Two-batch liar games on a general bounded channel

We consider an extension of the 2-person R\'enyi-Ulam liar game in which lies are governed by a channel $C$, a set of allowable lie strings of maximum length $k$. Carole selects $x\in[n]$, and Paul makes $t$-ary queries to uniquely determine $x$. In each of $q$ rounds, Paul weakly partitions $[n]=A_0\cup >... \cup A_{t-1}$ and asks for $a$ such that $x\in A_a$. Carole responds with some $b$, and if $a\neq b$, then $x$ accumulates a lie $(a,b)$. Carole's string of lies for $x$ must be in the channel $C$. Paul wins if he determines $x$ within $q$ rounds. We further restrict Paul to ask his questions in two off-line batches. We show that for a range of sizes of the second batch, the maximum size of the search space $[n]$ for which Paul can guarantee finding the distinguished element is $\sim t^{q+k}/(E_k(C)\binom{q}{k})$ as $q\to\infty$, where $E_k(C)$ is the number of lie strings in $C$ of maximum length $k$. This generalizes previous work of Dumitriu and Spencer, and of Ahlswede, Cicalese, and Deppe. We extend Paul's strategy to solve also the pathological liar variant, in a unified manner which gives the existence of asymptotically perfect two-batch adaptive codes for the channel $C$.Comment: 26 page

Minimum and maximum against k lies

A neat 1972 result of Pohl asserts that [3n/2]-2 comparisons are sufficient, and also necessary in the worst case, for finding both the minimum and the maximum of an n-element totally ordered set. The set is accessed via an oracle for pairwise comparisons. More recently, the problem has been studied in the context of the Renyi-Ulam liar games, where the oracle may give up to k false answers. For large k, an upper bound due to Aigner shows that (k+O(\sqrt{k}))n comparisons suffice. We improve on this by providing an algorithm with at most (k+1+C)n+O(k^3) comparisons for some constant C. The known lower bounds are of the form (k+1+c_k)n-D, for some constant D, where c_0=0.5, c_1=23/32=0.71875, and c_k=\Omega(2^{-5k/4}) as k goes to infinity.Comment: 11 pages, 3 figure

On the Multi-Interval Ulam-R\'enyi Game: for 3 lies 4 intervals suffice

We study the problem of identifying an initially unknown $m$-bit number by using yes-no questions when up to a fixed number $e$ of the answers can be erroneous. In the variant we consider here questions are restricted to be the union of up to a fixed number of intervals. For any $e \geq 1$ let $k_e$ be the minimum $k$ such that for all sufficiently large $m$, there exists a strategy matching the information theoretic lower bound and only using $k$-interval questions. It is known that $k_e = O(e^2)$. However, it has been conjectured that the $k_e = \Theta(e).$ This linearity conjecture is supported by the known results for small values of $e$. For $e\leq2$ we have $k_e = e.$ We extend these results to the case $e=3$. We show $k_3 \leq 4$ improving upon the previously known bound $k_3 \leq 10.$Comment: 31 pages, 5 figures, extension of the result to non-asymptotic strategie

Minimum average-case queries of q + 1 -ary search game with small sets

Given a search space S={1,2,...,n}, an unknown element x*∈S and fixed integers ℓ≥1 and q≥1, a q+1-ary ℓ-restricted query is of the following form: which one of the set {A 0,A 1,...,A q} is the x* in?, where (A 0,A 1,...,A q) is a partition of S and | Ai|≤ℓ for i=1,2,...,q. The problem of finding x* from S with q+1-ary size-restricted queries is called as a q+1-ary search game with small sets. In this paper, we consider sequential algorithms for the above problem, and establish the minimum number of average-case sequential queries when x* satisfies the uniform distribution on S. © 2011 Elsevier B.V. All rights reserved

Search when the lie depends on the target

The following model is considered. There is exactly one unknown element in the n-element set. A question is a partition of S into three classes: (A,L,B). If x ∈ A then the answer is "yes" (or 1), if x ∈ B then the answer is "no" (or 0), finally if x ∈ L then the answer can be either "yes" or "no". In other words, if the answer "yes" is obtained then we know that x ∈ A ∪ L while in the case of "no" answer the conclusion is x ∈ B ∪ L. The mathematical problem is to minimize the minimum number of questions under certain assumptions on the sizes of A,B and L. This problem has been solved under the condition |L| ≥ k by the author and Krisztián Tichler in previous papers for both the adaptive and non-adaptive cases. In this paper we suggest to solve the problem under the conditions |A| ≤ a, |B| ≤ b. We exhibit some partial results for both the adaptive and non-adaptive cases. We also show that the problem is closely related to some known combinatorial problems. Let us mention that the case b = n - a has been more or less solved in earlier papers. © Springer-Verlag Berlin Heidelberg 2013

On Nonadaptive Search Problem

2000 Mathematics Subject Classification: 91A46, 91A35.We consider nonadaptive search problem for an unknown element x from the set A = {1, 2, 3, . . . , 2^n}, n ≥ 3. For fixed integer S the questions are of the form: Does x belong to a subset B of A, where the sum of the elements of B is equal to S? We wish to find all integers S for which nonadaptive search with n questions finds x. We continue our investigation from [4] and solve the last remaining case n = 2^k , k ≥ 2