R. Greenlaw, H.J. Hoover, and W.L. Ruzzo. A compendium of problems complete for P . Technical Report 91-05-01, University of Washington, 1991.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....decorates a derivation tree of an attribute grammar. 6.2 The Borderline Since it is quite easy to simulate a Turing machine by an AG, it is easy to construct P complete problems using AGs. The list of known P complete problems is steadily growing (as can be seen in the following bibliographies: [GHR91a, GHR91b, GHR91c, MSS90]) We will restrict the dependency graphs and attribute functions somewhat such that we are able to perform the decoration in polylogarithmic time. Unfortunately also for a dependency graph that is a tree, there exists a P complete problem accompanying it. When we look at an expression over an ....

R. Greenlaw, H.J. Hoover, and W.L. Ruzzo. A compendium of problems complete for P . Technical Report 91-05-01, University of Washington, 1991.

....decorates a derivation tree of an attribute grammar. 6.2 The Borderline Since it is quite easy to simulate a Turing machine by an AG, it is easy to construct P complete problems using AGs. The list of known P complete problems is steadily growing (as can be seen in the following bibliographies: [GHR91a, GHR91b, GHR91c, MSS90]) We will restrict the dependency graphs and attribute functions somewhat such that we are able to perform the decoration in polylogarithmic time. Unfortunately also for a dependency graph that is a tree, there exists a P complete problem accompanying it. When we look at an expression over an ....

R. Greenlaw, H.J. Hoover, and W.L. Ruzzo. A compendium of problems complete for P . Technical Report 91-14, University of New Hampshire, 1991.

....decorates a derivation tree of an attribute grammar. 6.2 The Borderline Since it is quite easy to simulate a Turing machine by an AG, it is easy to construct P complete problems using AGs. The list of known P complete problems is steadily growing (as can be seen in the following bibliographies: [GHR91a, GHR91b, GHR91c, MSS90]) We will restrict the dependency graphs and attribute functions somewhat such that we are able to perform the decoration in polylogarithmic time. Unfortunately also for a dependency graph that is a tree, there exists a P complete problem accompanying it. When we look at an expression over an ....

R. Greenlaw, H.J. Hoover, and W.L. Ruzzo. A compendium of problems complete for P . Technical Report 91-11, University of Alberta, 1991.

....3 it follows that the CTP is in P . The P hardness follows from a reduction of the Circuit Value Problem (CVP) defined as: For a given Boolean circuit C and an input string x decide whether C on input x outputs 1, i.e. compute res C (x) It is well known that the CVP problem is P complete (see [GHR91]) Define the circuit CHAIN l (b) as a circuit with depth and size l that consists of l successing OR gates with input b. Let C and x be an instance of the CVP problem. Now define circuit C 0 : C OR CHAIN size(C) 1 (0) Note that timeC 0 (x) ae 1 timeC (x) if res C (x) 1 2 ....

R. Greenlaw, H. J. Hoover, W. L. Ruzzo, A Compendium of Problems Complete for P, Technical Report 91-05-01, University of Washington, 1991.

....1 (with right hand sides normalized to 1) can be output using log space. Each inequality is represented with a constant number of fractions chosen from f Sigma2 s 2 ; Gamman2 s 2 ; Sigma 2 s 2 2 s 2 1 ; n2 s 2 n2 s 2 1 g: 7 Surveys of P complete problems can be found in [HR85] [MSS89] 10 4. Positive Results Recall that we assumed that rationals are represented by writing their numerator and denominator in binary 8 . Given n and s, all these numbers can trivially be output using O(log ns) space. The theorem now easily follows. 2 4 Positive Results In this section ....

H.J. Hoover, W.L. Ruzzo. A compendium of problems complete for P. Technical Report, University of Washinton, 1986. References 17

....analogous fashion. The obvious question to explore is how the complexity classes incr POLYLOGTIME, incr POLYLOGSPACE, and incr LOGSPACE are related to the well known sequential complexity classes Our first intriguing result is that the commonly known P complete problems listed in [11] and [5] are incr POLYLOGTIME complete for the class P. These problems include the Circuit Value problem (CV) the Solvable Path System problem (SPS) Propositional Horn Satisfiability, and a host of other well known P complete problems. These problems are therefore as hard to solve incrementally as any ....

....the Circuit Value problem (CV) the Solvable Path System problem (SPS) Propositional Horn Satisfiability, and a host of other well known P complete problems. These problems are therefore as hard to solve incrementally as any other problem in P. Theorem 2 All P complete problems in [11] and [5] are incr POLYLOGTIME complete for the class P. Proof Sketch: We present the proof for the case of circuit value problem. Given any problem in P and an initial instance I 0 with jI 0 j n, we use the standard LOGSPACE reduction to create a circuit of size t(n) by t(n) where t(n) is a ....

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R. Greenlaw, H. J. Hoover, and W. L. Ruzzo, "A Compendium of Problems Complete for P," Department of Computer Science and Engineering, University of Washington, Technical Report TR-91-05-01, 1991.

....versions of the problems. Two such problems discussed in [6] are the Independent Set problem and the Clique problem. Usually the natural sequential algorithms for the lexicographic versions of such problems turn out to be P complete. For example, Lexicographically First Maximal Clique [8] and Lexicographically First Maximal Independent Set [4] are both P complete. In our study we define the ordered versions of various NP complete problems. Our definitions are algorithmic in nature and we obtain P completeness results for such problems. We find it interesting that the ordered ....

....we think it is interesting that the obvious approaches to LDS and HDS rely on LDVR and HDVR respectively. We showed that the ordered versions of all these problems were P complete, thus demonstrating that the problems are probably inherently sequential. The list of P complete problems given in [8] contains several lexicographic type problems. For example, the Lexicographically First Maximal Clique and the Lexicographically First Delta 1 Vertex Coloring are two such problems. We feel that our result showing that LexLDSM is NP complete provides an interesting contrast to these other ....

H. Hoover and L. Ruzzo. A compendium of problems complete for p. Draft, 1984.

....such that if x i receives an input from x j then i j. We will also restrict to circuits with not and or gates, where gates not have one input and one output and gates or have two inputs and at most two outputs. For such restricted circuits the circuit value problem remains P complete (Greenlaw et al. 1991). Thus we can assume that each x i in ff is chosen from the set ftrue; false; not(y i ) or(y i ; y 0 i ) y i ; y 0 i 2 fi 1; mg[f Gammam; Gammai Gamma1gg. Furthermore, we assume that if y i (y 0 i ) is negative, then x Gammay i (x Gammay 0 i ) is an or gate and y ....

Greenlaw, R., Hoover, H. J., and Ruzzo, W. L. (1991), "A compendium of problems complete for P," Technical Report TR 91-05-01, University of Washington.

....an analogous fashion. The obvious question to explore is how the complexity classes incr POLYLOGTIME, incr POLYLOGSPACE, and incr LOGSPACE are related to the well known sequential complexity classes Our first intriguing result is that the commonly known P complete problems listed in [18] and [10] are incr POLYLOGTIME complete for the class P. These problems include the Circuit Value problem (CV) the Solvable Path System problem (SPS) Propositional Horn Satisfiability, and a host of other well known P complete problems. These problems are therefore as hard to solve incrementally as any ....

....the Circuit Value problem (CV) the Solvable Path System problem (SPS) Propositional Horn Satisfiability, and a host of other well known P complete problems. These problems are therefore as hard to solve incrementally as any other problem in P. Theorem 3 All P complete problems in [18] and [10] are incr POLYLOGTIME complete for the class P. Proof : We present the proof for the case of circuit value problem. In this problem we are given a directed acyclic graph as the input. A node in this graph is either labeled an input node or an output node, or it corresponds to a gate in the ....

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R. Greenlaw, H. J. Hoover, and W. L. Ruzzo, "A Compendium of Problems Complete for P," Department of Computer Science and Engineering, University of Washington, Technical Report TR-91-05-01, 1991.

....such that if x i receives an input from x j then i j. We will also restrict to circuits with not and or gates, where gates not have one input and one output and gates or have two inputs and at most two outputs. For such restricted circuits the circuit value problem remains P complete [7]. We show how to simulate computation in a circuit ff by algorithm Greedy . The circuit ff is transformed in polylogarithmic time with polynomial number of processors into a weighted digraph. A certain edge will be chosen by Greedy if and only if the output of the circuit is true . The paths ....

R. Greenlaw, H. J. Hoover, and W. L. Ruzzo, A compendium of problems complete for P, Technical Report, University of Washington, 1991.

....theory, algebra, and graph theory. A partial list of such problems can be found in [8] and an up to date collection will appear in [15] Here we recall some of the examples which we will require later in our hardness proofs. We adopt the presentation format used by Greenlaw, Hoover, and Ruzzo [11]. 1 In fact, Cook and McKenzie use NC 1 circuits instead of AC 0 circuits. But with the exception of the reduction DFA BFS, where counting and thus at least TC 0 is needed, any of their reductions can be computed by AC 0 circuits. Directed Two Tree Accessibility (DTTA) Given: A ....

....to the complete subtree of T 1 rooted at v. NSTI is not as general as the Subtree Isomorphism (STI) Given: Two trees T 1 and T 2 . Problem: Determine whether there is subgraph of T 1 that is isomorphic to T 2 . STI is known to be in solvable in P RNC, but neither known to be Pcomplete [11] nor to be computable in NC. Our variant NSTI does not search for an arbitrary subgraph of T , but limits its quest to subgraphs that are complete subtrees subtended by particular nodes of T . Clearly, the NSTI question is an easier question, because the subgraphs relevant to NSTI have a very ....

R. Greenlaw, H. J. Hoover, and W. L. Ruzzo. A Compendium of Problems Complete for P . Oxford University Press, 1994.

....it is hardest to obtain a fast, efficient PRAM algorithm. Showing that any one of them was in NC would imply that all problems in P had fast, efficient PRAM algorithms. The first P complete problems were established in the early 1970s [136, 137, 164] Two recently published lists of such problems [110, 194] together contain around 250 problems. Those interested in P completeness results are strongly encouraged to consult [110] Two very simple P complete problems are the following. Subset Closure Given: A finite set X, a binary operation ffi on X, a subset S X, and an element x 2 X. To ....

....in P had fast, efficient PRAM algorithms. The first P complete problems were established in the early 1970s [136, 137, 164] Two recently published lists of such problems [110, 194] together contain around 250 problems. Those interested in P completeness results are strongly encouraged to consult [110]. Two very simple P complete problems are the following. Subset Closure Given: A finite set X, a binary operation ffi on X, a subset S X, and an element x 2 X. To determine: Whether x is contained in the smallest subset of X which contains S and is closed under ffi. Monotone Circuit Value ....

R Greenlaw, H J Hoover, and W L Ruzzo. A compendium of problems complete for P. Technical Report TR 91-05-01, Department of Computer Science, University of Washington, June 1991.

....made into NC algorithms or whether different fast parallel approximation algorithms do exist. 2. Several combinatorial problems in P turn out to be P complete so that it is unlikely that they can be exactly solved by an efficient parallel algorithm (see the list of P complete problems contained in [5]) It can thus be convenient to develop parallel approximation algorithms for these probems. 3. Parallel approximation algorithms may turn out to be more efficient than any known parallel exact algorithm. In other words, it can be useful to barter the quality of the solution with the efficiency of ....

R. Greenlaw, H.J. Hoover, and W.L. Ruzzo (1992), "A compendium of problems complete for P", University of Alberta, Computer Science Department, Technical Report 91-11.

....to the restrictions of Problem A.1.4, this version requires the circuit to be synchronous. That Part II: P Complete Problems ffl 49 is, each level in the circuit can receive its inputs only from gates on the preceding level. Problem: Does ff on input x 1 ; x n output 1 Reference: [GHR91] Hint: A proof is given in Section 5.2. The reduction is from AM2CVP. A.1.7 Planar Circuit Value Problem (PCVP) Given: An encoding of a planar Boolean circuit ff, that is one whose graph can be drawn in the plane with no edges crossing, plus inputs x 1 ; x n . Problem: Does ff on input ....

....labels for v i . A binary predicate P , where P ij (x; y) 1 if and only if the assignment of label x to v i is compatible with the assignment of label y to v j . A designated variable P 0 and a designated label f . Problem: Is there a valid assignment of the label f to P 0 in G Reference: [Kas86, GHR91] Hint: The original reduction is from the propositional Horn clause satisfiability problem, Problem A.6.3 [Kas86] The reduction we sketch is from NAND CVP. A variable is introduced for each circuit input. The variable must have label 1 (0) if the circuit input is true (false) We view each nand ....

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R. Greenlaw, H. Hoover, and W. L. Ruzzo. A compendium of problems complete for P , 1991. This work.

.... critical tree ranking O(log n) n 2 = log n CREW this paper super critical tree numbering O(log n) n 2 = log n CREW this paper 2 Super Critical Rankings and Numberings Background material on the class NC, P completeness theory, or parallel models of computation can be found, for example, in [3, 6, 9]. Let T be any tree. The notation T (u) is used to denote the subtree of T rooted at u. Let N = f0; 1; 2; g be the set of natural numbers and N = N Gamma f0g. If x 2 N then (x (m) x (1) denotes the m bit binary representation of x, where x (1) is the least significant ....

R. Greenlaw, H.J. Hoover, and W.L. Ruzzo. A compendium of problems complete for P . Technical Report TR 91-11, University of Alberta; TR 91-14, University of New Hampshire; and TR-91-05-01, University of Washington, 1991.

....to the restrictions of Problem A.1.4, this version requires the circuit to be synchronous. That Part II: P Complete Problems ffl 49 is, each level in the circuit can receive its inputs only from gates on the preceding level. Problem: Does ff on input x 1 ; x n output 1 Reference: [GHR91] Hint: A proof is given in Section 5.2. The reduction is from AM2CVP. A.1.7 Planar Circuit Value Problem (PCVP) Given: An encoding of a planar Boolean circuit ff, that is one whose graph can be drawn in the plane with no edges crossing, plus inputs x 1 ; x n . Problem: Does ff on input x ....

....labels for v i . A binary predicate P , where P ij (x; y) 1 if and only if the assignment of label x to v i is compatible with the assignment of label y to v j . A designated variable P 0 and a designated label f . Problem: Is there a valid assignment of the label f to P 0 in G Reference: [Kas86, GHR91] Hint: The original reduction is from the propositional Horn clause satisfiability problem, Problem A.6.3 [Kas86] The reduction we sketch is from NAND CVP. A variable is introduced for each circuit input. The variable must have label 1 (0) if the circuit input is true (false) We view each nand ....

[Article contains additional citation context not shown here]

R. Greenlaw, H. Hoover, and W. L. Ruzzo. A compendium of problems complete for P , 1991. This work.

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R. Greenlaw, H. J. Hoover, and W. L. Ruzzo. A compendium of problems complete for P. Technical Report, University of Washington, 1991.

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