P. Indyk. Personal communication.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....to higher dimensions, resulting in approximation ratios linear in the dimension . 4) NP Completeness results: We show that it is NP Complete to compute an optimal partitioning for the important SUM VAR metric in Example 2. We also extend the known result for the MAX SUM metric in Example 1 [7, 10] that it is NP Complete to compute a partitioning with a value less than two times the optimum to several new metrics. A preliminary version of the results in this paper appeared as part of [26] Our algorithms are based on the framework of Bronnimann and Goodrich in [5] and exploit an ....

....Fortran [13] and Vienna Fortran [6] and is in fact part of the HPF 2.0 language specification. NP Completeness: For the three dimensional case, Nicol [27] proved that it is NP Complete to find the best partitioning under the MAX SUM metric, using a reduction to Monotonic 3 SAT. Subsequently, [7, 10] independently proved the problem to be NP Complete in two dimensions, based on a reduction to the Balanced Complete Bipartite Subgraph problem. In fact, the argument implies that given a bound on the number of tiles, the optimal value of the MAX SUM metric cannot be approximated to within less ....

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M. Charikar, C. Chekuri, T. Feder, and R. Motwani. Personal communication, 1996.

....S i j j jS i 0 j. Such families are well known and have been used several times in cryptographic applications [11, 30, 21] In what follows, we x = 1=2 for simplicity, and will use the following (essentially optimal) result, non constructively proven by [19] and subsequently made ecient by [30, 25]. Theorem 1 ( 19, 30, 25] For any N and t, one can eciently construct a (t; 2 ) cover free collection of N subsets S 1 ; SN of U = f1; ug with jS i j = n for all i, satisfying u = t log N) and n = t log N) Since we assumed that t; log N = O(poly(k) we have u; n = ....

.... Such families are well known and have been used several times in cryptographic applications [11, 30, 21] In what follows, we x = 1=2 for simplicity, and will use the following (essentially optimal) result, non constructively proven by [19] and subsequently made ecient by [30, 25] Theorem 1 ([19, 30, 25]) For any N and t, one can eciently construct a (t; 2 ) cover free collection of N subsets S 1 ; SN of U = f1; ug with jS i j = n for all i, satisfying u = t log N) and n = t log N) Since we assumed that t; log N = O(poly(k) we have u; n = O(poly(k) as well. ....

P. Indyk. Personal communication.

....these connections, as well as the history of the development of pairwise independence, we expect that the concept of min wise independence will prove useful in many future applications. A preliminary version of this work has appeared in [9] Since then new constructions have been proposed by Indyk [18] and others [25] The use of min wise independent families for derandomization is discussed in [10] 2 Exact Min Wise Independence In this section, we provide bounds for the size of families that are exactly min wise independent. We begin by determining a lower bound, demonstrating that the size ....

....z=0 1)z n We have an upper bound on Pr(min #(X) #(x) for all pairwise independent families of permutations that is O(1 # k) based on a linear programming formulation of the problem. Subsequent to our original proof, Piotr Indyk suggested a simpler proof for this bound [18], so we do not present it here. 4.2 Linear Families, Upper and Lower Bounds We derive further bounds by considering specifically linear transformations. For instance, we show that the family of linear transformations is not even approximately min wise independent for any constant #. 23 Theorem ....

P. Indyk. Personal communication.

....results for graphs induced by d 2 . We generalize our results to graphs induced by points in d p ( d equipped with the l p norm) We prove a lower bound of d) on for graphs induced by d 1 ; an identical lower bound has recently been proved by Indyk for graphs induced by d 1 [14]. These two lower bounds and our upper bound for d 2 can be combined to obtain matching upper and lower bounds of (f(d; p) on , where f(d; p) d 1=p for 1 p 2 and f(d; p) d 1 1=p for p 2. We use our partitioning algorithm to obtain a O(f(d; p) log n) probabilistic ....

.... (d 1=p ) if 1 p 2 and (d 1 1=p ) if p 2. The upper bound follows from the partitioning algorithm for graphs embedded in d 2 , presented in Section 3.1. The lower bound follows by interpolating our lower bound for d 1 (Section 3. 2) and a recent lower bound for d 1 by Indyk [14]. Our upper bound is a significant improvement over the bound of O(log n) for general graphs. Specifically, if p = 2 then the dimension of the space can be reduced to O(log n) without distorting edge lengths by more than a constant factor[15] Thus for Euclidean graphs, it is possible to beat the ....

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P. Indyk. personal communication, April 1998.

....are similar; in other cases it may be a good approximation. Our task is to allocate a sequence of jobs to the machines in an on line fashion, while minimizing the maximum load of the machines. This problem was solved with a competitive ratio 8 by Aspnes et al. 2] Later, it was noticed by Indyk [6] that by randomizing properly the key parameter of the original algorithm the expected competitive ratio can be reduced to 2e. For the version of the problem where the speeds of all the machines are the same, Albers [1] proved a lower bound of 1:852 on the competitive ratio of deterministic ....

P. Indyk, personal communication.

....from a variety of application areas include [17, 23, 25, 24, 1, 3, 10] Here we review a selection of related work most relevant to us. Hardness Results. Hardness results exist only for a simple metric function, namely, MAX SUM ID [18] proved it to be NP hard for arbitrary partitions, and [11, 6] proved it to be NP hard for p Theta p partition. Our NP hardness results are inspired by the abovementioned results. However, the basic gadgets in our reductions are different from the ones in [18, 6] and we derive different nonapproximability bounds for various metrics. Algorithmic Results. ....

....function, namely, MAX SUM ID [18] proved it to be NP hard for arbitrary partitions, and [11, 6] proved it to be NP hard for p Theta p partition. Our NP hardness results are inspired by the abovementioned results. However, the basic gadgets in our reductions are different from the ones in [18, 6] and we derive different nonapproximability bounds for various metrics. Algorithmic Results. Dynamic programming has been used to find optimal results for hierarchical partitions in several contexts [2, 26, 18] However, our sparse hierarchy approach with provable guarantee appears to be new. For ....

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M. Charikar, C. Chekuri, T. Feder, and R. Motwani. Personal communication, 1996.

....metric) we introduce a new transformation, which B applies on y before computing its signature (see Section 4. 2) The most interesting property of this transformation is that it converts the LZ metric to Hamming metric; hence it may be helpful in nearest neighbor search problems for sequences [Ind99] The second round involves the actual exchange of documents via correcting the differences (see Section 5) Here A computes and sends B a deterministic ID for x which is unique among all strings that are in the d(x; y) neighborhood of x. By the use of known techniques in graph coloring or ....

Piotr Indyk. Personal communication, 1999.

....(a factor of 2 due to the load in all the rest of the phases except the last, and another factor of 2 due to imprecise approximation of OPT by ) Thus the competitive ratio is 4fi. We note that the factor of 4 can be replaces by e = 2:7: for restricted class of algorithms using randomization (see [27]) 3 Permanent tasks Tasks which start at arbitrary times but continue forever are called permanent. The situation in which all tasks are permanent is classified as permanent tasks. Otherwise, it is classified as temporary tasks. Permanent task is a special case of temporary one by assuming 1 ....

...., then algorithm Assign R never fails. Thus, the load on a machine never exceeds 2. If OPT is unknown in advance the doubling technique for implies that Assign R is 8 competitive. 7 Using randomized doubling it is possible to replace the deterministic 8 upper bound by 2e 5:436 expected value [27]. Recently it was shown by [16] that replacing the doubling by a more refined method improves the deterministic competitive ratio to 3 p 8 5:828 and the randomized variant to about 4:311. Also the lower bound is 2:438 for deterministic algorithms and 1:837 for randomized ones. Open problem ....

P. Indyk. Personal communication.

....obtained independently. Reference Bound Bokhari [B88] O(n 3 p) Anily Federgruen [AF91] O(n 2 p) Hansen Liu [HL92] O(n 2 p) Manne Sorevik [MS95] O(np log p) Choi Narahari [CN91] O(np) Olstad Manne [OM95] O(np) Nicol [N91] O(n p 2 log 2 n) Charikar, Chekuri Motwani [CCM96] O(n p 2 log 2 n) Han, Narahari Choi [HNC92] O(n p 1 ffl ) ffl 1 This paper O(n log n) Our result relies on a binary search over a space of O(n 2 ) items. However, at each test, an approximate median among these items is identified in only O(n) as opposed to O(n 2 ) time by ....

....bad (typically Omega ( p p) approximations. One such heuristic has been recently shown to have a performance guarantee of O( p p) by Halldorsson Manne [HM96] This is the currently best known approximation for this problem. Reference Result Grigni Manne [GM96] NP Hardness Charikar et al. [CCM96] APX Hardness Halldorsson Manne [HM96] O( p p) approximation This paper (Section 4) O(1) approximation We observe that using our result for the 2D ffi weight problem above, one can easily obtain an O( log n) 2 ) approximation algorithm for the 2D p Theta p partitioning problem under F . ....

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M. Charikar, C. Chekuri, and R. Motwani. Personal Communication, 1996.

....shown to be bad ( Omega (p) approximations. One such heuristic has been recently shown to have a performance guarantee of O( p p) by Halldorsson Manne [HM96] This is the currently best known approximation for this problem. Reference Result Grigni Manne [GM96] NP Hardness Charikar et al. [CCFM96] APX Hardness Halldorsson Manne [HM96] O( p p) approximation This paper (Section 4) O(1) approximation We observe that using our result for the 2D ffi weight problem above, one can easily obtain an O( log n) 2 ) approximation algorithm for the 2D p Theta p partitioning problem under F . ....

M. Charikar, C. Chekuri, T. Feder, and R. Motwani. Personal Communication, 1996.

..... 2) In other words we require that all the elements of any fixed set X have only an almost equal chance to become the minimum element of the image of X under #. Indyk has found a simple construction of approximately min wise independent permutations with useful properties for derandomization [9]. His results imply the following proposition. Proposition 1 [Indyk] There exists a constant c such that any c # wise independent family of permutations is approximately min wise independent with relative error #. Using the above proposition, an approximately min wise independent family can be ....

P. Indyk, personal communication.

....these connections, as well as the history of the development of pairwise independence, we expect that the concept of min wise independence will prove useful in many future applications. A preliminary version of this work has appeared in [9] Since then new constructions have been proposed by Indyk [18] and others [25] The use of min wise independent families for derandomization is discussed in [10] 2 Exact Min Wise Independence In this section, we provide bounds for the size of families that are exactly min wise independent. We begin by determining a lower bound, demonstrating that the size ....

.... 1)z n 1 2(k 1) # We have an upper bound on Pr(min #(X) #(x) for all pairwise independent families of permutations that is O(1 # k) based on a linear programming formulation of the problem. Subsequent to our original proof, Piotr Indyk suggested a simpler proof for this bound [18], so we do not present it here. 4.2 Linear Families, Upper and Lower Bounds We derive further bounds by considering specifically linear transformations. For instance, we show that the family of linear transformations is not even approximately min wise independent for any constant #. Theorem 11 ....

P. Indyk. Personal communication.

....these automata constructions are optimal from the descriptional complexity point of view. They can be exploited as a part of a more complex neural network design, for example, for the construction of a cyclic neural network, with O(2 n 2 ) neurons and edges, which computes any boolean function [9]. In section 8 we will introduce the concept of Hopfield languages as the languages that are recognized by the so called Hopfield acceptors (Hopfield neuromata) which are based on symmetric neural networks (Hopfield networks) Hopfield networks have been studied widely outside of the framework of ....

.... Delta Delta ; x ip of the input is in the buffer. The architecture of the neural string acceptor is depicted in figure 3 (instead of the above mentioned conjunction, the neuron reset is added to realize the negation of comparator conjunction to possibly terminate the clock) 2 Piotr Indyk [9] pointed out that the latter string acceptor construction from Theorem 7 can also be exploited for building a cyclic neural network, with O(2 n 2 ) neurons and edges, which computes any boolean function. In this case the binary vector of all function values is encoded into the string acceptor. ....

Indyk, P. 1995. Personal Communication.

....graphs induced by d 2 . We generalize our results to graphs induced by points in d p ( d equipped with the l p norm) We prove a lower bound of Omega Gamma d) on fi for graphs induced by d 1 ; an identical lower bound has recently been proved by Indyk for graphs induced by d 1 [14]. These two lower bounds and our upper bound for d 2 can be combined to obtain matching upper and lower bounds of Theta(f (d; p) on fi, where f(d; p) d 1=p for 1 p 2 and f(d; p) d 1 Gamma1=p for p 2. We use our partitioning algorithm to obtain a O(f(d; p) Delta log ....

....) if 1 p 2 and Theta(d 1 Gamma1=p ) if p 2. The upper bound follows from the partitioning algorithm for graphs embedded in d 2 , presented in Section 3.1. The lower bound follows by interpolating our lower bound for d 1 (Section 3. 2) and a recent lower bound for d 1 by Indyk [14]. Our upper bound is a significant improvement over the bound of O(log n) for general graphs. Specifically, if p = 2 then the dimension of the space can be reduced to O(log n) without distorting edge lengths by more than a constant factor[15] Thus for Euclidean graphs, it is possible to beat ....

[Article contains additional citation context not shown here]

P. Indyk. personal communication, April 1998.

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P. Indyk. Personal communication.

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M. Charikar. Personal communication, January 2000.

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P. Indyk. Personal communication, 2002.

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M. Charikar. Personal communication, January 2000.

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M. Charikar. Personal communication, January 2000.

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M. Charikar, C. Chekuri, T. Feder, and R. Motwani. Personal communication, 1996.

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P. Indyk. Personal communication, 2002.

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M. Charikar and P. Indyk. Personal communication, 1995.

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