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Network error correction, part I: Basic concepts and upper bounds
 Communications in Information and Systems
, 2006
"... Abstract. Error correction in existing pointtopoint communication networks is done on a linkbylink basis, which is referred to in this paper as classical error correction. Inspired by network coding, we introduce in this twopart paper a new paradigm called network error correction. The theory t ..."
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Abstract. Error correction in existing pointtopoint communication networks is done on a linkbylink basis, which is referred to in this paper as classical error correction. Inspired by network coding, we introduce in this twopart paper a new paradigm called network error correction. The theory thus developed subsumes classical algebraic coding theory as a special case. In Part I, we discuss the basic concepts and prove the network generalizations of the Hamming bound and the Singleton bound in classical algebraic coding theory. By studying a few elementary examples, the relation between network error correction and classical error correction is investigated.
Coding for Interactive Communication
 IN PROCEEDINGS OF THE 25TH ANNUAL SYMPOSIUM ON THEORY OF COMPUTING
, 1996
"... Let the input to a computation problem be split between two processors connected by a communication link; and let an interactive protocol ß be known by which, on any input, the processors can solve the problem using no more than T transmissions of bits between them, provided the channel is noiseless ..."
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Cited by 64 (3 self)
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Let the input to a computation problem be split between two processors connected by a communication link; and let an interactive protocol ß be known by which, on any input, the processors can solve the problem using no more than T transmissions of bits between them, provided the channel is noiseless in each direction. We study the following question: if in fact the channel is noisy, what is the effect upon the number of transmissions needed in order to solve the computation problem reliably? Technologically this concern is motivated by the increasing importance of communication as a resource in computing, and by the tradeoff in communications equipment between bandwidth, reliability and expense. We treat a model with random channel noise. We describe a deterministic method for simulating noiselesschannel protocols on noisy channels, with only a constant slowdown. This is an analog for general interactive protocols of Shannon's coding theorem, which deals only with data transmission, ...
Coping with errors in binary search procedures
, 1980
"... We consider the problem of identifying an unknown value x E (1, Z,..., n} using only comparisons of x to constants when as many as E of the comparisons may receive erroneous answers. For a continuous analogue of this problem we show that there is a unique strategy that is optimal in the worst case. ..."
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Cited by 18 (0 self)
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We consider the problem of identifying an unknown value x E (1, Z,..., n} using only comparisons of x to constants when as many as E of the comparisons may receive erroneous answers. For a continuous analogue of this problem we show that there is a unique strategy that is optimal in the worst case. This strategy for the continuous problem is then shown to yield a strategy for the original discrete problem that uses logsn + E. logslogsn + O(E. log&) comparisons in the worst case. This number is shown to be optimal even if arbitrary “YesNo ” questions are allowed. We show that a modified version of this search problem with errors is equivalent to the problem of finding the minimal root of a set of increasing functions. The modified version is then also shown to be of complexity logsn + E. logslogsn + O(E * log&).
Solution of Ulam's Searching Game with Three Lies or an Optimal Adaptive Strategy for Binary ThreeErrorCorrectingCodes
, 1998
"... In this paper we determine the minimal number of yesno queries needed to find an unknown integer between 1 and N if at most three of the answers are lies. This strategy is also an optimal adaptive strategy for binary threeerror correcting codes. 1 Introduction In 1976, Ulam [24] suggested an inte ..."
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Cited by 7 (0 self)
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In this paper we determine the minimal number of yesno queries needed to find an unknown integer between 1 and N if at most three of the answers are lies. This strategy is also an optimal adaptive strategy for binary threeerror correcting codes. 1 Introduction In 1976, Ulam [24] suggested an interesting twoperson search game in his autobiography (pp. 281282), which can be formalized as follows: Person 1 thinks of a number between one and one million. Person 2 is allowed to ask questions to which Person 1 is supposed to answer only yes or no. Person 2 asks for subsets of the set f1; : : : ; 1000000g. The difficulty is that Person 1 is allowed to lie l times. Now we want to know: How many questions does Person 2 have to ask in order to get the correct answer? The problem is solved in [20] for one lie. Solution for l = 2 and jX j = 10 6 can be find in [9]. Solution for jX j = 2 m and l = 2 is presented in [7] and its generalization to arbitrary jX j is given in [11]. The case of ...
ISIT’98 Plenary Lecture Report: Variations on the Theme of ‘Twenty Questions
 IEEE Information Theory Society Newsletter
, 1999
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Reliable Minimum Finding Comparator Networks
 Fundamenta Informaticae
, 2000
"... . We consider the problem of constructing reliable minimum finding networks built from unreliable comparators. In case of a faulty comparator inputs are directly output without comparison. Our main result is the first nontrivial lower bound on depths of networks computing minimum among n ? 2 items ..."
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. We consider the problem of constructing reliable minimum finding networks built from unreliable comparators. In case of a faulty comparator inputs are directly output without comparison. Our main result is the first nontrivial lower bound on depths of networks computing minimum among n ? 2 items in the presence of k ? 0 faulty comparators. We prove that the depth of any such network is at least max(dlog ne + 2k; log n + k log log n k+1 ). We also describe a network whose depth nearly matches the lower bound. The lower bounds should be compared with the first nontrivial upper bound O(log n + k log log n log k ) on the depth of kfault tolerant sorting networks that was recently derived by Leighton and Ma [6]. 1 Introduction Networks built from comparators are commonly used to perform such tasks as selection, sorting and merging. A comparator is a 2 input2 output device which sorts two items. Networks of minimum size, i.e. using the minimum number of comparators for a given tas...
Ulam’s pathological liar game with one halflie
, 2004
"... We introduce a dual game to Ulam’s liar game and consider the case of one halflie. In the original Ulam’s game, Paul attempts to isolate a distinguished element by disqualifying all but one of n possibilities with q yesno questions, while the responder Carole is allowed to lie a fixed number k of ..."
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Cited by 5 (3 self)
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We introduce a dual game to Ulam’s liar game and consider the case of one halflie. In the original Ulam’s game, Paul attempts to isolate a distinguished element by disqualifying all but one of n possibilities with q yesno questions, while the responder Carole is allowed to lie a fixed number k of times. In the dual game, Paul attempts to prevent disqualification of a distinguished element by “pathological ” liar Carole for as long as possible, given that a possibility associated with k+1 lies is disqualified. We consider the halflie variant in which Carole may only lie when the true answer is “no. ” We prove the equivalence of the dual game to the problem of covering the discrete hypercube with certain asymmetric sets. We define A ∗ 1 (q) for the case k = 1 to be the minimum number n such that Paul can prevent Carole from disqualifying all n elements in q rounds of questions, and prove that A ∗ 1 (q) ∼ 2q+1 /q.
On the complexity of function learning
, 1995
"... The majority of results in computational learning theory are concerned with concept learning, i.e. with the special case of function learning for classes of functions with range {0, 1}. Much less is known about the theory of learning functions with a larger fange such as Nor IR. In particular rela ..."
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Cited by 4 (2 self)
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The majority of results in computational learning theory are concerned with concept learning, i.e. with the special case of function learning for classes of functions with range {0, 1}. Much less is known about the theory of learning functions with a larger fange such as Nor IR. In particular relatively few results exist about he general structure of common models for function learning, and there are only very few nontrivial function classes for which positive learning results have been exhibited in any of these models. We introduce in this paper the notion of a binaly branching adversary tree for function learning, which allows us to give a somewhat surprising equivalent characterization f the optimal learning cost for learning a class of realvalued functions (in terms of a maxmin definition which does not invoive any "learning " model). Another general structural result of this paper elates the cost for learning a union of function classes to the learning costs for the individual function classes. Furthermore, we exhibit an efficient leaming algorithm for learning convex piecewise linear functions from Rd into IR. Previously, the class of linear functions from 1R d into R was the only class of functions with multidimensional domain that was known to be learnable within the rigorous framework of a formal model for online leaming. Finally we give a sufficient condition for an arbitrary class 5 ~ of functions from IR into R that allows us to learn the class of all functions that can be written as the pointwise maximum of k functions from 5 r. This allows us to exhibit a number of further nontrivial classes of functions from ~ into R for which there exist eflicient]earning algorithms.
The RényiUlam Pathological Liar Game with a Fixed Number of Lies
, 2004
"... The qround RényiUlam pathological liar game with k lies on the set [n]:= {1,..., n} is a 2player perfect information zero sum game. In each round Paul chooses a subset A ⊆ [n] and Carole either assigns 1 lie to each element of A or to each element of [n]\A. Paul wins if after q rounds there is at ..."
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Cited by 4 (4 self)
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The qround RényiUlam pathological liar game with k lies on the set [n]:= {1,..., n} is a 2player perfect information zero sum game. In each round Paul chooses a subset A ⊆ [n] and Carole either assigns 1 lie to each element of A or to each element of [n]\A. Paul wins if after q rounds there is at least one element with k or fewer lies. The game is dual to the original RényiUlam liar game for which the winning condition is that at most one element has k or fewer lies. We prove the existence of a winning strategy for Paul to the existence of a covering of the discrete hypercube with certain relaxed Hamming balls. Defining F ∗ k (q) to be the minimum n such that Paul can win the qround pathological liar game with k lies and initial set [n], we find F ∗ 1 (q) and F ∗ 2 (q) exactly. For fixed k we prove that F ∗ k (q) is within an absolute constant (depending only on k) of the sphere bound, 2q / ( q ≤k; this is already known to hold for the original RényiUlam liar game due to a result of J. Spencer.