Finding the Extrema of a Distributed Multiset

We consider the problem of finding the extrema of a distributed multiset in a ring, that is, of determining the minimum and the maximum values,x_minandx_max, of a multisetX= {x₀,x₂, ...,x_n−1} whose elements are drawn from a totally ordered universeUand stored at thenentities of a ring network. This problem is unsolvable if the ring size is not known to the entities, and it has complexity Θ(n²) in the case of asynchronous rings of known size. We show that, in synchronous rings of known size, this problem can always be solved inO((c+ logn) ·n) bits andO(n·c·x^1/c) time for any integerc> 0, wherex= Max{|x_min|, |x_max|}. The previous solutions requiredO(n²) bits and the same amount of time. Based on these results, we also present a bit-optimal solution to the problem of finding the multiplicity of the extrema.     

Computing a Context-Free Grammar-Generating Series

The parallel complexity of computing context-free grammar generating series is investigated. It is known that this problem is in DIV, but in terms of n^σ rather than n, where n is the index of the desired coefficient and σ is the grammar size. A new method is presented which is in DIV in terms of 2^2O(σ)·n. Evidence is provided that any direct application of elimination theory to this problem leads to a space and time resource factor that is nearly exponential in grammar size.     

Equivalence Problems for Circuits over Sets of Natural Numbers

We investigate the complexity of equivalence problems for {∪,∩,⁻,+,×}-circuits computing sets of natural numbers. These problems were first introduced by Stockmeyer and Meyer (1973). We continue this line of research and give a systematic characterization of the complexity of equivalence problems over sets of natural numbers. Our work shows that equivalence problems capture a wide range of complexity classes like NL, C ₌ L, P,Π₂^P, PSPACE, NEXP, and beyond. McKenzie and Wagner (2003) studied related membership problems for circuits over sets of natural numbers. Our results also have consequences for these membership problems: We provide an improved upper bound for the case of {∪,∩,⁻,+,×}-circuits.     

Two-way automata and length-preserving homomorphisms

Closure underlength-preserving homomorphisms is interesting because of its similarity tonondeterminism. We give a characterization of NP in terms of length-preserving homomorphisms and present related complexity results. However, we mostly study the case of two-way finite automata: Let l.p.hom[n state 2DFA] denote the class of languages that are length-preserving homomorphic images of languages recognized byn-state 2DFAs. We give a machine characterization of this class. We show that any language accepted by ann-state two-wayalternating finite automaton (2AFA), or by a l-pebble 2NFA, belongs to l.p.hom[O(n ²) state 2DFA]. Moreover, there are languages in l.p.hom[n state 2DFA] whose smallest accepting 2NFA has at leastc ⁿ states (for some constantc > 1). So for two-way finite automata, the closure under length-preserving homomorphisms is much more powerful than nondeterminism. We disprove two conjectures (of Meyer and Fischer, and of Chrobak) about the state-complexity of unary languages. Finally, we show that the equivalence problems for 2AFAs (resp. 1-pebble 2NFAs) are in PSPACE, and that the equivalence problem for 1-pebble 2AFAs is in ExpSPACE (thus answering a question of Jiang and Ravikumar); it was known that these problems are hard in these two classes. We also give a new proof that alternating 1-pebble machines recognize only regular languages (which was first proved by Goralčíket al.). This research was supported in part by N.S.F. Grant DMS 8702019.     

An exponential lower bound on the size of algebraic decision trees for Max

We prove an exponential lower bound on the size of any fixed degree algebraic decision tree for solving MAX, the problem of finding the maximum of n real numbers. This complements the n— 1 lower bound of [Rabin (1972)] on the depth of algebraic decision trees for this problem. The proof in fact gives an exponential lower bound on the size of the polyhedral decision problem MAX= for testing whether the j-th number is the maximum among a list of n real numbers. Previously, except for linear decision trees, no nontrivial lower bounds on the size of algebraic decision trees for any familiar problems are known. We also establish an interesting connection between our lower bound and the maximum number of minimal cutsets for any rank-d hypergraph on n vertices. Received: 15 April 1996     

An algorithm for Exact Satisfiability analysed with the number of clauses as parameter

We give an algorithm for Exact Satisfiability with polynomial space usage and a time bound of poly(L)⋅m!, where m is the number of clauses and L is the length of the formula. Skjernaa has given an algorithm for Exact Satisfiability with time bound poly(L)⋅m² but using exponential space. We leave the following problem open: Is there an algorithm for Exact Satisfiability using only polynomial space with a time bound of cm, where c is a constant and m is the number of clauses?     

Complexity of Weak Bisimilarity and Regularity for BPA and BPP

It is an open problem whether weak bisimilarity is decidable for Basic Process Algebra (BPA) and Basic Parallel Processes (BPP). A PSPACE lower bound for BPA and NP lower bound for BPP have been demonstrated by Stribrna. Mayr achieved recently a result, saying that weak bisimilarity for BPP is Π^p₂-hard. We improve this lower bound to PSPACE, moreover for the restricted class of normed BPP.Weak regularity (finiteness) of BPA and BPP is not known to be decidable either. In the case of BPP there is a Π^p₂-hardness result by Mayr, which we improve to PSPACE. No lower bound has previously been established for BPA. We demonstrate DP-hardness, which in particular implies both NP and co-NP-hardness.In each of the bisimulation/regularity problems we consider also the classes of normed processes.Note: full version of the paper appears as [18].     

On Space Efficient Two Dimensional Range Minimum Data Structures

The two dimensional range minimum query problem is to preprocess a static m by n matrix (two dimensional array) A of size N=m⋅n, such that subsequent queries, asking for the position of the minimum element in a rectangular range within A, can be answered efficiently. We study the trade-off between the space and query time of the problem. We show that every algorithm enabled to access A during the query and using a data structure of size O(N/c) bits requires Ω(c) query time, for any c where 1≤c≤N. This lower bound holds for arrays of any dimension. In particular, for the one dimensional version of the problem, the lower bound is tight up to a constant factor. In two dimensions, we complement the lower bound with an indexing data structure of size O(N/c) bits which can be preprocessed in O(N) time to support O(clog ² c) query time. For c=O(1), this is the first O(1) query time algorithm using a data structure of optimal size O(N) bits. For the case where queries can not probe A, we give a data structure of size O(N⋅min {m,log n}) bits with O(1) query time, assuming m≤n. This leaves a gap to the space lower bound of Ω(Nlog m) bits for this version of the problem.     

Approximate algorithms for the knapsack problem on parallel computers

Computing an optimal solution to the knapsack problem is known to be NP-hard. Consequently, fast parallel algorithms for finding such a solution without using an exponential number of processors appear unlikely. An attractive alternative is to compute an approximate solution to this problem rapidly using a polynomial number of processors. In this paper, we present an efficient parallel algorithm for finding approximate solutions to the 0–1 knapsack problem. Our algorithm takes an , 0 < < 1, as a parameter and computes a solution such that the ratio of its deviation from the optimal solution is at most a fraction of the optimal solution. For a problem instance having n items, this computation uses O(n^5/2/^3/2) processors and requires O(log³n + log²nlog(1/)) time. The upper bound on the processor requirement of our algorithm is established by reducing it to a problem on weighted bipartite graphs. This processor complexity is a significant improvement over that of other known parallel algorithms for this problem.     

How efficiently can room at the bottom be traded away for speed at the top?

Given exponential 2 ⁿ space, we know that an Adleman-Lipton computation can decide many hard problems – such as boolean formula and boolean circuit evaluation – in a number of steps that is linear in the problem size n. We wish to better understand the process of designing and comparing bio-molecular algorithms that trade away weakly exponential space to achieve as low a running time as possible, and to analyze the efficiency of their space and time utilization relative to those of their best extant classical/bio-molecular counterparts. We propose a randomized framework which augments that of the sticker model of Roweis et al. to provide an abstract setting for analyzing the space-time efficiency of both deterministic and randomized bio-molecular algorithms. We explore its power by developing and analyzing such algorithms for theCovering Code Creation (CCC) and k-SAT problems. In the process, we uncover new classical algorithms for CCC andk-SAT that, while exploiting the same space-time trade-off as the best previously known classical algorithms, are exponentially more efficient than them in terms of space-time product utilization. This work indicates that the proposed abstract bio-molecular setting for randomized algorithm design provides a logical tool of independent interest. This revised version was published online in June 2006 with corrections to the Cover Date.     

Exact Algorithms for Exact Satisfiability and Number of Perfect Matchings

We present exact algorithms with exponential running times for variants of n-element set cover problems, based on divide-and-conquer and on inclusion–exclusion characterizations. We show that the Exact Satisfiability problem of size l with m clauses can be solved in time 2 ^m l ^O(1) and polynomial space. The same bounds hold for counting the number of solutions. As a special case, we can count the number of perfect matchings in an n-vertex graph in time 2 ⁿ n ^O(1) and polynomial space. We also show how to count the number of perfect matchings in time O(1.732 ⁿ ) and exponential space. We give a number of examples where the running time can be further improved if the hypergraph corresponding to the set cover instance has low pathwidth. This yields exponential-time algorithms for counting k-dimensional matchings, Exact Uniform Set Cover, Clique Partition, and Minimum Dominating Set in graphs of degree at most three. We extend the analysis to a number of related problems such as TSP and Chromatic Number.     

The covering and boundedness problems for vector addition systems

New decision procedures for the covering and boundedness problems for vector addition systems are obtained. These procedures require at most space 2^cnlogn for some constant c. The procedures nearly achieve recently established lower bounds on the amount of space inherently required to solve these problems, and so are much more efficient than previously known non-primitive-recursive decision procedures.     

Derandomizing Arthur-Merlin Games and Approximate Counting Implies Exponential-Size Lower Bounds

We show that if Arthur-Merlin protocols can be derandomized, then there is a language computable in deterministic exponential-time with access to an NP oracle that requires circuits of exponential size. More formally, if every promise problem in prAM, the class of promise problems that have Arthur-Merlin protocols, can be computed by a deterministic polynomial-time algorithm with access to an NP oracle, then there is a language in E^NP that requires circuits of size Ω(2 ⁿ /n). The lower bound in the conclusion of our theorem suffices to construct pseudorandom generators with exponential stretch.     

On Exact Solutions for the Rectilinear Steiner Tree Problem Part I: Theoretical Results

The rectilinear Steiner tree problem asks for a shortest tree connecting given points in the plane with rectilinear distance. The best theoretically analyzed algorithms for this problem are based on dynamic programming and have a running time of O(n ² . . . 2.62 ⁿ ) (Ganley and Cohoon), resp. (Smith). The first algorithm can solve problems of size 27, the second one is highly impractical because of the large constant in the exponent. The best implementations perform poorly even on small problem instances; the best practical results can be reached using a Branch \& Bound approach (Salowe and Warme); this implementation can solve random problems of size 35 within a day, while the dynamic programming approach of Ganley and Cohoon can handle only 27 point examples. In this paper we improve the theoretical worst-case time bound to O(n ² . . . 2.38 ⁿ ) , for random problem instances we prove a running time of α ⁿ with a constant α < 2 . We have implemented our algorithms and can now solve problems of 40 points in a day using a provably good dynamic programming approach, and can solve problems of 55 points with a Branch \& Bound strategy. For exponential-time algorithms, this is an enormous improvement. Received April 2, 1997; revised January 5, 1998.     

A Neural Algorithm for the Maximum Clique Problem: Analysis,Experiments, and Circuit Implementation

{In this paper we design and analyze a neural approximation algorithm for the Maximum Clique problem. This algorithm, having as input an arbitrary undirected graph G = \langle V, E\rangle , constructs a finite sequence of Hopfield networks such that the attractor of the last network in the sequence represents a maximal clique of G . We prove that D(G) ⋅ |E ^{\rm c} | , where D(G) = max _{{i,j}\notin E} \min{d _i , d _j } , d _i is the degree of the vertex i of G , and |E ^{\rm c} | denotes the cardinality of the set of edges in the complement graph, is an upper bound to the number of the networks in the sequence. Some experiments made on the second DIMACS benchmark and on random graphs show that: 1. The quality of the solutions found by the algorithm is satisfactory. 2. The theoretical upper bound D(G) ⋅ |E ^{\rm c} | is quite pessimistic. For random graphs we propose an empirical formula that gives a better estimate of the number of networks in the sequence. Moreover, thanks to the simplicity of the algorithm, we are able to design a uniform family of circuits of small size (\approx 10n ² log ₂ n ) that implements it. The circuit, which solves the problems for graphs of at most 32 vertices, has then been programmed on FPGAs (Field Programmable Gate Arrays). An analysis in terms of size and time complexity is given. Received November 10, 1998; revised December 2000.     

Feedback equivalence for general control systems

We discuss the solution to the problem of local equivalence of control systems with n states and p controls in a neighbourhood of a generic point, under the Lie pseudo-group of local time independent feedback transformations. We have shown earlier that this problem is identical with the problem of simple equivalence of the time optimal variational problem. Here we indicate the way in which this identification may be used to obtain closed loop time critical controls for general systems. We show that the classification of general nonlinear systems depends on the classification of all n−p dimensional affine subspaces of the space of symmetric forms in p variables and that the case of control linear systems depends on the classification of all n−p dimensional affine subspaces of the space of skew forms in p variables. We show that in the latter case the G-structure is the prolongation of one determined by a state space transformation group. We give a complete list of normal forms for control linear systems in the case p=n−1.     

The number of limit cycles for a family of polynomial systems

In this paper, the number of limit cycles in a family of polynomial systems was studied by the bifurcation methods. With the help of a computer algebra system (e.g., Maple 7.0), we obtain that the least upper bound for the number of limit cycles appearing in a global bifurcation of systems (2.1) and (2.2) is 5n + 5 + (1 − (−1)ⁿ)/2 for c ≠ 0 and n for c ≡ 0.     

An Improved Stability Bound for Binary Exponential Backoff

Goodman, Greenberg, Madras and March gave a lower bound of n ^{-Ω(log n} ) for the maximum arrival rate for which the n -user binary exponential backoff protocol is stable. Thus, they showed that the protocol is stable as long as the arrival rate is at most n ^{-Ω(log n} ) . We improve the lower bound, showing that the protocol is stable for arrival rates up to O(n ^-(0.75+δ) ) , for any δ>0 . Received October 15, 1999, and in final form November 13, 2000. Online publication February 26, 2001.