Showing papers by "Christian Szegedy published in 2008"

Method, system, and computer program product for implementing incremental placement in electronics design

Abstract: Disclosed are a method, system, and computer program product for implementing incremental placement for an electronic design while predicting and minimizing the perturbation impact arising from incremental placement of electronic components. In some embodiments, an initial placement of an electronic design is identified, the abstract flow is computed, the target locations of various electronic components to be placed are identified, the relative ordering of electronic components are determined, and the placement is then legalized. Furthermore, in various embodiments, the method, system, or computer program product starts with an initial placement of an electronic design and derives a legal placement by using the incremental placement technique while minimizing the perturbation impact or the total quadratic movement of instances. In some embodiments, an augmented or incremental clumping technique based data structure is utilized for rapid and substantially exact perturbation prediction of effects of local incremental placement operations.

A class of problems for which cyclic relaxation converges linearly

Abstract: The method of cyclic relaxation for the minimization of a function depending on several variables cyclically updates the value of each of the variables to its optimum subject to the condition that the remaining variables are fixed. We present a simple and transparent proof for the fact that cyclic relaxation converges linearly to an optimum solution when applied to the minimization of functions of the form $f(x_{1},\ldots,x_{n})=\sum_{i=1}^{n}\sum_{j=1}^{n}a_{i,j}\frac{x_{i}}{x_{j}}+\sum_{i=1}^{n}(b_{i}x_{i}+\frac{c_{i}}{x_{i}})$ for a i,j ,b i ,c i ???0 with max?{min?{b 1,b 2,?,b n },min?{c 1,c 2,?,c n }}>0 over the n-dimensional interval [l 1,u 1]×[l 2,u 2]×???×[l n ,u n ] with 0

On the cost of optimal alphabetic code trees with unequal letter costs

Abstract: For some k>=2 let d=(d"1,d"2,...,d"k)@?R">"0^k. We denote the concatenation of k vectors a"1,a"2,...,a"[email protected][email protected]?"n">="0R^n by a"1a"2...a"k and use @e to denote the empty vector. We consider a recursively defined function D"d:@?"n">="0R^n->[email protected]?{-~} with D"d(@e)=-~, D"d((a))=a for [email protected]?R and D"d(a)=min{max{D"d(a"i)+d"i|[email protected][email protected]?k}|a=a"1a"2...a"k with a"[email protected][email protected]?m=0n-1R^m for [email protected][email protected]?k} for [email protected]?R^n with n>=2. The function D"d equals the cost of an optimal alphabetic code tree with unequal letter costs and the above recursion naturally generalizes a recursion studied by Kapoor and Reingold [S. Kapoor, E.M. Reingold, Optimum lopsided binary trees, J. Assoc. Comput. Mach. 36 (1989) 573-590]. If z(n) denotes the vector consisting of n>=0 zeros, then let f(@a)=max{[email protected]?N"0|D"d(z(i))@[email protected]} for @[email protected]?R. Let d=min{d"1,d"2,...,d"k} and D=max{d"1,d"2,...,d"k}. Our main result is that D"d(z(@?i=1nf(a"i)))@?D"d(a)@?D"d(z(@?i=1nf(a"i)))+6D-2d for a=(a"1,a"2,...,a"n)@?R">="0^n. This result is useful for the analysis of the asymptotic growth of D"d.