8 research outputs found
Randomized Algorithms for Determining the Majority on Graphs
Every node of an undirected connected graph is colored white or black. Adjacent nodes can be compared and the outcome of each comparison is either 0 (same color) or 1 (different colors). The aim is to discover a node of the majority color, or to conclude that there is the same number of black and white nodes. We consider randomized algorithms for this task and establish upper and lower bounds on their expected running time. Our main contribution are lower bounds showing that some simple and natural algorithms for this problem cannot be improved in general
Computing Majority with Triple Queries
Consider a bin containing balls colored with two colors. In a -query,
balls are selected by a questioner and the oracle's reply is related
(depending on the computation model being considered) to the distribution of
colors of the balls in this -tuple; however, the oracle never reveals the
colors of the individual balls. Following a number of queries the questioner is
said to determine the majority color if it can output a ball of the majority
color if it exists, and can prove that there is no majority if it does not
exist. We investigate two computation models (depending on the type of replies
being allowed). We give algorithms to compute the minimum number of 3-queries
which are needed so that the questioner can determine the majority color and
provide tight and almost tight upper and lower bounds on the number of queries
needed in each case.Comment: 22 pages, 1 figure, conference version to appear in proceedings of
the 17th Annual International Computing and Combinatorics Conference (COCOON
2011
Randomized strategies for the plurality problem
AbstractWe consider a game played by two players, Paul and Carol. At the beginning of the game, Carol fixes a coloring of n balls. At each turn, Paul chooses a pair of the balls and asks Carol whether the balls have the same color. Carol truthfully answers his question. Paulâs goal is to determine the most frequent (plurality) color in the coloring by asking as few questions as possible. The game is studied in the probabilistic setting when Paul is allowed to choose his next question randomly.We give asymptotically tight bounds both for the case of two colors and many colors. For the balls colored by k colors, we prove a lower bound Ω(kn) on the expected number of questions; this is asymptotically optimal. For the balls colored by two colors, we provide a strategy for Paul to determine the plurality color with the expected number of 2n/3+O(nlogn) questions; this almost matches the lower bound 2n/3âO(n)
On randomized algorithms for the majority problem
AbstractIn the majority problem, we are given n balls coloured black or white and we are allowed to query whether two balls have the same colour or not. The goal is to find a ball of majority colour in the minimum number of queries. The answer is known to be nâB(n) where B(n) is the number of 1âs in the binary representation of n. In this paper we study randomized algorithms for determining majority, which are allowed to err with probability at most Δ. We show that any such algorithm must have expected running time at least (23âo(1))n. Moreover, we provide a randomized algorithm which shows that this result is best possible. These extend a result of De Marco and Pelc [G. De Marco, A. Pelc, Randomized algorithms for determining the majority on graphs, Combin. Probab. Comput. 15 (2006) 823â834]