Abstract
It is known that the size of monotone arithmetic (+, ·) circuits can be exponentially decreased by allowing just one division “at the very end,” at the output gate. A natural question is: can the size of (+, ·) circuits be substantially reduced if we allow divisions “at the very beginning,” that is, if besides nonnegative real constants and variables x<sub>1</sub>, …, x<sub>n</sub>, the circuits can also use their reciprocals 1/x<sub>1</sub>, ..., 1/x<sub>n</sub> as inputs. We answer this question in the negative: the gain in circuit size is then always at most quadratic. Over tropical (min, +) and (max, +) semirings, division turns into subtraction; so, reciprocal inputs are then -x<sub>1</sub>, …, -x<sub>n</sub>. We give the same negative answer also for tropical circuits. The question of whether reciprocal inputs can substantially speed up tropical (min, +, max) circuits, remains open.
Publisher
Keldysh Institute of Applied Mathematics
Subject
Energy (miscellaneous),Energy Engineering and Power Technology,Renewable Energy, Sustainability and the Environment,Environmental Engineering,Pollution,Nuclear Energy and Engineering,Renewable Energy, Sustainability and the Environment,Environmental Chemistry,General Engineering,Energy Engineering and Power Technology,Fuel Technology,Nuclear Energy and Engineering,Renewable Energy, Sustainability and the Environment,Energy Engineering and Power Technology,Fuel Technology,Nuclear Energy and Engineering,Renewable Energy, Sustainability and the Environment,General Energy,Economics and Econometrics,General Energy,Energy Engineering and Power Technology,Building and Construction,Renewable Energy, Sustainability and the Environment,Energy Engineering and Power Technology,Fuel Technology,Nuclear Energy and Engineering,Renewable Energy, Sustainability and the Environment,Management, Monitoring, Policy and Law,Renewable Energy, Sustainability and the Environment,Geography, Planning and Development
Reference22 articles.
1. Кнут Д. Искусство программирования. Т. 2. Получисленные алгоритмы. — М.: Вильямс, 2004.
2. Левин А. Ю. Алгоритм кратчайшего соединения группы вершин графа // Докл. АН СССР. — 1971. — Т. 200(4). — С. 773–776.
3. Разборов А. А. Нижние оценки монотонной сложности логического перманента // Матем. заметки. — 1985. — Т. 37(6). — С. 887–900.
4. Alon N., Tarsi M. Colorings and orientations of graphs // Combinatorica. — 1992. — V. 12. — P. 125–134.
5. Bellman R. On a routing problem // Quarterly of Appl. Math. — 1958. — V. 16. — P. 87–90.