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SIAM J. on Computing

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1993

Volume 22, Issue 1, pp. 1-219


Implicit $O(1)$ Probe Search

Amos Fiat and Moni Naor

SIAM J. Comput. 22, pp. 1-10 (10 pages) | Cited 7 times

Online Publication Date: July 13, 2006

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Given a set of $n$ elements from the domain $\{ {1, \cdots ,m} \}$, this paper investigates how to arrange them in a table of size $n$, so that searching for an element in the table can be done in constant time. Yao [J. Assoc. Comput. Mach., 28(1981), pp. 615–628] has shown that this cannot be done when the domain is sufficiently large as a function of $n$.
This paper gives a constructive solution when the domain $m$ is polynomial in $n$, the number of elements, as well as a nonconstructive proof for $m$ no larger than exponential in ${\operatorname{poly}}(n)$. The authors improve upon a result of Yao and give better bounds on the maximum $m$ for which implicit $O(1)$ probe search can be done. The results are achieved by showing the tight relationship between hashing and certain encoding problems called rainbows.

Maintaining the $3$-Edge-Connected Components of a Graph On-Line

Zvi Galil and Giuseppe F. Italiano

SIAM J. Comput. 22, pp. 11-28 (18 pages) | Cited 10 times

Online Publication Date: July 13, 2006

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The problem of maintaining the $3$-edge-connected components of a graph undergoing repeated dynamic modifications, such as edge and vertex insertions, is studied. This paper shows how to answer the question of whether or not two vertices belong to the same $3$-edge-connected component of a connected graph that is undergoing only edge insertions. Any sequence of $q$ query and updates on an $n$-vertex graph can be performed in $O((n + q)\alpha (q,n))$ time.

On the Boundedness of Constant-Time-Maintainable Database Schemes

Héctor J. Hernández and Ke Wang

SIAM J. Comput. 22, pp. 29-45 (17 pages)

Online Publication Date: July 13, 2006

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Constant-time-maintainable database schemes are highly desirable with respect to constraint enforcement, since it is possible to determine whether any of their consistent states plus an inserted tuple is consistent in time independent of the state size. Several proper subclasses of constant-time-maintainable database schemes are known to be bounded with respect to dependencies and hence very desirable with respect to query answering. However, whether the whole class of constant-time-maintainable database schemes is bounded is not known for sure.
In this paper, it is proven that the entire class of constant-time-maintainable database schemes is bounded with respect to dependencies and thus very desirable with respect to query answering in the following cases: (1) only cover-embedded functional dependencies appear as constraints; (2) only equality-generating dependencies appear as constraints and the database scheme has a lossless join. In particular, it is shown that total projections of representative instances can be computed via unions of projections of simple chase join expressions. Since it is known how to optimize these expressions, it is possible to compute total projections optimally. These results show that the class of constant-time-maintainable database schemes is the largest class of database schemes, which are highly desirable with respect to both constraint enforcement and query answering. This class of schemes can be effectively recognized by known algorithms. The previously known largest class of database schemes with these desirable properties is the class of independent database schemes, which is a proper subclass of constant-time-maintainable schemes.

Tight Worst-Case Performance Bounds for Next-$k$-Fit Bin Packing

Weizhen Mao

SIAM J. Comput. 22, pp. 46-56 (11 pages) | Cited 5 times

Online Publication Date: July 13, 2006

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The bin packing problem is to pack a list of reals in $( {0,1} ]$ into unit-capacity bins using the minimum number of bins. Let $R[A]$ be the limiting worst value for the ratio ${{A(L)} / {L^ * }}$ as $L^ * $ goes to $\infty $, where $A(L)$ denotes the number of bins used in the approximation algorithm $A$, and $L^ * $ denotes the minimum number of bins needed to pack $L$. Obviously, $R[A]$ reflects the worst-case behavior of $A$. For Next-$k$-Fit($NkF$ for short, $k \geqslant 2$), which is a linear time approximation algorithm for bin packing, it was known that $1.7 + \frac{3}{{10(k - 1)}} \leqslant R[NkF] \leqslant 2$. In this paper, a tight bound $R[NkF] = 1.7 + \frac{3}{{10(k - 1)}}$ is proved.

A Note on the Complexity of a Simple Transportation Problem

Greg N. Frederickson

SIAM J. Comput. 22, pp. 57-61 (5 pages) | Cited 7 times

Online Publication Date: July 13, 2006

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Consider the problem of using a vehicle to transport $k$ objects one at a time between $s$ stations on a circular track. Let the cost of the transportation be the total distance traveled by the vehicle on the track. An $O(k + M(s,q))$ time algorithm is presented to find a minimum cost transportation, where $M(m,n)$ is the time to solve a minimum spanning tree problem on a graph with $m$ edges and $n$ vertices, and $q \leqslant \min \{ {k,s} \}$ is the number of strongly connected components in an associated balanced problem. Also, the minimum spanning tree problem on a graph with $m$ edges and $n$ vertices is reduced to a transportation problem on a linear track with $O(m)$ stations, $O(m)$ objects, and $O(n)$ strongly connected components in $O(m)$ time.

A Lower Bound on the Size of Shellsort Sorting Networks

Robert Cypher

SIAM J. Comput. 22, pp. 62-71 (10 pages) | Cited 1 time

Online Publication Date: July 13, 2006

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Shellsort is a sorting algorithm that is based on a set of parameters called increments. Shellsort has been used both as a sequential sorting algorithm and as a sorting network. The central result of this paper is that all Shellsort sorting networks based on monotonically decreasing increments require $\Omega (N\log ^2 {N / {\log \log N}})$ comparators. Previously, only the trivial $\Omega (N\log N)$ bound was known for this class of networks. The lower bound obtained in this paper nearly matches the upper bound of $O(N\log ^2 N)$ that was proven by Pratt.

A Note on Poset Geometries

Joel Friedman

SIAM J. Comput. 22, pp. 72-78 (7 pages) | Cited 1 time

Online Publication Date: July 13, 2006

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This note describes how varying the geometric representation of a poset can be applied to “poset balancing.” It is shown that the ${1 / 3}$, ${2 / 3}$ balancing property holds for a certain class of posets whose number of relations is sufficiently small, in a certain sense.

An $O(n)$ Algorithm for Determining the Subregion-Tree Representation of a Rectangular Dissection

Sukhamay Kundu

SIAM J. Comput. 22, pp. 79-101 (23 pages)

Online Publication Date: July 13, 2006

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A rectangular dissection is a partition of a rectangular space $R$ into $n \geqslant 1$ disjoint rectangles $\{ {r_1 ,r_2 , \cdots ,r_n } \}$. A $T_ * $-plan is a dissection that is obtained by repeated application of the (1) horizontal, (2) vertical, (3) left-spiral, and (4) right-spiral partitioning operations. Two common ways of representing a $T_ * $-plan are the wall representation $w(D)$ and the subregion-tree representation $t(D)$. It is known [S. Kundu, Comm. ACM, 31 (1988), pp. 752–763] that these two representations are equivalent in that one can be uniquely determined from the other. This paper presents an optimal $O(n)$ algorithm for constructing $t(D)$ from $w(D)$, which improves the previous bound of $O(n^2 )$ in [S. Kundu, Comm. ACM, 31 (1988), pp. 752–763]. The new algorithm is based on a domination relationship among the walls, which is defined here and represented by a digraph $G_w (D)$. The algorithm exploits the disjoint cycle property of $G_w (D)$ and the relationship between the tree $t(D)$ and the transitive reduction of the acyclic digraph obtained by merging the cycles of $G_w (D)$ into distinct nodes. The new method of constructing the tree $t(D)$ by means of the digraph $G_w (D)$ can be applied to an arbitrary class of dissections $D$ that are generated by a finite family of partitioning operations that satisfies certain natural restrictions. The complexity of the algorithm remains $O(n)$ for many such families.

Tally Versions of the Savitch and Immerman–Szelepcsényi Theorems for Sublogarithmic Space

Viliam Geffert

SIAM J. Comput. 22, pp. 102-113 (12 pages) | Cited 3 times

Online Publication Date: July 13, 2006

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It is shown that for each $s(n)$-space-bounded nondeterministic Turing machine recognizing a language $L \subseteq 1^ * $ there exists an equivalent deterministic $O(s^2 (n))$-space-bounded machine, and also a nondeterministic $O(s(n))$-space-bounded machine recognizing the complement of $L$, for any $s(n)$, independent of whether $s(n)$ is below $\log (n)$ or is space constructible. In other words, the Savitch [J. Comput. System Sci., 4(1970), pp. 177–192] and Immerman–Szelepcsényi [SIAM J. Comput., 17(1988), pp. 935–938], [Acts Inform., 26(1988), pp. 279–284] theorems can be extended to any space bound $s(n)$ for languages over a single-letter alphabet.

NV-Sequentiality: A Decidable Condition for Call-by-Need Computations in Term-Rewriting Systems

Michio Oyamaguchi

SIAM J. Comput. 22, pp. 114-135 (22 pages) | Cited 2 times

Online Publication Date: July 13, 2006

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In 1979 Huet and Levy introduced the class of sequential term-rewriting systems in which call-by-need computations are possible (without look-ahead) and defined the subclass called strongly sequential systems for which needed redexes in a given term are effectively found [chapter in Computational Logic: Essays in Honor of Alan Robinson, J.-L. Lassez and G. Plotkin, eds., MIT Press, Cambridge, MA, 1991]. This paper introduces a larger subclass that is a natural extension of strong sequentiality and is based on the analysis of both the left-hand sides and part of the right-hand sides (i.e., the nonvariable parts) of systems, whereas strong sequentiality is based on the analysis of left-hand sides alone. This new sequentiality is called NV-sequentiality. It is shown that (i) the class of NV-sequential systems properly includes the class of strongly sequential systems, (ii) there exists an algorithm for finding needed redexes for a given term when a system is NV-sequential, and (iii) it is decidable whether an arbitrary left-linear system is NV-sequential.

${\text{ASPACE}}(o(\log \log n))$ is Regular

Kazuo Iwama

SIAM J. Comput. 22, pp. 136-146 (11 pages) | Cited 2 times

Online Publication Date: July 13, 2006

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One of the common results of resource bounded Turing machines is the $\log \log n$ lower bound for the space usage of deterministic and nondeterministic Turing machines that accept nonregular languages. In this paper this result is extended to alternating Turing machines: It is proved that if $f(n) = o(\log \log n)$, then $f(n)$-space-bounded (off-line) alternating Turing machines can accept only regular sets. The problem has been open for a decade.

The Complexity of Malign Measures

Peter Bro Miltersen

SIAM J. Comput. 22, pp. 147-156 (10 pages) | Cited 1 time

Online Publication Date: July 13, 2006

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This paper analyzes the concept of malignness, which is the property of probability ensembles making the average case running time equal to the worst case running time for a class of algorithms. The author derives lower and upper bounds on the complexity of malign ensembles, which are tight for exponential time algorithms, and which show that no polynomial time computable malign ensemble exists for the class of polynomial time algorithms. Furthermore, it is shown that for no class of superlinear algorithms a polynomial time samplable malign ensemble exists, unless every language in $P$ has an expected polynomial time constructor.

Scan-First Search and Sparse Certificates: An Improved Parallel Algorithm for $k$-Vertex Connectivity

Joseph Cheriyan, Ming-Yang Kao, and Ramakrishna Thurimella

SIAM J. Comput. 22, pp. 157-174 (18 pages) | Cited 9 times

Online Publication Date: July 13, 2006

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Given a graph $G = (V,E)$, a certificate of $k$-vertex connectivity is an edge subset $E' \subset E$ such that the subgraph $(V,E')$ is $k$-vertex connected if and only if $G$ is $k$-vertex connected. Let $n$ and $m$ denote the number of vertices and edges. A certificate is called sparse if it contains $O(kn)$ edges.
For undirected graphs, this paper introduces a graph search called the scan-first search, and shows that a certificate with at most $k(n - 1)$ edges can be computed by executing scan-first search $k$ times in sequence on subgraphs of $G$. For each of the parallel, distributed, and sequential models of computation, the complexity of scan-first search matches the best complexity of any graph search on that model. In particular, the parallel scan-first search runs in $O(\log n)$ time using $C(n,m)$ processors on a CRCW PRAM, where $C(n,m)$ is the number of processors needed to find a spanning tree in each connected component in $O(\log n)$ time, and the parallel certificate algorithm runs in $O(k\log n)$ time using $C(n,m)$ processors. The parallel certificate algorithm can be employed to test the $k$-vertex connectivity of an undirected graph in $O(k^2 \log n)$ time using $knC(n,kn)$ processors on a CRCW PRAM. For all combinations of $n$, $m$, and $k > 3$, both the running time and the number of processors either improve on or match those of all known deterministic parallel algorithms.
This paper also obtains an online algorithm for computing an undirected graph certificate with at most $2kn$ edges, and a sequential algorithm for computing a directed graph certificate with at most $2k^2 n$ edges.

Decomposing Finite-Valued Transducers and Deciding Their Equivalence

Andreas Weber

SIAM J. Comput. 22, pp. 175-202 (28 pages) | Cited 3 times

Online Publication Date: July 13, 2006

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In this paper finite-valued finite transducers are investigated in connection with their inner structure. The following results are shown: A finite-valued nondeterministic generalized sequential machine (NGSM) $M$ can be effectively decomposed into finitely many single-valued NGSMs $M_1 , \ldots ,M_N $ such that the transduction realized by $M$ is the union of the transductions realized by $M_1 , \ldots ,M_N $. Using this decomposition, the equivalence of finite-valued NGSMs is decidable in deterministic double exponential time. By reduction, both results can be generalized to normalized finite transducers.

One More Occurrence of Variables Makes Satisfiability Jump from Trivial to NP-Complete

Jan Kratochvíl, Petr Savický, and Zsolt Tuza

SIAM J. Comput. 22, pp. 203-210 (8 pages) | Cited 4 times

Online Publication Date: July 13, 2006

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A Boolean formula in a conjunctive normal form is called a $(k,s)$ – formula if every clause contains exactly $k$ variables and every variable occurs in at most $s$ clauses. The $(k,s)$–${\text{SAT}}$ problem is the SATISFIABILITY problem restricted to $(k,s)$–formulas. It is proved that for every $k \geqslant 3$ there is an integer $f(k)$ such that $(k,s)$–${\text{SAT}}$ is trivial for $s \leqslant f(k)$ (because every $(k,s)$–formula is satisfiable) and is NP-complete for $s \geqslant f(k) + 1$. Moreover, $f(k)$ grows exponentially with $k$, namely, $\lfloor {{{2^k } / {ek}}} \rfloor \leqslant f(k) \leqslant 2^{k - 1} - 2^{k - 4} - 1$ for $k \geqslant 4$.

Rounds in Communication Complexity Revisited

Noam Nisan and Avi Wigderson

SIAM J. Comput. 22, pp. 211-219 (9 pages) | Cited 9 times

Online Publication Date: July 13, 2006

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The $k$-round two-party communication complexity was studied in the deterministic model by [P. H. Papadimitriou and M. Sipser, Proc. of the 14th STOC, 1982, pp. 330–337] and [P. Duris, Z. Galil, and G. Schnitger, Proc. of the 16th STOC, 1984, pp. 81–91] and in the probabilistic model by [A. C. Yao, Proc. of the 24th FOCS, 1983, pp. 420–428] and [B. Halstenberg and R. Reischuk, Proc. of the 20th STOC, 1988, pp. 162–172]. This paper presents new lower bounds that give (1) randomization is more powerful than determinism in $k$-round protocols, and (2) an explicit function which exhibits an exponential gap between its $k$ and $(k - 1)$-round randomized complexity.
This paper also studies the three-party communication model, and exhibits an exponential gap in 3-round protocols that differ in the starting player.
Finally, this paper shows new connections of these questions to circuit complexity, that motivate further work in this direction.
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