Anomaly Detection and Three Anomalous Coins Problem
01 Jan 2016-pp 303-320
TL;DR: Representing coins as any data items, an algorithm to determine three false coins out of n given coins is introduced and the objective is to solve the problem in minimum number of comparisons with the help of an equal arm balance.
Abstract: Counterfeit coin problem has been considered for a very long time and is a topic of great significance in Mathematics as well as in Computer Science. In this problem, out of n given coins, one or more false coins (the coins are classified as false because of their different weight from a standard coin) are present which have the same appearance as the other coins. The word counterfeit or anomalous means something deviated from the standard one. In this respect, finding out these anomalous objects from a given set of data items is of utmost importance in data learning problem. Thus, representing coins as any data items, we have introduced an algorithm to determine three false coins out of n given coins. In addition, our objective is to solve the problem in minimum number of comparisons with the help of an equal arm balance.
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TL;DR: The purpose of this paper is to indicate a systematic way in which the theory of dynamic programming can be used to provide a computational solution to the determination of optimal and suboptimal testing policies.
Abstract: The problem of ascertaining the minimum number of weighings which suffice to determine the defective coin in a set of N coins of the same appearance, given an equal arm balance and the information that there is precisely one defective coin present, is well known. A large number of ingenious solutions exist, some based upon sequential procedures and some not. The problem in the case where there are known to be two or more defective coins is far more complex because we cannot draw any simple definite conclusions at the end of a single test. We shall analyze this in detail in the following paper. The purpose of this paper is to indicate a systematic way in which the theory of dynamic programming can be used to provide a computational solution to the determination of optimal and suboptimal testing policies. We shall illustrate this by means of some numerical results obtained using a digital computer.
35 citations
TL;DR: In this article, the problem of determining the minimum number of counterfeit (heavier) coins in a set of n coins of the same appearance, given a balance scale and the information that there are exactly two heavier coins present, was considered.
32 citations
TL;DR: Several variations of the balancing problem have been solved before, all of which have been presented in this paper, and all of these problems are solved by the original Schell solver, which is the only one known to us.
Abstract: Since such weighing problems are today as much a part of the tradition of recreational mathematics as magic squares and mobius bands, it is interesting to note that they date only from that problem in 1945. The classic works of Loyd, Ball, Dudeney and Kraitchik contain no such problems. The responses to Schell's problem, a flurry of papers in the Monthly, Scripta Mathematica, and the Mathematical Gazette, contain no mention of earlier publications which might be relevant. Thus, it is apparent that this class of extremely natural and appealing puzzles is a recent invention, not an old chestnut which "crops up from time to time to puzzle and infuriate new generations of solvers" as someone wrote in 1961 (when such puzzles were just fifteen years old!). That early spate of papers, appearing in 1945 and the next few years with a speed unheard of in these days of publication backlogs, solved, resolved, and generalized the original problem in all directions. In this paper I present several variations of the balancing problem, all of which have been solved before. The method of solution given here may be original. We will always be dealing with a set of coins, of identical appearance, and a beam balance. We are interested in minimizing the maximum number of weighings which may be required to find the odd coin. The method of solution chosen may require solution of several types of weighing problems simultaneously, because a problem may change character after a weighing. For example, after a single use of the beam we have some coins which we know to be genuine (those on the beam, if it balances, those left off it if it does not). Thus, after one weighing we have a problem of a different type than the original one. We shall break all of the problems dealt with into two classes: those in which the counterfeit coin is known to be underweight and those in which it is only known to be of a different weight than the genuine coins. At the very end of the paper we will examine the possibility of no counterfeit coin. For the time being, we will assume that in every case exactly one coin is not genuine. Sometimes a "standard" coin, known to be the correct weight, will be provided. We begin with the first class of problems.
19 citations
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TL;DR: A polynomial time randomized algorithm to find the hidden weighted graph G when the number of edges in G is known to be at most $m\geq 2$ and the weight of each edge $e$ satisfies $\ga \leq |w(e)$ for fixed constants $\ga, \gb>0$.
Abstract: We consider the problem of finding edges of a hidden weighted graph using a certain type of queries. Let $G$ be a weighted graph with $n$ vertices. In the most general setting, the $n$ vertices are known and no other information about $G$ is given. The problem is finding all edges of $G$ and their weights using additive queries, where, for an additive query, one chooses a set of vertices and asks the sum of the weights of edges with both ends in the set. This model has been extensively used in bioinformatics including genom sequencing. Extending recent results of Bshouty and Mazzawi, and Choi and Kim, we present a polynomial time randomized algorithm to find the hidden weighted graph $G$ when the number of edges in $G$ is known to be at most $m\geq 2$ and the weight $w(e)$ of each edge $e$ satisfies $\ga \leq |w(e)|\leq \gb$ for fixed constants $\ga, \gb>0$. The query complexity of the algorithm is $O(\frac{m \log n}{\log m})$, which is optimal up to a constant factor.
6 citations
10 Jun 2011
TL;DR: Algorithms for solving the counterfeit coin problem for any given number n of coins are developed, based on the existing classical solution for the eight coins problem (with slight modification) for larger values of n, where n is a power of two beyond eight, as two and four being base cases.
Abstract: Eight coins problem is a well-known problem in mathematics as well as in computer science. In this problem eight coins are given, say A, B, C, D, E, F, G, and H, and we are told that only one is counterfeit (or false), as it has a different weight than each of the others. We want to determine which coin it is, making use of an equal arm balance. At the same time we want to identify the counterfeit coin using a minimum number of comparisons and determine whether the false coin is heavier or lighter than each of the remaining. In this paper, we develop algorithms for solving the counterfeit coin problem for any given number n of coins. The first algorithm is in essence based on the existing classical solution for the eight coins problem (with slight modification) for larger values of n, where n is a power of two beyond eight, as two and four being base cases. Then we develop an algorithm for solving n coins problem, where n is even but not power of two, i.e., the numbers are six, ten, 12, 14, 18, 20, etc. At the end, we have extended the same to solve the counterfeit coin problem for odd number of coins as well.
4 citations