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UDK 681.3.06
Alexander Marchenko and Mikhail Udovikhin
A TOPOLOGICAL ALGORITHM OF MULTIWAY HYPERGRAPH
PARTITIONING
A chip may contain tens of millions of transistors. Layout of the entire circuit cannot be handled in a flat mode due to the limitation of memory space as well as a computation power available. This problem can be solved by using partitioning as a first step in physical design phase.
Partitioning is a complex problem which is NP-complete. There are several well known heuristics such as Kernigan-Lin [1] and Fiduccia-Mattheyes [2] which produce good results as well as the best known approach hMetis [3] - a multilevel partitioning algorithm. All these algorithms need to know the number of parts on which graph to be partitioned. Usually this problem is solved manually.
In this paper we propose an algorithm of multiway hypergraph partitioning that does not need predefined number of subgraphs. The approach comprises the following steps which can be repeated recursively:
♦ mapping hypergraph to metric space by introducing special distance between its nodes;
♦ building a set of subgraphs having {2, 3,..., n-1} nodes where each subgraph has minimum average distance between its nodes;
♦ computation of Maximum of the Average Distance Increment (MADI);
♦ introducing a separability criterion based on MADI;
♦ separation an appropriate subgraph in accordance with separability criterion.
Известия ТРТУ
Тематический выпуск
The output of the algorithm includes a number of subgraphs and the following quality metrics used in hMetis:
♦ Hyperedge Cut - is the number of the hyperedges that span multiple partitions;
♦ Sum of External Degrees - is defined as a number of hyperedges, that are incident but not fully inside this partition;
♦ Scaled Cost and Absorption that were calculated in accordance with [3].
The testing proposed approach shows its effectiveness in comparison with hMetis program package.
LITERATURE
1. Kernigan and B.W,. S.Lin ,”An efficient heuristic procedure for partitioning graphs”, Bell Syst. Tech. J., vol.49, pp.291-307, Feb.1970.
2. Fiduccia and C.M., R.M.Mattheyes, “A linear time heuristic for improving network partitions”, in Proc. IEEE-ACM Design Automation Conf., 1982. pp.175-181.
3. Karypis G.and Kumar V., “Multilevel k-way hypergraph partitioning”, in Proc. IEEE- ACM Design Automation Conf., 1999. pp. 343-348.
УДК 519.14
H.A. Надолинский ТРЕХМЕРНАЯ ГРАФИКА В ОБУЧАЮЩИХ СИСТЕМАХ
,
, , -ве является трехмерная графика. Все чаще и чаще трехмерная графика используется ив образовательных системах по химии, биологии, астрономии. Наглядное представление в совокупности с высоким качеством изображения предоставляет возможность осознания и усвоения материала.
Представленный доклад опирается на созданную обучающую программу по , , всеми ее компонентами.
Для максимального восприятия, трехмерная сцена должна содержать в себе , , .
Моделирование такого рода сцен сопряжено с некоторыми проблемами, основными из которых являются:
♦ Моделирование поверх ности астероидов и планет.
♦ Алгоритмы удаления невидимых поверхностей, использующиеся по мере надобности в процессе выполнения программы.
Моделирование поверхностей астероидов и планет может осуществляться не, -пользованием фрактальной геометрии. Как известно, фрактальное изображение создается применением некоторой операции к геометрическому объекту и последующим многократным применением той же операции к полученному результату. Чтобы сгенерировать фрактальную планету, достаточно разбить сферу на равно, , поднимая или опуская середину каждого ребра на случайную высоту. Соединив линиями середины трех сторон, мы получим четыре треугольника. Если же теперь
