P-Complete Geometric Problems

Atallah, Mikhail J.; Callahan, Paul B.; Goodrich, Michael T.

doi:10.1142/s0218195993000282

Cited by 14 publications

(10 citation statements)

References 18 publications

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“…Interestingly enough computing the lexicographically first matching for full convex bipartite graphs is in NC, in contraposition with the results given in [1] which show that many problems defined through a lexicographically first procedure in the plane are P-complete. It is an interesting problem to show whether all these problems fall in NC when they are convex.…”

Section: Open Problemssupporting

confidence: 57%

Parallel Algorithms for Two Processors Precedence Constraint Scheduling

Serna

2008

Encyclopedia of Algorithms

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In the general form of multiprocessor precedence scheduling problems a set of n tasks to be executed on m processors is given. Each task requires exactly one unit of execution time and can run on any processor. A directed acyclic graph specifies the precedence constraints where an edge from task x to task y means task x must be completed before task y begins. A solution to the problem is a schedule of shortest length indicating when each task is started. The work of Jung, Serna, and Spirakis provides a parallel algorithm (on a PRAM machine) that solves the above problem for the particular case that m = 2, that is where there are two parallel processors.The two processor precedence constraint scheduling problem is defined by a directed acyclic graph (dag) G = (V, E). The vertices of the graph represent unit time tasks, and the edges specify precedence constraints among the tasks. If there is an edge from node x to node y then x is an immediate predecessor of y. Predecessor is the transitive closure of the relation immediate predecessor, and successor is its symmetric counterpart. A two processor schedule is an assignment of the tasks to time units 1, . . . , t so that each task is assigned exactly one time unit, at most two tasks are assigned to the same time unit, and if x is a predecessor of y then x is assigned to a lower time unit than y. The length of the schedule is t. A schedule having minimum length is an optimal schedule. Thus the problem is the following: Name Two processor precedence constraint scheduling Input A directed acyclic graph Output A minimum length schedule preserving the precedence constraints.Preliminaries The algorithm assume that tasks are partitioned into levels as follows: (i) Every task will be assigned to only one level (ii) Tasks having no successors will be assigned to level 1 and (iii) For each level i, all tasks which are immediate predecessors of tasks in level i will be assigned to level i + 1.Clearly topological sort will accomplish the above partition, and this can be done by an NC algorithm that uses O(n 3 ) processors and O(log n) time, see [3]. Thus, from now on, it is assumed that a level partition is given as part of the input. For sake of convenience two special tasks, t 0 and t * are added, in such a way that the original graph could be taught as the graph formed by all tasks that are successors of t 0 and predecessors of t * . Thus t 0 is a predecessor of all tasks in the system

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Section: Open Problemssupporting

confidence: 57%

Parallel Algorithms for Two Processors Precedence Constraint Scheduling

Serna

2008

Encyclopedia of Algorithms

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“…with the help of [ACG93,GaOv95,BrKi98]. Moreover also Φ in turn had been refined: PTOLEMY introduced additional so-called epicycles and deferents, located and rotating on the originally earth-centered spheres.…”

Section: General Eudoxus/aristotle; Ptolemy Copernicus and Keplermentioning

confidence: 99%

Physically-relativized Church–Turing Hypotheses: Physical foundations of computing and complexity theory of computational physics

Ziegler

2009

Applied Mathematics and Computation

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Abstract. We turn 'the' Church-Turing Hypothesis from an ambiguous source of sensational speculations into a (collection of) sound and well-defined scientific problem(s):Examining recent controversies, and causes for misunderstanding, concerning the state of the ChurchTuring Hypothesis (CTH), suggests to study the CTH relative to an arbitrary but specific physical theory-rather than vaguely referring to "nature" in general. To this end we combine (and compare) physical structuralism with (models of computation in) complexity theory. The benefit of this formal framework is illustrated by reporting on some previous, and giving one new, example result(s) of computability and complexity in computational physics.

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“…Consider Case 1, that is, the robots current position is now p hm+1, 3 . Then it is easy to verify that L continues by a forward traversal through c hm+1 until p hm+1,16 is reached.…”

Section: The Geometric Lexicographic Dead End Path In P (I)mentioning

confidence: 99%

“…Since it is still unvisited, the robot enters it. Now we again have the following situation: All vertices to the left of p hm+1,2 are unvisited, the vertex with smallest weight visible from p hm+1,2 is p hm+1, 3 . To the right of p hm+1,2 the only possibilities are main truth setting vertices and by Property 2 the only one which is visible is p j2,6 (p j2,11 , respectively).…”

Section: The Geometric Lexicographic Dead End Path In P (I)mentioning

confidence: 99%

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A P-Completeness Result for Visibility Graphs of Simple Polygons

Dietel

Hecker

2000

MLQ - Math. Log. Quart.

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For each vertex of a simple polygon P an integer valued weight is given. We consider the path p1, p2, . . . , pk in P which is created according to the following strategy: p1 is a designated start vertex s and pi+1 is obtained by choosing the vertex with smallest weight among all vertices visible from pi and different from p1, p2, . . . , pi. If there is no such vertex the path is finished. This path is called geometric lexicographic dead end path. We shall prove the problem of determining whether a distinguished vertex t of P is on the geometric lexicographic dead end path or not to be P-complete.Mathematics Subject Classification: 68U05, 03D15, 68Q15.

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P-Complete Geometric Problems

Cited by 14 publications

References 18 publications

Parallel Algorithms for Two Processors Precedence Constraint Scheduling

Parallel Algorithms for Two Processors Precedence Constraint Scheduling

Physically-relativized Church–Turing Hypotheses: Physical foundations of computing and complexity theory of computational physics

A P-Completeness Result for Visibility Graphs of Simple Polygons

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