Suppose that there are terminals on two concentric circles Cin and Cout, with Cin inside of Cout. A set of two-terminal nets is given and the routing area is the annular region between the two circles. In this paper, we present an O(n2) time algorithm for testing whether the given net set is two-layer routable, where n is the number of nets. Applying this algorithm repeatedly, we can find, in O(n3) time, a maximal subset of nets which is two-layer routable.

It is known that the problem of finding a largest common subgraph is NP-hard for general graphs even if the number of input graphs is two. It is also known that the problem can be solved in polynomial time if the input is restricted to two trees. In this paper, a randomized parallel (an RNC) algorithm for finding a largest common subtree of two trees is presented. The dynamic tree contraction technique and the RNC minimum weight perfect matching algorithm are used to obtain the RNC algorithm. Moreover, an efficient NC algorithm is presented in the case where input trees are of bounded vertex degree. It works in O(log(n1)log(n2)) time using O(n1n2) processors on a CREW PRAM, where n1 and n2 denote the numbers of vertices of input trees. It is also proved that the problem is NP-hard if the number of input trees is more than two. The three dimensional matching problem, a well known NP-complete problem, is reduced to the problem of finding a largest common subtree of three trees.

Circuit complexity of a Boolean function is defined to be the minimum number of gates in circuits computing the function. In general, the circuit complexity is established by deriving two types of bounds on the complexity. On one hand, an upper bound is derived by showing a circuit, of the size given by the bound, to compute a function. On the other hand, a lower bound is established by proving that a function can not be computed by any circuit of the size. There has been much success in obtaining good upper bounds, while in spite of much efforts few progress has been made toward establishing strong lower bounds. In this paper, after surveying general results concerning circuit complexity for Boolean functions, we explain recent results about lower bounds, focusing on the method of approximation.

A chordal ring network is a processor network on which n processors are arranged to a ring with additional chords. We study a distributed leader election algorithm on chordal ring networks and present trade-offs between the message complexity and the number of chords at each processor and between the message complexity and the length of chords as follows:For every d(1dlog* n1) there exists a chordal ring network with d chords at each processor on which the message complexity for leader election is O(n(log(d1)nlog* n)).For every d(1dlog* n1) there exists a chordal ring network with log(d1)nd1 chords at each processor on which the message complexity for leader election is O(dn).For every m(2mn/2) there exists a chordal ring network whose chords have at most length m such that the message complexity for leader election is O((n/m)log n).

A maximal l-diameter tree cover of a graph G(V,E) is a spanning subgraph C(V,EC) of G such that each connected component of C is a tree, C contains no path with more than l edges, and adding any edge in EEC to C yields either a path of length l1 or a cycle. For every function f from positive integers to positive integers, the maximal f-diameter tree cover prolem (MDTC(f) problem for short) is to find a maximal f(n)-diameter tree cover of G, given an n-node graph G. In this paper, we give two parallel algorithms for the MDTC(f) problem. The first algorithm can be implemented in time O(TMSP(n,f(n))log2n) using polynomial number of processors on an EREW PRAM, where TMSP(n,f(n) is the time needed to find a maximal set of vertex disjoint paths of length f(n) in a given n-node graph using polynomial number of processors on an EREW PRAM. We then show that if suitable restrictions are imposed on the input graph and/or on the magnitude of f, then TMSP(n,f(n))O(logkn) for some constant k and thus, for such cases, we obtain an NC algorithm for the MDTC(f) problem. The second algorithm runs in time O(n log2n/{f(n)1}) using polynomial number of processors on an EREW PRAM. Thus if f(n)Ω(n/logkn) for some kO, we obtain an NC algorithm for the MDTC(f) problem.