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Computing the visibility polygon using few variables
 In Proceedings of the 22nd International Symposium on Algorithms and Computation, volume 7014 of Lecture Notes in Computer Science
, 2011
"... Abstract. We present several algorithms for computing the visibility polygon of a simple polygon P from a viewpoint inside the polygon, when the polygon resides in readonly memory and only few working variables can be used. The first algorithm uses a constant number of variables, and outputs the ve ..."
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Abstract. We present several algorithms for computing the visibility polygon of a simple polygon P from a viewpoint inside the polygon, when the polygon resides in readonly memory and only few working variables can be used. The first algorithm uses a constant number of variables, and outputs the vertices of the visibility polygon in O(nr̄) time, where r ̄ denotes the number of reflex vertices of P that are part of the output. The next two algorithms use O(log r) variables, and output the visibility polygon in O(n log r) randomized expected time or O(n log2 r) deterministic time, where r is the number of reflex vertices of P. 1
ConstantWorkSpace Algorithm for a Shortest Path in a Simple Polygon
"... Abstract. We present two spaceefficient algorithms. First, we show how to report a simple path between two arbitrary nodes in a given tree. Using a technique called “computing instead of storing”, we can design a naive quadratictime algorithm for the problem using only constant work space, i.e., O ..."
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Abstract. We present two spaceefficient algorithms. First, we show how to report a simple path between two arbitrary nodes in a given tree. Using a technique called “computing instead of storing”, we can design a naive quadratictime algorithm for the problem using only constant work space, i.e., O(log n) bits in total for the work space, where n is the number of nodes in the tree. Then, another technique “controlled recursion” improves the time bound to O(n 1+ε) for any positive constant ε. Second, we describe how to compute a shortest path between two points in a simple ngon. Although the shortest path problem in general graphs is NLcomplete, this constrained problem can be solved in quadratic time using only constant work space. 1
MemoryConstrained Algorithms for Simple Polygons
, 2011
"... A constantworkspace algorithm has readonly access to an input array and may use only O(1) additional words of O(log n) bits, where n is the size of the input. We show that we can find a triangulation of a plane straightline graph with n vertices in O(n²) time. We also consider preprocessing a sim ..."
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A constantworkspace algorithm has readonly access to an input array and may use only O(1) additional words of O(log n) bits, where n is the size of the input. We show that we can find a triangulation of a plane straightline graph with n vertices in O(n²) time. We also consider preprocessing a simple ngon, which is given by the ordered sequence of its vertices, for shortest path queries when the space constraint is relaxed to allow s words of working space. After a preprocessing of O(n²) time, we are able to solve shortest path queries between any two points inside the polygon in O(n²/s) time.
Spacetime tradeoffs for stackbased algorithms
, 2013
"... In memoryconstrained algorithms we have readonly access to the input, and the number of additional variables is limited. In this paper we introduce the compressed stack technique, a method that allows to transform algorithms whose space bottleneck is a stack into memoryconstrained algorithms. Give ..."
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Cited by 5 (2 self)
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In memoryconstrained algorithms we have readonly access to the input, and the number of additional variables is limited. In this paper we introduce the compressed stack technique, a method that allows to transform algorithms whose space bottleneck is a stack into memoryconstrained algorithms. Given an algorithm A that runs in O(n) time using a stack of length Θ(n), we can modify it so that it runs in O(n 2 /2 s) time using a workspace of O(s) variables (for any s ∈ o(log n)) or O(n log n / log p) time using O(p log n / log p) variables (for any 2 ≤ p ≤ n). We also show how the technique can be applied to solve various geometric problems, namely computing the convex hull of a simple polygon, a triangulation of a monotone polygon, the shortest path between two points inside a monotone polygon, 1dimensional pyramid approximation of a 1dimensional vector, and the visibility profile of a point inside a simple polygon. Our approach exceeds or matches the bestknown results for these problems in constantworkspace models (when they exist), and gives a tradeoff between the size of the workspace and running time. To the best of our knowledge, this is the first general framework for obtaining memoryconstrained algorithms.
Spaceefficient algorithms for visibility problems in simple polygon. Eprint arXiv:1204.2634
, 2012
"... Abstract. Given a simple polygon P consisting of n vertices, we study the problem of designing spaceefficient algorithms for computing (i) the visibility polygon of a point inside P, (ii) the weak visibility polygon of a line segment inside P and (iii) the minimum link path between a pair of points ..."
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Abstract. Given a simple polygon P consisting of n vertices, we study the problem of designing spaceefficient algorithms for computing (i) the visibility polygon of a point inside P, (ii) the weak visibility polygon of a line segment inside P and (iii) the minimum link path between a pair of points inside P. For problem (i) two algorithms are proposed. The first one is an inplace algorithm where the input array may be lost. It uses only O(1) extra space apart from the input array. The second one assumes that the input is given in a readonly array, and it needs O( n) extra space. The time complexity of both the algorithms are O(n). For problem (ii), we have assumed that the input polygon is given in a readonly array. Our proposed algorithm runs in O(n2) time using O(1) extra space. For problem (iii) the time and space complexities of our proposed algorithm are O(kn) and O(1) respectively; k is the length (number of links) in a minimum link path between the given pair of points. 1
Minimum Enclosing Circle with Few Extra Variables
"... Asano et al. [JoCG 2011] proposed an open problem of computing the minimum enclosing circle of a set of n points in R2 given in a readonly array in subquadratic time. We show that Megiddo’s prune and search algorithm for computing the minimum radius circle enclosing the given points can be tailore ..."
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Asano et al. [JoCG 2011] proposed an open problem of computing the minimum enclosing circle of a set of n points in R2 given in a readonly array in subquadratic time. We show that Megiddo’s prune and search algorithm for computing the minimum radius circle enclosing the given points can be tailored to work in a readonly environment in O(n1+ɛ) time using O(log n) extra space, where ɛ is a positive constant less than 1. As a warmup, we first solve the same problem in an inplace setup in linear time with O(1) extra space.
A Static Optimality Transformation with Applications to Planar Point Location
, 2012
"... Over the last decade, there have been several data structures that, given a planar subdivision and a probability distribution over the plane, provide a way for answering point location queries that is finetuned for the distribution. All these methods suffer from the requirement that the query distr ..."
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Over the last decade, there have been several data structures that, given a planar subdivision and a probability distribution over the plane, provide a way for answering point location queries that is finetuned for the distribution. All these methods suffer from the requirement that the query distribution must be known in advance. We present a new data structure for point location queries in planar triangulations. Our structure is asymptotically as fast as the optimal structures, but it requires no prior information about the queries. This is a 2d analogue of the jump from Knuth’s optimum binary search trees (discovered in 1971) to the splay trees of Sleator and Tarjan in 1985. While the former need to know the query distribution, the latter are statically optimal. This means that we can adapt to the query sequence and achieve the same asymptotic performance as an optimum static structure, without needing any additional information. 1
ConstantWorkspace Algorithms for Visibility Problems in the Plane
, 2013
"... In the constantworkspace model, the input is given as a readonly array which allows random access and the output is to be produced on a writeonly array as a stream. In addition to that, only a constant number of variables are available, independent on the size of the input. Most ordinary algorit ..."
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In the constantworkspace model, the input is given as a readonly array which allows random access and the output is to be produced on a writeonly array as a stream. In addition to that, only a constant number of variables are available, independent on the size of the input. Most ordinary algorithms for geometric problems make heavy use on the construction of smart data structures such as doublylinked lists, heaps, and search trees which enable fast processing. In the constantworkspace model such data structures are not available due to the small amount of memory. Instead we need to access the input repeatedly. We try to minimize the number of accesses in order to make the algorithms as efficient as possible. In this thesis, we present new algorithms for visibility problems in the plane using constant workspace. We devise an O(n2)time algorithm computing the circular visibility region of a polygon with n vertices from a given point within the polygon. Next, we present an O(n)time algorithm to compute the visible part of one edge from another edge in a polygon. Using that algorithm, we describe an algorithm
Spaceefficient Algorithms for Empty Space Recognition among a Point Set in 2D and 3D
, 2011
"... In this paper, we consider the problem of designing inplace algorithms for computing the maximum area empty rectangle of arbitrary orientation among a set of points in 2D, and the maximum volume empty axisparallel cuboid among a set of points in 3D. If n points are given in an array of size n, the ..."
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In this paper, we consider the problem of designing inplace algorithms for computing the maximum area empty rectangle of arbitrary orientation among a set of points in 2D, and the maximum volume empty axisparallel cuboid among a set of points in 3D. If n points are given in an array of size n, the worst case time complexity of our proposed algorithms for both the problems is O(n³); both the algorithms use O(1) extra space in addition to the array containing the input points.