When Can You Fold a Map?

We explore the following problem: given a collection of creases on a piece of paper, each assigned a folding direction of mountain or valley, is there a flat folding by a sequence of simple folds? There are several models of simple folds; the simplest one-layer simple fold rotates a portion of paper about a crease in the paper by +-180 degrees. We first consider the analogous questions in one dimension lower -- bending a segment into a flat object -- which lead to interesting problems on strings. We develop efficient algorithms for the recognition of simply foldable 1D crease patterns, and reconstruction of a sequence of simple folds. Indeed, we prove that a 1D crease pattern is flat-foldable by any means precisely if it is by a sequence of one-layer simple folds. Next we explore simple foldability in two dimensions, and find a surprising contrast: ``map'' folding and variants are polynomial, but slight generalizations are NP-complete. Specifically, we develop a linear-time algorithm for deciding foldability of an orthogonal crease pattern on a rectangular piece of paper, and prove that it is (weakly) NP-complete to decide foldability of (1) an orthogonal crease pattern on a orthogonal piece of paper, (2) a crease pattern of axis-parallel and diagonal (45-degree) creases on a square piece of paper, and (3) crease patterns without a mountain/valley assignment.Comment: 24 pages, 19 figures. Version 3 includes several improvements thanks to referees, including formal definitions of simple folds, more figures, table summarizing results, new open problems, and additional reference

The Lazy Bureaucrat Scheduling Problem

We introduce a new class of scheduling problems in which the optimization is performed by the worker (single ``machine'') who performs the tasks. A typical worker's objective is to minimize the amount of work he does (he is ``lazy''), or more generally, to schedule as inefficiently (in some sense) as possible. The worker is subject to the constraint that he must be busy when there is work that he can do; we make this notion precise both in the preemptive and nonpreemptive settings. The resulting class of ``perverse'' scheduling problems, which we denote ``Lazy Bureaucrat Problems,'' gives rise to a rich set of new questions that explore the distinction between maximization and minimization in computing optimal schedules.Comment: 19 pages, 2 figures, Latex. To appear, Information and Computatio

Searching on a Tape

Coordinated Science Laboratory was formerly known as Control Systems LaboratoryNational Science Foundation / ECS-84-10902 and CCR-87-14565Amoco Foundatio

TABLET: The Personal Computer of the Year 2000

Coordinated Science Laboratory was formerly known as Control Systems LaboratorySRC / 87-DP-109NASA / NAG 1-61

The Stony Brook Algorithm Repository

A computer science professor at the State University of New York maintains this online repository, which serves "as a comprehensive collection of algorithm implementations for over seventy of the most fundamental problems in combinatorial algorithms." The algorithms are implemented in a variety of programming languages, from C++ to FORTRAN. Seven general categories are listed to facilitate finding a particular implementation. The code for each algorithm was gathered from many different sources and is now housed in one spot on this site, making it easy for programmers to use pre-written code for common problems rather than having to write their own

The data science design manual

e-Book available, please log-in on Member Area to access or contact our librarian.xvii, 445 p