R. L. Rivest and J. Vuillemin. On recognizing graph properties from adjacency matrices. TCS, 3:371-384, 1976.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....used for a worst case input. A basic problem studied for this model is the complexity of monotone graph properties. Here the input is a graph (on n vertices) and the function is the indicator of a monotone (i.e. upward closed) family of such graphs A. A fundamental result of Rivest and Vuillemin [RV76] is that every such function requires ) queries. In other words, the trivial algorithm of querying the whole input is not worse than the best algorithm by more than a constant factor. We assume throughout that properties A are nontrivial, meaning A contains some, but not all graphs. ....

R. Rivest and J. Vuillemin, On recognizing graph properties from adjacency matrices, TCS 3, pp. 371-384, 1976.

....or adding one edge. Thus, the C 5 is bicritically imperfect and Strategy 1 does also work for n = 5 with choosing G n = C 5 . We are also able to settle the case n = 6 by providing some special strategy. For n = 8 we were able to prove elusiveness by making use of a result of Rivest and Vuillemin [10] and by doing some computer search. Details for the case n = 6 nd n = 8 are given in a forthcoming paper. There we also hope to settle the case n = 7 which is currently the only case for which it is not known whether perfectness is elusive. ....

R.L. Rivest and J. Vuillemin, On recognizing graph properties from adjacency matrices, Theor. Comput. Sci. 3 (1976/77) pp. 371-384

....to an earlier version of the famous Anderaa Rosenberg conjecture, which stated that any nontrivial graph property that is invariant under graph isomorphism requires n 2 ) probes of the adjacency matrix to test. Anderaa disproved this version in 1975 using a di erent counterexample (see [18]) but conjectured that it held for monotone properties (those that cannot change from false to true when edges are deleted) This was later veri ed by Rivest and Vuillemin [18] Homework 2 1: Extendible Arrays The array doubling idea is folklore. A generalization of this idea has been ....

....2 ) probes of the adjacency matrix to test. Anderaa disproved this version in 1975 using a di erent counterexample (see [18] but conjectured that it held for monotone properties (those that cannot change from false to true when edges are deleted) This was later veri ed by Rivest and Vuillemin [18]. Homework 2 1: Extendible Arrays The array doubling idea is folklore. A generalization of this idea has been described by Overmars in [16] for the design of dynamic data structures. 2: Binomial Heaps A proof that binomial heaps support merging in amortized constant time was given in [10] ....

R. L. Rivest and J. Vuillemin. On recognizing graph properties from adjacency matrices. Theoretical Computer Science, 3:371-384, 1976/77.

.... of the above conjecture if we replace non evasive with the respective weaker requirements ( denotes the reduced Euler characteristic of a simplicial complex) These generalized conjectures all hold once again for prime power numbers of vertices by a theorem of Rivest and Vuillemin [13] (see also [11] Yet, for non prime power numbers there are counterexamples known to all of the generalized conjectures with the exception of the Evasiveness Conjecture for simplicial complexes which still remains open. There is an abundance of (non trivial) Q acyclic vertex homogeneous ....

R. L. Rivest and J. Vuillemin. On recognizing graph properties from adjacency matrices. Theor. Comput. Sci. 3, 371-384 (1976).

....2.3 shows that there exists a quantum black box network that computes f exactly with cost at most D XOR (f ) The following lemma establishes a limit on how much we can expect PARIT Y to help simplify the computation of a function f . It is an extension of a result of Rivest and Vuillemin [10] for standard decision trees. Lemma 17 Let f be a Boolean function on 0, 1 N . If D PARIT Y (f) # d, then 2 N d divides f 1 (1) Proof. Each leaf of the decision tree corresponds to a set of inputs: those inputs for which the computation terminates at that leaf. These sets ....

Ronald L. Rivest and Jean Vuillemin. On recognizing graph properties from adjacency matrices. Theoret. Comput. Sci., 3:371--384, 1976.

....of [Bo] and section I.2 of [Bj] Most of the work has concentrated on the important special case of determining whether or not a graph satis es a monotone graph property when one is permitted to check one edge at a time. The general problem we have presented above seems to have rst appeared in [RV1]. Partially supported by the National Science Foundation and the National Security Agency. The author would like to thank the referees for their very thoughtful comments. Typeset by A M S T E X 1 In [KSS] it was shown that there is a fundamental relationship between the evasiveness of the ....

R.L. Rivest and J. Vuillemin, On recognizing graph properties from adjacency matrices, Theor. Comp. Sci, 3 (1976), 371-384.

....monotone properties (for suciently large n) Rosenberg [7] attributes to Karp the following conjecture which, remarkably, remains open even today. Karp Conjecture: Every nontrivial monotone graph property is evasive. As a rst step towards a resolution of this conjecture, Rivest and Vuillemin [6] proved that such properties have complexity at least n 2 =16, thereby settling the Aanderaa Rosenberg conjecture [7] of an n 2 ) complexity lower bound. The next big advance was the work of Kahn, Saks and Sturtevant [4] where an interesting topological approach was used to prove that the ....

Rivest, R.L., Vuillemin, J. On recognizing graph properties from adjacency matrices, Theoret. Comput. Sci., 3 (1976), 371-384.

.... deterministic complexity of f is D(f) min A2A(f) max w2f0;1g n C(A; w) Clearly, for every function f on n variables, D(f) n: The first important result about the deterministic decision tree complexity was a linear lower bound for monotone graph properties obtained by Rivest and Vuillemen [8]. In another interesting development Nisan [7] proved that the CREW PRAM complexity of any function f is Theta(log D(f) As usual, we can also permit randomization in this computational model. A randomized decision tree algorithm P on n variables is a distribution over all deterministic ....

R. Rivest and J. Vuillemen (1978), On recognizing graph properties from adjacency matrices, Theoretical Computer Science 3, 371-384.

....For example, we can apply it to estimate the gap between the actual solution to MSA and its lower bound. Property testing and spot checking. In the last two decades there has been significant work on estimating the complexity of checking graph properties. In particular, Rivest and Vuillemin [21] showed that the problem of deciding of any non trivial monotone graph property requires inspection of at least Omega Gamma n 2 ) edges; the same bound is conjectured to hold for randomized algorithms as well. Very few non trivial graph properties have subquadratic complexity. The notion of ....

R.L. Rivest, J. Vuillemin, "On recognizing graph properties from adjacency matrices", Theoretical Computer Science 3 (1976), pp. 371-384.

....conjecture) says that all non constant monotone graph properties P are evasive. This conjecture is still open; see [27] for an overview. The conjecture has been proved for graphs where the number of vertices is a prime power [25] but the best known general bound is D(P ) 2 Omega Gamma N) [35,25,26]. This bound also follows from a degree bound by Dodis and Khanna [11, Theorem 2] Theorem 22 (Dodis Khanna) If P is a non constant monotone graph property, then deg(P ) 2 Omega Gamma N) Corollary 7 If P is a non constant monotone graph property, then D(P ) 2 Omega Gamma N) and QE (P ) 2 ....

R. Rivest and S. Vuillemin. On recognizing graph properties from adjacency matrices. Theoretical Computer Science, 3:371--384, 1976.

....with n vertices must use at least log(n=2) n=2 = Omega# n log n) bits. 2 1.4.2 Time Lower Bounds There has been some previous work on lower bounds to determine monotone graph properties given an adjacency matrix encoding. See, for example, Rivest and 11 Vuillemin s and Kahn et al. s work [RV76, KSS84] This work is not directly applicable to s t connectivity, since their bounds apply only to global graph properties (properties that are independent of vertex numbering) So, for example, their work gives an Omega# n 2 ) time lower bound to determine whether a graph contains a pair of ....

R. C. Rivest and J. Vuillemin. On recognizing graph properties from adjacency matrices. Theoretical Computer Science, 3(3):371--384, December 1976.

....on n variables, all Boolean functions on n variables being equally likely. Let P be a certain property of Boolean functions. If the probability that a f has property P is p n and lim n 1 p n = 1, then we say a random Boolean function has property P . 2 Theorem 1.1.1. R. Rivest and J. Vuillemin [30]) A random Boolean function is evasive. The theorem suggests that there are so many evasive functions that we should look for a class of functions and prove uniformly that its members are all evasive. Transitive functions. The automorphism group of a Boolean function f is the group Aut(f) of ....

....Aut(f) such that x = y. Roughly speaking a function is transitive if we cannot distinguish its variables. This class is quite wide: it includes the symmetric functions and graph properties. The importance of this class is shown by the following theorem. Theorem 1.1.2. R. Rivest and J. Vuillemin [30]) Let n be a prime power, and f be a transitive function on n variables. If f(0; 0) 6= f(1; 1) then f is evasive. It is known [18] that the theorem becomes false if we do not assume n to be a prime power. A Boolean function is monotone if changing the value of a variable from 0 ....

R. Rivest and S. Vuillemin, On recognizing graph properties from adjacency matrices, Theor. Comp. Sci. 3(1976), 371--384.

....following question: What are the properties that make a boolean function completely evasive 1 Various researchers have used the terms elusive and exhaustive to denote the same concept. 1 A rather simple counting argument (see section 2) shows that most boolean functions are completely evasive[Rivest and Vuillemin, 1976]. Nevertheless, it is important that we understand the nature of evasiveness. In this way, we can define general conditions for evasiveness, or nonevasiveness. If a boolean function satisfies conditions for complete evasiveness, then we know that it cannot be computed from a proper subset of its ....

....leaves in which they get accepted, in which case you encode the whole decision tree ) However one needs to trade information for complexity: the proofs become very obscure, when multivariate polynomials are involved and it is not clear whether the results are substantially stronger. Theorem 2. 13 [Rivest and Vuillemin, 1976; Best et al. 1974] If P is nonevasive (C(P) k, k n) then (1 z) d Gammak j P 1 (z) Proof: Since C(P) k there is an optimal decision tree A for P , such that every path in A has length at most k. This implies that at most k bits are checked in any path from the root to a leaf. To ....

[Article contains additional citation context not shown here]

R. L. Rivest and J. Vuillemin, "On recognizing Graph Properties from Adjacency Matrices," Theoretical Computer Science, 3:371--384, 1976.

....to an earlier version of the famous Anderaa Rosenberg conjecture, which stated that any nontrivial graph property that is invariant under graph isomorphism requires n 2 ) probes of the adjacency matrix to test. Anderaa disproved this version in 1975 using a di erent counterexample (see [18]) but conjectured that it held for monotone properties (those that cannot change from false to true when edges are deleted) This was later veri ed by Rivest and Vuillemin [18] Homework 2 1: Extendible Arrays The array doubling idea is folklore. A generalization of this idea has been ....

....2 ) probes of the adjacency matrix to test. Anderaa disproved this version in 1975 using a di erent counterexample (see [18] but conjectured that it held for monotone properties (those that cannot change from false to true when edges are deleted) This was later veri ed by Rivest and Vuillemin [18]. Homework 2 1: Extendible Arrays The array doubling idea is folklore. A generalization of this idea has been described by Overmars in [16] for the design of dynamic data structures. 2: Binomial Heaps A proof that binomial heaps support merging in amortized constant time was given in [10] ....

R. L. Rivest and J. Vuillemin. On recognizing graph properties from adjacency matrices. Theoretical Computer Science, 3:371-384, 1976/77.

....as we shall elaborate later, many graph properties have very fast property testing algorithms whose query complexities do not depend at all on the size of the graph. This should be put in contrast to known lower bounds on the complexity of exactly deciding graph properties. Rivest and Vuillemin [RV76] showed that any deterministic procedure for deciding any non trivial monotone N vertex graph property must examine Omega Gamma N 2 ) entries in the adjacency matrix representing the graph, thus resolving the Aanderaa Rosenberg Conjecture [Ros73] The query complexity of randomized decision ....

R. L. Rivest and J. Vuillemin. On recognizing graph properties from adjacency matrices. Theoretical Computer Science, 3:371--384, 1976. 38

....Science, Rehovot, Israel y The Hebrew University, Jerusalem, Israel z The Hebrew University, Jerusalem, Israel. Partially supported by Israel American Binational Science Foundation Grant No. 87 00082 1 The first major result for this model was the linear lower bound of Rivest and Viullemin [RV78] for the class of monotone graph properties, proving the AanderaaRosenberg conjecture. A conjecture that an Omega Gamma n) lower bound applies to this class even if we allow randomization, is attributed to Karp. This has been proven for a few special monotone graph properties, but the best ....

R. Rivest and S. Viullemin. On recognizing graph properties from adjacency matrices. Theoretical Computer Science, 3:371--384, 1978.

....is also investigated, we show separation results between these models. 1 Introduction The model of Boolean decision tree is a fundamental model in the theory of Boolean functions. It provides a very important measure of complexity and it has been investigated thoroughly in the literature [23, 20, 13, 24, 16, 18, 7]. In the basic model only queries of single variables are allowed. Several generalizations have been considered in the past [6, 17, 7] Recently [7] investigated the complexity of such a model when general threshold functions are allowed as queries. We are interested in case were the set of ....

R. Rivest and S. Viullemin. On recognizing graph properties from adjacency matrices. Theoretical Computer Science, 3:371--384, 1978.

....conjecture) says that all non constant monotone graph properties P are evasive. This conjecture is still open; see [LY94] for an overview. The conjecture has been proved for graphs where the number of vertices is a prime power [KSS84] but the best known general bound is D(P ) 2 Omega Gamma N) RV76, KSS84, Kin88] This bound also follows from a degree bound by Dodis and Khanna [DK99] Theorem 22 (Dodis Khanna) deg(P ) 2 Omega Gamma N) for all non constant monotone graph properties P . Corollary 7 D(P ) 2 Omega Gamma N) and QE (P ) 2 Omega Gamma N) for all non constant monotone graph ....

R. Rivest and S. Vuillemin. On recognizing graph properties from adjacency matrices. Theoretical Computer Science, 3:371--384, 1976.

....the surveys in [Bo] and [Bj] Most of the work has concentrated on the important special case of determining whether or not a graph satisfies a monotone graph property when one is permitted to check one edge at a time. The general problem we have presented above seems to have first appeared in [RV1]. In [KSS] it was shown that there is a fundamental relationship between the evasiveness of the subcomplex M and its topology. Partially supported by the National Science Foundation and the National Security Agency Typeset by A M S T E X 1 Theorem 1 [KSS] If M is nonevasive then M is ....

R.L. Rivest and J. Vuillemin, On recognizing graph properties from adjacency matrices, Theor. Comp. Sci, 3 (1976), 371--384.

....B play optimally. The property P is called evasive if B can force A to ask all the jSj possible questions. If P is considered as a Boolean function, the complexity of P is a lower bound for the time any algorithm recognizing P must take in the worst case on any model of sequential machine [8]. A well studied special case of this problem is, when the considered set S is the set of all possible edges of an n vertex graph. The relation between this concept of recognition complexity of graph properties and the computer representation of graphs is discussed in [9] In this context The ....

R. L. Rivest and J. Vuillemin, On recognizing graph properties from adjacency matrices, Theor. Comp. Science 3 (1976) 371-384.

....the recognition complexity of P . A property is called elusive, if B can force A to ask all possible jSj questions. If P is considered as a Boolean function, the complexity of P is a lower bound for the time any algorithm recognizing P must take in the worst case on any model of sequential machine [10]. A famous and well studied special case of this game is, when S is regarded as the set of possible edges of a graph on n vertices, i.e. P is a property of graphs [2] 6] 7] 8] 9] The relation between this concept of recognition complexity of graph properties and the computer ....

R. L. Rivest and J. Vuillemin, On recognizing graph properties from adjacency matrices, Theor. Comp. Science 3 (1976) 371-384.

....For example, we can apply it to estimate the gap between the actual solution to MSA and its lower bound. Property testing and spot checking. In the last two decades there has been significant work on estimating the complexity of checking graph properties. In particular, Rivest and Vuillemin [17] showed that deciding of any non trivial monotone graph property requires inspection of at least Omega Gamma n 2 ) edges; the same bound is conjectured to hold for randomized algorithms as well. Very few non trivial graph properties have subquadratic complexity. The notion of approximate ....

R.L. Rivest, J. Vuillemin, "On recognizing graph properties from adjacency matrices", Theoretical Computer Science 3 (1976), pp. 371-384.

....Klauck, inspired by an earlier version of this paper, which gives the first total function with quantum classical gap in the zero error model of communication complexity. Finally, a class of black box problems that has received wide attention concerns the determination of monotone graph properties [35, 22, 24, 17]. Consider a directed graph on n vertices. It has n(n Gamma 1) possible edges and hence can be represented by a black box of n(n Gamma 1) binary variables, where each variable indicates whether or not a specific edge is present. A nontrivial monotone graph property is a property of such a graph ....

....of the graph, and monotone. Clearly, n(n Gamma 1) is an upper bound on the number of queries required to compute such properties. The Aanderaa Karp Rosenberg or evasiveness conjecture states that D(P ) n(n Gamma 1) for all P . The best known general lower bound is Omega Gamma n 2 ) [35, 22, 24]. It has also been conjectured that R 0 (P ) 2 Omega Gamma n 2 ) for all P , but the current best bound is only Omega Gamma n 4=3 ) 17] A natural question is whether or not quantum algorithms can determine monotone graph properties more efficiently. We show that they can. Firstly, in the ....

[Article contains additional citation context not shown here]

R. Rivest and S. Vuillemin. On recognizing graph properties from adjacency matrices. Theoretical Computer Science, 3:371--384, 1976.

....of the notion of deciding the graph property P which has received much attention in the last two decades [30] In the classical problem there are no margins of error, and one is required to accept all graphs having property P and reject all graphs which lack it. In 1975 Rivest and Vuillemin [35] resolved the Aanderaa Rosenberg Conjecture [32] showing that any deterministic procedure for deciding any non trivial monotone N vertex graph property must examine Omega Gamma N 2 ) entries in the adjacency matrix representing the graph. The query complexity of randomized decision procedures ....

R. L. Rivest and J. Vuillemin. On recognizing graph properties from adjacency matrices. Theoretical Computer Science, 3:371--384, 1976.

.... [2] came up with a more general method (compared to Kuty lowski [8] that delivers lower bounds for Boolean functions in terms of their degree [2, 13] Their main result is CREW(f) OE(deg f) Since almost all functions have degree n and every Boolean function can be computed in OE(n) 1 steps [2, 12], nearly all functions have time complexity OE(n) plus an additive constant of at most one. We show the corresponding result CREW(f) dlog(deg f)e 1 by a similar proof technique. Critical functions [1] that are those functions for which there exists an input such that flipping any single input ....

....5. CREW(f) dlog ne 3 for all f 2 Bn . Proof. Sketch) For every possible input w, there will be one processor that knows after dlog ne 2 steps whether x = w. This processor writes the result in step dlog ne 3 into the first memory cell. ut Almost all Boolean functions have degree n [2, 12]. We conclude that CREW(f) log n c for some c 2 f1; 2; 3g and for almost all Boolean functions f . 4 Lower and upper bounds for critical functions A critical function is a Boolean function for which an input I exists such that flipping any single bit of I changes the output. The OR function ....

R. L. Rivest and J. Vuillemin. On recognizing graph properties from adjacency matrices. TCS, 3:371--384, 1976.

....a function with restricted domain in [BCW98] and bounded error gaps in [ASTV98, Raz99] This result includes a new lower bound in classical communication complexity. Finally, a class of black box problems that has received wide attention concerns the determination of monotone graph properties [RV76, KSS84, Kin88, Haj91] Consider a directed graph on n vertices (our results also hold for undirected graphs) It has n(n Gamma 1) possible edges and hence can be represented by a black box of n(n Gamma 1) binary variables, where each variable indicates whether or not a specific edge is present. ....

....of the graph, and monotone. Clearly, n(n Gamma 1) is an upper bound on the number of queries required to compute such properties. The Aanderaa Karp Rosenberg or evasiveness conjecture states that D(P ) n(n Gamma 1) for all P . The best known general lower bound is Omega Gamma n 2 ) RV76, KSS84, Kin88] It has also been conjectured that R 0 (P ) 2 Omega Gamma n 2 ) for all P , but the current best bound is only Omega Gamma n 4=3 ) Haj91] A natural question is whether or not quantum algorithms can determine monotone graph properties more efficiently. We show that they can. ....

[Article contains additional citation context not shown here]

R. Rivest and S. Vuillemin. On recognizing graph properties from adjacency matrices. Theoretical Computer Science, 3:371--384, 1976.

....is also investigated, we show separation results between these models. 1 Introduction The model of Boolean decision tree is a fundamental model in the theory of Boolean functions. It provides a very important measure of complexity and it has been investigated thoroughly in the literature [1, 2, 3, 4, 5, 6, 7]. In the basic model only queries of single variables are allowed. Several generalizations have been considered in the past [8] Recently [7] investigated the complexity of such a model when general threshold functions are allowed as queries. We are interested in case were the set of queries is ....

R. Rivest and S. Viullemin. On recognizing graph properties from adjacency matrices. Theoretical Computer Science, 3:371--384, 1978.

....By Lemma 9, it suOEces to show that Q 0 (XOR) is 1. Theorem 1 provides an algorithm accomplishing this. The following lemma establishes a limit on how much we can expect this technique to help simplify the computation of a function f . It is an extension of a result of Rivest and Vuillemin [RV76] for standard decision trees. Lemma 11 Let f be a Boolean function on f0; 1g N . If D PARITY (f) d, then 2 N Gammad divides jf Gamma1 (1)j. Proof Each leaf of the decision tree corresponds to a set of inputs: those inputs for which the computation terminates at that leaf. These ....

Ronald L. Rivest and Jean Vuillemin. On recognizing graph properties from adjacency matrices. Theoret. Comput. Sci., 3:371384, 1976.

....using the results of Alon et al. 1] property P which has received much attention in the last two decades [29] In the classical problem there are no margins of error, and one is required to accept all graphs having property P and reject all graphs which lack it. In 1975 Rivest and Vuillemin [33] resolved the Aanderaa Rosenberg Conjecture [34] showing that any deterministic procedure for deciding any non trivial monotone N vertex graph property must examine Omega Gamma N 2 ) entries in the adjacency matrix representing the graph. The query complexity of randomized decision ....

R. L. Rivest and J. Vuillemin. On recognizing graph properties from adjacency matrices. Theoretical Computer Science, 3:371--384, 1976.

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R. L. Rivest and J. Vuillemin. On recognizing graph properties from adjacency matrices. TCS, 3:371-384, 1976.

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R. L. Rivest and J. Vuillemin. On recognizing graph properties from adjacency matrices. Theoretical Computer Science, 3:371--384, 1976.

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R. L. Rivest and J. Vuillemin. On recognizing graph properties from adjacency matrices. Theoretical Comput. Sci., 3(3):371--384, 1976.

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