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SIAM J. on Computing

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1994

Volume 23, Issue 3, pp. 449-669


Top-Bottom Routing around a Rectangle is as Easy as Computing Prefix Minima

Omer Berkman, Joseph JáJá, Sridhar Krishnamurthy, Ramakrishna Thurimella, and Uzi Vishkin

SIAM J. Comput. 23, pp. 449-465 (17 pages)

Online Publication Date: July 31, 2006

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A new parallel algorithm for the prefix minima problem is presented for inputs drawn from the range of integers $[1..s]$. For an input of size $n$, it runs in $O(\log \log \log s)$ time and $O(n)$ work (which is optimal). A faster algorithm is presented for the special case $s = n$; it runs in $O(\log ^ * n)$ time with optimal work. Both algorithms are for the Priority concurrent-read concurrent-write parallel random access machine (CROW PRAM). A possibly surprising outcome of this work is that, whenever the range of the input is restricted, the prefix minima problem can be solved significantly faster than the $\Omega (\log \log n)$ time lower bound in a decision model of parallel computation, as described by Valiant [SIAM J. Comput., 4 (1975), pp. 348–355].
The top-bottom routing problem, which is an important subproblem of routing wires around a rectangle in two layers, is also considered. It is established that, for parallel (and hence for serial) computation, the problem of top-bottom routing is no harder than the prefix minima problem with $s = n$, thus giving an $O(\log ^ * n)$ time optimal parallel algorithm for the top-bottom routing problem. This is one of the first nontrivial problems to be given an optimal parallel algorithm that runs in sublogarithmic time.

Faster Approximation Algorithms For the Unit Capacity Concurrent Flow Problem with Applications to Routing and Finding Sparse Cuts

Philip Klein, Serge Plotkin, Clifford Stein, and Éva Tardos

SIAM J. Comput. 23, pp. 466-487 (22 pages) | Cited 15 times

Online Publication Date: July 31, 2006

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This paper describes new algorithms for approximately solving the concurrent multicommodity flow problem with uniform capacities. These algorithms are much faster than algorithms discovered previously. Besides being an important problem in its own right, the uniform-capacity concurrent flow problem has many interesting applications. Leighton and Rao used uniform-capacity concurrent flow to find an approximately “sparsest cut” in a graph and thereby approximately solve a wide variety of graph problems, including minimum feedback arc set, minimum cut linear arrangement, and minimum area layout. However, their method appeared to be impractical as it required solving a large linear program. This paper shows that their method might be practical by giving an $O(m^2 \log m)$ expected-time randomized algorithm for their concurrent flow problem on an $m$-edge graph. Raghavan and Thompson used uniform-capacity concurrent flow to solve approximately a channel width minimization problem in very large scale integration. An $O(k^{{3 / 2}} (m + n\log n)$ expected-time randomized algorithm and an $O(k\min \{ n,k\} (m + n\log n)\log k)$ deterministic algorithm is given for this problem when the channel width is $\Omega (\log n)$, where $k$ denotes the number of wires to be routed in an $n$-node, $m$-edge network.

The Extended Low Hierarchy is an Infinite Hierarchy

Ming-Jye Sheu and Timothy J. Long

SIAM J. Comput. 23, pp. 488-509 (22 pages)

Online Publication Date: July 31, 2006

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Balcázar, Book, and Schöning introduced the extended low hierarchy based on the $\Sigma $-levels of the polynomial-time hierarchy as follows: for $k \geqslant 1$, level $k$ of the extended low hierarchy is the set $EL_k^{P,\Sigma } = \{ A|\Sigma _k^P (A) \subseteq \Sigma _{k - 1}^P (A \oplus {\text{SAT}})\} $. Allender and Hemachandra and Long and Sheu introduced refinements of the extended low hierarchy based on the $\Delta $- and $\Theta $-levels, respectively, of the polynomial-time hierarchy: for $k \geqslant 2$, level $k$, $EL_k^{P,\Delta } = \{ A|\Delta _k^P (A) \subseteq \Delta _{k - 1}^P (A \oplus {\text{SAT}})\} $ and $EL_k^{P,\Theta } = \{ A|\Theta _k^P (A) \subseteq \Theta _{k - 1}^P (A \oplus {\text{SAT}})\} $. This paper shows that the extended low hierarchy is properly infinite by showing, for $k \geqslant 2$, that $EL_k^{P,\Sigma } \subsetneq EL_{k + 1}^{P,\Theta } \subsetneq EL_{k + 1}^{P,\Delta } \subsetneq EL_{k + 1}^{P,\sum } $. The proofs use the circuit lower bound techniques of Håstad and Ko. As corollaries to the constructions, for $k \geqslant 2$, oracle sets $B_k $, $C_k $, and $D_k $, such that ${\operatorname{PH}}(B_k ) = \Sigma _k^P (B_k ) \supsetneq \Delta _k^P (B_k )$, ${\operatorname{PH}}(C_k ) = \Delta _k^P (C_k ) \supsetneq \Theta _k^P (C_k )$, and ${\operatorname{PH}}(D_k ) = \Theta _k^P (D_k ) \supsetneq \sum _{k - 1}^P (D_k )$ are obtained.

Complexity of Network Reliability and Optimal Resource Placement Problems

Donald B. Johnson and Larry Raab

SIAM J. Comput. 23, pp. 510-519 (10 pages)

Online Publication Date: July 31, 2006

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A fundamental problem of distributed system design in an existing network where components can fail is finding an optimal location at which to place a resource. This paper proves exactly how hard this placement problem is under the measure of data availability. Specifically, it shows that the optimal placement problem for availability is #$P$-complete, a measure of intractability at least as severe as $NP$-completeness. To obtain these results, the environment in which a distributed system operates is modelled by a probabilistic graph, which is a set of fully reliable vertices representing sites and a set of edges representing communication links, each operational with a rational probability. Finding the optimal placement in a probabilistic graph is proved to be #$P$-complete by giving a sequence of Turing reductions from #Satisfiability. This result is generalized to networks in which each site and each link has an independent, rational operational probability and to networks in which all the sites or all the links have fixed, uniform operational probabilities. Given the anticipated computational difficulty of finding an exact solution, the requirements for effective, practical approximation methods are discussed.

Polynomial Algorithms for Hamiltonian Cycle in Cocomparability Graphs

Jitender S. Deogun and George Steiner

SIAM J. Comput. 23, pp. 520-552 (33 pages) | Cited 7 times

Online Publication Date: July 31, 2006

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Finding a Hamiltonian cycle in a graph is one of the classical NP-complete problems. Complexity of the Hamiltonian problem in permutation graphs has been a well-known open problem. In this paper the authors settle the complexity of the Hamiltonian problem in the more general class of cocomparability graphs. It is shown that the Hamiltonian cycle existence problem for cocomparability graphs is in $P$. A polynomial time algorithm for constructing a Hamiltonian path and cycle is also presented. The approach is based on exploiting the relationship between the Hamiltonian problem in a cocomparability graph and the bump number problem in a partial order corresponding to the transitive orientation of its complementary graph.

Packet Transmission in a Noisy-Channel Ring Network

Victor Pestien, S. Ramakrishnan, and Dilip Sarkar

SIAM J. Comput. 23, pp. 553-562 (10 pages) | Cited 1 time

Online Publication Date: July 31, 2006

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Assume that $n$ stations, each with a buffer to hold only one packet at a time, are connected as a ring and that data packets are transmitted counterclockwise. A station will attempt to transmit a packet to the next station only if (i) it has a packet to send and (ii) the next station’s buffer is empty. The communication cannels connecting the stations are noisy, and there is a fixed probability $p(0 < p < 1)$ of error-free transmission of a packet from one station to the next in one attempt.
An exact expression for the long-run average time for a packet to go around the ring is derived. (A special case of this answers a question raised by Berman and Simon in [Proc. 20th ACM Symp. on Theory of Computing, ACM Press, 1988, pp. 66–77].) For fixed $n$ and $p$, the throughput of the system is maximum when the number of packets is an integer closest to ${n / 2}$.

Subquadratic Simulations of Balanced Formulae by Branching Programs

Jin-Yi Cai and Richard J. Lipton

SIAM J. Comput. 23, pp. 563-572 (10 pages)

Online Publication Date: July 31, 2006

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This paper considers Boolean formulae and their simulations by bounded width branching programs. It is shown that every balanced Boolean formula of size $s$ can be simulated by a constant width (width 5) branching program of length $s^{1.811 \ldots } $. A lower bound for the translational cost from formulae to permutation branching programs is also presented.

Linear Time Algorithms and NP-Complete Problems

Etienne Grandjean

SIAM J. Comput. 23, pp. 573-597 (25 pages) | Cited 4 times

Online Publication Date: July 31, 2006

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This paper defines and studies a computational model (a random access machine with powerful input/output instructions), and shows that the classes ${\text{DLINEAR}}$ and ${\text{NLINEAR}}$ of problems computable in deterministic (respectively, nondeterministic) linear time in this model of computation are robust and powerful. In particular, ${\text{DLINEAR}}$ includes most of the concrete problems commonly regarded as computable in linear time (such as graph problems, topological sorting, strong connectivity, etc.). Most combinatorial NP-complete problems are in ${\text{NLINEAR}}$. The interest of ${\text{NLINEAR}}$ class is enhanced by the fact that some natural NP-complete problems, for example, “reduction of incompletely specified automata” $({\text{RISA}})$, are ${\text{NLINEAR}}$-complete (consequently, ${\text{NLINEAR}} \ne {\text{ DLINEAR}}$ if and only if ${\text{RISA}} \notin {\text{DLINEAR}}$). This notion strengthens NP-completeness, as this paper argues that propositional satisfiability is not ${\text{NLINEAR}}$ complete.

Digital Search Trees Again Revisited: The Internal Path Length Perspective

Peter Kirschenhofer, Helmut Prodinger, and Wojciech Szpankowski

SIAM J. Comput. 23, pp. 598-616 (19 pages) | Cited 5 times

Online Publication Date: July 31, 2006

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This paper studies the asymptotics of the variance for the internal path length in a symmetric digital search tree under the Bernoulli model. This problem has been open until now. It is proved that the variance is asymptotically equal to $N \cdot 0.26600 + N \cdot \delta (\log _2 N)$, where $N$ is the number of stored records and $\delta (x)$ is a periodic function of mean zero and a very small amplitude. This result completes a series of studies devoted to the asymptotic analysis of the variances of digital tree parameters in the symmetric case. In order to prove the previous result a number of nontrivial problems concerning analytic continuations and some others of a numerical nature had to be solved. In fact, some of these techniques are motivated by the methodology introduced in an influential paper by Flajolet and Sedgewick.

Improved Approximation Algorithms for Shop Scheduling Problems

David B. Shmoys, Clifford Stein, and Joel Wein

SIAM J. Comput. 23, pp. 617-632 (16 pages) | Cited 31 times

Online Publication Date: July 31, 2006

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In the job shop scheduling problem, there are $m$ machines and $n$ jobs. A job consists of a sequence of operations, each of which must be processed on a specified machine, and the aim is to complete all jobs as quickly as possible. This problem is strongly .$\mathcal{NP}$-hard even for very restrictive special cases. The authors give the first randomized and deterministic polynomial-time algorithms that yield polylogarithmic approximations to the optimal length schedule. These algorithms also extend to the more general case where a job is given not by a linear ordering of the machines on which it must be processed but by an arbitrary partial order. Comparable bounds can also be obtained when there are $m'$ types of machines, a specified number of machines of each type, and each operation must be processed on one of the machines of a specified type, as well as for the problem of scheduling unrelated parallel machines subject to chain precedence constraints.

Randomized Algorithms for Binary Search and Load Balancing on Fixed Connection Networks with Geometric Applications

John H. Reif and Sandeep Sen

SIAM J. Comput. 23, pp. 633-651 (19 pages) | Cited 1 time

Online Publication Date: July 31, 2006

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There are now a number of fundamental problems in computational geometry that have optimal algorithms on PRAM models. This paper presents randomized parallel algorithms that execute on an $n$-processor butterfly interconnection network in $O(\log n)$ time for the following problems of input size $n$: trapezoidal decomposition, visibility, triangulation, and two-dimensional convex hull. These algorithms involve tackling some of the very basic problems, like binary search and load balancing, that are taken for granted in PRAM models. Apart from a two-dimensional convex hull algorithm, these are the first nontrivial geometric algorithms that attain this performance on fixed connection networks. These techniques use a number of ideas from Flashsort that have to be modified to handle more difficult situations; it seems likely that they will have wider applications.

Communication-Space Tradeoffs for Unrestricted Protocols

Paul Beame, Martin Tompa, and Peiyuan Yan

SIAM J. Comput. 23, pp. 652-661 (10 pages) | Cited 2 times

Online Publication Date: July 31, 2006

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This paper introduces communicating branching programs and develops a general technique for demonstrating communication-space tradeoffs for pairs of communicating branching programs. This technique is then used to prove communication-space tradeoffs for any pair of communicating branching programs that hashes according to a universal family of hash functions. Other tradeoffs follow from this result. As an example, any pair of communicating Boolean branching programs that computes matrix-vector products over ${\text{GF}}(2)$ requires communication-space product $\Omega (n^2 )$, provided the space used is $o({n / {\log n}})$. These are the first examples of communication-space tradeoffs on a completely general model of communicating processes.

A New Insight into the Coffman–Graham Algorithm

Bertrand Braschi and Denis Trystram

SIAM J. Comput. 23, pp. 662-669 (8 pages) | Cited 5 times

Online Publication Date: July 31, 2006

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The approximate solution of the $m$-machine problem is addressed. The Lam–Sethi worst-case analysis of the Coffman–Graham algorithm is set up to be partly incorrect. A slightly different context is set up to correct and complete this analysis. It is shown that the makespan of a schedule computed by an extended Coffman–Graham algorithm is lower than or at worst equal to $({{2 - 2} / m})\omega _{{\text{opt}}} - {{(m - 3)} / m}$, where $\omega _{{\text{opt}}} $ is the minimal makespan of a schedule.
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