Sorting Permutations with Finite-Depth Stacks

Sorting organizes information for optimal usage and is desirable in many different fields. Noted computer scientist Donald Knuth first considered using stacks of infinite depth as a powerful means to sort data. We extend this work to consider stack-sortable permutations using stacks of specified finite depths. We characterize patterns that sortable permutations must avoid and derive a handy enumeration formula. Further generalizations include the introduction of multiple stacks and the analysis of the resulting counting sequences

Simulation Modeling and Analysis of Coal Shipping Operations

Computer simulations are increasingly powerful and realistic models for complex real-world scenarios, and our project applies this technology to model a coal transportation case study. Given a baseline scenario of fourteen carriers transporting coal from three U.S. locations to four international locations, we optimize operations in terms of product flow, time required for shipments, and total operation costs. Implementing the case study\u27s factors into modular code, we introduce several potential changes to current operations and develop specific scenarios. Further, in analyzing these scenarios we test for robustness and sensitivity, by changing values such as demand and bad weather occurrences, and noting how well the model responds. We ultimately gain a better intuition of the factors at play, identify optimizations, and develop a more efficient configuration. Also, we note several areas of potential improvement and suggest several directions for future work. Finally, taking advantage of modern graphical software, we present the optimized scenario in an animated interface, including a 3D view of the model and real-time data charts. While delving into complex data to reach the desired results, our model is accessible to a broad audience and presents an intriguing glimpse into the future of computational modeling

Sorting Permutations with Finite-Depth Stacks

Sorting organizes information for optimal usage, and our work examines the mathematics behind sorting with stacks. In 1968, Donald Knuth showed that a permutation is sortable in an infinite-depth stack if and only if it avoids the pattern 231; Knuth also enumerated these permutations. Twenty-five years later, Julian West extended these ideas to permutations sortable with 2 consecutive stacks. We continue this work by limiting the stack(s) to a finite depth. In particular, we completely characterize permutations sortable through a single finite-depth stack and derive a handy enumeration formula. We also apply our pattern characterization and enumeration techniques to permutations that are sortable after k-passes through a finite-depth stack

Structural Rounding: Approximation Algorithms for Graphs Near an Algorithmically Tractable Class

We develop a framework for generalizing approximation algorithms from the structural graph algorithm literature so that they apply to graphs somewhat close to that class (a scenario we expect is common when working with real-world networks) while still guaranteeing approximation ratios. The idea is to edit a given graph via vertex- or edge-deletions to put the graph into an algorithmically tractable class, apply known approximation algorithms for that class, and then lift the solution to apply to the original graph. We give a general characterization of when an optimization problem is amenable to this approach, and show that it includes many well-studied graph problems, such as Independent Set, Vertex Cover, Feedback Vertex Set, Minimum Maximal Matching, Chromatic Number, (l-)Dominating Set, Edge (l-)Dominating Set, and Connected Dominating Set. To enable this framework, we develop new editing algorithms that find the approximately-fewest edits required to bring a given graph into one of a few important graph classes (in some cases these are bicriteria algorithms which simultaneously approximate both the number of editing operations and the target parameter of the family). For bounded degeneracy, we obtain an O(r log{n})-approximation and a bicriteria (4,4)-approximation which also extends to a smoother bicriteria trade-off. For bounded treewidth, we obtain a bicriteria (O(log^{1.5} n), O(sqrt{log w}))-approximation, and for bounded pathwidth, we obtain a bicriteria (O(log^{1.5} n), O(sqrt{log w} * log n))-approximation. For treedepth 2 (related to bounded expansion), we obtain a 4-approximation. We also prove complementary hardness-of-approximation results assuming P != NP: in particular, these problems are all log-factor inapproximable, except the last which is not approximable below some constant factor 2 (assuming UGC)

Optimally Sorting Evolving Data

We give optimal sorting algorithms in the evolving data framework, where an algorithm\u27s input data is changing while the algorithm is executing. In this framework, instead of producing a final output, an algorithm attempts to maintain an output close to the correct output for the current state of the data, repeatedly updating its best estimate of a correct output over time. We show that a simple repeated insertion-sort algorithm can maintain an O(n) Kendall tau distance, with high probability, between a maintained list and an underlying total order of n items in an evolving data model where each comparison is followed by a swap between a random consecutive pair of items in the underlying total order. This result is asymptotically optimal, since there is an Omega(n) lower bound for Kendall tau distance for this problem. Our result closes the gap between this lower bound and the previous best algorithm for this problem, which maintains a Kendall tau distance of O(n log log n) with high probability. It also confirms previous experimental results that suggested that insertion sort tends to perform better than quicksort in practice

Benchmarking treewidth as a practical component of tensor-network--based quantum simulation

Tensor networks are powerful factorization techniques which reduce resource requirements for numerically simulating principal quantum many-body systems and algorithms. The computational complexity of a tensor network simulation depends on the tensor ranks and the order in which they are contracted. Unfortunately, computing optimal contraction sequences (orderings) in general is known to be a computationally difficult (NP-complete) task. In 2005, Markov and Shi showed that optimal contraction sequences correspond to optimal (minimum width) tree decompositions of a tensor network's line graph, relating the contraction sequence problem to a rich literature in structural graph theory. While treewidth-based methods have largely been ignored in favor of dataset-specific algorithms in the prior tensor networks literature, we demonstrate their practical relevance for problems arising from two distinct methods used in quantum simulation: multi-scale entanglement renormalization ansatz (MERA) datasets and quantum circuits generated by the quantum approximate optimization algorithm (QAOA). We exhibit multiple regimes where treewidth-based algorithms outperform domain-specific algorithms, while demonstrating that the optimal choice of algorithm has a complex dependence on the network density, expected contraction complexity, and user run time requirements. We further provide an open source software framework designed with an emphasis on accessibility and extendability, enabling replicable experimental evaluations and future exploration of competing methods by practitioners.Comment: Open source code availabl