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David Pritchard

Publications and source records attributed to David Pritchard.

At least 19 recordsLinked to original sources
arXiv2026

Discovering Misconceptions and Misunderstandings From Administrations of Research-Designed Multiple Choice Instruments

Misconceptions are "alternate hypotheses" that are incorrect according to established theories of how the world works. Often held with confidence by students, they are relatively context-insensitive, can seem like common-sense views, and are noted for being resistant to remediation using traditional instruction. To find misconceptions in Newtonian mechanics, we analyze ~34,000 administrations of the pioneering Force Concept Inventory using a flexible multidimensional item-response model for multiple-choice data. In contrast to most earlier work, we allow answer choices within each question to have different directions in the multidimensional space of student knowledge, essential for concept inventories in which distractors often codify distinct misconceptions. We uncover 22 robust, partly-overlapping dimensions whose distractors share a coherent theme identifiable with a misconception or misunderstanding. Motivated by the realization that many mirror previously-accepted theories of mechanics, we broadly sort these by historical era: Ancient (learned by infants but codified by Greeks), Medieval (reactions and extensions of Aristotelian ideas), and Post-Newtonian (including known modern misconceptions as well as two which appear novel). We also present a simple approach for computing "misconception scores" for students and classes. Examining these scores before and after instruction reveals surprisingly varied patterns of remediation in our sample: some misconceptions persist largely unchanged by instruction, while others are better remediated in below- or above-average students. In general, we find that many misconceptions are poorly remediated for students of average or lower ability. We hope our work will serve as a guide for developing, evaluating, and improving interventions for these while providing physics instructors with a valuable tool for class-level formative assessment.

physics.ed-ph↗
arXiv2023

Pendent Steady Rivulets: From Lubrication to Bifurcation

We consider the shape of the free surface of steady pendent rivulets beneath a planar substrate. We formulate the governing equations in terms of two closely related dynamical systems and use classical phase-plane techniques to develop the bifurcation structure of the problem. Our results explain why lubrication theory is unable to capture this bifurcation structure for pendent rivulets, although it is successful in the related problem of sessile rivulets.

physics.flu-dyn↗
arXiv2016

Generalized Hypergraph Matching via Iterated Packing and Local Ratio

In $k$-hypergraph matching, we are given a collection of sets of size at most $k$, each with an associated weight, and we seek a maximum-weight subcollection whose sets are pairwise disjoint. More generally, in $k$-hypergraph $b$-matching, instead of disjointness we require that every element appears in at most $b$ sets of the subcollection. Our main result is a linear-programming based $(k-1+\tfrac{1}{k})$-approximation algorithm for $k$-hypergraph $b$-matching. This settles the integrality gap when $k$ is one more than a prime power, since it matches a previously-known lower bound. When the hypergraph is bipartite, we are able to improve the approximation ratio to $k-1$, which is also best possible relative to the natural LP. These results are obtained using a more careful application of the \emph{iterated packing} method. Using the bipartite algorithmic integrality gap upper bound, we show that for the family of combinatorial auctions in which anyone can win at most $t$ items, there is a truthful-in-expectation polynomial-time auction that $t$-approximately maximizes social welfare. We also show that our results directly imply new approximations for a generalization of the recently introduced bounded-color matching problem. We also consider the generalization of $b$-matching to \emph{demand matching}, where edges have nonuniform demand values. The best known approximation algorithm for this problem has ratio $2k$ on $k$-hypergraphs. We give a new algorithm, based on local ratio, that obtains the same approximation ratio in a much simpler way.

cs.DS↗
arXiv2015

Frequency Distribution of Error Messages

Which programming error messages are the most common? We investigate this question, motivated by writing error explanations for novices. We consider large data sets in Python and Java that include both syntax and run-time errors. In both data sets, after grouping essentially identical messages, the error message frequencies empirically resemble Zipf-Mandelbrot distributions. We use a maximum-likelihood approach to fit the distribution parameters. This gives one possible way to contrast languages or compilers quantitatively.

cs.SE↗
arXiv2012

CS Circles: An In-Browser Python Course for Beginners

Computer Science Circles is a free programming website for beginners that is designed to be fun, easy to use, and accessible to the broadest possible audience. We teach Python since it is simple yet powerful, and the course content is well-structured but written in plain language. The website has over one hundred exercises in thirty lesson pages, plus special features to help teachers support their students. It is available in both English and French. We discuss the philosophy behind the course and its design, we describe how it was implemented, and we give statistics on its use.

cs.CY↗
arXiv2012

Cover-Decomposition and Polychromatic Numbers

A colouring of a hypergraph's vertices is polychromatic if every hyperedge contains at least one vertex of each colour; the polychromatic number is the maximum number of colours in such a colouring. Its dual, the cover-decomposition number, is the maximum number of disjoint hyperedge-covers. In geometric hypergraphs, there is extensive work on lower-bounding these numbers in terms of their trivial upper bounds (minimum hyperedge size and degree); our goal here is to broaden the study beyond geometric settings. We obtain algorithms yielding near-tight bounds for three families of hypergraphs: bounded hyperedge size, paths in trees, and bounded VC-dimension. This reveals that discrepancy theory and iterated linear program relaxation are useful for cover-decomposition. Finally, we discuss the generalization of cover-decomposition to sensor cover.

math.CO↗
arXiv2012

On Approximating String Selection Problems with Outliers

Many problems in bioinformatics are about finding strings that approximately represent a collection of given strings. We look at more general problems where some input strings can be classified as outliers. The Close to Most Strings problem is, given a set S of same-length strings, and a parameter d, find a string x that maximizes the number of "non-outliers" within Hamming distance d of x. We prove this problem has no PTAS unless ZPP=NP, correcting a decade-old mistake. The Most Strings with Few Bad Columns problem is to find a maximum-size subset of input strings so that the number of non-identical positions is at most k; we show it has no PTAS unless P=NP. We also observe Closest to k Strings has no EPTAS unless W[1]=FPT. In sum, outliers help model problems associated with using biological data, but we show the problem of finding an approximate solution is computationally difficult.

cs.DS↗
arXiv2011

Counting large distances in convex polygons

In a convex n-gon, let d[1] > d[2] > ... denote the set of all distances between pairs of vertices, and let m[i] be the number of pairs of vertices at distance d[i] from one another. Erdos, Lovasz, and Vesztergombi conjectured that m[1] + ... + m[k] <= k*n. Using a new computational approach, we prove their conjecture when k <= 4 and n is large; we also make some progress for arbitrary k by proving that m[1] + ... + m[k] <= (2k-1)n. Our main approach revolves around a few known facts about distances, together with a computer program that searches all distance configurations of two disjoint convex hull intervals up to some finite size. We thereby obtain other new bounds such as m[3] <= 3n/2 for large n.

math.CO↗
arXiv2011

An LP with Integrality Gap 1+epsilon for Multidimensional Knapsack

In this note we study packing or covering integer programs with at most k constraints, which are also known as k-dimensional knapsack problems. For any integer k > 0 and real epsilon > 0, we observe there is a polynomial-sized LP for the k-dimensional knapsack problem with integrality gap at most 1+epsilon. The variables may be unbounded or have arbitrary upper bounds. In the packing case, we can also remove the dependence of the LP on the cost-function, yielding a polyhedral approximation of the integer hull. This generalizes a recent result of Bienstock on the classical knapsack problem.

cs.DM↗
arXiv2010

k-Edge-Connectivity: Approximation and LP Relaxation

In the k-edge-connected spanning subgraph problem we are given a graph (V, E) and costs for each edge, and want to find a minimum-cost subset F of E such that (V, F) is k-edge-connected. We show there is a constant eps > 0 so that for all k > 1, finding a (1 + eps)-approximation for k-ECSS is NP-hard, establishing a gap between the unit-cost and general-cost versions. Next, we consider the multi-subgraph cousin of k-ECSS, in which we purchase a multi-subset F of E, with unlimited parallel copies available at the same cost as the original edge. We conjecture that a (1 + Theta(1/k))-approximation algorithm exists, and we describe an approach based on graph decompositions applied to its natural linear programming (LP) relaxation. The LP is essentially equivalent to the Held-Karp LP for TSP and the undirected LP for Steiner tree. We give a family of extreme points for the LP which are more complex than those previously known.

cs.DM↗
arXiv2010

Efficient Divide-and-Conquer Implementations Of Symmetric FSAs

A deterministic finite-state automaton (FSA) is an abstract sequential machine that reads the symbols comprising an input word one at a time. An FSA is symmetric if its output is independent of the order in which the input symbols are read, i.e., if the output is invariant under permutations of the input. We show how to convert a symmetric FSA A into an automaton-like divide-and-conquer process whose intermediate results are no larger than the size of A's memory. In comparison, a similar result for general FSA's has been long known via functional composition, but entails an exponential increase in memory size. The new result has applications to parallel processing and symmetric FSA networks.

cs.FL↗
arXiv2010

Edge Cover Colouring Versus Minimum Degree in Multigraphs

An edge colouring of a multigraph can be thought of as a partition of the edges into matchings (a matching meets each vertex at most once). Analogously, an edge cover colouring is a partition of the edges into edge covers (an edge cover meets each vertex at least once). We aim to determine a tight lower bound on the maximum number of parts in an edge cover colouring as a function of the minimum degree delta, which would be an analogue of Shannon's theorem from 1949 on edge-colouring multigraphs. We are able to give a lower bound that is tight except when delta=9 or delta is odd and > 12; in these non-tight cases the best upper and lower bounds differ by one.

math.CO↗
arXiv2010

Fast Computation of Small Cuts via Cycle Space Sampling

We describe a new sampling-based method to determine cuts in an undirected graph. For a graph (V, E), its cycle space is the family of all subsets of E that have even degree at each vertex. We prove that with high probability, sampling the cycle space identifies the cuts of a graph. This leads to simple new linear-time sequential algorithms for finding all cut edges and cut pairs (a set of 2 edges that form a cut) of a graph. In the model of distributed computing in a graph G=(V, E) with O(log V)-bit messages, our approach yields faster algorithms for several problems. The diameter of G is denoted by Diam, and the maximum degree by Delta. We obtain simple O(Diam)-time distributed algorithms to find all cut edges, 2-edge-connected components, and cut pairs, matching or improving upon previous time bounds. Under natural conditions these new algorithms are universally optimal --- i.e. a Omega(Diam)-time lower bound holds on every graph. We obtain a O(Diam+Delta/log V)-time distributed algorithm for finding cut vertices; this is faster than the best previous algorithm when Delta, Diam = O(sqrt(V)). A simple extension of our work yields the first distributed algorithm with sub-linear time for 3-edge-connected components. The basic distributed algorithms are Monte Carlo, but they can be made Las Vegas without increasing the asymptotic complexity. In the model of parallel computing on the EREW PRAM our approach yields a simple algorithm with optimal time complexity O(log V) for finding cut pairs and 3-edge-connected components.

cs.DC↗
arXiv2010

Diameter Bounds for Planar Graphs

The inverse degree of a graph is the sum of the reciprocals of the degrees of its vertices. We prove that in any connected planar graph, the diameter is at most 5/2 times the inverse degree, and that this ratio is tight. To develop a crucial surgery method, we begin by proving the simpler related upper bounds (4(V-1)-E)/3 and 4V^2/3E on the diameter (for connected planar graphs), which are also tight.

math.CO↗
arXiv2010

Hypergraphic LP Relaxations for Steiner Trees

We investigate hypergraphic LP relaxations for the Steiner tree problem, primarily the partition LP relaxation introduced by Koenemann et al. [Math. Programming, 2009]. Specifically, we are interested in proving upper bounds on the integrality gap of this LP, and studying its relation to other linear relaxations. Our results are the following. Structural results: We extend the technique of uncrossing, usually applied to families of sets, to families of partitions. As a consequence we show that any basic feasible solution to the partition LP formulation has sparse support. Although the number of variables could be exponential, the number of positive variables is at most the number of terminals. Relations with other relaxations: We show the equivalence of the partition LP relaxation with other known hypergraphic relaxations. We also show that these hypergraphic relaxations are equivalent to the well studied bidirected cut relaxation, if the instance is quasibipartite. Integrality gap upper bounds: We show an upper bound of sqrt(3) ~ 1.729 on the integrality gap of these hypergraph relaxations in general graphs. In the special case of uniformly quasibipartite instances, we show an improved upper bound of 73/60 ~ 1.216. By our equivalence theorem, the latter result implies an improved upper bound for the bidirected cut relaxation as well.

cs.DM↗
arXiv2010

Limits of Approximation Algorithms: PCPs and Unique Games (DIMACS Tutorial Lecture Notes)

These are the lecture notes for the DIMACS Tutorial "Limits of Approximation Algorithms: PCPs and Unique Games" held at the DIMACS Center, CoRE Building, Rutgers University on 20-21 July, 2009. This tutorial was jointly sponsored by the DIMACS Special Focus on Hardness of Approximation, the DIMACS Special Focus on Algorithmic Foundations of the Internet, and the Center for Computational Intractability with support from the National Security Agency and the National Science Foundation. The speakers at the tutorial were Matthew Andrews, Sanjeev Arora, Moses Charikar, Prahladh Harsha, Subhash Khot, Dana Moshkovitz and Lisa Zhang. The sribes were Ashkan Aazami, Dev Desai, Igor Gorodezky, Geetha Jagannathan, Alexander S. Kulikov, Darakhshan J. Mir, Alantha Newman, Aleksandar Nikolov, David Pritchard and Gwen Spencer.

cs.CC↗
arXiv2010

Approximability of Sparse Integer Programs

The main focus of this paper is a pair of new approximation algorithms for certain integer programs. First, for covering integer programs {min cx: Ax >= b, 0 <= x <= d} where A has at most k nonzeroes per row, we give a k-approximation algorithm. (We assume A, b, c, d are nonnegative.) For any k >= 2 and eps>0, if P != NP this ratio cannot be improved to k-1-eps, and under the unique games conjecture this ratio cannot be improved to k-eps. One key idea is to replace individual constraints by others that have better rounding properties but the same nonnegative integral solutions; another critical ingredient is knapsack-cover inequalities. Second, for packing integer programs {max cx: Ax <= b, 0 <= x <= d} where A has at most k nonzeroes per column, we give a (2k^2+2)-approximation algorithm. Our approach builds on the iterated LP relaxation framework. In addition, we obtain improved approximations for the second problem when k=2, and for both problems when every A_{ij} is small compared to b_i. Finally, we demonstrate a 17/16-inapproximability for covering integer programs with at most two nonzeroes per column.

cs.DS↗