Discrete Applied Mathematics 159 (2011) 478–483
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Discrete Applied Mathematics
journal homepage: www.elsevier.com/locate/dam
Q -ary Rényi–Ulam pathological liar game with one lie
Kun Meng
a,b,∗
, Chuang Lin
b
, Wen An Liu
c
, Yang Yang
a
a
School of Information Engineering, University of Science and Technology Beijing, Beijing 100083, People’s Republic of China
b
Department of Computer Science and Technology, Tsinghua University, Beijing 100084, People’s Republic of China
c
College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, People’s Republic of China
a r t i c l e i n f o
Article history:
Received 28 March 2010
Received in revised form 28 December 2010
Accepted 29 December 2010
Available online 19 January 2011
Keywords:
Rényi–Ulam game
Pathological liar game
State
Character
Worst case
a b s t r a c t
The m-round q-ary Rényi–Ulam pathological liar game with e lies, referred to as the game
[q, e, n; m]
∗
, is considered. Two players, say Paul and Carole, fix nonnegative integers m, n,
q and e. In each round, Paul splits [n] := {1, 2, . . . , n} into q subsets, and Carole chooses
one subset as her answer and assigns 1 lie to all elements except those in her answer.
Paul wins, after m rounds, if there exists at least one element assigned with e or fewer
lies. Let f
∗
(q, e, n) be the maximum value of m such that Paul can certainly win the game
[q, e, n; m]
∗
. This paper gives the exact value of f
∗
(q, 1, n) for n ≥ q
q−1
and presents a tight
bound on f
∗
(q, 1, n) for n < q
q−1
.
© 2011 Elsevier B.V. All rights reserved.
1. Introduction
The original q-ary Rényi–Ulam liar game with e lies, heneceforth referred to as the game [q, e, n; m], has been studied
extensively (see [1,6,16,17]). Two players, say Paul and Carole, fix integers m, n, q and e. Carole thinks of an integer
x
∗
∈ [n] := {1, 2, . . . , n}, and Paul has to determine x
∗
by asking q-ary queries T of the type ‘‘which one among the sets
{T
1
, T
2
, . . . , T
q
} does x
∗
belong to’’? We indicate such a q-ary query by {T
1
: T
2
: . . . : T
q
}, where [n] = T
1
∪ T
2
. . . ∪ T
q
and T
i
∩ T
j
= ∅ when i ̸= j. Carole is allowed to lie at most e times. We say that Paul has a winning strategy for the game
[q, e, n; m] if he can certainly determine x
∗
in at most m rounds.
The original liar game corresponds to the code with feedback [2,6]. Let f (q, n, e) = min{m|. Paul has a winning strategy
for the game [q, e, n; m]}. It is one of the most important problems to determine the value of f (q, e, n). Pelc [14], Guzicki [10]
and Deppe [5] determined the exact value of f (2, e, n) for e = 1, e = 2 and e = 3, respectively. The exact values of f (3, 1, n)
and f (3, 2, n) were determined by Pelc [15] and Liu et al. [12]. Malinowski [13] (also see Aigner [1]) determined the exact
values of f (q, 1, n). Cicalese and Vaccaro [4] determined the exact value of f (q, 2, n) for n = q
i
. For more results, refer to
the comprehensive survey literature [6,11,16].
The q-ary Rényi–Ulam pathological liar game with e lies, which is the dual of the original liar game and is also referred
to as the game [q, e, n; m]
∗
, was formulated by Ellis and Yan [9] in 2004. Two players, say Paul and Carole, fix nonnegative
integers m, n, q and e. In each round, Paul chooses a query, and Carole chooses one subset as her answer and assigns 1 lie to
all elements except those in her answer. We say that Paul has a winning strategy for the game if, after m rounds, there must
exist at least one element associated with e or fewer lies.
The pathological liar game has a reformulation in terms of covering codes which is of interest in adaptive
coding theory [3,9]. Let f
∗
(q, e, n) = max{m|Paul has a winning stategy for the game [q, e, n; m]
∗
} and g
∗
(q, e, m) =
∗
Corresponding author at: School of Information Engineering, University of Science and Technology Beijing, Beijing 100083, People’s Republic of China.
Tel.: +86 10 62783596.
E-mail address: mengkun1024@163.com (K. Meng).
0166-218X/$ – see front matter © 2011 Elsevier B.V. All rights reserved.
doi:10.1016/j.dam.2010.12.021