Bengt J. Nilsson | Malmö University (original) (raw)
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Papers by Bengt J. Nilsson
We present a fast algorithm for computing a watchman route in a simple polygon that is at most a ... more We present a fast algorithm for computing a watchman route in a simple polygon that is at most a constant factor longer than the shortest watchman route. The algorithm runs in O(n log n) time as compared to the best known algorithm that computes a shortest watchman route which ...
Nordic Journal of Computing, 1999
In this paper we investigate parallel searching on m concurrent rays. We assume that a target t i... more In this paper we investigate parallel searching on m concurrent rays. We assume that a target t is located somewhere on one of the rays and that a group of m point robots has to reach t. Furthermore, we assume that the robots have no way of communicating over distance. Given a strategy S we are interested in the competitive ratio which is defined as the ratio of the time needed by the robots using S and the time needed if the location of t is known in advance. If a lower bound on the distance to the target is known, then there is a simple strategy which achieves a competitive ratio of 9 --- independent of m. We show that even in the case m = 2 there is a lower bound of 9 on the competitive ratio for two large classes of strategies. Moreover, we show that a lower bound of 9 for m = 2 implies a lower bound of 9 for m ? 2 --- as is to be expected. If the minimum distance to the target is not known in advance, then we show a lower bound on the competitive ratio of 1 + 2(k + 1) k+1 =k k ...
Proceedings of the 7th ACM conference on Recommender systems - RecSys '13, 2013
We study the problem of optimizing recommendation systems for e-commerce sites. We consider in pa... more We study the problem of optimizing recommendation systems for e-commerce sites. We consider in particular a combinatorial solution to this optimization based on the well known Maximum Coverage problem that asks for the k sets (products) that cover the most elements from a ground set (consumers). This formulation provides an abstract model for what k products should be recommended to maximize the probability of consumer purchase. Unfortunately, Maximum Coverage is NP-complete but an efficient approximation algorithm exists based on the Greedy methodology.
Exploring a polygon is the problem of a robot that does not have a map of its surroundings to see... more Exploring a polygon is the problem of a robot that does not have a map of its surroundings to see the complete polygon. In other words, for the robot to construct a map of the polygon. Exploration can be viewed as an online problem. Typical for online problems is that the solution method must make decisions based on past events but without knowledge about the future. In our case the robot does not have complete information about the environment. Competitive analysis can be used to measure the performance of methods solving online problems. The competitive factor of such a method is the ratio between the method's performance and the performance of the best method having full knowledge about the future. We prove a 5/3-competitive strategy for exploring a simple rectilinear polygon in the L1 metric. This improves the previous factor two bound of Deng, Kameda and Papadimitriou.
A polygon P is x-monotone if any line orthogonal to the x-axis has a simply connected intersectio... more A polygon P is x-monotone if any line orthogonal to the x-axis has a simply connected intersection with P. A set G of points inside P or on the boundary of P is said to guard the polygon if every point inside P or on the boundary of P is seen by a point in G. An interior guard can lie anywhere inside or on the boundary of the polygon. Using a reduction from Monotone 3SAT, we prove that interior guarding a monotone polygon is NP-hard. Because interior guards can be placed anywhere inside the polygon, a clever gadget is introduced that forces interior guards to be placed at very specific locations.
Proceedings ICCI `92: Fourth International Conference on Computing and Information, 1992
In this paper we consider the problem of computing an optimum set of watchmen routes in a histogr... more In this paper we consider the problem of computing an optimum set of watchmen routes in a histogram. A watchman, in the terminology of art galleries, is a mobile guard and in this version we want to minimize the length of the longest route in the solution. We give an O(n 2 log n) time algorithm to compute the MinMax optimum set of m watchmen in a histogram polygon and we extend the algorithm to solve the Weighted MinMax problem under the same time bound.
Lecture Notes in Computer Science, 2015
Lecture Notes in Computer Science, 1991
The problem of finding the diameter of a simple polygon has been studied extensively in recent ye... more The problem of finding the diameter of a simple polygon has been studied extensively in recent years. O(n log n) time upper bounds have been given for computing the geodesic diameter and the link diameter for a polygon.
Lecture Notes in Computer Science, 1997
Lecture Notes in Computer Science, 1999
In this paper we investigate parallel searching on m concurrent rays. We assume that atarget t is... more In this paper we investigate parallel searching on m concurrent rays. We assume that atarget t is located somewhere on one of the rays and that a group of m point robots has toreach t. Furthermore, we assume that the robots have no way of communicating over distance.Given a strategy S we are interested in the competitive ratio which is defined as the ratio ofthe time needed by the robots using S and the time needed if the location of t is known inadvance.If a lower bound on the distance...
Lecture Notes in Computer Science, 1999
We study the approximation complexity of certain kinetic variants of the Traveling Salesman Probl... more We study the approximation complexity of certain kinetic variants of the Traveling Salesman Problem (TSP) where we consider instances in which each point moves with a fixed constant speed in a fixed direction. We prove the following results:
Lecture Notes in Computer Science, 1997
A watchman route in a polygon P is a route inside P such that each point in the interior of P is ... more A watchman route in a polygon P is a route inside P such that each point in the interior of P is visible from at least one point along the route. The objective of the shortest watchman route problem is to minimize the length of the watchman route for a given polygon. In 1991 Chin and Ntafos claimed an O(n 4 ) algorithm, solving the shortest watchman route problem for simple polygons, given a starting point of the route. Later, improvements of this result were presented by Tan, Hirata and Inagaki, decreasing the timebound to O(n 2 ). We prove that the time bound analyses of these algorithms are erroneous and that their true time bound is (2 n ). Furthermore, a modi cation to the latest algorithm is given, restoring its time bound to O(n 2 ).
Lecture Notes in Computer Science, 2006
We consider the following problem from swarm robotics: given one or more "awake" robots in some m... more We consider the following problem from swarm robotics: given one or more "awake" robots in some metric space M , wake up a set of "asleep" robots. 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. We investigate offline and online versions of this problem and give a 2-competitive strategy and a lower bound of 2 in the case when M is discrete and the objective is to minimize the total movement cost. We also study the case when M is continuous and show a lower bound of 7/3 when the objective is to minimize the time when the last robot awakens.
A clique clustering of a graph is a partitioning of its vertices into disjoint cliques. The quali... more A clique clustering of a graph is a partitioning of its vertices into disjoint cliques. The quality of a clique clustering is measured by the total number of edges in its cliques. We consider the online variant of the clique clustering problem, where the vertices of the input graph arrive one at a time. At each step, the newly arrived vertex forms a singleton clique, and the algorithm can merge any existing cliques in its partitioning into larger cliques, but splitting cliques is not allowed. We give an online algorithm with competitive ratio 15.645 and we prove a lower bound of 6 on the competitive ratio, improving the previous respective bounds of 31 and 2.
Lecture Notes in Computer Science, 1992
In this paper we survey some results in restricted orientation computational geometry. The aim is... more In this paper we survey some results in restricted orientation computational geometry. The aim is to embed our own results into a more general context. We discuss methods for making object queries, computing shortest paths, and questions on restricted orientation convexity. Furthermore, we give an optimal algorithm for computing shortest paths when three arbitrary orientations are allowed for path finks and obstacle edges.
In this paper we investigate parallel searching on m concurrent rays. We assume that atarget t is... more In this paper we investigate parallel searching on m concurrent rays. We assume that atarget t is located somewhere on one of the rays and that a group of m point robots has toreach t. Furthermore, we assume that the robots have no way of communicating over distance.Given a strategy S we are interested in the competitive ratio which is defined as the ratio ofthe time needed by the robots using S and the time needed if the location of t is known inadvance.If a lower bound on the distance...
Theoretical Computer Science, 2006
Journal of Phytopathology, 1969
We present a fast algorithm for computing a watchman route in a simple polygon that is at most a ... more We present a fast algorithm for computing a watchman route in a simple polygon that is at most a constant factor longer than the shortest watchman route. The algorithm runs in O(n log n) time as compared to the best known algorithm that computes a shortest watchman route which ...
Nordic Journal of Computing, 1999
In this paper we investigate parallel searching on m concurrent rays. We assume that a target t i... more In this paper we investigate parallel searching on m concurrent rays. We assume that a target t is located somewhere on one of the rays and that a group of m point robots has to reach t. Furthermore, we assume that the robots have no way of communicating over distance. Given a strategy S we are interested in the competitive ratio which is defined as the ratio of the time needed by the robots using S and the time needed if the location of t is known in advance. If a lower bound on the distance to the target is known, then there is a simple strategy which achieves a competitive ratio of 9 --- independent of m. We show that even in the case m = 2 there is a lower bound of 9 on the competitive ratio for two large classes of strategies. Moreover, we show that a lower bound of 9 for m = 2 implies a lower bound of 9 for m ? 2 --- as is to be expected. If the minimum distance to the target is not known in advance, then we show a lower bound on the competitive ratio of 1 + 2(k + 1) k+1 =k k ...
Proceedings of the 7th ACM conference on Recommender systems - RecSys '13, 2013
We study the problem of optimizing recommendation systems for e-commerce sites. We consider in pa... more We study the problem of optimizing recommendation systems for e-commerce sites. We consider in particular a combinatorial solution to this optimization based on the well known Maximum Coverage problem that asks for the k sets (products) that cover the most elements from a ground set (consumers). This formulation provides an abstract model for what k products should be recommended to maximize the probability of consumer purchase. Unfortunately, Maximum Coverage is NP-complete but an efficient approximation algorithm exists based on the Greedy methodology.
Exploring a polygon is the problem of a robot that does not have a map of its surroundings to see... more Exploring a polygon is the problem of a robot that does not have a map of its surroundings to see the complete polygon. In other words, for the robot to construct a map of the polygon. Exploration can be viewed as an online problem. Typical for online problems is that the solution method must make decisions based on past events but without knowledge about the future. In our case the robot does not have complete information about the environment. Competitive analysis can be used to measure the performance of methods solving online problems. The competitive factor of such a method is the ratio between the method's performance and the performance of the best method having full knowledge about the future. We prove a 5/3-competitive strategy for exploring a simple rectilinear polygon in the L1 metric. This improves the previous factor two bound of Deng, Kameda and Papadimitriou.
A polygon P is x-monotone if any line orthogonal to the x-axis has a simply connected intersectio... more A polygon P is x-monotone if any line orthogonal to the x-axis has a simply connected intersection with P. A set G of points inside P or on the boundary of P is said to guard the polygon if every point inside P or on the boundary of P is seen by a point in G. An interior guard can lie anywhere inside or on the boundary of the polygon. Using a reduction from Monotone 3SAT, we prove that interior guarding a monotone polygon is NP-hard. Because interior guards can be placed anywhere inside the polygon, a clever gadget is introduced that forces interior guards to be placed at very specific locations.
Proceedings ICCI `92: Fourth International Conference on Computing and Information, 1992
In this paper we consider the problem of computing an optimum set of watchmen routes in a histogr... more In this paper we consider the problem of computing an optimum set of watchmen routes in a histogram. A watchman, in the terminology of art galleries, is a mobile guard and in this version we want to minimize the length of the longest route in the solution. We give an O(n 2 log n) time algorithm to compute the MinMax optimum set of m watchmen in a histogram polygon and we extend the algorithm to solve the Weighted MinMax problem under the same time bound.
Lecture Notes in Computer Science, 2015
Lecture Notes in Computer Science, 1991
The problem of finding the diameter of a simple polygon has been studied extensively in recent ye... more The problem of finding the diameter of a simple polygon has been studied extensively in recent years. O(n log n) time upper bounds have been given for computing the geodesic diameter and the link diameter for a polygon.
Lecture Notes in Computer Science, 1997
Lecture Notes in Computer Science, 1999
In this paper we investigate parallel searching on m concurrent rays. We assume that atarget t is... more In this paper we investigate parallel searching on m concurrent rays. We assume that atarget t is located somewhere on one of the rays and that a group of m point robots has toreach t. Furthermore, we assume that the robots have no way of communicating over distance.Given a strategy S we are interested in the competitive ratio which is defined as the ratio ofthe time needed by the robots using S and the time needed if the location of t is known inadvance.If a lower bound on the distance...
Lecture Notes in Computer Science, 1999
We study the approximation complexity of certain kinetic variants of the Traveling Salesman Probl... more We study the approximation complexity of certain kinetic variants of the Traveling Salesman Problem (TSP) where we consider instances in which each point moves with a fixed constant speed in a fixed direction. We prove the following results:
Lecture Notes in Computer Science, 1997
A watchman route in a polygon P is a route inside P such that each point in the interior of P is ... more A watchman route in a polygon P is a route inside P such that each point in the interior of P is visible from at least one point along the route. The objective of the shortest watchman route problem is to minimize the length of the watchman route for a given polygon. In 1991 Chin and Ntafos claimed an O(n 4 ) algorithm, solving the shortest watchman route problem for simple polygons, given a starting point of the route. Later, improvements of this result were presented by Tan, Hirata and Inagaki, decreasing the timebound to O(n 2 ). We prove that the time bound analyses of these algorithms are erroneous and that their true time bound is (2 n ). Furthermore, a modi cation to the latest algorithm is given, restoring its time bound to O(n 2 ).
Lecture Notes in Computer Science, 2006
We consider the following problem from swarm robotics: given one or more "awake" robots in some m... more We consider the following problem from swarm robotics: given one or more "awake" robots in some metric space M , wake up a set of "asleep" robots. 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. We investigate offline and online versions of this problem and give a 2-competitive strategy and a lower bound of 2 in the case when M is discrete and the objective is to minimize the total movement cost. We also study the case when M is continuous and show a lower bound of 7/3 when the objective is to minimize the time when the last robot awakens.
A clique clustering of a graph is a partitioning of its vertices into disjoint cliques. The quali... more A clique clustering of a graph is a partitioning of its vertices into disjoint cliques. The quality of a clique clustering is measured by the total number of edges in its cliques. We consider the online variant of the clique clustering problem, where the vertices of the input graph arrive one at a time. At each step, the newly arrived vertex forms a singleton clique, and the algorithm can merge any existing cliques in its partitioning into larger cliques, but splitting cliques is not allowed. We give an online algorithm with competitive ratio 15.645 and we prove a lower bound of 6 on the competitive ratio, improving the previous respective bounds of 31 and 2.
Lecture Notes in Computer Science, 1992
In this paper we survey some results in restricted orientation computational geometry. The aim is... more In this paper we survey some results in restricted orientation computational geometry. The aim is to embed our own results into a more general context. We discuss methods for making object queries, computing shortest paths, and questions on restricted orientation convexity. Furthermore, we give an optimal algorithm for computing shortest paths when three arbitrary orientations are allowed for path finks and obstacle edges.
In this paper we investigate parallel searching on m concurrent rays. We assume that atarget t is... more In this paper we investigate parallel searching on m concurrent rays. We assume that atarget t is located somewhere on one of the rays and that a group of m point robots has toreach t. Furthermore, we assume that the robots have no way of communicating over distance.Given a strategy S we are interested in the competitive ratio which is defined as the ratio ofthe time needed by the robots using S and the time needed if the location of t is known inadvance.If a lower bound on the distance...
Theoretical Computer Science, 2006
Journal of Phytopathology, 1969