SIAM Digital Library
 
 
 

SIAM J. on Computing

All Journal content published prior to 1997 is part of LOCUS.

Search Issue | RSS Feeds RSS
Previous Issue

1986

Volume 15, Issue 4, pp. 903-1194


The Risch Differential Equation Problem

J. H. Davenport

SIAM J. Comput. 15, pp. 903-918 (16 pages) | Cited 1 time

Online Publication Date: July 31, 2006

Full Text: | Download PDF

Show Abstract
We propose a new algorithm, similar to Hermite’s method for the integration of rational functions, for the resolution of Risch differential equations in closed form, or proving that they have no resolution. By requiring more of the presentation of our differential fields (in particular that the exponentials be weakly normalised), we can avoid the introduction of arbitrary constants which have to be solved for later.
We also define a class of fields known as exponentially reduced, and show that solutions of Risch differential equations which arise from integrating in these fields satisfy the “natural” degree constraints in their main variables, and we-conjecture (after Risch and Norman) that this is true in all variables.

Data Structures for Retrieval on Square Grids

Martin David Katz and Dennis J. Volper

SIAM J. Comput. 15, pp. 919-931 (13 pages) | Cited 1 time

Online Publication Date: July 31, 2006

Full Text: | Download PDF

Show Abstract
Families of data structures are presented for retrieval of the sum of values of points within a half plane or polygon, given that the points are at integral coordinates $(N \times N)$ in the plane. Fredman has shown that the problem has a lower bound of $\Omega (N^{{2 / 3}} )$ for intermixed updates and retrievals. When the points are not restricted to integral coordinates, Edelsbrunner and Welzl have shown a retrieval time of $O(N^{ \approx 1.39} )$ (update time$ = O(N^2 \log N)$). One of the data structures presented here permits intermixed updates and retrievals in $O(N^{{2 / {\log N}}})$.
We store multiple, rotated data structures to match against the query. Rotation appears to be an effective method for trading-off update time against retrieval time for geometric problems. We also present constructions for efficient retrieval of triangles and polygons. For our data structures, the expected complexity when the points are uniformly distributed is less than the worst case complexity when the points are at integral coordinates.

Communication Complexity of Computing the Hamming Distance

King F. Pang and Abbas El Gamal

SIAM J. Comput. 15, pp. 932-947 (16 pages) | Cited 1 time

Online Publication Date: July 31, 2006

Full Text: | Download PDF

Show Abstract
Let ${\bf x},{\bf y} \in \{ 0,1\} ^n $. Persons $A$ and $B$ are given ${\bf x}$ and ${\bf y}$ respectively. They communicate in order that both find the Hamming Distance $d_H^n ({\bf x},{\bf y})$. Three communication models, viz, deterministic, $\varepsilon $-error and $\varepsilon $-randomized, are considered. Let $C(d_H^n )$, $C_\varepsilon (d_H^n )$ and $D_\varepsilon (d_H^n )$ be the respective minimum number of bits that must be communicated under the three models. It is shown that \[ n + \log (n + 1 - \sqrt n ) \leqq C\left( {d_H^n } \right) \leqq n + \lceil {\log (n + 1)} \rceil . \] It is also shown that both $C_\varepsilon (d_H^n )$ and $D_\varepsilon (d_H^n )$ are lower bounded by $\Omega (n)$, thus solving an open problem posed by Yao.

Improved Bounds for Matroid Partition and Intersection Algorithms

William H. Cunningham

SIAM J. Comput. 15, pp. 948-957 (10 pages) | Cited 11 times

Online Publication Date: July 31, 2006

Full Text: | Download PDF

Show Abstract
We give bounds on total lengths of augmenting paths in standard implementations of the matroid partition and intersection algorithms, and indicate how these observations can be used to improve the running times in certain applications. For example, for the matroid intersection algorithm on two $r$ by $n$ matrices the running time is shown to be $O(nr^2 \log r)$. We also give improved versions of the two algorithms, when running times are measured in terms of calls to an independence oracle. For example, there is a matroid partition algorithm on $O(n)$$n$-element matroids using $O(n^{2.5} )$ independence tests.

An Inhomogeneity in the Structure of Karp Degrees

Klaus Ambos–Spies

SIAM J. Comput. 15, pp. 958-963 (6 pages)

Online Publication Date: July 31, 2006

Full Text: | Download PDF

Show Abstract
We show that there is a recursive nonzero Karp (polynomial time many-one) degree which is not supremum of a minimal pair, i.e. of an incomparable pair of degrees with infimum 0, the degree of polynomial time computable sets. By existence of minimal pairs, this implies that there are nonisomorphic initial segments of Karp degrees.

Heaps on Heaps

Gaston H. Gonnet and J. Ian Munro

SIAM J. Comput. 15, pp. 964-971 (8 pages) | Cited 16 times

Online Publication Date: July 31, 2006

Full Text: | Download PDF

Show Abstract
As part of a study of the general issue of complexity of comparison based problems, as well as interest in the specific problem, we consider the task of performing the basic priority queue operations on a heap. We show that in the worst case: $\lg \lg n \pm O(1)$ comparisons are necessary and sufficient to insert an element into a heap. (This improves the previous upper and lower bounds of $\lg n$ and $O(1)$.) $\lg n + \log ^ * n \pm O(1)$ comparisons are necessary and sufficient to replace the maximum in a heap. (This improves the previous upper and lower bounds of $2\lg n$ and $\lg n$.) $1.625n + O(\lg n\log ^ * n)$ comparisons are sufficient to create a heap. $1.37 \ldots n$ comparisons are necessary not only in the worst case but also on the average.
Here lg indicates the logarithm base 2 and $\log ^ * $ denotes the iterated logarithm or number of times the logarithm base 2 may be taken before the quantity is at most 0.

On Approximations and Incidence in Cylindrical Algebraic Decompositions

David Prill

SIAM J. Comput. 15, pp. 972-993 (22 pages) | Cited 4 times

Online Publication Date: July 31, 2006

Full Text: | Download PDF

Show Abstract
Let $P \subset \mathbb{Z}[x_1 , \cdots ,x_r ]$ be a finite set. This paper describes and analyzes a variant of the algorithm of Collins and others for decomposing $\mathbb{R}^r $ into semi-algebraic cells so that the value of each $f \in P$ has constant sign (positive, negative, or zero) on the points of each cell. The version here has several advantages: 1. The boundary of each cell is a disjoint union of lower-dimensional cells. For each bounded cell $\alpha $ the pair $(\bar \alpha ,\alpha )$ is homeomorphic to a closed ball and its interior. 2. An algorithm is presented which for fixed $r$ computes incidence of cells in polynomial time. 3. A priori estimates of the accuracy of approximations of roots of polynomials required in order to determine the combinatorial structure of the cell complex are given. This avoids computation in algebraic number fields.

Log Depth Circuits for Division and Related Problems

Paul W. Beame, Stephen A. Cook, and H. James Hoover

SIAM J. Comput. 15, pp. 994-1003 (10 pages) | Cited 20 times

Online Publication Date: July 31, 2006

Full Text: | Download PDF

Show Abstract
We present optimal depth Boolean circuits (depth $O(\log n)$) for integer division, powering, and multiple products. We also show that these three problems are of equivalent uniform depth and space complexity. In addition, we describe an algorithm for testing divisibility that is optimal for both depth and space.

On the Validity of the Direct Sum Conjecture

Joseph Ja’Ja’ and Jean Takche

SIAM J. Comput. 15, pp. 1004-1020 (17 pages) | Cited 4 times

Online Publication Date: July 31, 2006

Full Text: | Download PDF

Show Abstract
The direct sum conjecture states that the multiplicative complexity of disjoint sets of bilinear computations is the sum of their separate multiplicative complexities. This conjecture is known to hold for only a few specialized cases. In this paper, we establish its validity for large classes of computations. One such class can be defined as follows. Let $S_1 $ be a set of $r$$m \times n$ bilinear forms, and let $S_2 $ be a different set of $s$$p \times q$ bilinear forms. Then, if $2 \in \{ r,m,n,s,p,q\} $, we show that the direct sum conjecture holds over any field. The proof involves some nontrivial facts from linear algebra and relies on the theory of invariant polynomials. This result also settles the multiplicative complexity of pairs of bilinear forms over any field with large enough cardinality. It is also shown that the direct sum conjecture is true for the case when $r = mn - 2$.

On the Single-Operation Worst-Case Time Complexity of the Disjoint Set Union Problem

Norbert Blum

SIAM J. Comput. 15, pp. 1021-1024 (4 pages) | Cited 7 times

Online Publication Date: July 31, 2006

Full Text: | Download PDF

Show Abstract
We give an algorithm for the disjoint set union problem, within the class of algorithms defined by Tarjan, which has $O(\log n/\log \log n)$ single-operation time complexity in the worst case. Also we define a class of algorithms for the disjoint set union problem, which includes the class of algorithms defined by Tarjan. We prove that any algorithm from this class has at least $\Omega ({{\log n} / {\log \log n}})$ single-operation time complexity in the worst case.

Ranking and Unranking of AVL-Trees

Liwu Li

SIAM J. Comput. 15, pp. 1025-1035 (11 pages) | Cited 1 time

Online Publication Date: July 31, 2006

Full Text: | Download PDF

Show Abstract
In this paper, we consider the problem of generating, ranking, and unranking of AVL-trees with n leaves. We represent AVL-trees by integer-pair sequences, called LDP-sequences. Then we propose a linear ordering among these sequences, i.e., among the AVL-trees. The problem of ranking is to determine the order number (rank) of a given tree in this ordering, unranking means constructing the tree of a given rank. The main result is that ranking and unranking can be done in $0(n\log ^2 n)$ and $0(n\log ^3 n)$ time, respectively, after a preprocessing step that takes $0(n^2 \log n)$ time.

A Simple Parallel Algorithm for the Maximal Independent Set Problem

Michael Luby

SIAM J. Comput. 15, pp. 1036-1053 (18 pages) | Cited 53 times

Online Publication Date: July 31, 2006

Full Text: | Download PDF

Show Abstract
Two basic design strategies are used to develop a very simple and fast parallel algorithms for the maximal independent set (MIS) problem. The first strategy consists of assigning identical copies of a simple algorithm to small local portions of the problem input. The algorithm is designed so that when the copies are executed in parallel the correct problem output is produced very quickly. A very simple Monte Carlo algorithm for the MIS problem is presented which is based upon this strategy. The second strategy is a general and powerful technique for removing randomization from algorithms. This strategy is used to convert the Monte Carlo algorithm for this MIS problem into a simple deterministic algorithm with the same parallel running time.

Finding a Maximum Clique in an Arbitrary Graph

Egon Balas and Chang Sung Yu

SIAM J. Comput. 15, pp. 1054-1068 (15 pages) | Cited 54 times

Online Publication Date: July 31, 2006

Full Text: | Download PDF

Show Abstract
We describe a new type of branch and bound procedure for finding a maximum clique in an arbitrary graph $G = (V,E)$. The two main ingredients, both of $O(|V| + |E|)$ time complexity, are (i) an algorithm for finding a maximal triangulated induced subgraph of $G$; and (ii) an algorithm for finding a maximal $k$-chromatic induced subgraph of $G$. We discuss computational experience on randomly generated graphs with up to 400 vertices and 30,000 edges.

Average Case Analysis of Marking Algorithms

D. S. Hirschberg and L. L. Larmore

SIAM J. Comput. 15, pp. 1069-1074 (6 pages)

Online Publication Date: July 31, 2006

Full Text: | Download PDF

Show Abstract
The Lindstrom marking algorithm uses bounded workspace. Its time complexity is $0(n^2 )$ in all cases, but it has been assumed that the average case time complexity is $0(n\log n)$. It is proven that the average case time complexity is $\Theta (n^2 )$ for a wide variety of probability distributions. Similarly, the average size of the Wegbreit bit stack is shown to be $\Theta (n)$.

Searching in Trees, Series-Parallel and Interval Orders

U. Faigle, L. Lovász, R. Schrader, and Gy. Turán

SIAM J. Comput. 15, pp. 1075-1084 (10 pages) | Cited 5 times

Online Publication Date: July 31, 2006

Full Text: | Download PDF

Show Abstract
Linial and Saks [2] have shown that $O(\log N)$ evaluations of an order preserving map $f:p \to \mathbb{R}$ are necessary and sufficient to determine whether $\alpha \in f(P)$, where $N$ is the number of ideals of $N$ and $\alpha \in \mathbb{R}$ is a given real number. In this paper, we investigate the problem of how to perform the evaluations so that Linial and Saks’ bound is guaranteed, and solve the problem for the classes of interval and series-parallel orders and hence, in particular, for rooted trees. We observe that the greedy-type binary search algorithm, which is optimal for chains, already need not be optimal for general rooted trees. We furthermore discuss the computational complexity of the general search problem and obtain results indicating that the general problem might be hard.

Processor-Shared Time-Sharing Models in Heavy Traffic

Donald P. Gaver and Patricia A. Jacobs

SIAM J. Comput. 15, pp. 1085-1100 (16 pages) | Cited 8 times

Online Publication Date: July 31, 2006

Full Text: | Download PDF

Show Abstract
Probability models are presented for computer systems with processor-shared (time-sliced) service discipline. The response (sojourn) time of an arriving job that requires $T$ units of processing time is shown to be approximately Gaussian/normal under moderately heavy traffic conditions, e.g. when the number of terminals is large.

Some Observations about the Randomness of Hard Problems

Dung T. Huynh

SIAM J. Comput. 15, pp. 1101-1105 (5 pages) | Cited 3 times

Online Publication Date: July 31, 2006

Full Text: | Download PDF

Show Abstract
In this note we investigate some connections between hard languages and random languages. We show that there exist languages that are both hard and random. We also show that every EXPTIME-hard language is polynomial-time weakly random.

Probabilistic Analysis of Two Heuristics for the $3$-Satisfiability Problem

Ming-Te Chao and John Franco

SIAM J. Comput. 15, pp. 1106-1118 (13 pages) | Cited 19 times

Online Publication Date: July 31, 2006

Full Text: | Download PDF

Show Abstract
An algorithm for the 3-Satisfiability problem is presented and a probabilistic analysis is performed. The analysis is based on an instance distribution which is parameterized to simulate a variety of sample characteristics. The algorithm assigns values to variables appearing in a given instance of 3-Satisfiability, one at a time, using the unit clause heuristic and a maximum occurring literal selection heuristic; at each step a variable is chosen randomly from a subset of variables which is usually large. The algorithm runs in polynomial time and it is shown that the algorithm finds a solution to a random instance of 3-Satisfiability with probability bounded from below by a constant greater than zero for a range of parameter values. The heuristics studied here can be used to select variables in a Backtrack algorithm for 3-Satisfiability. Experiments have shown that for about the same range of parameters as above the Backtrack algorithm using the heuristics finds a solution in polynomial average time.

Worst Case Bound of an LRF Schedule for the Mean Weighted Flow-Time Problem

Tsuyoshi Kawaguchi and Seiki Kyan

SIAM J. Comput. 15, pp. 1119-1129 (11 pages) | Cited 30 times

Online Publication Date: July 31, 2006

Full Text: | Download PDF

Show Abstract
This paper studies the problem of scheduling a set of $n$ independent tasks on $m$ identical processors so as to minimize mean weighted flow-time. The problem is known to be NP-complete for $m \geqq 2$ and to be NP-complete in the strong sense for m arbitrary.
The worst case behavior of a heuristic algorithm which requires time $O(n\log n)$ is investigated, and it is shown that the mean weighted flow-time obtained by the algorithm does not exceed ${{(\sqrt 2 + 1)} / 2} \cong 1.0207$ times that of an optimal schedule. Moreover the bound ${{(\sqrt 2 + 1)} / 2}$ is best possible.

On Maintaining Dynamic Information in a Concurrent Environment

Udi Manber

SIAM J. Comput. 15, pp. 1130-1142 (13 pages) | Cited 4 times

Online Publication Date: July 31, 2006

Full Text: | Download PDF

Show Abstract
This paper considers the amount of cooperation required for independent asynchronous processes to share a simple dynamic data structure. We present a scheme for designing efficient concurrent algorithms to add and remove elements from a shared pool of elements. The efficiency is measured mainly by the number of non-local operations that a process may have to make. Non-local operations may involve writing into a shared variable, locking, or sending a message, hence they introduce interference (or require cooperation). We derive upper and lower bounds on the interference in the worst case. Applications to distributed computation are also discussed.

Sums of Divisors, Perfect Numbers and Factoring

Eric Bach, Gary Miller, and Jeffrey Shallit

SIAM J. Comput. 15, pp. 1143-1154 (12 pages) | Cited 1 time

Online Publication Date: July 31, 2006

Full Text: | Download PDF

Show Abstract
Let $N$ be a positive integer, and let $\sigma (N)$ denote the sum of the divisors of $N$ (e.g. $\sigma (6) = 1 + 2 + 3 + 6 = 12$). We show computing $\sigma (N)$ is equivalent to factoring $N$ in the following sense: there is a random polynomial time algorithm that, given $\sigma (N)$, produces the prime factorization of $N$, and $\sigma (N)$ can be computed in polynomial time given the factorization of $N$.
We show that the same result holds for $\sigma _k (N)$, the sum of the $k$th powers of divisors of $N$
We give three new examples of problems that are in Gill’s complexity class BPP: perfect numbers, multiply perfect numbers, and amicable pairs. These are the first “natural” sets in BPP that are not obviously in RP.

Completion of a Set of Rules Modulo a Set of Equations

Jean-Pierre Jouannaud and Hélène Kirchner

SIAM J. Comput. 15, pp. 1155-1194 (40 pages) | Cited 9 times

Online Publication Date: July 31, 2006

Full Text: | Download PDF

Show Abstract
Abstract Church-Rosser properties are first presented, depending on an arbitrary relation $R$, an equivalence relation $E$ and a reduction relation $R^E $ used to compute normal forms of $R$ modulo $E$. Terminating rewriting systems operating on equational congruence classes of terms of a free algebra are then considered. In this framework, the Church–Rosser property is proved decidable for a very general reduction relation which may take into account the left-linearity of rules for efficiency reasons, under the only assumption of existence of a complete and finite unification algorithm for the underlying equational theory, whose congruence classes are assumed to be finite. This extends previous results by Lankford and Ballantyne, Peterson and Stickel, Huet, Jouannaud. A general completion procedure for mixed sets of rules and equations is then presented that generalizes and improves Peterson and Stickel’s one. In addition to computing a Church–Rosser set of rules when it terminates, it yields a semi-decision procedure for testing equality when it runs forever. Finally a post-processor is described that yields a Church–Rosser set of inter-reduced rules. All proofs, including the correctness proof of our completion algorithm, are based on the powerful proof technique of multiset induction.
Close

Your Recent History Recent History Help

Recently Viewed

  1. The Generalized Graetz Problem in Finite Domains
  2. Self-Similar Voiding Solutions of a Single Layered Model of Folding Rocks
  3. Geometry-Driven Charge Accumulation in Electrokinetic Flows between Thin, Closely Spaced Laminates
  4. A Generalized Birkhoff–Rott Equation for Two-dimensional Active Scalar Problems
  5. Lyapunov Functions and Global Stability for Age-Structured HIV Infection Model
© SIAM By using SIAM Publications Online you agree to abide by the Terms and Conditions of Use. Banner art adapted from a figure by Hinke M. Osinga and Bernd Krauskopf (University of Bristol, UK).
close