Log Mod Research Articles

The Yorke-Pianigiani measure and the asymptotic law on the limit Cantor set of expanding systems

Consider an expanding map T:A contained in Rn to Rn with TA contains/implies A strictly. Pianigiani and Yorke have shown that under some suitable conditions there exists a conditionally invariant probability measure mu f, with density f satisfying Pf= alpha f for the Perron-Frobenius operator P. We prove that the conditional probability measure of staying in A when the evolution occurs with probability mu f, i.e. mu f(B intersection T-nA mod T-nA) to n to infinity nu (B) for any Borel set B contained in/implied by A, is a T-invariant probability measure nu on the limit Cantor set K= intersection n>or=0T-nA which is Gibbsian with potential log mod detDTx mod .

Bottleneck Steiner trees in the plane

A Steiner tree with maximum-weight edge minimized is called a bottleneck Steiner tree (BST). The authors propose a Theta ( mod rho mod log mod rho mod ) time algorithm for constructing a BST on a point set rho , with points labeled as Steiner or demand; a lower bound, in the linear decision tree model, is also established. It is shown that if it is desired to minimize further the number of used Steiner points, then the problem becomes NP-complete. It is shown that when locations of Steiner points are not fixed the problem remains NP-complete; however, if the topology of the final tree is given, then the problem can be solved in Theta ( mod rho mod log mod rho mod ) time. The BST problem can be used, for example, in VLSI layout, communication network design, and (facility) location problems.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

On Sullivan's conformal measures for rational maps of the Riemann sphere

For any rational map T:C to C the topological pressure P(t) of the function -t log mod T' mod is well defined, although its Julia set may contain critical points. The authors extend the class of Sullivan's (1982, 1983) conformal measures on Julia sets to the class of measures which are conformal except on some finite set and show that the minimal exponent for which such measures exist, the supremum of Hausdorff dimensions of ergodic T-invariant measures with positive entropy and the minimal zero of the function P(t) are equal. This result may be regarded as a version of the Bowen-McCluskey-Manning formula. A large subclass of rational maps is found for which the same statement is true for conformal measures in the sense of Sullivan.

Partial factoring: an efficient algorithm for approximating two-terminal reliability on complete graphs

An efficient network reliability approximation algorithm for two-terminal undirected complete graphs is presented. The method recursively applies a factoring theorem of network reliability in conjunction with series-parallel reliability-preserving reductions. After each pivot, one of the resulting subproblems is further factored, while the other is approximated. The final approximate solution is given in terms of an upper and a lower bound. In the worse case, the algorithm requires O( mod V mod /sup 2/ log mod V mod ) time and O( mod V/sup 2/ mod ) space. For 128-node complete graphs, the method requires less than 16 s on an IBM AT personal computer. The algorithm's mean error bound is both theoretically and empirically analyzed. The mean difference between the upper and lower bounds is guaranteed to be less than 10/sup -14/ for component reliabilities uniformly distributed between 0.99 and 1.00. Empirically, this difference is always less than 10/sup -17/. >

Fault-tolerant distributed algorithm for election in complete networks

Electron in asynchronous complete networks is considered for the case in which the links may fail intermittently and undetectably. Undetectable link failure is consistent with the standard model of distributed algorithms in which link delays can be arbitrary but finite. An algorithm is given that correctly solves the problem when the links fail before or during the execution of the algorithm. If n is the number of nodes in the network, f the maximum number of fault links, and r a design parameter, the algorithm uses no more than O(nrf+(nr/(r-1) log(n/(r-1)f))) messages, runs in time O(n/(r-1)) f, and uses at most O(log mod T mod ) bits per message, where mod T mod is the cardinality of the set of node identifiers.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>