Logspace Algorithm Research Articles

On commutator length in free groups

Let F be a free group. We present for arbitrary g\in\mathbb{N} a LOGSPACE (and thus polynomial time) algorithm that determines whether a given w\in F is a product of at most g commutators; and more generally, an algorithm that determines, given w\in F , the minimal g such that w may be written as a product of g commutators (and returns \infty if no such g exists). This algorithm also returns words x_1,y_1,\dots,x_g,y_g such that w=[x_1,y_1]\dots[x_g,y_g] . These algorithms are also efficient in practice. Using them, we produce the first example of a word in the free group whose commutator length decreases under taking a square. This disproves in a very strong sense a conjecture by Bardakov.

Forbidden Patterns for FO2 Alternation Over Finite and Infinite Words

We consider two-variable first-order logic FO2 and its quantifier alternation hierarchies over both finite and infinite words. Our main results are forbidden patterns for deterministic automata (finite words) and for Carton-Michel automata (infinite words). In order to give concise patterns, we allow the use of subwords on paths in finite graphs. This concept is formalized as subword-patterns. For certain types of subword-patterns there exists a non-deterministic logspace algorithm to decide their presence or absence in a given automaton. In particular, this leads to NL algorithms for deciding the levels of the FO2 quantifier alternation hierarchies. This applies to both full and half levels, each over finite and infinite words. Moreover, we show that these problems are NL-hard and, hence, NL-complete.

Min/Max-Poly Weighting Schemes and the NL versus UL Problem

For a graph G ( V , E ) (| V | = n ) and a vertex s ∈ V , a weighting scheme ( W : E ↦ Z + ) is called a min-unique (resp. max-unique ) weighting scheme if, for any vertex v of the graph G , there is a unique path of minimum (resp. maximum) weight from s to v , where weight of a path is the sum of the weights assigned to the edges. Instead, if the number of paths of minimum (resp. maximum) weight is bounded by n c for some constant c , then the weighting scheme is called a min-poly (resp. max-poly ) weighting scheme. In this article, we propose an unambiguous nondeterministic log-space (UL) algorithm for the problem of testing reachability graphs augmented with a min-poly weighting scheme. This improves the result in Reinhardt and Allender [2000], in which a UL algorithm was given for the case when the weighting scheme is min-unique . Our main technique involves triple inductive counting and generalizes the techniques of Immerman [1988], Szelepcsényi [1988], and Reinhardt and Allender [2000], combined with a hashing technique due to Fredman et al. [1984] (also used in Garvin et al. [2014]). We combine this with a complementary unambiguous verification method to give the desired UL algorithm. At the other end of the spectrum, we propose a UL algorithm for testing reachability in layered DAGs augmented with max-poly weighting schemes. To achieve this, we first reduce reachability in layered DAGs to the longest path problem for DAGs with a unique source, such that the reduction also preserves the max-unique and max-poly properties of the graph. Using our techniques, we generalize the double inductive counting method in Limaye et al. [2009], in which the UL algorithm was given for the longest path problem on DAGs with a unique sink and augmented with a max-unique weighting scheme. An important consequence of our results is that, to show NL = UL, it suffices to design log-space computable min-poly (or max-poly ) weighting schemes for layered DAGs.

Solving the canonical representation and Star System Problems for proper circular-arc graphs in logspace

We present a logspace algorithm that constructs a canonical intersection model for a given proper circular-arc graph, where canonical means that isomorphic graphs receive identical models. This implies that the recognition and the isomorphism problems for these graphs are solvable in logspace. For the broader class of concave-round graphs, which still possess (not necessarily proper) circular-arc models, we show that a canonical circular-arc model can also be constructed in logspace. As a building block for these results, we design a logspace algorithm for computing canonical circular-arc models of circular-arc hypergraphs. This class of hypergraphs corresponds to matrices with the circular ones property, which play an important role in computational genomics. Our results imply that there is a logspace algorithm that decides whether a given matrix has this property.Furthermore, we consider the Star System Problem that consists in reconstructing a graph from its closed neighborhood hypergraph. We show that this problem is solvable in logarithmic space for the classes of proper circular-arc, concave-round, and co-convex graphs.Note that solving a problem in logspace implies that it is solvable by a parallel algorithm of the class AC1. For the problems under consideration, at most AC2 algorithms were known earlier.

Interval graph representation with given interval and intersection lengths

We consider the problem of finding interval graph representations that additionally respect given interval lengths and/or pairwise intersection lengths, which are represented as weight functions on the vertices and edges, respectively. Pe'er and Shamir (1997 [25]) proved that the problem is NP-complete if only the former are given. For the case when both are given, Fulkerson and Gross (1965 [8]) gave an O(n2) time algorithm; we improve this to O(n+m) time and supplement it with a logspace algorithm. For the case when only the latter are given, we give both an O(nm) time algorithm and a logspace algorithm. In all these bounds, n is the number of vertices and m is the number of edges in the input graph.Complementing their hardness result, Pe'er and Shamir give a polynomial-time algorithm for the case that the input graph has a unique interval ordering of its maximal cliques. For such graphs, their algorithm computes an interval representation (if it exists) that respects a given set of distance inequalities between the interval endpoints. We observe that deciding if such a representation exists is NL-complete.

S-T connectivity on digraphs with a known stationary distribution

We present a deterministic logspace algorithm for solving S-T Connectivity on directed graphs if: (i) we are given a stationary distribution of the random walk on the graph in which both of the input vertices s and t have nonnegligible probability mass and (ii) the random walk which starts at the source vertex s has polynomial mixing time. This result generalizes the recent deterministic logspace algorithm for S-T Connectivity on undirected graphs [Reingold, 2008]. It identifies knowledge of the stationary distribution as the gap between the S-T Connectivity problems we know how to solve in logspace ( L ) and those that capture all of randomized logspace ( RL ).

Interval Graphs: Canonical Representations in Logspace

We present a logspace algorithm for computing a canonical labeling, in fact, a canonical interval representation, for interval graphs. To achieve this, we compute canonical interval representations of interval hypergraphs. This approach also yields a canonical labeling of convex graphs. As a consequence, the isomorphism and automorphism problems for these graph classes are solvable in logspace. For proper interval graphs we also design logspace algorithms computing their canonical representations by proper and by unit interval systems.

Derandomized Constructions of k-Wise (Almost) Independent Permutations

Constructions of k-wise almost independent permutations have been receiving a growing amount of attention in recent years. However, unlike the case of k-wise independent functions, the size of previously constructed families of such permutations is far from optimal. This paper gives a new method for reducing the size of families given by previous constructions. Our method relies on pseudorandom generators for space-bounded computations. In fact, all we need is a generator, that produces “pseudorandom walks” on undirected graphs with a consistent labelling. One such generator is implied by Reingold’s log-space algorithm for undirected connectivity (Reingold/Reingold et al. in Proc. of the 37th/38th Annual Symposium on Theory of Computing, pp. 376–385/457–466, 2005/2006). We obtain families of k-wise almost independent permutations, with an optimal description length, up to a constant factor. More precisely, if the distance from uniform for any k tuple should be at most δ, then the size of the description of a permutation in the family is $O(kn+\log \frac{1}{\delta})$.

Undirected connectivity in log-space

We present a deterministic , log-space algorithm that solves st-connectivity in undirected graphs. The previous bound on the space complexity of undirected st-connectivity was log 4/3 (⋅) obtained by Armoni, Ta-Shma, Wigderson and Zhou (JACM 2000). As undirected st-connectivity is complete for the class of problems solvable by symmetric, nondeterministic, log-space computations (the class SL), this algorithm implies that SL = L (where L is the class of problems solvable by deterministic log-space computations). Independent of our work (and using different techniques), Trifonov (STOC 2005) has presented an O (log n log log n )-space, deterministic algorithm for undirected st-connectivity. Our algorithm also implies a way to construct in log-space a fixed sequence of directions that guides a deterministic walk through all of the vertices of any connected graph. Specifically, we give log-space constructible universal-traversal sequences for graphs with restricted labeling and log-space constructible universal-exploration sequences for general graphs.

The diameter of randomly perturbed digraphs and some applications

The central observation of this paper is that if εn random arcs are added to any n-node strongly connected digraph with bounded degree then the resulting graph has diameter 𝒪(lnn) with high probability. We apply this to smoothed analysis of algorithms and property testing. Smoothed Analysis: Recognizing strongly connected digraphs is a basic computational task in graph theory. Even for digraphs with bounded degree, it is NL-complete. By XORing an arbitrary bounded degree digraph with a sparse random digraph R ∼ 𝔻n,ε/n we obtain a “smoothed” instance. We show that, with high probability, a log-space algorithm will correctly determine if a smoothed instance is strongly connected. We also show that if NL ⫅̸ almost-L then no heuristic can recognize similarly perturbed instances of (s,t)-connectivity. Property Testing: A digraph is called k-linked if, for every choice of 2k distinct vertices s1,…,sk,t1,…,tk, the graph contains k vertex disjoint paths joining sr to tr for r = 1,…,k. Recognizing k-linked digraphs is NP-complete for k ≥ 2. We describe a polynomial time algorithm for bounded degree digraphs, which accepts k-linked graphs with high probability, and rejects all graphs that are at least εn arcs away from being k-linked. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2007

Progress in Computational Complexity Theory

We briefly survey a number of important recent achievements in Theoretical Computer Science (TCS), especially Computational Complexity Theory. We will discuss the PCP Theorem, its implications to inapproximability on combinatorial optimization problems; space bounded computations, especially deterministic logspace algorithm for undirected graph connectivity problem; deterministic polynomial-time primality test; lattice complexity, worst-case to average-case reductions; pseudorandomness and extractor constructions; and Valiant's new theory of holographic algorithms and reductions.

L-Printable Sets

A language is if there is a logspace algorithm which, on input 1n, prints all members in the language of length n. Following the work of Allender and Rubinstein [SIAM J. Comput., 17 (1988), pp. 1193--1202] on P-printable sets, we present some simple properties of the sets. This definition of L-printable is robust and allows us to give alternate characterizations of the sets in terms of tally sets and Kolmogorov complexity. In addition, we show that a regular or context-free language is if and only if it is sparse, and we investigate the relationship between sets, L-rankable sets (i.e., sets A having a logspace algorithm that, on input x, outputs the number of elements of A that precede x in the standard lexicographic ordering of strings), and the sparse sets in L. We prove that under reasonable complexity-theoretic assumptions, these three classes of sets are all different. We also show that the class of sets of small generalized Kolmogorov space complexity is exactly the class of sets that are L-isomorphic to tally languages.

A fast randomized LOGSPACE algorithm for graph connectivity

We study the relationship between undirected graph reachability and graph connectivity, in the context of randomized LOGSPACE algorithms. Aleluinas et al. [2] show that graph reachability (checking whether there is a path connecting vertices J and ℸ) can be decided in logarithmic space and polynomial time, by starting a random walk at J, and checking whether ℸ is hit within some time limit. The randomized algorithm has one-sided error (with small probability, it fails to determine that J and ℸ are connected). The reachability algorithm may be used in order to decide (with one-sided error) whether a graph is connected, by running it n − 1 times, each time with a different target vertex ℸ. This increases the running time by a factor of n. In this paper we give an alternative randomized LOGSPACE algorithm for graph connectivity. Its running time varies between O( n 2) steps and O( n 3) steps, depending on the structure of the input graph. This matches the fastest known RLOGSPACE algorithm for reachability, up to a constant factor. Our algorithm has two-sided error.

RL $$ \subseteq $$ SC

We show that any randomized logspace algorithm (running in polynomial time with bounded two-sided error) can be simulated deterministically in polynomial time andO(log2 n) space. This puts RL in SC, “Steve's Class” In particular, we get a polynomial time,O(log2 n) space algorithm for thest-connectivity problem on undirected graphs.Subject classifications. 68Q10, 68Q15, 68Q25.

Boolean Functions, Invariance Groups, and Parallel Complexity

This paper studies the invariance groups ${\bf S}(f)$ of boolean functions $f \in {\bf B}_n $ (i.e., $f:\{ 0,1\} ^n \to \{ 0,1\} $) on n variables, i.e., the set of all permutations on n elements which leave f invariant. After building intuition by presenting several examples that suggest relations between algebraic properties of groups and computational complexity of languages, necessary and sufficient conditions are given via Polya’s cycle index for an arbitrary finite permutation group to be of the form $S(f)$, for some $f \in {\bf B}_n $. It is shown that asymptotically “almost all” boolean functions have trivial invariance groups. For cyclic groups $G \leqq {\bf S}_n $ a logspace algorithm for determining whether the given group is of the form ${\bf S}(f)$, for some $f \in {\bf B}_n $ is given. The applicability of group theoretic techniques in the study of the parallel complexity of languages is demonstrated. For any language L let $L_n $ be the characteristic function of the set of all strings in L...

Recognition of deterministic ETOL languages in logarithmic space

It is shown that if G is a deterministic ETOL system, there is a nondeterministic log space algorithm to determine membership in L(G). Consequently, every deterministic ETOL language is recognizable in polynomial time. As a corollary, all context-free languages of finite index, and all Indian parallel languages are recognizable within the same bounds.

Recognition of Deterministic ET0L Languages in Polynomial Time

It is shown that if G is a deterministic ETOL system, there is a nondeterministic log space algorithm to determine membership in L(G). Consequently every deterministic ETOL language is recognizable in polynomial time. As a corollary, all context-free languages of finite index, and all indian parallel languages are recognizable within the same bounds.