatorial group testing regimen.

cs.DS↗

arXiv1999

Geometric Thickness of Complete Graphs

We define the geometric thickness of a graph to be the smallest number of layers such that we can draw the graph in the plane with straight-line edges and assign each edge to a layer so that no two edges on the same layer cross. The geometric thickness lies between two previously studied quantities, the (graph-theoretical) thickness and the book thickness. We investigate the geometric thickness of the family of complete graphs, K_n. We show that the geometric thickness of K_n lies between ceiling((n/5.646) + 0.342) and ceiling(n/4), and we give exact values of the geometric thickness of K_n for n <= 12 and n in {15,16}. We also consider the geometric thickness of the family of complete bipartite graphs. In particular, we show that, unlike the case of complete graphs, there are complete bipartite graphs with arbitrarily large numbers of vertices for which the geometric thickness coincides with the standard graph-theoretical thickness.

math.CO↗