In this paper we consider a more complicated real-world problem originating in the steel industry. The bins are inhomogeneous sheets with impurities. We assume that each impure area is rectangle. For each bin we are given a set of impurities, size, and location of each impurity into the bin. As a consequence now the bins are not identical anymore and the number of bins is finite. Moreover, we i...

The balls-and-bins exercise that we did in Homework 1 is also useful for modeling Hashing. A hash function h from a universe U = [0, 1, .., m− 1] into a range [0, ..., n− 1] can be thought of as a way of placing items from the universe into n bins. The collection of bins is called a hash table. We can model the distribution of items in bins with the same distribution as m balls placed randomly ...

The 2-dimensional Bin Packing problem is a generalization of the classical Bin Packing problems and is defined as follows: Given a collection of rectangles specified by their width and height, the goal is to pack these into minimum number of square bins of unit size. Recently, the problem was proved to be APX-hard even in the asymptotic case, i.e. when the optimum solutions requires a large num...

Balls-into-bins games describe in an abstract setting several multiple-choice scenarios, and allow for a systematic and unified theoretical treatment. In the process that we consider, there are n bins, initially empty, and m = bcnc balls. Each ball chooses independently and uniformly at random k ≥ 3 bins. We are looking for an allocation such that each ball is placed into one of its chosen bins...

ONLINE BIN STRETCHING is a semi-online variant of bin packing in which the algorithm has to use the same number of bins as an optimal packing, but is allowed to slightly overpack the bins. The goal is to minimize the amount of overpacking, i.e., the maximum size packed into any bin. We give an algorithm for ONLINE BIN STRETCHING with a stretching factor of 1.5 for any number of bins. We build o...

In the variable-sized online bin packing problem, one has to assign items to bins one by one. The bins are drawn from some xed set of sizes, and the goal is to minimize the sum of the sizes of the bins used. We present the rst unbounded space algorithms for this problem. We also show the rst lower bounds on the asymptotic performance ratio. The case where bins of two sizes, 1 and 2 (0; 1), are ...

Multiple-choice allocation algorithms have been studied intensively over the last decade. These algorithms have several applications in the areas of load balancing, routing, resource allocation and hashing. The underlying idea is simple and can be explained best in the balls-and-bins model: Instead of assigning balls (jobs, requests, or keys) simply at random to bins (machines, servers, or posi...

This paper presents the CBIR based on bins approach. It introduces a new idea of partitioning the histogram into three parts using Centre of gravity. This partitioning leads to generation of 27 bins. In this work we have tried to reduce the feature vector dimension to just 27 bins out of 256 histogram bins. This paper elaborates the bins approach using linear (LP) and centre of gravity (CG) bas...