In case of a future pandemic, the mobility dynamics of a city can be controlled by intervening in the mobility patterns of people. Instead of hard quarantine policies, incentives can be designed that are compatible with people's preferences. At first, we distinguish mobility from the different types of locations for which distance matters. We match these types of locations in a way that maximizes the natural preference of people to visit the locations. We investigate different approaches for matching locations, such as retail and educational services, while considering people's preferences. We show that satisfying the preferences of the entire city is a computationally hard problem. Approximation algorithms are proposed in which the penalty for preference violation is bounded. We propose a fast approximation algorithm that focuses on the penalty value of locations, and we propose a more computationally heavy approximation that focuses on user penalty with a specific scheme of user allocation to locations. Additionally, we investigated higher-order matching of locations and the complexity of urban partitioning. We tested our approach in Euclidean space and network space. Finally, we show that applying such mobility restrictions can reduce the transmission rate, and we extract cells whose people can be incentivized to fulfill their needs based on the proposed algorithms, slowing down a future pandemic and preventing potential superspreading events.

We present a new class of problems where the goal is to select a “fair” subgraph H of a given graph G = (V,E), such that H decomposes into many small components. A subgraph H c G is (P,d) fair if every vertex v ϵ P has the same degree d in H, where P c V and d > 0 are input parameters. These problems arise when the goal is to allow individuals to equally participate in activities in such a way that the connected components within an interaction graph, which models potential interactions among people, are of the smallest possible size, so that the spread of the contagion, and the difficulty of contact tracing in case of infection, is minimized. Within a preference graph that models the set of preferred choices for each individual when selecting among available options of where to conduct any particular type of activity (e.g., which gym to attend), we seek to compute the fair subgraph of assignments of individuals to these options, so that the number of people in each connected component (“interaction community”) of the resulting subgraph is minimized, and everyone is given the same number of options for every activity. We show that the fair subgraph selection problem is NP-hard, even for very restricted versions. We then formulate the problem as an integer program, and give a polynomial time computable lower bound on the optimal solution.

We present the first universal reconfiguration algorithm for transforming a modular robot between any two facet-connected square-grid configurations using pivot moves. More precisely, we show that five extra “helper” modules (“musketeers”) suffice to reconfigure the remaining n modules between any two given configurations. Our algorithm uses $$O(n^2)$$ pivot moves, which is worst-case optimal. Previous reconfiguration algorithms either require less restrictive “sliding” moves, do not preserve facet-connectivity, or for the setting we consider, could only handle a small subset of configurations defined by a local forbidden pattern. Configurations with the forbidden pattern do have disconnected reconfiguration graphs (discrete configuration spaces), and indeed we show that they can have an exponential number of connected components. But forbidding the local pattern throughout the configuration is far from necessary, as we show that just a constant number of added modules (placed to be freely reconfigurable) suffice for universal reconfigurability. We also classify three different models of natural pivot moves that preserve facet-connectivity, and show separations between these models.

Motivated by indoor localization by tripwire lasers, we study the problem of cutting a polygon into small-size pieces, using the chords of the polygon. Several versions are considered, depending on the definition of the "size" of a piece. In particular, we consider the area, the diameter, and the radius of the largest inscribed circle as a measure of the size of a piece. We also consider different objectives, either minimizing the maximum size of a piece for a given number of chords, or minimizing the number of chords that achieve a given size threshold for the pieces. We give hardness results for polygons with holes and approximation algorithms for multiple variants of the problem.

Due to increasing concerns of data privacy, databases are being encrypted before they are stored on an untrusted server. To enable search operations on the encrypted data, searchable encryption techniques have been proposed. Representative schemes use order-preserving encryption (OPE) for supporting efficient Boolean queries on encrypted databases. Yet, recent works showed the possibility of inferring plaintext data from OPE-encrypted databases, merely using the order-preserving constraints, or combined with an auxiliary plaintext dataset with similar frequency distribution. So far, the effectiveness of such attacks is limited to single-dimensional dense data (most values from the domain are encrypted), but it remains challenging to achieve it on high-dimensional datasets (e.g., spatial data), which are often sparse in nature. In this paper, for the first time, we study data inference attacks on multi-dimensional encrypted databases (with 2-D as a special case). We formulate it as a 2-D order-preserving matching problem and explore both unweighted and weighted cases, where the former maximizes the number of points matched using only order information and the latter further considers points with similar frequencies. We prove that the problem is NP-hard, and then propose a greedy algorithm, along with a polynomial-time algorithm with approximation guarantees. Experimental results on synthetic and real-world datasets show that the data recovery rate is significantly enhanced compared with the previous 1-D matching algorithm.

A determinacy race occurs if two or more logically parallel instructions access the same memory location and at least one of them tries to modify its content. Races are often undesirable as they can lead to nondeterministic and incorrect program behavior. A data race is a special case of a determinacy race which can be eliminated by associating a mutual-exclusion lock with the memory location in question or allowing atomic accesses to it. However, such solutions can reduce parallelism by serializing all accesses to that location. For associative and commutative updates to a memory cell, one can instead use a reducer, which allows parallel race-free updates at the expense of using some extra space. More extra space usually leads to more parallel updates, which in turn contributes to potentially lowering the overall execution time of the program. We start by asking the following question. Given a fixed budget of extra space for mitigating the cost of races in a parallel program, which memory locations should be assigned reducers and how should the space be distributed among those reducers in order to minimize the overall running time? We argue that under reasonable conditions the races of a program can be captured by a directed acyclic graph (DAG), with nodes representing memory cells and arcs representing read-write dependencies between cells. We then formulate our original question as an optimization problem on this DAG. We concentrate on a variation of this problem where space reuse among reducers is allowed by routing every unit of extra space along a (possibly different) source to sink path of the DAG and using it in the construction of multiple (possibly zero) reducers along the path. We consider two different ways of constructing a reducer and the corresponding duration functions (i.e., reduction time as a function of space budget). We generalize our race-avoiding space-time tradeoff problem to a discrete resource-time tradeoff problem with general non-increasing duration functions and resource reuse over paths of the given DAG. For general DAGs, we show that even if the entire DAG is available offline the problem is strongly NP-hard under all three duration functions, and we give approximation algorithms for solving the corresponding optimization problems. We also prove hardness of approximation for the general resource-time tradeoff problem and give a pseudo-polynomial time algorithm for series-parallel DAGs.

In this paper we study the following problem: given a set of m sensors that collectively cover a set of n target points with heterogeneous coverage requirements (target j needs to be covered every fjslots), how to schedule the sensor duty cycles such that all coverage requirements are satisfied and the maximum number of sensors turned on at any time slot is minimized. The problem models varied real-world applications in which sensing tasks exhibit high discrepancy in coverage requirements - critical locations often need to be covered much more frequently. We provide multiple algorithms with best approximation ratio of O (log n + log m) for the maximum number of sensors to turn on, and bi-criteria algorithm with (α, β)-approximation factors with high probability, where the number of sensors turned on is an α = O(δ(log (n) + log(m))/β)-approximation of the optimal (satisfying all requirements) and the coverage requirement is a β-approximation; δ is the approximation ratio achievable in an appropriate instance of set multi-cover. When the sensor coverage exhibits extra geometric properties, the approximation ratios can be further improved. We also evaluated our algorithms via simulations and experiments on a camera testbed. The performance improvement (energy saving) is substantial compared to turning on all sensors all the time, or a random scheduling baseline.

Given $n$ pairs of points, $\mathcal{S} = \{\{p_1, q_1\}, \{p_2, q_2\}, \dots, \{p_n, q_n\}\}$, in some metric space, we study the problem of two-coloring the points within each pair, red and blue, to optimize the cost of a pair of node-disjoint networks, one over the red points and one over the blue points. In this paper we consider our network structures to be spanning trees, traveling salesman tours or matchings. We consider several different weight functions computed over the network structures induced, as well as several different objective functions. We show that some of these problems are NP-hard, and provide constant factor approximation algorithms in all cases.

We study the r-gather clustering problem in a mobile and distributed setting. In this problem, nodes must be clustered into groups of at least r nodes each, and the goal is to minimize the diameter of the clusters. This notion of clustering is motivated by protecting user anonymity in location-based services or trajectory publication. Prior works on r-gather problems are centralized and cannot be easily adapted to the mobile setting. We describe a distributed algorithm that produces compact clusters, within an approximation factor 4 of the minimum cluster diameter possible. The algorithm can run on the mobile nodes and access points at the network edge locally, and can handle node mobility, rapidly switching cluster memberships as needed. The distributed approach naturally comes with the advantage of greater resilience and stability. Additionally, we show that it achieves local optimality; i.e., from the point of view of any particular node, the solution is nearly as favorable as possible, irrespective of the global configuration. We also show how to cluster trajectories with dynamic re-groupings. Further, we improve the theoretical hardness results for the problem in the Euclidean setting.

We show how to preprocess a polygonal domain with a fixed starting point s in order to answer efficiently the following queries: Given a point q, how should one move from s in order to see q as soon as possible? This query resembles the well-known shortest-path-to-a-point query, except that the latter asks for the fastest way to reach q, instead of seeing it. Our solution methods include a data structure for a different generalization of shortest-path-to-a-point queries, which may be of independent interest: to report efficiently a shortest path from s to a query segment in the domain.

Wireless communication systems, such as RFIDs and wireless sensor networks, are increasingly being used in security-sensitive applications, e.g. credit card transactions or monitoring patient health in hospitals. Wireless jamming by transmitting artificial noise, which is traditionally used as an offensive technique for disrupting communication, has recently been explored as a means of protecting sensitive communication from eavesdroppers.

Security-sensitive applications, such as patient health monitoring and credit card transactions, are increasingly utilizing wireless communication systems, RFIDs, wireless sensor networks, and other wireless communication systems. The use of interference-emitting jammers to protect these sensitive communications has been recently explored in the literature, and has shown high potential. In this paper we consider optimization problems relating to the temporal distributions of jammers' activity, and the suitable coding regimes used for communication. Solving the joint problem optimally enables comprehensive security in space, at a low power consumption and low communication overhead. The joint optimization of jamming in space and time is driven by a new framework that uses the bit-error probability as a measure of communication quality. Under this framework, we show how to guarantee information-theoretic security within a geographic region, and with increased flexibility to tailor the coding regime to the problem's geometry. We present efficient algorithms for different settings, and provide simulations for various scenarios using the bit-error probability functions. These simulations demonstrate the efficiency of the scheme. We believe that our scheme can lead to practical, economical and scalable solutions for providing another layer of protection of sensitive data, in cases where encryption schemes are limited or impractical.

We study a facility location problem motivated by requirements pertaining to the distribution of charging stations for electric vehicles: Place a minimum number of battery charging stations at a subset of nodes of a network, so that battery-powered electric vehicles will be able to move between destinations using "t-spanning" routes, of lengths within a factor t > 1 of the length of a shortest path, while having sufficient charging stations along the way. We give constant-factor approximation algorithms for minimizing the number of charging stations, subject to the t-spanning constraint. We study two versions of the problem, one in which the stations are required to support a single ride (to a single destination), and one in which the stations are to support multiple rides through a sequence of destinations, where the destinations are revealed one at a time.

We consider the problem of finding a maximum number of disjoint paths for unit disks moving amidst static or dynamic obstacles. For the static case we give efficient exact algorithms, based on adapting the "continuous uppermost path" paradigm. As a by-product, we establish a continuous analogue of Menger's Theorem. (In this extended abstract we only state these results.)

We consider a class of geometric facility location problems in which the goal is to determine a set X of disks given by their centers (tj) and radii (rj) that cover a given set of demand points Y∈R2 at the smallest possible cost. We consider cost functions of the form Εjf(rj), where f(r)=rα is the cost of transmission to radius r. Special cases arise for α=1 (sum of radii) and α=2 (total area); power consumption models in wireless network design often use an exponent α>2. Different scenarios arise according to possible restrictions on the transmission centers tj, which may be constrained to belong to a given discrete set or to lie on a line, etc.We obtain several new results, including (a) exact and approximation algorithms for selecting transmission points tj on a given line in order to cover demand points Y∈R2; (b) approximation algorithms (and an algebraic intractability result) for selecting an optimal line on which to place transmission points to cover Y; (c) a proof of NP-hardness for a discrete set of transmission points in R2 and any fixed α>1; and (d) a polynomial-time approximation scheme for the problem of computing a minimum cost covering tour (MCCT), in which the total cost is a linear combination of the transmission cost for the set of disks and the length of a tour/path that connects the centers of the disks.

We study the problem of covering a set of points or polyhedra in R3 with two axis-aligned boxes in order to minimize a function of the measures of the two boxes, such as the sum or the maximum of their volumes. This 2-box cover problem arises naturally in the construction of bounding volume hierarchies, as well as in shape approximation and clustering. Existing algorithms solve the min-max version of the exact problem in quadratic time. Our results are more general, addressing min-max, min-sum and other versions. Our results give the first approximation schemes for the problem, which run in nearly linear time, as well as some new exact algorithms. We give (1+e)-approximation algorithms for minimizing the maximum or sum of volumes (or surface areas, diameters, widths, or girths) of the two boxes in R3. We investigate also the problem of computing balanced coverings, in which each box covers at least a fraction of the input objects, and we discuss the application to constructing provably-good bounding volume hierarchies of polyhedra. We also generalize our results to higher dimension.

We give the first algorithmic study of a class of "covering tour" problems related to the geometric traveling salesman problem: Find a polygonal tour for a cutter so that it sweeps out a specified region ("pocket") in order to minimize a cost that depends mainly on the number of turns. These problems arise naturally in manufacturing applications of computational geometry to automatic tool path generation and automatic inspection systems, as well as arc routing ("postman") problems with turn penalties. We prove the NP-completeness of minimum-turn milling and give efficient approximation algorithms for several natural versions of the problem, including a polynomial-time approximation scheme based on a novel adaptation of the m-guillotine method.

In the "freeze-tag" problem, we are given a swarm of n sleeping (frozen or inactive) robots and a single awake (active) robot. The goal is to awaken all robots in the shortest possible time. A robot is awakened when an active robot "touches" it. The goal is to compute an optimal awakening schedule such that all robots are awake by time t/sup */, for the smallest possible value of t/sup */. We devise and test a variety of heuristic strategies on geometric and network datasets. Our experiments show that all of the strategies perform acceptably well, with the simple greedy strategy performing particularly well. A theoretical analysis of the greedy strategy gives a tight approximation bound of /spl Theta/(/spl radic/logn) for points in the plane. We show more generally a tight performance bound of /spl Theta/((logn)/sup 1-1/d/) in d dimensions. The geometric case contrasts with the case of general metric spaces, where greedy is known to have a /spl Theta/(logn) approximation factor, and no method is known to achieve an approximation factor of o(logn).

No abstract available.

In the Freeze-Tag Problem, the objective is to awaken a set of "asleep" robots, starting with only one "awake" robot. A robot awakens a sleeping robot by moving to the sleeping robot's position. When a robot awakens, it is available to assist in awakening other slumbering robots. The objective is to compute an optimal awakening schedule/ such that all robots are awake by time t*, for the smallest possible value of t*. Because of its resemblance to the children's game of freeze-tag, this problem has been called Freeze-Tag Problem (FTP).A particularly intriguing aspect of the FTP is that any algorithm that is not purposely unproductive yields an O(log n)-approximation, while no o(log n)-approximation algorithms are known for general metric spaces.This paper presents an O(1)-approximation algorithm for the FTP in unweighted graphs, in which there is one asleep robot at each node. We show that this version of the FTP is NP-hard.We generalize our methods to the case in which there are multiple robots at each node and edges are unweighted; we obtain a θ(∗log n)-approximation in this case. In the case of weighted edges, our methods yield an O((L/d)log n)-approximation algorithm, where L is the length of the longest edge and d is the diameter of the graph.

The Computational Plant or Cplant is a commodity-based supercomputer under development at Sandia National Laboratories. This paper describes resource-allocation strategies to achieve processor locality for parallel jobs in Cplant and other supercomputers. Users of Cplant and other Sandia supercomputers submit parallel jobs to a job queue. When a job is scheduled to run, it is assigned to a set of processors. To obtain maximum throughput, jobs should be allocated to localized clusters of processors to minimize communication costs and to avoid bandwidth contention caused by overlapping jobs. This paper introduces new allocation strategies and performance metrics based on space-filling curves and one dimensional allocation strategies. These algorithms are general and simple. Preliminary simulations and Cplant experiments indicate that both space-filling curves and one-dimensional packing improve processor locality compared to the sorted free list strategy previously used on Cplant. These new allocation strategies are implemented in the new release of the Cplant System Software, Version 2.0, phased into the Cplant systems at Sandia by May 2002.

A set V and a collection of (possibly nondisjoint) subsets are given. Also given is a real matrix describing distances between elements of V. A cover is a subset of V containing at least one representative from each subset. The multiple-choice minimum-diameter problem is to select a cover of minimum diameter. The diameter is defined as the maximum distance between any pair of elements in the cover. The multiple-choice dispersion problem, which is closely related, asks us to maximize the minimum distance between any pair of elements in the cover. The problems are NP-hard. We present polynomial time algorithms for approximating special cases and generalizations of these basic problems, and we prove in other cases that no such algorithms exist (assuming P ≠ NP). © 2000 John Wiley & Sons, Inc.

We study the problem of computing a Hamiltonian tour (cycle) or path on a set of points in order to maximize the minimum edge length in the tour or path. This "maximum scatter" traveling salesperson problem (TSP) is closely related to the bottleneck TSP and is motivated by applications in manufacturing (e.g., sequencing of rivet operations) and medical imaging. In this paper, we give the first algorithmic study of these problems, including complexity results, approximation algorithms, and exact algorithms for special cases. In an attempt to model more accurately the real problems that arise in practice, we also generalize the basic problem to consider a more general measure of "scatter" in which points on a tour or path should be far not only from their immediate predecessor and successor, but also from other near-neighbors along the tour or path.

WC study a variety of geometric network optimization prob lcms on a set of points, in which we are given a resource bound, a, on the total length of the network, and our ob jcctivc is to maximize the number of points visited (or the total "value" of points visited), In particular, we resolve the well-publicized open problem on the approximabiity of the rooted "orienteering problem" for the case in which the sites are given as points in the plane and the network required is a cycle.We obtain a 2approximation for this problem, We also obtain approximation algorithms for variants of this problem in which the network required is a tree (S-approximation) or a path Q-approximation).No prior approximation bounds were known for any of these problems,We also obtain improved approximation algorithms for geometric instances of the unrooted orienteering problem, where we obtain a 2-approximation for both the cycle and tree versions of the problem on points in the plane, as well as a G-approximation for the tree version in edge-weighted graphs, E'urther, we study generalizations of the basic orienteering problem, to the case of multiple roots, sites that are polygonnl regions, etc., where we again give the first known approximation results.Our methods are based on some new tools which may be of interest in their own right: ( 1) some new results on m-'Department of Applied Mathematics and Statistics, State Univcrsitv of New York.Stonv Brook.NY 11794-3600: aat~o6smb .ounyob.

A fundamental problem in computer vision is identifying which of a given set of geometric models is present in animage.Reconsider anapproach to model recognition basedon computing efficient strategies (decision trees) for 'probing" a scanned image of a typeset document, in order to perform fast and effective optical character recognition (OCR).We consider a "proben to be a simply computed local operator that can be applied to discriminate between two sets of possible models.By carefully constructing effective probes, and assembling them into a geometric decision tree, we have devised, implemented, and compared a variety of methods to perform OCR.In this paper, we present algorithms for probing strategies and decision tree construction, and we report experiment al results on the effectiveness of theae algorithms in identifying English characters and numerals in scanned images of printed pages of text.These algorithms are implemented as part of a system used by a document processing company (Syngen Corp.).

We consider the problem of verifying a simple polygon in the plane using “test points”. A test point is a geometric probe that takes as input a point in Euclidean space, and returns “+” if the point is inside the object being probed or “−” if it is outside. A verification procedure takes as input a description of a target object, including its location and orientation, and it produces a set of test points that are used to verify whether a test object matches the description. We give a procedure for verifying an n-sided, non-degenerate, simple target polygon using 5n test points. This testing strategy works even if the test polygon has n + 1 vertices, and we show a lower bound of 3n + 1 test points for this case. We also give algorithms using O(n) test points for simple polygons that may be degenerate and for test polygons that may have up to n + 2 vertices. All of these algorithms work for polygons with holes. We also discuss extensions of our results to higher dimensions.

We consider a delivery problem on a network in which nodes have supplies or demands for certain products and arcs have lengths satisfying the triangle inequality. A vehicle of infinite capacity travels through the network, carrying products to their destinations, and is limited in that it can carry only a single type of product at a time. The general problem asks for a shortest delivery route of all products from their origin to their destination. Here, we consider certain restrictions on the delivery paths allowed and compare the quality of the solution of the unrestricted problem to that of the restricted one. Both the general and restricted problems are NP-hard, and we discuss approximation algorithms. We also give a constant factor approximation algorithm for the Clustered Traveling Salesman Problem. © 1997 John Wiley & Sons, Inc.

A fundamental problem in model-based computer vision is that of identifying which of a given set of geometric models is present in an image. Considering a “probe” to be an oracle that tells us whether or not a model is present at a given point, we study the problem of computing efficient strategies (“decision trees”) for probing an image, with the goal to minimize the number of probes necessary (in the worst case) to determine which single model is present. We show that a ⌈lg k ⌉ height binary decision tree always exists for k polygonal models (in fixed position), provided (1) they are non-degenerate (do not share boundaries) and (2) they share a common point of intersection. Further, we give an efficient algorithm for constructing such decision trees when the models are given as a set of polygons in the plane. We show that constructing a minimum height tree is NP-complete if either of the two assumptions is omitted. We provide an efficient greedy heuristic strategy and show that, in the general case, it yields a decision tree whose height is at most ⌈lg n ⌉ times that of an optimal tree. Finally, we discuss some restricted cases whose special structure allows for improved results.

The authors consider the problem of finding a shortest polygonal path from s to t within a simple polygon P, subject to the restriction that the path have at most k links (edges). They give an algorithm to compute a k-link path with length at most (1 + epsilon ) times the length of a shortest k-link path, for any error tolerance epsilon >0. The algorithm runs in time O(n/sup 3/k/sup 3/ log (Hk/ epsilon /sup 1/k/)), where N is the largest integer coordinate among the n vertices of P. They also study the more general problem of approximating shortest k-link paths in polygons with holes. In this case, they give an algorithm that returns a path with at most 2k links and length at most that of a shortest k-link path; the running time is O(kE/sup 2/), where E is the number of edges in the visibility graph. Finally, they study the bicriteria path problem in which the two criteria are link length and 'total turn' (the integral of mod Delta theta mod along a path). They obtain in an exact polynomial-time algorithm for polygons with holes.>

We consider several cases of the point matching problem in which we are to find a transformation of a set of n points such that each transformed point lies in one of n given pairwise-disjoint “noise regions.” We prove upper and lower bounds on the number of possible matches, under a variety of types of transformations (rotations, translations, similarity) and noise regions (circles, squares, polygons). We also give efficient algorithms for computing the set of all possible matches, along with a corresponding transformation that realizes each match. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

Article Arrangements of segments that share endpoints: single face results Share on Authors: Esther M. Arkin School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NYView Profile , Dan Halperin Department of Computer Science, School of Mathematical Sciences, Tel Aviv University Department of Computer Science, School of Mathematical Sciences, Tel Aviv UniversityView Profile , Klara Kedem Department of Computer Science, School of Mathematical Sciences, Tel Aviv University Department of Computer Science, School of Mathematical Sciences, Tel Aviv UniversityView Profile , Joseph S. B. Mitchell School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NYView Profile , Nir Naor Department of Computer Science, School of Mathematical Sciences, Tel Aviv University Department of Computer Science, School of Mathematical Sciences, Tel Aviv UniversityView Profile Authors Info & Claims SCG '91: Proceedings of the seventh annual symposium on Computational geometryJune 1991 Pages 324–333https://doi.org/10.1145/109648.109684Online:01 June 1991Publication History 1citation220DownloadsMetricsTotal Citations1Total Downloads220Last 12 Months3Last 6 weeks0 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteGet Access

Certain problems related to the length of cycles and paths modulo a given integer are studied. Linear-time algorithms are presented that determine whether all cycles in an undirected graph are of length P mod Q and whether all paths between two specified nodes are of length P mod Q , for fixed integers P . Q . These results are compared to those for directed graphs.

A method for comparing polygons that is a metric, invariant under translation, rotation, and change of scale, reasonably easy to compute, and intuitive is presented. The method is based on the L/sub 2/ distance between the turning functions of the two polygons. It works for both convex and nonconvex polygons and runs in time O(mn log mn), where m is the number of vertices in one polygon and n is the number of vertices in the other. Some examples showing that the method produces answers that are intuitively reasonable are presented.>

We study the class of problems associated with the detection and computation of monotone paths among a set of disjoint obstacles. We give an Ο(nE) algorithm for finding a monotone path (if one exists) between two points in the plane in the presence of polygonal obstacles. (Here, E is the size of the visibility graph defined by the n vertices of the obstacles.) If all of the obstacles are convex, we prove that there always exists a monotone path between any two points s and t. We give an Ο(n log n) algorithm for finding such a path for any s and t, after an initial Ο(E + n log n) preprocesing. We introduce the notions of “monotone path map”, and “shortest monotone path map” and give algorithms to compute them. We apply our results to a class of separation and assembly problems, yielding polynomial-time algorithms for planning an assembly sequence (based on separations by single translations) of arbitrary polygonal parts in two dimensions.