Lower bounds for maximal and convex layers problems

In this paper we derive tight lower bounds for the maximal and convex layers problems in the plane. Our lower bound proofs for the maxima problem and convex hull problem are simpler than those previously known. We also obtain an (nlog n) lower bound for the maximal depth problem, and the convex depth problem, when the points are given in sorted order of their x-coordinates.     

Lower bounds for convex programmes

An important aspect associated with the solution of any mathematical programme; is a sensitivity analysis applied to the various parameters of the programme. For the case of linear programmes, the notion of sensitivity analysis is fully developed. However, for non-linear programmes there is a scarcity of useful results. It is the purpose of this paper to establish lower bounds on the solution of a convex programme when the parameters of the programme are allowed to vary. Such bounds may be useful, for example, in establishing the minimal effects of inflation on optimal cost estimations.     

Lower bounds for polynomial evaluation and interpolation problems

We show that there is a set of pointsp ₁,p ₂,...,p _n such that any arithmetic circuit of depthd for polynomial evaluation (or interpolation) at these points has size $$\Omega \left( {\frac{{n\log n}}{{\log (2 + d/\log n}}} \right).$$ Moreover, for circuits of sub-logarithmic depthd, we obtain a lower bound of Ω(dn ^1+1/d ) on its size.     

Lower bounds for decision problems in imaginary,norm-Euclidean quadratic integer rings

We prove lower bounds for the complexity of deciding several relations in imaginary, norm-Euclidean quadratic integer rings, where computations are assumed to be relative to a basis of piecewise-linear operations. In particular, we establish lower bounds for deciding coprimality in these rings, which yield lower bounds for gcd computations. In each imaginary, norm-Euclidean quadratic integer ring, a known binary-like gcd algorithm has complexity that is quadratic in our lower bound.     

Lower bounds for the majority communication complexity of various graph accessibility problems

We investigate the probabilistic communication complexity (more exactly, the majority communication complexity), of the graph accessibility problem (GAP) and its counting versions MOD _k -GAP,k ≥ 2. Due to arguments concerning matrix variation ranks and certain projection reductions, we prove that, for any partition of the input variables, GAP and MOD _m -GAP have majority communication complexity Ω,(n), wheren denotes the number of nodes of the graph under consideration.     

Lower bounds for context-free grammars

Ellul, Krawetz, Shallit and Wang prove an exponential lower bound on the size of any context-free grammar generating the language of all permutations over some alphabet. We generalize their method and obtain exponential lower bounds for many other languages, among them the set of all squares of given length, and the set of all words containing each symbol at most twice.     

Lower bounds for asynchronous consensus

Impossibility results and best-case lower bounds are proved for the number of message delays and the number of processes required to reach agreement in an asynchronous consensus algorithm that tolerates non-Byzantine failures. General algorithms exist that achieve these lower bounds in the normal case, when the response time of non-faulty processes and the transmission delay of messages they send to one another are bounded. Our theorems allow algorithms to do better in certain exceptional cases, and such algorithms are presented. Two of these exceptional algorithms may be of practical interest.     

Lower bounds for monotone span programs

Span programs provide a linear algebraic model of computation. Lower bounds for span programs imply lower bounds for formula size, symmetric branching programs, and contact schemes. Monotone span programs correspond also to linear secret-sharing schemes. We present a new technique for proving lower bounds for monotone span programs. We prove a lower bound of (m ^2.5) for the 6-clique function. Our results improve on the previously known bounds for explicit functions.     

Lower bounds for the polynomial calculus

We show that polynomial calculus proofs (sometimes also called Groebner proofs) of the pigeonhole principle must have degree at least over any field. This is the first non-trivial lower bound on the degree of polynomial calculus proofs obtained without using unproved complexity assumptions. We also show that for some modifications of , expressible by polynomials of at most logarithmic degree, our bound can be improved to linear in the number of variables. Finally, we show that for any Boolean function in n variables, every polynomial calculus proof of the statement “ cannot be computed by any circuit of size t,” must have degree . Loosely speaking, this means that low degree polynomial calculus proofs do not prove . Received: January 15, 1997.     

Lower bounds for Bayes error estimation

We give a short proof of the following result. Let (X,Y) be any distribution on N×{0,1}, and let (X₁,Y₁),...,(X_n,Y_n) be an i.i.d. sample drawn from this distribution. In discrimination, the Bayes error L*=inf_gP{g(X)≠Y} is of crucial importance. Here we show that without further conditions on the distribution of (X,Y), no rate-of-convergence results can be obtained. Let φ_n(X₁,Y₁,...,X_n,Y_n ) be an estimate of the Bayes error, and let {φ_n(.)} be a sequence of such estimates. For any sequence {a_n} of positive numbers converging to zero, a distribution of (X,Y) may be found such that E{|L*-φ_n(X₁,Y ₁,...,X_n,Y_n)|}⩾a_n often converges infinitely     

Lower bounds for set intersection queries

We consider the followingset intersection reporting problem. We have a collection of initially empty sets and would like to process an intermixed sequence ofn updates (insertions into and deletions from individual sets) andq queries (reporting the intersection of two sets). We cast this problem in thearithmetic model of computation of Fredman [F1] and Yao [Ya2] and show that any algorithm that fits in this model must take time (q+nq) to process a sequence ofn updates andq queries, ignoring factors that are polynomial in logn. We also show that this bound is tight in this model of computation, again to within a polynomial in logn factor, improving upon a result of Yellin [Ye]. Furthermore, we consider the caseq=O(n) with an additional space restriction. We only allow the use ofm memory locations, wherem n^3/2. We show a tight bound of (n²/m^1/3) for a sequence ofn operations, again ignoring the polynomial in logn factors.     

Decision problems for convex languages

We examine decision problems for various classes of convex languages, previously studied by Ang and Brzozowski, originally under the name “continuous languages”. We can decide whether a language L is prefix-, suffix-, factor-, or subword-convex in polynomial time if L is represented by a DFA, but these problems become PSPACE-complete if L is represented by an NFA. If a regular language is not convex, we find tight upper bounds on the length of the shortest words demonstrating this fact, in terms of the number of states of an accepting DFA. Similar results are proved for some subclasses of convex languages: the prefix-, suffix-, factor-, and subword-closed languages, and the prefix-, suffix-, factor-, and subword-free languages. Finally, we briefly examine these questions where L is represented by a context-free grammar.     

Lower bounds on treespan

Recently Fomin, Heggernes and Telle [Algorithmica 41 (2004) 73] introduced the notion of the treespan of a graph as a natural extension of the well-known bandwidth. They motivate this new concept from different viewpoints involving graph searching, tree decompositions and elimination orderings. In the present paper we prove several lower bounds on the treespan.     

Lower bounds for the smallest singular value

Let A = (a_ij) be an n × n complex matrix. Suppose that G(A), the undirected graph of A, has no isolated vertex. Let E be the set of edges of G(A). We prove that the smallest singular value of A, σ_n, satisfies: σ_n ≥ min σ_ij | (i, j) ∈ E, where g_ij ≡ a_i + a_j − [(a_i − a_j)² + (r_i + c_i)(r_j + c_j)]^1/2/2 with a_i ≡ |a_ii| and r_i,c_i are the i^th deleted absolute row sum and column sum of A, respectively. The result simplifies and improves that of Johnson and Szulc: σ_n ≥ min_i≠j σ_ij. (See [1].)     

Lower bounds for polynomials with algebraic coefficients

We give a method, based on algebraic geometry, to show lower bounds for the complexity of polynomials with algebraic coefficients. Typical examples are polynomials with coefficients which are roots of unity, such as

$\text{Σ}\text{j=1}\text{d}\text{e}{}^{\mathrm{<mtext>2\pi i</mtext><mtext>i</mtext>}}\text{X}{}^{\mathrm{i}}$

and

$\text{Σ}\text{j=i}\text{d}\text{e}{}^{\mathrm{<mtext>2\pi i</mtext><mtext>pi</mtext>}}\text{X}{}^{\mathrm{j}}$

where p_j is the jth prime number.We apply the method also to systems of linear equations.     

Lower bounds for on-line two-dimensional packing algorithms

Summary Many problems, such as cutting stock problems and the scheduling of tasks with a shared resource, can be viewed as two-dimensional bin packing problems. Using the two-dimensional packing model of Baker, Coffman, and Rivest, a finite list L of rectangles is to be packed into a rectangular bin of finite width but infinite height, so as to minimize the total height used. An algorithm which packs the list in the order given without looking ahead or moving pieces already packed is called an on-line algorithm. Since the problem of finding an optimal packing is NP-hard, previous work has been directed at finding approximation algorithms. Most of the approximation algorithms which have been studied are on-line except that they require the list to have been previously sorted by height or width. This paper examines lower bounds for the worst-case performance of on-line algorithms for both non-preordered lists and for lists preordered by increasing or decreasing height or width.This author's work was supported by the Joint Services Electronics Program (U.S. Army, U.S. Navy and U.S. Air Force) under Contract DAAG-29-78-C-0016     

Lower bounds for the cycle detection problem

Given a function f over a domain and an element x in the domain, the cycle detection problem is to find a repetition in the sequence of values x, f(x), f(f(x)), f³(x),…, if one exists. This paper investigates lower bounds on the number of function evaluations needed when there is a bound on the amount of memory available. For certain restricted classes of algorithms which use two memory locations optimality is achieved. A summary of the major results appears in the final section.     

Lower bounds for sorting on mesh-connected architectures

Summary Lower bounds for sorting on mesh-connected arrays of processors are presented. For sorting N=n₁ n ₂...n _r elements on an n ₁×n₂×... ×n _r array 2(n ₁+...+n _r–1)+n _r data interchange steps are needed asymptotically. For two dimensions these bounds are asymptotically best possible provided that n ₁ and n ₂ are powers of 2. In this case the generalized s ²-way merge sort of Thompson and Kung turns out to be asymptotically optimal. The minimal asymptotic bound of 2 2N interchange steps can be obtained only by sorting algorithms suitable for N/2×2N meshes. For r3 dimensions an analysis of aspect-ratios also demonstrates that there exist mesh-connected architectures which are better suited for sorting than simple r-dimensional cubes.This work was done at the Institut für Informatik und Praktische Mathematik, University of Kiel, Federal Republic of Germany