Theory and Algorithms Commons^™

Optimal Hypergraph Connectivity With Cut Queries, Hang Liao

Dartmouth College Ph.D Dissertations

Finding connected components in undirected hypergraphs—hypergraph connectivity—is a fundamental problem in computer science. It can be framed as a special case of Symmetric Submodular Function Minimization (SSFM), where the objective is to determine if the non-trivial minimizer is zero. This thesis develops an optimal algorithm for hypergraph connectivity within the $\CUT$ query model, where an algorithm probes a subset of vertices to learn the weight of the hyperedges ``cut" by that partition.

Our approach is constructive, culminating in an optimal algorithm for the general problem by first developing the necessary tools for two foundational subproblems. The main contributions of this …

A Steiner Tree Vc Set System In Minor-Free (Di)Graphs, Eli Friedman

Computer Science Senior Theses

We propose a set system of maximum-covering minimum-density partial Steiner trees for planar and minor-free graphs. We show that this system has VC dimension at most h-1 for edge-weighted K_h-minor-free graphs, both directed and undirected. We also consider its geometric interpretation as a range space, proving it to be piercing.

In addition, we demonstrate how one can form a junction tree set system of bounded VC dimension from such Steiner trees. This is motivated by refining the junction tree set cover approach used in Chekuri and Jain's polylogarithmic approximation algorithm for Directed Steiner Forest in planar graphs [CJ25].

Property Testing Ai: An Efficient Frontier, Paul Sopher Lintilhac

Dartmouth College Ph.D Dissertations

In this dissertation, we take a step towards addressing the major problem of a lack of standardized and rigorous approaches to testing and evaluation of AI systems. Taking inspiration from both the fields of Property Testing and Property Based Testing (for programs), we develop a novel taxonomy of partially overlapping classes of properties of AI systems, including simple properties, compound properties, higher order properties, data relation properties, and architecture-utility properties. We argue that this taxonomy categorizes a diverse set of AI traits -- including accuracy, fairness, robustness, monotonicity, point-wise and global privacy properties, sensitivity, and more -- according to the …

Achieving Domain-Independent Certified Robustness Via Knowledge Continuity, Alan Wenyuan Sun

Computer Science Senior Theses

We present knowledge continuity, a novel definition inspired by Lipschitz continuity which aims to certify the robustness of neural networks across input domains (such as continuous and discrete domains in vision and language, respectively). Most existing approaches that seek to certify robustness, especially Lipschitz continuity, lie within the continuous domain with norm and distribution-dependent guarantees. In contrast, our proposed definition yields certification guarantees that depend only on the loss function and the intermediate learned metric spaces of the neural network. These bounds are independent of domain modality, norms, and distribution. We further demonstrate that the expressiveness of a model …

On Adaptivity And Randomness For Streaming Algorithms, Manuel Stoeckl

Dartmouth College Ph.D Dissertations

A streaming algorithm has a limited amount of memory and reads a long sequence (data stream) of input elements, one by one, and computes an output depending on the input. Such algorithms may be used in an online fashion, producing a sequence of intermediate outputs corresponding to the prefixes of the data stream. Adversarially robust streaming algorithms are required to give correct outputs with a desired probability even when the data stream is adaptively generated by an adversary that can see all intermediate outputs of the algorithm. This thesis binds together research on a variety of problems related to the …

Disentangling Cyclic Causality: An Instance-Based Framework For Causal Discovery, Chase A. Yakaboski

Dartmouth College Ph.D Dissertations

Correlation does not imply causation" is one of the fundamental principles taught in science, emphasizing that associations between variables do not necessarily indicate causality. Yet, over the past three decades, extensive research has begun to challenge this perspective by developing sophisticated methods to differentiate causal from correlative relationships. This research suggests that correlations often involve a blend of confounded and causal interactions, which, given certain assumptions, can be disentangled to uncover actionable insights and deepen our understanding of physical, biological, and societal systems.

Accurately discovering causal relationships from data amidst cyclic dynamics remains a challenging open problem in causality research. …

Say That Again: The Role Of Multimodal Redundancy In Communication And Context, Brandon Javier Dormes

Cognitive Science Senior Theses

With several modes of expression, such as facial expressions, body language, and speech working together to convey meaning, social communication is rich in redundancy. While typically relegated to signal preservation, this study investigates the role of cross-modal redundancies in establishing performance context, focusing on unaided, solo performances. Drawing on information theory, I operationalize redundancy as predictability and use an array of machine learning models to featurize speakers' facial expressions, body poses, movement speeds, acoustic features, and spoken language from 24 TEDTalks and 16 episodes of Comedy Central Stand-Up Presents. This analysis demonstrates that it is possible to distinguish between these …

Cyclic Mixed-Radix Dense Gray Codes, Jessica Cheng

Computer Science Senior Theses

A Gray code is a sequence of n binary integers in the range 0 to n-1 that has the Gray-code property: each integer in the sequence differs from the integer before it in a single digit. Gray codes have many applications, ranging from rotary encoders to Boolean circuit minimization. We refer to Gray codes where the first and last
codewords in the sequence fulfill the Gray-code property as cyclic. Additionally, we refer to a Gray code as dense if the sequence of n numbers consists of a permutation of ⟨0, 1, . . . , n − 1⟩. This thesis …

Space-Efficient Algorithms And Verification Schemes For Graph Streams, Prantar Ghosh

Dartmouth College Ph.D Dissertations

Structured data-sets are often easy to represent using graphs. The prevalence of massive data-sets in the modern world gives rise to big graphs such as web graphs, social networks, biological networks, and citation graphs. Most of these graphs keep growing continuously and pose two major challenges in their processing: (a) it is infeasible to store them entirely in the memory of a regular server, and (b) even if stored entirely, it is incredibly inefficient to reread the whole graph every time a new query appears. Thus, a natural approach for efficiently processing and analyzing such graphs is reading them as …

The Behaviors Of Bert Attention Heads In Stereotype Detection, Joseph H. Hajjar

Dartmouth College Master’s Theses

We are living in the age of information, where it has become increasingly easy to share ideas, news, and content which are seen by an increasingly large number of people. This increasing scope of the increasing amount of data that is being shared lends itself to the question: how can we determine whether what we are reading promotes a stereotype? Previous work has applied transformer based models in this domain yielding impressive performance, but few studies exist interpreting the nature of attention heads in this task. Our work explores the feature encoding and extraction behaviors of attention heads in transformer …

A Machine-Verified Proof Of Linearizability For A Queue Algorithm, Ugur Yavuz

Dartmouth College Master’s Theses

Proofs of linearizability are typically intricate and lengthy, and readers may find it difficult to verify their correctness. We present a unique technique for producing proofs of linearizability that are fully verifiable by a mechanical proof system, thereby eliminating the need for any manual verification. Specifically, we reduce the burden of proving linearizable object implementations correct to the proof of a particular invariant whose correctness can be shown inductively. Noting that the latter is a task that many proof systems (such as the TLA+ Proof System we chose to work with) are well-suited to handle, this technique allows us to …

Counting And Sampling Small Structures In Graph And Hypergraph Data Streams, Themistoklis Haris

Dartmouth College Undergraduate Theses

In this thesis, we explore the problem of approximating the number of elementary substructures called simplices in large k-uniform hypergraphs. The hypergraphs are assumed to be too large to be stored in memory, so we adopt a data stream model, where the hypergraph is defined by a sequence of hyperedges.

First we propose an algorithm that (ε, δ)-estimates the number of simplices using O(m^1+1/k / T) bits of space. In addition, we prove that no constant-pass streaming algorithm can (ε, δ)- approximate the number of simplices using less than O( m ^1+1/k / T ) bits of space. Thus …

Object Manipulation With Modular Planar Tensegrity Robots, Maxine Perroni-Scharf

Dartmouth College Undergraduate Theses

This thesis explores the creation of a novel two-dimensional tensegrity-based mod- ular system. When individual planar modules are linked together, they form a larger tensegrity robot that can be used to achieve non-prehensile manipulation. The first half of this dissertation focuses on the study of preexisting types of tensegrity mod- ules and proposes different possible structures and arrangements of modules. The second half describes the construction and actuation of a modular 2D robot com- posed of planar three-bar tensegrity structures. We conclude that tensegrity modules are suitably adapted to object manipulation and propose a future extension of the modular 2D …

Improving Structure Mcmc For Bayesian Networks Through Markov Blanket Resampling, Chengwei Su, Mark E. Borsuk

Dartmouth Scholarship

Algorithms for inferring the structure of Bayesian networks from data have become an increasingly popular method for uncovering the direct and indirect influences among variables in complex systems. A Bayesian approach to structure learning uses posterior probabilities to quantify the strength with which the data and prior knowledge jointly support each possible graph feature. Existing Markov Chain Monte Carlo (MCMC) algorithms for estimating these posterior probabilities are slow in mixing and convergence, especially for large networks. We present a novel Markov blanket resampling (MBR) scheme that intermittently reconstructs the Markov blanket of nodes, thus allowing the sampler to more effectively …

Trip: Tracking Rhythms In Plants, An Automated Leaf Movement Analysis Program For Circadian Period Estimation, Kathleen Greenham, Ping Lou, Sara E. Remsen, Hany Farid, C Robertson Mcclung

Dartmouth Scholarship

Background: A well characterized output of the circadian clock in plants is the daily rhythmic movement of leaves. This process has been used extensively in Arabidopsis to estimate circadian period in natural accessions as well as mutants with known defects in circadian clock function. Current methods for estimating circadian period by leaf movement involve manual steps throughout the analysis and are often limited to analyzing one leaf or cotyledon at a time.

Methods: In this study, we describe the development of TRiP (Tracking Rhythms in Plants), a new method for estimating circadian period using a motion estimation algorithm that can …

Derivation Of A Novel Efficient Supervised Learning Algorithm From Cortical-Subcortical Loops, Ashok Chandrashekar, Richard Granger

Dartmouth Scholarship

Although brain circuits presumably carry out powerful perceptual algorithms, few instances of derived biological methods have been found to compete favorably against algorithms that have been engineered for specific applications. We forward a novel analysis of a subset of functions of cortical-subcortical loops, which constitute more than 80% of the human brain, thus likely underlying a broad range of cognitive functions. We describe a family of operations performed by the derived method, including a non-standard method for supervised classification, which may underlie some forms of cortically dependent associative learning. The novel supervised classifier is compared against widely used algorithms for …

A Subgroup Algorithm To Identify Cross-Rotation Peaks Consistent With Non-Crystallographic Symmetry, Ryan H. Lilien, Chris Bailey-Kellogg, Amy C. Anderson, Bruce R. Donald

Dartmouth Scholarship

Molecular replacement (MR) often plays a prominent role in determining initial phase angles for structure determination by X-ray crystallography. In this paper, an efficient quaternion-based algorithm is presented for analyzing peaks from a cross-rotation function in order to identify model orientations consistent with proper non-crystallographic symmetry (NCS) and to generate proper NCS-consistent orientations missing from the list of cross-rotation peaks. The algorithm, CRANS, analyzes the rotation differences between each pair of cross-rotation peaks to identify finite subgroups. Sets of rotation differences satisfying the subgroup axioms correspond to orientations compatible with the correct proper NCS. The CRANS algorithm was first …

The Online Median Problem, Ramgopal R. Mettu, C. Greg Plaxton

Dartmouth Scholarship

We introduce a natural variant of the (metric uncapacitated) k-median problem that we call the online median problem. Whereas the k-median problem involves optimizing the simultaneous placement of k facilities, the online median problem imposes the following additional constraints: the facilities are placed one at a time, a facility cannot be moved once it is placed, and the total number of facilities to be placed, k, is not known in advance. The objective of an online median algorithm is to minimize the competitive ratio, that is, the worst-case ratio of the cost of an online placement to …

Approximation Techniques For Average Completion Time Scheduling, Chandra Chekuri, Rajeev Motwani, Balas Natarajan, Clifford Stein

Dartmouth Scholarship

We consider the problem of nonpreemptive scheduling to minimize average ( weighted) completion time, allowing for release dates, parallel machines, and precedence constraints. Recent work has led to constant-factor approximations for this problem based on solving a preemptive or linear programming relaxation and then using the solution to get an ordering on the jobs. We introduce several new techniques which generalize this basic paradigm. We use these ideas to obtain

improved approximation algorithms for one-machine scheduling to minimize average completion time with release dates. In the process, we obtain an optimal randomized on-line algorithm for the same problem that beats …

Asymptotically Tight Bounds For Performing Bmmc Permutations On Parallel Disk Systems, Thomas H. Cormen, Thomas Sundquist, Leonard F. Wisniewski

Dartmouth Scholarship

This paper presents asymptotically equal lower and upper bounds for the number of parallel I/O operations required to perform bit-matrix-multiply/complement (BMMC) permutations on the Parallel Disk Model proposed by Vitter and Shriver. A BMMC permutation maps a source index to a target index by an affine transformation over GF(2), where the source and target indices are treated as bit vectors. The class of BMMC permutations includes many common permutations, such as matrix transposition (when dimensions are powers of 2), bit-reversal permutations, vector-reversal permutations, hypercube permutations, matrix reblocking, Gray-code permutations, and inverse Gray-code permutations. The upper bound improves upon the asymptotic …

Fast Discrete Polynomial Transforms With Applications To Data Analysis For Distance Transitive Graphs, J. R. Driscoll, D. M. Healy, D. N. Rockmore

Dartmouth Scholarship

Let $\poly = \{P_0,\dots,P_{n-1}\}$ denote a set of polynomials with complex coefficients. Let $\pts = \{z_0,\dots,z_{n-1}\}\subset \cplx$ denote any set of {\it sample points}. For any $f = (f_0,\dots,f_{n-1}) \in \cplx^n$, the {\it discrete polynomial transform} of f (with respect to $\poly$ and $\pts$) is defined as the collection of sums, $\{\fhat(P_0),\dots,\fhat(P_{n-1})\}$, where $\fhat(P_j) = \langle f,P_j \rangle = \sum_{i=0}^{n-1} f_iP_j(z_i)w(i)$ for some associated weight function w. These sorts of transforms find important applications in areas such as medical imaging and signal processing.

In this paper, we present fast algorithms for computing discrete orthogonal polynomial transforms. For a system …

Low-Degree Spanning Trees Of Small Weight, Samir Khuller, Balaji Raghavachari, Neal Young

Dartmouth Scholarship

Given n points in the plane, the degree-K spanning-tree problem asks for a spanning tree of minimum weight in which the degree of each vertex is at most K. This paper addresses the problem of computing low-weight degree-K spanning trees for $K > 2$. It is shown that for an arbitrary collection of n points in the plane, there exists a spanning tree of degree 3 whose weight is at most 1.5 times the weight of a minimum spanning tree. It is shown that there exists a spanning tree of degree 4 whose weight is at most 1.25 times …

Improved Algorithms For Bipartite Network Flow, Ravindra K. Ahuja, James B. B. Orlin, Clifford Stein, Robert E. Tarjan

Dartmouth Scholarship

In this paper, network flow algorithms for bipartite networks are studied. A network G = (V,E) is called bipartite if its vertex set V can be partitioned into two subsets V_1 and V_2 such that all edges have one endpoint in V_1 and the other in $V_2 $. Let $n = |V|, n_1 = |V_1 | , n_2 = |V_2 |, m = |E| and assume without loss of generality that n_1 \leqslant n_2. A bipartite network is called unbalanced if n_1 \ll n_2 $ and balanced otherwise. (This notion is necessarily imprecise.) It is shown that several maximum flow …