Linear Time Reduction Research Articles

A historical note on the complexity of scheduling problems

In 1972 E.M. Livshits and V.I. Rublinetsky published a paper in Russian, in which they presented linear-time reductions of the partition problem to a number of scheduling problems. Unaware of complexity theory, they argued that, since partition is not known to have a simple algorithm, one cannot expect to find simple algorithms for these scheduling problems either. Their work did not go completely unnoticed, but it received little recognition. We describe the approach and review the results.

Teaching of the harmonic oscillator damped by a constant force: The use of analogy and experiments

A description of oscillations damped by a constant force, despite the importance of the phenomenon, is missing from secondary (i.e., high-school) and university physics textbooks. In this paper, a simple explanation of this type of energy loss is given, using the analogy of a vertical spring-mass system. The predicted result, a linear time reduction in the amplitude of oscillation, can be confirmed in the classroom with the demonstration outlined here. In addition, the possibility of extending the experiment to a laboratory exercise at the university level is described. During the exercise, students distinguish different energy loss mechanisms in harmonic oscillations and measure the values of physical quantities associated with energy loss using a simple analytical procedure.

Hindsight Optimization for Hybrid State and Action MDPs

Hybrid (mixed discrete and continuous) state and action Markov Decision Processes (HSA-MDPs) provide an expressive formalism for modeling stochastic and concurrent sequential decision-making problems. Existing solvers for HSA-MDPs are either limited to very restricted transition distributions, require knowledge of domain-specific basis functions to achieve good approximations, or do not scale. We explore a domain-independent approach based on the framework of hindsight optimization (HOP) for HSA-MDPs, which uses an upper bound on the finite-horizon action values for action selection. Our main contribution is a linear time reduction to a Mixed Integer Linear Program (MILP) that encodes the HOP objective, when the dynamics are specified as location-scale probability distributions parametrized by Piecewise Linear (PWL) functions of states and actions. In addition, we show how to use the same machinery to select actions based on a lower-bound generated by straight line plans. Our empirical results show that the HSA-HOP approach effectively scales to high-dimensional problems and outperforms baselines that are capable of scaling to such large hybrid MDPs.

Set Intersection and Sequence Matching with mismatch counting

In the classical pattern-matching problem, one is given a text and a pattern both of which are sequences of letters. The requirement is to find all occurrences of the pattern in the text. We studied two modifications of the classical problem, where each letter in the text and pattern is a set (Set Intersection Matching problem) or a sequence (Sequence Matching problem). Two “letters” are found to match if the intersection of the corresponding sets is not empty or if the two sequences have a common element in the same index. We first show that the two problems are similar by establishing a linear time reduction between them. We then show the first known non-trivial and efficient algorithms for these problems, when the maximum set/sequence size d is small. The first is a Monte Carlo randomized algorithm for Set Intersection Matching, that takes Θ(4dnlog⁡nlog⁡m) time, where n and m are the lengths of the text and the pattern, respectively; the failure probability is less than 1n2. This algorithm can also be used, with slight modifications, when up to k mismatches is allowed. In addition, it can be used to maintain an approximation of factor 1±ϵ of the mismatch count in Θ(1ϵ24dnlog⁡nlog⁡m) time; the failure probability is bounded by 1n. The second is a deterministic algorithm for Set Intersection Matching that can be used to count the number of matches at each index of the text in a total running time Θ(∑i=1d(σi)nlog⁡m)=O(σdnlog⁡m), where σ is the size of the alphabet. The third algorithm, also deterministic, solves the Sequence Matching problem in Θ(4dnlog⁡m) time.

ScaffoldScaffolder: solving contig orientation via bidirected to directed graph reduction.

The contig orientation problem, which we formally define as the MAX-DIR problem, has at times been addressed cursorily and at times using various heuristics. In setting forth a linear-time reduction from the MAX-CUT problem to the MAX-DIR problem, we prove the latter is NP-complete. We compare the relative performance of a novel greedy approach with several other heuristic solutions. Our results suggest that our greedy heuristic algorithm not only works well but also outperforms the other algorithms due to the nature of scaffold graphs. Our results also demonstrate a novel method for identifying inverted repeats and inversion variants, both of which contradict the basic single-orientation assumption. Such inversions have previously been noted as being difficult to detect and are directly involved in the genetic mechanisms of several diseases. http://bioresearch.byu.edu/scaffoldscaffolder. paulmbodily@gmail.com Supplementary data are available at Bioinformatics online.

Tree [formula omitted]-spanners in outerplanar graphs via supply demand partition

A treet-spanner of an unweighted graph G is a spanning tree T such that for every two vertices their distance in T is at most t times their distance in G. Given an unweighted graph G and a positive integer t as input, the Treet-spanner problem is to compute a tree t-spanner of G if one exists. This decision problem is known to be NP-complete even in the restricted class of unweighted planar graphs. We present a linear-time reduction from Treet-spanner in outerplanar graphs to the supply–demand tree partition problem. Based on this reduction, we obtain a linear-time algorithm to solve Treet-spanner in outerplanar graphs. Consequently, we show that the minimum value of t for which an input outerplanar graph on n vertices has a tree t-spanner can be found in O(nlogn) time.

Refining discordant gene trees.

BackgroundEvolutionary studies are complicated by discordance between gene trees and the species tree in which they evolved. Dealing with discordant trees often relies on comparison costs between gene and species trees, including the well-established Robinson-Foulds, gene duplication, and deep coalescence costs. While these costs have provided credible results for binary rooted gene trees, corresponding cost definitions for non-binary unrooted gene trees, which are frequently occurring in practice, are challenged by biological realism.ResultWe propose a natural extension of the well-established costs for comparing unrooted and non-binary gene trees with rooted binary species trees using a binary refinement model. For the duplication cost we describe an efficient algorithm that is based on a linear time reduction and also computes an optimal rooted binary refinement of the given gene tree. Finally, we show that similar reductions lead to solutions for computing the deep coalescence and the Robinson-Foulds costs.ConclusionOur binary refinement of Robinson-Foulds, gene duplication, and deep coalescence costs for unrooted and non-binary gene trees together with the linear time reductions provided here for computing these costs significantly extends the range of trees that can be incorporated into approaches dealing with discordance.

Replacement Paths via Row Minima of Concise Matrices

Matrix $M$ is {\em $k$-concise} if the finite entries of each column of $M$ consist of $k$ or less intervals of identical numbers. We give an $O(n+m)$-time algorithm to compute the row minima of any $O(1)$-concise $n\times m$ matrix. Our algorithm yields the first $O(n+m)$-time reductions from the replacement-paths problem on an $n$-node $m$-edge undirected graph (respectively, directed acyclic graph) to the single-source shortest-paths problem on an $O(n)$-node $O(m)$-edge undirected graph (respectively, directed acyclic graph). That is, we prove that the replacement-paths problem is no harder than the single-source shortest-paths problem on undirected graphs and directed acyclic graphs. Moreover, our linear-time reductions lead to the first $O(n+m)$-time algorithms for the replacement-paths problem on the following classes of $n$-node $m$-edge graphs (1) undirected graphs in the word-RAM model of computation, (2) undirected planar graphs, (3) undirected minor-closed graphs, and (4) directed acyclic graphs.

On the Complexity of Strongly Connected Components in Directed Hypergraphs

We study the complexity of some algorithmic problems on directed hypergraphs and their strongly connected components (SCCs). The main contribution is an almost linear time algorithm computing the terminal strongly connected components (i.e. SCCs which do not reach any components but themselves). "Almost linear" here means that the complexity of the algorithm is linear in the size of the hypergraph up to a factor alpha(n), where alpha is the inverse of Ackermann function, and n is the number of vertices. Our motivation to study this problem arises from a recent application of directed hypergraphs to computational tropical geometry. We also discuss the problem of computing all SCCs. We establish a superlinear lower bound on the size of the transitive reduction of the reachability relation in directed hypergraphs, showing that it is combinatorially more complex than in directed graphs. Besides, we prove a linear time reduction from the well-studied problem of finding all minimal sets among a given family to the problem of computing the SCCs. Only subquadratic time algorithms are known for the former problem. These results strongly suggest that the problem of computing the SCCs is harder in directed hypergraphs than in directed graphs.

A Phase Transition in Energy-Filtered RNA Secondary Structures

In this article we study the effect of energy parameters on minimum free energy (mfe) RNA secondary structures. Employing a simplified combinatorial energy model that is only dependent on the diagram representation and is not sequence-specific, we prove the following dichotomy result. Mfe structures derived via the Turner energy parameters contain only finitely many complex irreducible substructures, and just minor parameter changes produce a class of mfe structures that contain a large number of small irreducibles. We localize the exact point at which the distribution of irreducibles experiences this phase transition from a discrete limit to a central limit distribution and, subsequently, put our result into the context of quantifying the effect of sparsification of the folding of these respective mfe structures. We show that the sparsification of realistic mfe structures leads to a constant time and space reduction, and that the sparsification of the folding of structures with modified parameters leads to a linear time and space reduction. We, furthermore, identify the limit distribution at the phase transition as a Rayleigh distribution.

ON COMPUTING GEODESICS IN BAUMSLAG–SOLITAR GROUPS

We introduce the peak normal form for elements of the Baumslag–Solitar groups BS(p,q). This normal form is very close to the length-lexicographical normal form, but more symmetric. Both normal forms are geodesic. This means the normal form of an element u-1v yields the shortest path between u and v in the Cayley graph. For horocyclic elements the peak normal form and the length-lexicographical normal form coincide. The main result of this paper is that we can compute the peak normal form in polynomial time if p divides q. As consequence we can compute geodesic lengths in this case. In particular, this gives a partial answer to Question 1 in [4] For arbitrary p and q it is possible to compute the peak normal form (length-lexicographical normal form resp.) also the for elements in the horocyclic subgroup and, more generally, for elements which we call hills. This approach leads to a linear time reduction of the problem of computing geodesics to the problem of computing geodesics for Britton-reduced words where the t-sequence starts with t-1 and ends with t.

Description and analysis of a bottom-up DFA minimization algorithm

We establish linear-time reductions between the minimization of a deterministic finite automaton (DFA) and the conjunction of 3 subproblems: the minimization of a strongly connected DFA, the isomorphism problem for a set of strongly connected minimized DFAs, and the minimization of a connected DFA consisting in two strongly connected components, both of which are minimized. We apply this procedure to minimize, in linear time, automata whose nontrivial strongly connected components are cycles.

A Tight Lower Bound for Computing the Diameter of a 3D Convex Polytope

The diameter of a set P of n points in ℝ d is the maximum Euclidean distance between any two points in P. If P is the vertex set of a 3-dimensional convex polytope, and if the combinatorial structure of this polytope is given, we prove that, in the worst case, deciding whether the diameter of P is smaller than 1 requires Ω(nlog n) time in the algebraic computation tree model. It shows that the O(nlog n) time algorithm of Ramos for computing the diameter of a point set in ℝ3 is optimal for computing the diameter of a 3-polytope. We also give a linear time reduction from Hopcroft’s problem of finding an incidence between points and lines in ℝ2 to the diameter problem for a point set in ℝ7.

Linear time reduction of large kinetic mechanisms with directed relation graph: n-Heptane and iso-octane

The algorithm of directed relation graph recently developed for skeletal mechanism reduction was extended to overall linear time operation, thereby greatly facilitating the computational effort in mechanism reduction, particularly for those involving large mechanisms. Together with a two-stage reduction strategy and using the kinetic responses of autoignition and perfectly stirred reactor (PSR) with extensive parametric variations as the criteria in eliminating unimportant species, a detailed 561-species n-heptane mechanism and a detailed 857-species iso-octane mechanism were successfully reduced to skeletal mechanisms consisting of 188 and 233 species, respectively. These skeletal mechanisms were demonstrated to mimic well the performance of the detailed mechanisms, not only for the autoignition and PSR systems based on which the reduced mechanisms were developed but also for the independent system of jet-stirred reactor. It was further observed that the accuracy of calculated species concentrations was equivalently bounded by the user-specified error threshold value and that the reduction time for a single reaction state is only about 50 ms for the large iso-octane mechanism.

For completeness, sublogarithmic space is no space

It is shown that for any class C closed under linear-time reductions, the complete sets for C under sublogarithmic reductions are also complete under 2DFA reductions, and thus are isomorphic under first-order reductions.

A Randomized Algorithm for Triangulating a Simple Polygon in Linear Time

We describe a randomized algorithm for computing the trapezoidal decomposition of a simple polygon. Its expected running time is linear in the size of the polygon. By a well-known and simple linear time reduction, this implies a linear time algorithm for triangulating a simple polygon. Our algorithm is considerably simpler than Chazelle’s [3] celebrated optimal deterministic algorithm. The new algorithm can be viewed as a combination of Chazelle’s algorithm and of simple nonoptimal randomized algorithms due to Clarkson et al. [6], [7], [9] and to Seidel [20]. As in Chazelle’s algorithm, it is indispensable to include a bottom-up preprocessing phase, in addition to the actual top-down construction. An essential new idea is the use of random sampling on subchains of the initial polygonal chain, rather than on individual edges as is normally done.

On the sorting-complexity of suffix tree construction

The suffix tree of a string is the fundamental data structure of combinatorial pattern matching. We present a recursive technique for building suffix trees that yields optimal algorithms in different computational models. Sorting is an inherent bottleneck in building suffix trees and our algorithms match the sorting lower bound. Specifically, we present the following results. (1) Weiner [1973], who introduced the data structure, gave an optimal 0 ( n )-time algorithm for building the suffix tree of an n -character string drawn from a constant-size alphabet. In the comparison model, there is a trivial Ο( n log n )-time lower bound based on sorting, and Weiner's algorithm matches this bound. For integer alphabets, the fastest known algorithm is the O ( n log n )time comparison-based algorithm, but no super-linear lower bound is known. Closing this gap is the main open question in stringology. We settle this open problem by giving a linear time reduction to sorting for building suffix trees. Since sorting is a lower-bound for building suffix trees, this algorithm is time-optimal in every alphabet mode. In particular, for an alphabet consisting of integers in a polynomial range we get the first known linear-time algorithm. (2) All previously known algorithms for building suffix trees exhibit a marked absence of locality of reference, and thus they tend to elicit many page faults (I/Os) when indexing very long strings. They are therefore unsuitable for building suffix trees in secondary storage devices, where I/Os dominate the overall computational cost. We give a linear-I/O reduction to sorting for suffix tree construction. Since sorting is a trivial I/O-lower bound for building suffix trees, our algorithm is I/O-optimal.

Computational complexity via programming languages: constant factors do matter

The purpose of this work is to promote a programming-language approach to studying computability and complexity, with an emphasis on time complexity. The essence of the approach is: a programming language, with semantics and complexity measure, can serve as a computational model that has several advantages over the currently popular models and in particular the Turing machine. An obvious advantage is a stronger relevance to the practice of programming. In this paper we demonstrate other advantages: certain proofs and constructions that are hard to do precisely and clearly with Turing machines become clearer and easier in our approach, and sometimes lead to finer results. In particular, we prove several time hierarchy theorems, for deterministic and non-deterministic time complexity which show that, in contrast with Turing machines, constant factors do matter in this framework. This feature, too, brings the theory closer to practical considerations. The above result suggests that this framework may be appropriate for studying low complexity classes, such as linear time. As an example we give a problem complete for non-deterministic\/ linear time under deterministic linear-time reductions. Finally, we consider some extensions and modifications of our programming language and their effect on time complexity results.

Time-Complexity of the Word Problem for Semigroups and the Higman Embedding Theorem

The following algebraic characterization of the computational complexity of the word problem for finitely generated semigroups is proved, in the form of a refinement of the Higman Embedding Theorem: Let S be a finitely generated semigroup whose word problem has nondeterministic time complexity T (where T is a function on the positive integers which is superadditive, i.e. T(n+m) ≥T(n)+T(m)). Then S can be embedded in a finitely presented semigroup H in which the derivation distance between any two equivalent words x and y (and hence the isoperimetric function) is O (T(∣x∣+∣y∣)2). Moreover, there is a conjunctive linear-time reduction from the word problem of H to the word problem of S, so the word problems of S and H have the same nondeterministic time complexity (and also the same deterministic time complexity). Thus, a finitely generated semigroup S has a word problem in NTIME(T) (or in DTIME(To)) iff S is embeddable into a finitely presented semigroup H whose word problem is in NTIME(T) (respectively in DTIME(To)). In the other direction, if a finitely generated semigroup S is embeddable in a finitely presented semigroup H with isoperimetric function ≤ D (where D(n) ≥ n), then the word problem of S has nondeterministic time complexity O(D). The word problem of a finitely generated semigroup S is in NP (or more generally, in NTIME((T) O(1) )) iff S can be embedded in a finitely presented semigroup H with polynomial (respectively (T) O(1) ) isoperimetric function. An algorithmic problem L is in NP (or more generally, in NTIME((T) O(1) )) iff L is reducible (via a linear-time one-to-one reduction) to the word problem of a finitely presented semigroup with polynomial (respectively (T) O(1) ) isoperimetric function. In essence, this shows: (1) Finding embeddings into finitely presented semigroups or groups is an algebraic analogue of nondeterministic algorithm design; (2) the isoperimetric function is an algebraic analogue of nondeterministic time complexity.

Recognizing s-star polygons

We consider the problem of recognizing star-polygons under staircase visibility ( s-visibility). We show that the s-visibilitypolygon from a point inside a simple orthogonal polygon of n sides can be computed in O( n) time. When the polygon contains holes the algorithm runs in O( n log n) time, which we prove to be optimal by linear time reduction from sorting. We present an algorithm for computing the s-kernel of a polygon in O( n) time for simple orthogonal polygons and in O( n 2) time for orthogonal polygons with holes; both complexities are optimal in the worst case. Finally, we report the main result of this paper: we show that even though the s-kernel of a polygon with holes may have Ω(n 2) components it is possible to recognize such polygons in O( n log n) time.