ABI]Alon, N., L. Babai, A. Itai, A fast and simple randomized parallel algorithm for the [ maximal independent set problem, J. of Algorithms, 7(1986), pp. 567-583.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the same sparsi cation techniques of the preceding sections to augment I to a connected dominating set of size at most 3 jI j, which gives (essentially) a 12approximation. It is well known that an MIS can be computed in O(log n) communication rounds in our distributed model using randomization [1, 11]. 5 Open Problems and Summary of results It would be nice to develop a more direct and computationally simpler method to obtain connected dominating sets with good stretch properties. One way to achieve this would be to solve the following problem. Let G = A; B; E) be a bipartite network. A ....

N. Alon, L. Babai, and A. Itai. A Fast and Simple Randomized Parallel Algorithm for the Maximal Independent Set Problem. J. Algorithms, 7:567-583, 1986.

....To get our PRF under the sf DDH assumption (i.e. in groups were regular DDH might be false) it sufficing to construct a 4 wise indepepndent encoding C . Naturally, the goal is to make L as close to as possible. Such encodings come up quite often in the theory of derandomization (see [ABI86, AS00] and are closely related to coding theory. In our case, the well known construction is very simple and efficient, so we present it in a self contained manner. Let us now view any element of x 2 f0; 1g as an element of the field GF (2 ) which can be represented as an ....

Noga Alon, Laszlo Babai, and Alon Itai. A fast and simple randomized parallel algorithm for the maximal independent set problem. Journal of Algorithms, 7:567--583, 1986.

....Basic Research in Computer Science, Centre of the Danish National Research Foundation. 1 Introduction During the last years there is a growing interest in techniques for removing randomness from parallel (and sequential) algorithms. These techniques were originated by [7, 8] and generalized in [1, 2, 4, 6, 9, 10, 11]. The approach usually followed can be summarized as follows: The random variables which are considered are defined over a smaller probability space, specially designed, containing only a polynomial number of sample points. In that space, the random variables are only k wise independent (for ....

N. Alon, L. Babai and A. Itai, A Fast and Simple Randomized Parallel Algorithm for the Maximal Independent Set Problem, Journal of Algorithms 7 (1986) 567-583.

....and other randomized rounding algorithms. Randomized rounding of linear programs and the method of pessimistic estimators has since been quite popular in the design of approximation algorithms (see [MNR95] Returning to the context of parallel algorithmic derandomization, Alon, Babai, and Itai [ABI86] gave optimal constructions of k wise independent binary random variables. However, it was soon realized that O(1) wise independent binary random variable constructions were inadequate to derandomize many parallel algorithms; n random variables that are roughly O(log n) wise independent seemed to ....

N. Alon, L. Babai, and A. Itai. A fast and simple randomized parallel algorithm for the maximal independent set problem. Journal of Algorithms, 7:567--583, 1986.

....are fairly tight. In fact, the above inequality automatically gives a logspace algorithm to obtain a 2 approximation to K(oe) since one can easily compute F (oe) when oe appears in a stream. We assume that the permutation is presented in its inverted form, i.e. in the form of an array oe[1]; oe[n] such that the i th entry of the array oe[i] u if and only if oe(u) i. We use square brackets or parentheses to indicate the semantics for the use of oe. A list L = L[1] L[n] drawn from a universe [m] is a sequence of n ordered elements such that for each i, L[i] 2 ....

....a stream. We assume that the permutation is presented in its inverted form, i.e. in the form of an array oe[1] oe[n] such that the i th entry of the array oe[i] u if and only if oe(u) i. We use square brackets or parentheses to indicate the semantics for the use of oe. A list L = L[1]; L[n] drawn from a universe [m] is a sequence of n ordered elements such that for each i, L[i] 2 [m] Though a list is not necessarily a permutation, the notion of inversions in a list can still be defined. Let K(L) denote the number of pairs hi; ji such that i j and L[i] L[j] Given ....

[Article contains additional citation context not shown here]

N. Alon, L. Babai, and A. Itai. A fast and simple randomized parallel algorithm for the maximal independent set problem. J. Algorithms, 7(4):567--583, 1986.

....a subset S of V that does not contain any edge. S is a maximal independent set (MIS) if no proper superset of S is an IS. It is easy to find an MIS sequentially, but efficient parallel algorithms appear much harder. There has been much work on finding MISs in parallel in (hyper)graphs (see, e.g. [1, 4, 7, 9, 17, 15, 16, 18, 21]) Since an MIS can be much smaller than a maximum independent set (as in the case of a star graph) it is also of much interest to find ISs that have a guaranteed size. What is known in this context Recall that given a hypergraph H = V; E) the degree of a vertex is the number of edges that it ....

....graph families for which ff 2 (G) Theta(n) and n 2 = 2m n) is just Theta(1) see item (i) in Section 1.2. Recently, the second author developed an NC algorithm that finds an IS of size (1 Gamma o(1) Delta ff 2 (G) where the o(1) term goes to zero as n 1 [24] Alon, Babai and Itai [1] studied the MIS problem on hypergraphs: they gave an NC algorithm that finds an IS of size c(n k =m) 1= k Gamma1) 0 c 1, in k uniform hypergraphs with k 2 being any constant. The work of Karger and Koller [13] generalized this to arbitrary k. 1.2 Main Results We give an RNC algorithm ....

[Article contains additional citation context not shown here]

N. Alon, L. Babai and A. Itai. A fast and simple randomized parallel algorithm for the maximal independent set problem. J. of Algorithms, 7, pp. 567--583, 1986.

....random variables rather than totally independent ones. The generation of k wise independent and approximately k wise independent random variables has been well studied [Jof74, CG89, NN93, EGL 98, CRS00] Early applications of k wise independence in derandomization can be found in [KW85, Lub86, ABI86, Lub93, BR91] Very recently Klivans and Spielman [KS01] gave a randomness ecient method for testing if a polynomial is identically zero. In all of these algorithms a reduction in randomness is traded for suboptimal performance. 1.1 Our Results In this paper we address the issue of reducing ....

N. Alon, L. Babai, and A. Itai. A fast and simple randomized parallel algorithm for the maximal independent set problem. J. Algor., 7:567-583, 1986.

.... considering E[ Z Gamma E[Z] k ] or similar expectations, which look at the Z i k or fewer at a time (via linearity of expectation) The main application of this has been that the Z i s can then be sampled using few random bits, yielding a de randomization pseudo randomness result (e.g. 1 [3, 22, 8, 23, 24, 29]) Our results show that such ideas can in fact be used to show that some structures exist This is one of our main contributions. However, while many applications of Lemma 1.1 have been constructivized (Beck [5] Alon [1] our MIP result is only existential. For PIPs and CIPs, we present a ....

N. Alon, L. Babai, and A. Itai, A fast and simple randomized parallel algorithm for the maximal independent set problem, Journal of Algorithms, 7 (1986), pp. 567-- 583.

.... generation of k wise independent and approximately k wise independent random variables has been well studied [Jof74, CG89, NN93, EGL 98, CRS00] Several results are known on derandomizing randomized algorithms that use k wise independence to obtain deterministic algorithms (see [KW85, Lub86, ABI86, Lub93, BR91] Very recently Klivans and Spielman [KS01] gave a randomness ecient method for testing if a multivariate polynomial is identically zero. In all of these algorithms a reduction in randomness is traded for an acceptable increase in the running time 2 . 1.1 Our Results In this ....

N. Alon, L. Babai, and A. Itai. A fast and simple randomized parallel algorithm for the maximal independent set problem. J. of Algorithms, 7:567-583, 1986.

....8 We also give an explicit natural property that requires a large number of quantum queries to test. Theorem 5.2 The range of a d wise independent pseudorandom generator requires (d 1) 2 quantum queries to test for any odd d n= log n 1. We will make use of the following lemma: Lemma 5. 3 (see [ABI86] Suppose n = 2 k 1 and d = 2t 1 n. Then there exists a uniform probability space of size 2(n 1) t and d wise independent random variables 1 , n over each of which takes the values 0 and 1 with probability 1=2. The proof of Lemma 5.3 is constructive and the construction ....

N. Alon, L. Babai, and A. Itai. A fast and simple randomized parallel algorithm for the maximal independent set problem. Journal of Algorithms, 7:567-583, 1986.

....on Randomization Methods in Algorithm Design. 1 1 Introduction Constructing pseudorandom permutation families is often more difficult than constructing pseudorandom function families. For example, there are polynomial size constructions of k wise independent function families for constant k [8, 9, 1, 12]. On the other hand, although there are polynomial size 3 wise independent permutation families (see, e.g. 14] there are only exponential size constructions known for higher k. In fact, the only subgroups of the symmetric group that are 6 wise independent are the alternating group and the ....

N. Alon, L. Babai, and A. Itai. A fast and simple randomized parallel algorithm for the maximal independent set problem. Journal of Algorithms (7), 1986, pp. 567--583.

....range of X i is A i . In other words, each X i is uniformly distributed in A i , and any k of the X i s are mutually independent. Our (random) output is S = fX 1 ; X l g. Methods to construct such k wise independent random variables using k log n random bits are well known: see, e.g. [ABI, Lub]. Extracting One Block. The function B: B has 2 parameters: l, the size of the output, and k, the amount of independence used. 1. INPUT: x 2 f0; 1g n ; y 2 f0; 1g t (where t = k log n) 2. Use y to choose a set fi 1 : i l g ae f1 : ng of size l using k wise independence, as described ....

N. Alon, L. Babai, and A. Itai, "A fast and simple randomized parallel algorithm for the maximal independent set problem," Journal of Algorithms, 7:567--583, 1986.

....with respect to v. 3.1.1 Constructing a small set of distinguishers A natural approach to the problem of reducing the number of random bits in a randomized algorithm is to show that limited independence of the random variables suffices to assure a high probability of success. See e.g. [5, 36, 38]) However, in our case, it is not clear from the proof of Proposition 3.1 how many vectors r remain distinguishers with respect to the vector v when the entries of R are not completely independent. Moreover, an example can be constructed in which if the entries of r are chosen pairwise ....

....we will need n random variables such that: 1. Each random variable is uniformly distributed in f1; ng. 2. Every subset of the random variables of cardinality at most c is independent. There are known methods of generating such random variables that use only O(c log n) random bits. 38] [5], 17] We first assume that l, the precise number of non zero elements in v, is known to be in the range [k : 2k Gamma 1] A two step process is applied. 1. The vector v is replaced by a new vector v 0 = v 0 1 ; v 0 n ) that contains only c (for some constant) non zero ....

[Article contains additional citation context not shown here]

N. Alon, L. Babai and A. Itai, A fast and simple randomized parallel algorithm for the maximal independent set problem, Journal of Algorithms, Vol 7 (1986) , pp. 567-583.

....set (the MIS problem) then the problem becomes polynomial. In fact, a maximal independent set can be easily found sequentially in a linear time. Karp and Wigderson [15] showed that MIS is in NC. Starting with their work, a number of parallel algorithms have been proposed to solve this problem [2], 10] 11] 17] 18] Currently, the most ecient algorithm is presented in [11] it runs in O(log 3 n) time on O( n m) log n) processors. A common drawback of all NC algorithms for MIS mentioned above is that occasionally they can nd too small a set. Any graph with a large independent ....

N. Alon, L. Babai, and A. Itai, A fast and simple randomized parallel algorithm for the maximal independent set problem, J. Algorithms, 7 (1986), pp. 567-583.

....on a linear number of processors under the CRCW PRAM model of computation. In the EREW PRAM model, it runs in O(log 2 n) time. The deterministic version of the algorithm uses O(n 2 m) processors. A still di erent probabilistic algorithm for MIS was described by Alon, Babai, and Itai in [4]. Since there is a trivial sequential algorithm that runs in linear time, every ecient algorithm for MIS must use at most a linear number of processors (up to a polylogarithmic factor) A deterministic algorithm that uses a linear number of processors was proposed by Goldberg in [12] its running ....

N. Alon, L. Babai, and A. Itai, A fast and simple randomized parallel algorithm for the maximal independent set problem, J. Algorithms, 7 (1986), pp. 567-583.

....In fact it was even conjectured that the MIS problem does not belong to NC [17] The rst NC algorithm for the MIS problem was presented in [9] but especially Luby s algorithm from [13] received a lot of attention. It was the rst example of the so called derandomization technique [6,2] see [1,3,15,5] for further applications. Roughly speaking the derandomization technique is based on the transformation of a randomized NC algorithm (which is easier to design) into a deterministic NC algorithm by simulating the randomized algorithm in parallel for several possible outcomes of its random ....

N. Alon, L. Babai, and A. Itai. A fast and simple randomized parallel algorithm for the maximal independent set problem. Journal of Algorithms, 7:567-583, 1986.

....[Kar84] Computing or approximating the size of the lexicographically first maximal independent set is also P complete; see Section 6.2. Karp and Wigderson gave the first NC algorithm for finding a maximal independent set [KW85] subsequently improved by Luby [Lub86] by Alon, Babai and Itai [ABI86], and by Goldberg and Spencer [GS89] These algorithms do not compute the lexicographically first maximal independent set. Part II: P Complete Problems ffl 51 A.2.2 Lexicographically First Maximal Clique (LFMC) Given: An undirected graph G with an ordering on the vertices and a designated vertex ....

N. Alon, L. Babai, and A. Itai. A fast and simple randomized parallel algorithm for the maximal independent set problem. Journal of Algorithms, 7:567--583, 1986.

....of random bits. We refer the reader to [Nis96] for a survey on this method. With respect to polynomial time randomized computation, there are two fundamental approaches to derandomization: the method of conditional probabilities [ES73, Spe87, Rag88] and constructing small sized sample spaces [KW84, Lub85, ABI86, AKS87, NN90]. In the former approach, we search for a good point in a large sample space by improving certain conditional probabilities (or expectations) in an adaptive manner; in the latter method, we construct a sample space of polynomial size while guaranteeing the existence of a good point so that we ....

....space of polynomial size while guaranteeing the existence of a good point so that we could accomplish finding such a point by exhaustive search. The most commonly used tools for constructing small sized sample spaces are: k wise independent hashing [CW79] sample spaces with limited independence [KW84, Lub85, ABI86], sample spaces with small bias [NN90] and expander graphs [AKS87] Remark: Here we have assumed that the polynomial time randomized computation has one side error, in which case, finding a good sample point is sufficient for derandomization. In fact, for almost all the problems that are known ....

[Article contains additional citation context not shown here]

N. Alon, L. Babai and A. Itai. A fast and simple randomized parallel algorithm for the Maximal Independent Set Problem. J. Algorithms 7:567-583, 1986.

....pseudorandom permutations; explicit constructions. 1 Introduction Constructing pseudorandom permutation families is often more di#cult than constructing pseudorandom function families. For example, there are polynomial size constructions of k wise independent function families for constant k [5, 6, 1, 8]. On the other hand, although there are polynomial size 3 wise independent permutation families (see, e.g. 10] there are only exponential size constructions known for higher k. In fact, the only subgroups of the symmetric group that are 6 wise independent are the alternating group and the ....

N. Alon, L. Babai, and A. Itai. A fast and simple randomized parallel algorithm for the maximal independent set problem. Journal of Algorithms (7), 1986, pp. 567--583.

....d. Most of the previous work has focused on constructing approximations to such distributions. Joffe [9] first demonstrated a construction of a joint distribution of n d wise independent random variables with an essential sample space of cardinality O(n d ) Luby [10] and Alon, Babai, and Itai [1] generalize Joffe s construction to non uniform distributions. In many cases, these constructions only approximately satisfy the required constraints; that is, the distributions are d wise independent, but the probabilities Pr(X i = b) may differ from the corresponding probabilities in the ....

....it cannot be used to convert RNC algorithms into NC ones. In Section 4 we show an example of how our technique can be applied to de randomization of algorithms. We discuss the problem of finding a large independent set in a d uniform hypergraph. The underlying randomized algorithm, described in [1], was de randomized in the same paper for fixed values of d. It was later derandomized also for d = O(polylogn) in [6] and [11] We show how this algorithm can be derandomized for any d. We point out that a sequential deterministic polynomial time solution for the independent set problem in ....

[Article contains additional citation context not shown here]

N. Alon, L. Babai, and A. Itai, "A fast and simple randomized parallel algorithm for the maximal independent set problem," Journal of Algorithms, 7 (1986) 567--583.

....problem is a typical maximality problem, that is to find a maximal vertex induced subgraph that satisfies a specified graph property. Since Karp and Wigderson showed that the MIS problem is in the class NC [11] much work has been devoted to the study of parallel complexity of maximality problems [13, 1, 6, 5, 3, 17, 18]. On the other hand, the lexicographically first maximal independent set (LFMIS) problem is a typical P complete problem [2] and P completeness of the lexicographically first maximal subgraph (LFMS) problems for some properties was shown [14, 17] see also [15, 7] for a comprehensive reference) ....

....1 reducibility in [2] we use the log space reducibility simply as in [14] A function F 0 is said to be P complete if F 0 is in P and for each F in P there are log space computable functions f and g such that F (x) g(F 0 (f(x) for all inputs. It is well known that the MIS problem is in NC [11, 1, 13, 6, 5, 12], and the LFMIS problem is one of the fundamental P complete problems [14, 15, 7] Recall that the EREW PRAM is the parallel model where the processors operate synchronously and share a common memory, but no two of them are allowed simultaneous access to a memory cell (whether the access is for ....

N. Alon, L. Babai, and A. Itai. A Fast and Simple Randomized Parallel Algorithm for the Maximal Independent Set Problem. Journal of Algorithms, 7:567--583, 1986.

....done while visiting the International Computer Science Institute and while at Carnegie Mellon University. 0 1 Introduction The problem of constructing small sample spaces that approximate the independent distribution on n random variables has received considerable attention recently (cf. [7, 1, 17, 2, 4]) The primary motivation for this line of research is that random variables that are approximately independent suffice for the analysis of many interesting randomized algorithm, and hence, constructing a small probability space that approximates the independent distribution yields a way to ....

....U n;2 (or any other joint distribution of n non degenerate random variables) Previously known approximations to U n;m are of three forms. ffl k wise independent approximations: constructions of sample spaces of size maxfn; mg k that are k wise independent approximations for U n;m are given in [7, 1]. ffl ffl approximations: constructions of sample spaces of size poly(n=ffl) that are ffl approximations for U n;2 are given in [17, 2] ffl (k; ffl) approximations: constructions of sample spaces of size poly( k log n) ffl) that are (k; ffl) approximations for U n;2 can be derived from the ....

[Article contains additional citation context not shown here]

Alon, N., Babai, L., Itai, A., "A Fast and Simple Randomized Parallel Algorithm for the Maximal Independent Set Problem", Journal of Algorithms, 7, pp. 567--583, 1986.

....computation has attracted a great deal of attention in the theory of algorithms. Some problems, easy to solve sequentially in polynomial time, became non trivial questions in parallel computation. The maximal independent set problem was one of them. The problem is now known to be in NC ( KW] Lu] [ABI], GS] Other important problems like maximum matching (matching of maximum size) are still not known to be in NC. There exist methods ( Lo1] KUW] MVV] which show that matching is in RNC (random parallel polylog time) in fact, in Las Vegas NC [Ka] One relaxation of the maximum matching ....

N. Alon, L. Babai and A. Itai, A fast and simple randomized parallel algorithm for the maximal independent set problem, Journal of Algorithms, 7 (1986), pp. 567-583.

....[80] where local search is combined with randomized heuristic. Their computational results indicated that their approach was effective in finding large cliques of randomly generated graphs. A different implementation of a randomized algorithm for the maximum independent set problem can be found in [65]. Advanced search heuristics. Local search algorithms are only capable of finding local solutions of an optimization problem. In the past few years, many powerful variations of the basic local search procedure have been developed which try to avoid this problem, many of which are inspired from ....

N. Alon, L. Babai, and Itai, A.: `A fast and simple randomized parallel algorithm for the maximal independent set problem', J. Algorithms 7 (1986), 567--583.

....done while visiting the International Computer Science Institute and while at Carnegie Mellon University. 0 1 Introduction The problem of constructing small sample spaces that approximate the independent distribution on n random variables has received considerable attention recently (cf. [7, 1, 19, 2, 4]) The primary motivation for this line of research is that random variables that are approximately independent suffice for the analysis of many interesting randomized algorithm, and hence, constructing a small probability space that approximates the independent distribution yields a way to ....

....U n;2 (or any other joint distribution of n non degenerate random variables) Previously known approximations to U n;m are of three forms. ffl k wise independent approximations: constructions of sample spaces of size maxfn; mg k that are k wise independent approximations for U n;m are given in [7, 1]. ffl ffl approximations: constructions of sample spaces of size poly(n=ffl) that are ffl approximations for U n;2 are given in [19, 2] ffl (k; ffl) approximations: constructions of sample spaces of size poly( k log n) ffl) that are (k; ffl) approximations for U n;2 can be derived from the ....

[Article contains additional citation context not shown here]

Alon, N., Babai, L., Itai, A., "A Fast and Simple Randomized Parallel Algorithm for the Maximal Independent Set Problem", Journal of Algorithms, 7, pp. 567--583, 1986.

....of Depth 2 Circuits Michael Luby Boban Velickovi c y Avi Wigderson z Abstract We describe deterministic algorithms which for a given depth 2 circuit F approximate the probability that on a random input F outputs a specific value ff. Our approach gives an algorithm which for a given GF[2] multivariate polynomial p and given ffl 0 approximates the number of zeros of p within a multiplicative factor 1 ffl. The algorithm runs in time exp(exp(O( p log(n=ffl) where n is the size of the circuit. We also obtain an algorithm which given a DNF formula F and ffl 0 ....

....of 1 ffl and runs in time exp(O( log(n=ffl) 4 ) 1 Introduction This paper deals with the problem of approximating the accepting probability of general depth 2 boolean circuits. Examples of boolean functions which can be computed by such circuits are DNF formulas, polynomials over GF[2], polynomials over other small fields, threshold functions, etc. There are easy probabilistic algorithms which for a given circuit F and a real parameter ffl 0 approximate the probability that on a random input F evaluates to 0. The algorithm simply chooses N = O(ln(1=ffi) ffl 2 ) assignments ....

[Article contains additional citation context not shown here]

Alon, N., Babai, L., Itai, A., "A Fast and Simple Randomized Parallel Algorithm for the Maximal Independent Set Problem", Journal of Algorithms, 7, pp. 567--583, 1986.

....a specific class has a good bound on the expected size of the largest bucket is to build a class specifically designed to have such good property. One immediate such result is obtained by looking at the class of d degree polynomials over finite fields, where d = c log n= log log n (see, e.g. [ABI86]. It is easy to see that this class maps each d elements of the domain independently to the range, and thus, the bound that applies to the class of all functions also applies to this class. We can combine this with the following well known construction, which is usually called collapsing the ....

N. Alon, L. Babai and A. Itai, A fast and simple randomized parallel algorithm for the maximal independent set problem. J. 20 Algorithms 7 (1986) 567--583.

....than fully independent. It follows that one can obtain an algorithmic version of the proofs of Theorem 2.1 and Theorem 2.3 by checking all points of a small sample space that supports n 4 wise independent random variables i as needed. Constructions of such spaces with O(n 2 ) points appear in [1] (see also [2] providing the required ecient, deterministic algorithm. The proof in Section 3 provides no explicit example of a tournament T on n vertices that admits no realization of quality 212 Such examples are given by the quadratic residue tournaments. For a prime p 3 ( mod 4) the ....

N. Alon, L. Babai and A. Itai, A fast and simple randomized parallel algorithm for the maximal independent set problem, J. Algorithms 7(1986), 567-583.

....that a speci c class has a good bound on the expected size of the largest bucket is to build a class speci cally designed to have such good property. One immediate such result is obtained by looking at the class H of d degree polynomials over nite elds, where d = c log n= log log n (see, e.g. [ABI86]. It is easy to see that this class maps each d elements of the domain independently to the range, and thus, the bound that applies to the class of all functions also applies to this class. We can combine this with the following well known construction, found in, e.g. FKS84] and sometimes ....

N. Alon, L. Babai and A. Itai, A fast and simple randomized parallel algorithm for the maximal independent set problem. J. Algorithms 7 (1986) 567-583.

....0 in our random graph G(n; p) where the maximality is with respect to containment. Such a set can be found in NC (i.e. in polylogarithmic time, using a polynomial number of parallel processors) using any of the known NC algorithms for the maximal independent set problem (see, e.g. 10] 11] [1]) The rest of the algorithm only has to find perfect matchings in appropriately defined graphs, and this can be done in (randomized) NC by the results of [9] or [12] Thus, the H factors whose existence is guaranteed almost surely in Theorems 1.1 and 1.2 can actually be found, almost surely, ....

N. Alon, L. Babai and A. Itai, A fast and simple randomized parallel algorithm for the maximal independent set problem, J. Algorithms 7 (1986), 567-583.

....any maximal set of internally disjoint paths of length 4 between each two members of A 0 is of size at least 6m. Such a set can be easily found in NC by applying one of the many known NC algorithms for finding a maximal independent set in an appropriately defined graph (see, e.g. 20] 21] [4]) The graph considered here is of course the one whose set of vertices are all possible paths of length 4 with the required endpoints where two are adjacent if they share an internal vertex) Once the 14 sets of 6m paths between each pair are given, we can find the required copy of H as ....

N. Alon, L. Babai and A. Itai, A fast and simple randomized parallel algorithm for the maximal independent set problem, J. Algorithms 7 (1986), 567-583.

No context found.

ABI]Alon, N., L. Babai, A. Itai, A fast and simple randomized parallel algorithm for the [ maximal independent set problem, J. of Algorithms, 7(1986), pp. 567-583.

No context found.

N. Alon, L. Babai, and A. Itai. A fast and simple randomized parallel algorithm for the maximal independent set problem. Journal of Algorithms, 7:567--583, 1986.

No context found.

N. Alon, L. Babai and A. Itai, A fast and simple randomized parallel algorithm for the maximal independent set problem, J. Algorithms, vol. 7(4), 1986, pp. 567-583.

No context found.

N. Alon, L. Babai, and A. Itai. A fast and simple randomized parallel algorithm for the maximal independent set problem. J. Algorithms, 7(4):567-- 583, Dec. 1986.

No context found.

N. Alon, L. Babai, and A. Itai. A fast and simple randomized parallel algorithm for the maximal independent set problem. J. Algorithms, 7(4):567--583, 1986.

No context found.

. N. Alon, L. Babai, A. Itai. A fast and simple randomized parallel algorithm for the maximal independent set problem. J. of Algorithms 7, 567-583(1986).

No context found.

N. Alon, L. Babai, A. Itai. A fast and simple randomized parallel algorithm for the maximal independent set problem. J. of Algorithms 7, 567-583(1986).

No context found.

N. Alon, L. Babai, A. Itai. A fast and simple randomized parallel algorithm for the maximal independent set problem. J. of Algorithms 7, 567-583(1986).

No context found.

N. Alon, L. Babai and A. Itai, A fast and simple randomized parallel algorithm for the maximal independent set problem, Journal of Algorithms 7 (1986), pp. 567--583.

No context found.

N. Alon, L. Babai, and A. Itai. A fast and simple randomized parallel algorithm for the maximal independent set problem. J. Algorithms, 7:567--583, 1986.

No context found.

N. Alon, L. Babai, and A. Itai. A fast and simple randomized parallel algorithm for the maximal independent set problem. Journal of Algorithms, 7:567--583, 1986.

No context found.

N. Alon, L. Babai, and A. Itai, "A fast and simple randomized parallel algorithm for the maximal independent Journal of Algorithms, 7 (1986) 567--583.

No context found.

Noga Alon, Laszlo Babai, and Alon Itai. A fast and simple randomized parallel algorithm for the maximal independent set problem. J. Algorithms, 7(4):567--583, 1986.

No context found.

N. Alon, L. Babai, and A. Itai. A fast and simple randomized parallel algorithm for the maximal independent set problem. Journal of Algorithms, 7 (1986), pp. 567--583.

No context found.

N Alon, L. Babai, A. Itai, \A fast and simple randomized parallel algorithm for the Maximal Independent Set Problem", Journal of Algorithms, Vol. 7, 1986, pp. 567-583.

No context found.

N. Alon, L. Babai, and A. Itai. A fast and simple randomized parallel algorithm for the maximal independent set problem. Journal of Algorithms, 7:567--583, 1986.

No context found.

N. Alon, L. Babai, and A. Itai. A fast and simple randomized parallel algorithm for the maximal independent set problem. Journal of Algorithms, 7:567--583, 1986.

No context found.

Alon, N., L. Babai, A. Itai, "A Fast and Simple Randomized Parallel Algorithm for the Maximal Independent Set Problem", Journal of Algorithms, 7, pp. 567--583, 1986.

No context found.

Alon, N., Babai, L., Itai, A., "A Fast and Simple Randomized Parallel Algorithm for the Maximal Independent Set Problem", Journal of Algorithms, 7, pp. 567--583, 1986.

First 50 documents  Next 50

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC