4,773 research outputs found
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 , a set of allowable lie strings of maximum length
. Carole selects , and Paul makes -ary queries to uniquely
determine . In each of rounds, Paul weakly partitions and asks for such that . Carole responds with some
, and if , then accumulates a lie . Carole's string of
lies for must be in the channel . Paul wins if he determines within
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 for which Paul can guarantee finding the
distinguished element is as ,
where is the number of lie strings in of maximum length . 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 .Comment: 26 page
Improving the Sphere-Packing Bound for Binary Codes over Memoryless Symmetric Channels
A lower bound on the minimum required code length of binary codes is
obtained. The bound is obtained based on observing a close relation between the
Ulam's liar game and channel coding. In fact, Spencer's optimal solution to the
game is used to derive this new bound which improves the famous Sphere-Packing
Bound.Comment: 5 pages,3 figures, Presented at the Forty-Seventh Annual Allerton
Conference on Communication, Control, and Computing, Sep. 200
Deterministic and Probabilistic Binary Search in Graphs
We consider the following natural generalization of Binary Search: in a given
undirected, positively weighted graph, one vertex is a target. The algorithm's
task is to identify the target by adaptively querying vertices. In response to
querying a node , the algorithm learns either that is the target, or is
given an edge out of that lies on a shortest path from to the target.
We study this problem in a general noisy model in which each query
independently receives a correct answer with probability (a
known constant), and an (adversarial) incorrect one with probability .
Our main positive result is that when (i.e., all answers are
correct), queries are always sufficient. For general , we give an
(almost information-theoretically optimal) algorithm that uses, in expectation,
no more than queries, and identifies the target correctly with probability at
leas . Here, denotes the
entropy. The first bound is achieved by the algorithm that iteratively queries
a 1-median of the nodes not ruled out yet; the second bound by careful repeated
invocations of a multiplicative weights algorithm.
Even for , we show several hardness results for the problem of
determining whether a target can be found using queries. Our upper bound of
implies a quasipolynomial-time algorithm for undirected connected
graphs; we show that this is best-possible under the Strong Exponential Time
Hypothesis (SETH). Furthermore, for directed graphs, or for undirected graphs
with non-uniform node querying costs, the problem is PSPACE-complete. For a
semi-adaptive version, in which one may query nodes each in rounds, we
show membership in in the polynomial hierarchy, and hardness
for
Economic and Organizational Issues in Alaska Water Quality Management
The work upon which this report (Proj. A-029-ALAS) is based was supported by funds provided
by the United States Department of the Interior, Office of Water Resources Research, as
authorized under the Water Resources Act of 1964
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