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.

Some diffusive and recurrence properties of Lorentz Lattice Gas Cellular Automata (LLGCA) have been expensively studied in terms of the densities of some of the left/right static/flipping mirrors/rotators. In this paper, for any combination S of these well known scatters, we study the computational complexity of the following problem which we call PERIODICITY on the S-model: given a finite configuration that distributes only those scatters in S, whether a particle visits the starting position periodically or not. Previously, the flipping mirror model and the occupied flipping rotator model have been shown unbounded, i.e. the process is always diffusive [17]. On the other hand, PERIODICITY is shown PSPACE-complete in the unoccupied flipping rotator model [21]. In this paper, we show that PERIODICITY is PSPACE-compete in any S-model that is neither occupied, unbounded, nor static. Particularly, we prove that PERIODICITY in any unoccupied and bounded model containing flipping mirror is PSPACE-complete.

This paper deals with the maximum-weight 2-path packing problem (M2PP), which is the problem of computing a set of vertex-disjoint paths of length 2 in a given edge-weighted complete graph so that the total weight of edges in the paths is maximized. Previously, Hassin and Rubinstein gave a randomized cubic-time approximation algorithm for M2PP which achieves an expected ratio of - ε ≈ 0.5223 - ε for any constant ε > 0. We refine their algorithm and derandomize it to obtain a deterministic cubic-time approximation algorithm for the problem which achieves a better ratio (namely, 0.5265 - ε for any constant ε>0).

The maximal linear forest problem is to find, given a graph G = (V, E), a maximal subset of V that induces a linear forest. Three parallel algorithms for this problem are presented. The first one is randomized and runs in O(log n) expected time using n2 processors on a CRCW PRAM. The second one is deterministic and runs in O(log 2n) timeusing n4 processors on an EREW PRAM. The last one is deterministic and runs in O(log 5n) time using n3 processors on an EREW PRAM. The results put the problem in the class NC.

The notion of alternating context-free grammar (ACFG for short) was introduced by Moriya in 1989. In this paper, we study the relationships between some complexity classes and the classes of languages generated by restricted types of ACFG's. Two restricted types of ACFG's considered are linear ACFG's and ε-free ACFG's. For an ACFG G, let Lleft (G) denote the language of terminal strings generated by leftmost derivations in G. Let {Lleft (G)G is an ε-free ACFG} and ACFLlinear{Lleft (G)G is a linear ACFG's}. The main results of the present paper are as follows:(1) the class of languages that are log-space many-one reducible to languages in ACFLlinear is equivalent to P, and(2) the class of languages that are log-space many-one reducible to languages in is equivalent to PSPACE.

It is a well-known and useful problem to find a matching in a given graph G whose size is at most a given parameter k and whose weight is maximized (over all matchings of size at most k in G). In this paper, we consider two natural extensions of this problem. One is to find t disjoint matchings in a given graph G whose total size is at most a given parameter k and whose total weight is maximized, where t is a (small) constant integer. Previously, only the special case where t=2 was known to be fixed-parameter tractable. In this paper, we show that the problem is fixed-parameter tractable for any constant t. When t=2, the time complexity of the new algorithm is significantly better than that of the previously known algorithm. The other is to find a set of vertex-disjoint paths each of length 1 or 2 in a given graph whose total length is at most a given parameter k and whose total weight is maximized. As interesting applications, we further use the algorithms to speed up several known approximation algorithms (for related NP-hard problems) whose approximation ratio depends on a fixed parameter 0<ε<1 and whose running time is dominated by the time needed for exactly solving the problems on graphs in which each connected component has at most ε-1 vertices.