International Journal of Foundations of Computer Science

The longest path problem is the one that finds a longest path in a given graph. While the graph classes in which the Hamiltonian path problem can be solved efficiently are widely investigated, few graph classes are known to be solved efficiently for the longest path problem. Among those, for trees, a simple linear time algorithm for the longest path problem is known. We first generalize the algorithm, and show that the longest path problem can be solved efficiently for some tree-like graph classes by this approach. We next propose two new graph classes that have natural interval representations, and show that the longest path problem can be solved efficiently on these classes.

There are several known undecidable problems for 3 × 3 integer matrices the proof of which use a reduction from the Post Correspondence Problem (PCP). We establish new lower bounds in the number of matrices for the mortality, zero in the left upper corner, vector reachability, matrix reachability, scalar reachability and freeness problems. Also, we give a short proof for a strengthened result due to Bell and Potapov stating that the membership problem is undecidable for finitely generated matrix semigroups R ⊆ ℤ^4×4 whether or not kI₄ ∈ R for any given |k| > 1. These bounds are obtained by using the Claus instances of the PCP.

We investigate deterministically simulating (i.e., solving the membership problem for) nondeterministic finite automata (NFA), relying solely on the NFA's resources (states and transitions). Unlike the standard NFA simulation, involving an algorithm which stores at each step all the states reached nondeterministically while reading the input, we consider deterministic finite automata (DFA) with lookahead, which choose the “right” NFA transitions based on a fixed number of input symbols read ahead. This concept, known as lookahead delegation, arose in a formal study of web services composition and its subsequent practical applications. Here we answer several related questions, such as “when is lookahead delegation possible?” and “how hard is it to find a delegator with a given lookahead buffer size?”. In particular, we show that only finite languages have the property that all their NFA have delegators. This implies, among others, that delegation is a machine property, rather than a language property. We also prove that the existence of lookahead delegators for unambiguous NFA is decidable, thus partially solving an open problem. Finally, we show that finding delegators (even for a given buffer size) is hard in general, and is more efficient for unambiguous NFA, and we give an algorithm and a compact characterization for NFA delegation in general.

We show that every function of several variables on a finite set of k elements with n > k essential variables has a variable identification minor with at least n − k essential variables. This is a generalization of a theorem of Salomaa on the essential variables of Boolean functions. We also strengthen Salomaa's theorem by characterizing all the Boolean functions f having a variable identification minor that has just one essential variable less than f.

Threshold decryption allows a message encrypted under a public key to be read only when a quorum of users cooperate to decrypt the ciphertext. However, such threshold decryption scheme does not apply well in the situation where all the users have their own public/private key pairs, but not share any private key associated with a public key, such as mobile ad hoc network featured by its dynamic character. An immediate way to achieve threshold decryption in this situation is to split the message into pieces, then encrypt these pieces under the public keys of different users. However, it is not efficient. In this paper, we propose an efficient identity based multi-receiver threshold decryption scheme that could be applied efficiently in the above situation. We also define the security notions and prove the security in random oracle model. At last, we add the broadcast feature to the scheme, such that a message could be broadcast to any number of groups.

The star graph S_n, proposed by [1], has many advantages over the n-cube. It is shown in [2] that when a large number of vertices are deleted from S_n, the resulting graph can have at most two components, one of which is small. In this paper, we show that Cayley graphs generated by transpositions have this property.

The APPLICATION PLACEMENT PROBLEM (APP, for short) arises in hosting platforms: clusters of servers that are used for hosting large, distributed applications such as Internet services. Hosting platforms imply a business relationship between an entity called the platform provider and a number of entities called the application providers. The latter pay the former for the resources on the hosting platform, in return for which, the former provides guarantees on resource availability for the applications. This implies that a hosting platform should host only applications for which it has sufficient resources. The objective of the APP is to maximize the number of applications that can be hosted on the platform while satisfying their resource requirements. The complexity of the APP is studied here, with the following results. The general APP is NP-hard; indeed, even restricted versions of the APP may not admit polynomial-time approximation schemes. However, several significant variants of the online version of the APP admit efficient approximation algorithms.

In this research, a framework is proposed for answering the question: whether a dominant sequence exists for the multiple self-contained input streams destined for a computation provider running the requested applications concurrently. The framework is extended to find an optimal semidominant sequence for multiple input streams. This paper also shows that an existent universal dominant sequence can be formed by permuting a set of dominant input streams. In this paper, the neighbor-connected graph is first proposed to model the sequence problem of multiple input streams. Then, an algorithm with time complexity O(nLⁿ) is proposed to determine the existence of a dominant data-receiving sequence, where L is the maximal length of the input streams plus one and n is the number of input streams. Examples of applying the framework to solve the sequence problems for several computation-intensive applications are also given. Furthermore, the framework is extended to find an optimal semidominant sequence using an algorithm with time complexity O(nLⁿlogδ*), where δ* is the maximal degree that the sequence diverges from all the optimal subsequences. Finally, the property of a universal dominant sequence is studied and the performance of the sequence is compared with those of other sequences.

A theory of representation of semiautomata by canonical words and equivalences was developed in [7]. That work was motivated by trace-assertion specifications of software modules, but its focus was entirely on the underlying mathematical model. In the present paper we extend that theory to automata with Moore and Mealy outputs, and show how to apply the extended theory to the specification of modules. In particular, we present a unified view of the trace-assertion methodology, as guided by our theory. We illustrate this approach, and some specific issues, using several nontrivial examples. We include a discussion of finite versus infinite modules, methods of error handling, some awkward features of the trace-assertion method, and a comparison to specifications by automata. While specifications by trace assertions and automata are equivalent in power, there are cases where one approach appears to be more natural than the other. We conclude that, for certain types of system modules, formal specification by automata, as opposed to informal state machines, is not only possible, but practical.

In this paper we study a generalization of the classical notions of bordered and unbordered words, motivated by DNA computing. DNA strands can be viewed as finite strings over the alphabet {A, G, C, T}, and are used in DNA computing to encode information. Due to the fact that A is Watson-Crick complementary to T and G to C, DNA single strands that are Watson-Crick complementary can bind to each other or to themselves in either intended or unintended ways. One of the structures that is usually undesirable for biocomputation, since it makes the affected DNA string unavailable for future interactions, is the hairpin: If some subsequences of a DNA single string are complementary to each other, the string will bind to itself forming a hairpin-like structure. This paper studies a mathematical formalization of a particular case of hairpins, the Watson-Crick bordered words. A Watson-Crick bordered word is a word with the property that it has a prefix that is Watson-Crick complementary to its suffix. More generally, we investigate the notion of θ-bordered words, where θ is a morphic or antimorphic involution. We show that the set of all θ-bordered words is regular, when θ is an antimorphic involution and the set of all θ-bordered words is context-sensitive when θ is a morphic involution. We study the properties of θ-bordered and θ-unbordered words and also the relation between θ-bordered and θ-unbordered words and certain type of involution codes.

Traditional message passing based checkpointing and rollback recovery algorithms perform well for tightly coupled systems. In wide area distributed systems these algorithms may suffer from large overhead due to message passing delay and network traffic. Mobile agents offer an attractive option for designing checkpointing schemes for wide area distributed systems. Network topology is assumed to be arbitrary. Processes are mobile agent enabled. When a process wants to take a checkpoint, it just creates one mobile agent. Concurrent initiations by multiple processes are allowed. Synchronization and creation of a consistent global state (CGS) for checkpointing is managed by the mobile agent(s). In the worst case, for k concurrent initiations among n processes, checkpointing algorithm requires a total of O(kn) hops by all the mobile agents. A mobile agent carries O(n/k) (on the average) size data.

The process of gene assembly in ciliates, an ancient group of organisms, is one of the most complex instances of DNA manipulation known in any organism. Three molecular operations Id, hi, and dlad have been postulated for the gene assembly process. We propose in this paper a mathematical model for contextual variants of Id and dlad on strings: recombinations can be done only if certain contexts are present. We prove that the proposed model is Turing-universal.