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SIAM J. on Computing

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1986

Volume 15, Issue 1, pp. 1-315


Integration in Finite Terms with Special Functions: The Logarithmic Integral

G. W. Cherry

SIAM J. Comput. 15, pp. 1-21 (21 pages) | Cited 5 times

Online Publication Date: August 02, 2006

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Since R. Risch published an algorithm for calculating symbolic integrals of elementary functions in 1969 (Traps. Amer. Math. Soc., 139 (1969), pp. 167–189), there has been an interest in extending his methods to include nonelementary functions. In this paper, we use the framework of differential algebra to make precise the notion of integration in terms of elementary functions and logarithmic integrals. Basing our work on a recent extension of Liouville’s theorem on integration in finite terms, we then describe a decision procedure for determining if a given element in a transcendental elementary field has an integral which can be written in terms of elementary functions and logarithmic integrals. This algorithm first examines the structure of the integrand in order to limit the logarithmic integrals which could appear in the integral to a finite number. This allows us to write a general expression for the integral and then use techniques similar to those employed by Risch to calculate the undetermined parts.

An Amortized Analysis of Insertions into AVL-Trees

Kurt Mehlhorn and Athanasios Tsakalidis

SIAM J. Comput. 15, pp. 22-33 (12 pages) | Cited 2 times

Online Publication Date: August 02, 2006

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We analyse the amortized behavior of AVL-trees under sequences of insertions. We show that the total rebalancing cost (=balance changes) for a sequence of $n$ arbitrary insertions is at most $2.618n$. For random insertions the bound is improved to $2.26n$. We also show that the probability that $t$ or more balance changes are required decreases exponentially with $t$.

The Signature of a Plane Curve

Joseph O’Rourke

SIAM J. Comput. 15, pp. 34-51 (18 pages) | Cited 4 times

Online Publication Date: August 02, 2006

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The signature of a plane curve $\Gamma $ associated with every point $p$ of $\Gamma $ the length of $\Gamma $ to the left of or on the line tangent to $\Gamma $ at $p$. The signature has properties that make it a useful tool for pattern recognition: it discards the location, orientation, and scale, and “slant” in special cases, but preserves symmetries. Its integral is a measure of convexity. This paper explores the theoretical properties of this concept. It is shown that in the special case of closed rectilinear curves, the signature retains enough information to permit exact reconstruction of the curve. Computing the signature and reconstructing curves from their signatures are interesting computational problems; time complexity bounds on these problems are presented. Several challenging open questions are posed.

Self-Adjusting Heaps

Daniel Dominic Sleator and Robert Endre Tarjan

SIAM J. Comput. 15, pp. 52-69 (18 pages) | Cited 16 times

Online Publication Date: August 02, 2006

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In this paper we explore two themes in data structure design: amortized computational complexity and self adjustment. We are motivated by the following observations. In most applications of data structures, we wish to perform not just a single operation but a sequence of operations, possibly having correlated behavior. By averaging the running time per operation over a worst-case sequence of operations, we can sometimes obtain an overall time bound much smaller than the worst-case time per operation multiplied by the number of operations. We call this kind of averaging amortization.
Standard kinds of data structures, such as the many varieties of balanced trees, are specifically designed so that the worst-case time per operation is small. Such efficiency is achieved by imposing an explicit structural constraint that must be maintained during updates, at a cost of both running time and storage space. However, if amortized running time is the complexity measure of interest, we can guarantee efficiency without maintaining a structural constraint. Instead, during each access or update operation we adjust the data structure in a simple, uniform way. We call such a data structure self adjusting.
In this paper we develop the skew heap, a self-adjusting form of heap related to the leftist heaps of Crane and Knuth. (What we mean by a heap has also been called a “priority queue” or a “mergeable heap”.) Skew heaps use less space than leftist heaps and similar worst-case-efficient data structures and are competitive in running time, both in theory and in practice, with worst-case structures. They are also easier to implement. We derive an information-theoretic lower bound showing that skew heaps have minimum possible amortized running time, to within a constant factor, on any sequence of certain heap operations.

The Complexity of Languages Generated by Attribute Grammars

Joost Engelfriet

SIAM J. Comput. 15, pp. 70-86 (17 pages) | Cited 2 times

Online Publication Date: August 02, 2006

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A string-valued attribute grammar (SAG) has a semantic domain of strings over some alphabet, with concatenation as basic operation. It is shown that the output language (i.e., the range of the translation) of a SAG is log-space reducible to a context-free language.

Upper and Lower Time Bounds for Parallel Random Access Machines without Simultaneous Writes

Stephen Cook, Cynthia Dwork, and Rüdiger Reischuk

SIAM J. Comput. 15, pp. 87-97 (11 pages) | Cited 42 times

Online Publication Date: August 02, 2006

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One of the frequently used models for a synchronous parallel computer is that of a parallel random access machine, where each processor can read from and write into a common random access memory. Different processors may read the same memory location at the same time, but simultaneous writing is disallowed. We show that even if we allow nonuniform algorithms, an arbitrary number of processors, and arbitrary instruction sets, $\Omega (\log n)$ is a lower bound on the time required to compute various simple functions, including sorting $n$ keys and finding the logical “or” of $n$ bits. We also prove a surprising time upper bound of $.72\log _2 n$ steps for these functions, which beats the obvious algorithms requiring $\log _2 n$ steps.If simultaneous writes are allowed, there are simple algorithms to compute these functions in a constant number of steps.

The Boyer–Moore–Galil String Searching Strategies Revisited

Alberto Apostolico and Raffaele Giancarlo

SIAM J. Comput. 15, pp. 98-105 (8 pages) | Cited 19 times

Online Publication Date: August 02, 2006

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Based on the Boyer–Moore–Galil approach, a new algorithm is proposed which requires a number of character comparisons bounded by 2n, regardless of the number of occurrences of the pattern in the textstring. Preprocessing is only slightly more involved and still requires a time linear in the pattern size.

Efficient Simulations among Several Models of Parallel Computers

Friedhelm Meyer auf der Heide

SIAM J. Comput. 15, pp. 106-119 (14 pages) | Cited 5 times

Online Publication Date: August 02, 2006

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A parallel computer (PC) with fixed communication network is called fair if the degree of this network is bounded, otherwise it is called unfair. In a PC with predictable communication each processor can precompute the addresses of the processors it wants to communicate with in the next $t$ steps in $O(t)$ steps. For an arbitrary $\varepsilon > 0$ we define fair PC’s $M$ and $M'$ with $O(n^{1 + \varepsilon } )$ processors each. $M(M')$ can simulate each unfair PC with predictable communication and $O(\log (n))$ storage locations per processor (each fair PC) with $n$ processors with constant time loss. $M'$ improves a result from [Acts Informatics, 19 (1983), pp. 269–296] where a time loss of $O(\log \log (n))$ was achieved. Assuming some reasonable properties of simulations we finally prove a lower bound $\Omega (\log (n))$ for the time loss of a fair PC which can simulate each unfair PC. Applying fast sorting or packet switching algorithms (Proc.15th Annual ACM Symposiums on Theory of Computing, Boston, 1983, pp. 1–9; 10–16; Proc. ACM Symposiums on Principles of Distributed Computing, Ottawa, 1982) one sees easily that this bound is asymptotically tight.

An $O(EV\log V)$ Algorithm for Finding a Maximal Weighted Matching in General Graphs

Zvi Galil, Silvio Micali, and Harold Gabow

SIAM J. Comput. 15, pp. 120-130 (11 pages) | Cited 13 times

Online Publication Date: August 02, 2006

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We define two generalized types of a priority queue by allowing some forms of changing the priorities of the elements in the queue. We show that they can be implemented efficiently. Consequently, each operation takes $O(\log n)$ time. We use these generalized priority queues to construct an $O(EV\log V)$ algorithm for finding a maximal weighted matching in general graphs.

Optimal Termination Protocols for Network Partitioning

Francis Chin and K. V. S. Ramarao

SIAM J. Comput. 15, pp. 131-144 (14 pages) | Cited 1 time

Online Publication Date: August 02, 2006

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We address the problem of maintaining the distributed database consistency in presence of failures while maximizing the database availability. Network Partitioning is a failure which partitions the distributed system into a number of parts, no part being able to communicate with any other. Formalizations of various notions in this context are developed and two measures for the performances of protocols in presence of a network partitioning are introduced. A general optimality theory is developed for two classes of protocols—centralized and decentralized. Optimal protocols are produced in all cases.

Computational Complexity: On the Geometry of Polynomials and a Theory of Cost: II

M. Shub and S. Smale

SIAM J. Comput. 15, pp. 145-161 (17 pages) | Cited 3 times

Online Publication Date: August 02, 2006

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This paper deals with traditional algorithms, Newton’s method and a higher order generalization due to Euler. These iterations schemes and their modifications have had a great success in solving nonlinear systems of equations. We give some understanding of this phenomenon by giving estimates of efficiency. The problem we focus on is that of finding a zero of a complex polynomial.

A Provably Good Algorithm for the Two Module Routing Problem

Brenda S. Baker

SIAM J. Comput. 15, pp. 162-188 (27 pages) | Cited 1 time

Online Publication Date: August 02, 2006

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In the Mead–Conway design methodology for LSI, modules are designed and then connected by wires to form larger modules in a hierarchical fashion. It would be helpful to have a design aid that would do the routing automatically and be guaranteed of coming within some fixed percentage of the size of an optimal routing. With this goal in mind, we investigate the problem of routing two-terminal nets between two modules of the same width but possibly different heights, assuming that the sides are aligned vertically. The terminals may lie on any of the sides of either module. Wires must be routed according to the “Manhattan” reserved-layer model, in which all wires must lie on a rectilinear grid, and wires running the same direction must be separated by at least unit distance. Finding an optimal routing for this problem is NP-hard, where the measure of performance is the perimeter of the bounding box around the whole routing region. We describe an algorithm whose worst-case performance is asymptotically at most 19/10 times that of an optimal routing. The algorithm runs in $O(n\log n)$ time, where $n$ is the number of nets.
One of the problems encountered in routing is how to evaluate a routing when the optimal routing is not available for comparison. The techniques given here can be used to calculate lower bounds on the size of an optimal routing. Thus, these techniques may be useful in evaluating routings produced by methods other than the algorithm in this paper.

Alphabetic Minimax Trees of Degree at Most $t$

D. Coppersmith, M. M. Klawe, and N. J. Pippenger

SIAM J. Comput. 15, pp. 189-192 (4 pages) | Cited 2 times

Online Publication Date: August 02, 2006

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Problems in circuit fan-out reduction motivate the study of constructing various types of weighted trees that are optimal with respect to maximum weighted path length. An upper bound on the maximum weighted path length and an efficient construction algorithm will be presented for trees of degree at most $t$, along with their implications for circuit fan-out reduction.

On Shortest Paths in Polyhedral Spaces

Micha Sharir and Amir Schorr

SIAM J. Comput. 15, pp. 193-215 (23 pages) | Cited 41 times

Online Publication Date: August 02, 2006

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We consider the problem of computing the shortest path between two points in two- or three-dimensional space bounded by polyhedral surfaces. In the 2-D case the problem is easily solved in time $O(n^2 \log n)$. In the general 3-D case the problem is quite hard to solve, and is not even discrete; we present a doubly-exponential procedure for solving the discrete subproblem of determining the sequence of boundary edges through which the shortest path passes. Finally we consider a favorable special case of the 3-D shortest path problem, namely that of finding the shortest path between two points along the surface of a convex polyhedron, and solve it in time $O(n^3 \log n)$.

An Efficient Algorithm for Generating Linear Transformations in a Shuffle-Exchange Network

T. Etzion and A. Lempel

SIAM J. Comput. 15, pp. 216-221 (6 pages) | Cited 1 time

Online Publication Date: August 02, 2006

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This paper presents an algorithm for generating all the permutations defined by linear transformations on a shuffle-exchange network of $2^n $ processors in $2n - 1$ passes. The proposed algorithm generates any such permutation in $O(n\log ^2 n)$ elementary steps. The subclass of bit-permutations is generated in $O(n)$ steps.

Variable Sized Bin Packing

D. K. Friesen and M. A. Langston

SIAM J. Comput. 15, pp. 222-230 (9 pages) | Cited 35 times

Online Publication Date: August 02, 2006

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In the classical bin packing problem one seeks to pack a list of pieces in the minimum space using unit capacity bins. This paper addresses the more general problem in which a fixed collection of bin sizes is allowed. Three efficient approximation algorithms are described and analyzed. They guarantee asymptotic worst-case performance bounds of 2, ${3 / 2}$ and ${4 / 3}$.

Logarithmic Depth Circuits for Algebraic Functions

John H. Reif

SIAM J. Comput. 15, pp. 231-242 (12 pages) | Cited 12 times

Online Publication Date: August 02, 2006

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This paper describes circuits for computation of a large class of algebraic functions on polynomials, power series, and integers, for which, it has been a long standing open problem to compute in depth less than $\Omega (\log n)^2 $.
Algebraic circuits assume unit cost for elemental addition and multiplication. This paper describes $O(\log n)$ depth algebraic circuits which given as input the coefficients of $n$ degree polynomials (over an appropriate ring), compute the product of $n^{O(1)} $ polynomials, the symmetric functions, as well as division and interpolation of real polynomials. Also described are $O(\log n)$ depth algebraic circuits which are given as input the first $n$ coefficients of a power series (over an appropriate ring) compute the product of $n^{O(1)} $ power series, as well as division, reciprocal and reversion of real power series.
Furthermore this paper describes boolean circuits of depth $O(\log n(\log \log n))$ which, given $n$-bit binary numbers, compute the product of $n$ numbers and integer division. As corollaries, we get boolean circuits of the same depth for evaluating, within accuracy $2^{ - n} $, polynomials, power series, and elementary functions such as (fixed) powers, roots, exponentiations, logarithm, sine and cosine.
All these circuits have constant indegree, polynomial size, and may be uniformly constructed by a deterministic Turing machine with space $O(\log n)$.

Algebraic Computations of Scaled Padé Fractions

Stanley Cabay and Dong-Koo Choi

SIAM J. Comput. 15, pp. 243-270 (28 pages) | Cited 11 times

Online Publication Date: August 02, 2006

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Two companion algorithms are developed for constructing Pade fractions along an off-diagonal path of the Padé table for a function ${{ - A(z)} / {B(z)}}$, where $A(z)$ and $B(z)$ are formal power series over a field.
One of the algorithms computes the first $n$ Padé fractions along the off-diagonal in time $O(n^2 )$. When $A(z)$ and $B(z)$ are finite power series (i.e., polynomials), it is shown that the algorithm is equivalent to Euclid’s extended algorithm for computing greatest common divisors.
The other algorithm, a generalization of the first, proceeds along the off-diagonal in quadratic steps, and is of complexity $O(n\log ^2 n)$. When $A(z)$ and $B(z)$ are polynomials, the second algorithm becomes a fast Euclid’s extended algorithm for computing greatest common divisors. The algorithm is of the same complexity as other fast greatest common divisor methods, but its iterative nature provides a practical advantage during implementation.
The algorithms may also be used for computing Padé fractions along an anti-diagonal path of the Padé table. The fast algorithm is of the same complexity as other fast algorithms for anti-diagonal computations. However, it has the advantage of being able to determine easily any specific Padé fraction along the anti-diagonal.

Constructing Belts in Two-Dimensional Arrangements with Applications

H. Edelsbrunner and E. Welzl

SIAM J. Comput. 15, pp. 271-284 (14 pages) | Cited 17 times

Online Publication Date: August 02, 2006

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For $H$ a set of lines in the Euclidean plane, $A(H)$ denotes the induced dissection, called the arrangement of $H$. We define the notion of a belt in $A(H)$, which is bounded by a subset of the edges in $A(H)$, and describe two algorithms for constructing belts. All this is motivated by applications to a host of seemingly unrelated problems including a type of range search and finding the minimum area triangle with the vertices taken from some finite set of points.

Average Case Complete Problems

Leonid A. Levin

SIAM J. Comput. 15, pp. 285-286 (2 pages) | Cited 10 times

Online Publication Date: August 02, 2006

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Many interesting combinatorial problems were found to be NP-complete. Since there is little hope to solve them fast in the worst case, researchers look for algorithms which are fast just “on average”. This matter is sensitive to the choice of a particular NP-complete problem and a probability distribution of its instances. Some of these tasks were easy and some not. But one needs a way to distinguish the “difficult on average” problems. Such negative results could not only save “positive” efforts but may also be used in areas (like cryptography) where hardness of some problems is a frequent assumption. It is shown below that the Tiling problem with uniform distribution of instances has no polynomial “on average” algorithm, unless every NP-problem with every simple probability distribution has it. It is interesting to try to prove similar statements for other NP-problems which resisted so far “average case” attacks.

The Ultimate Planar Convex Hull Algorithm?

David G. Kirkpatrick and Raimund Seidel

SIAM J. Comput. 15, pp. 287-299 (13 pages) | Cited 40 times

Online Publication Date: August 02, 2006

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We present a new planar convex hull algorithm with worst case time complexity $O(n\log H)$ where $n$ is the size of the input set and $H$ is the size of the output set, i.e. the number of vertices found to be on the hull. We also show that this algorithm is asymptotically worst case optimal on a rather realistic model of computation even if the complexity of the problem is measured in terms of input as well as output size. The algorithm relies on a variation of the divide-and-conquer paradigm which we call the “marriage-before-conquest” principle and which appears to be interesting in its own right.

Computing the Largest Empty Rectangle

B. Chazelle, R. L. Drysdale, and D. T. Lee

SIAM J. Comput. 15, pp. 300-315 (16 pages) | Cited 17 times

Online Publication Date: August 02, 2006

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We consider the following problem: Given a rectangle containing $N$ points, find the largest area subrectangle with sides parallel to those of the original rectangle which contains none of the given points. If the rectangle is a piece of fabric or sheet metal and the points are flaws, this problem is finding the largest-area rectangular piece which can be salvaged. A previously known result [13] takes $O(N^2 )$ worst-case and $O(N\log ^2 N)$ expected time. This paper presents an $O(N\log ^3 N)$ time, $O(N\log N)$ space algorithm to solve this problem. It uses a divide-and-conquer approach similar to the ones used by Bentley [1] and introduces a new notion of Voronoi diagram along with a method for efficient computation of certain functions over paths of a tree.
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