IWOCA 2010: Combinatorial Algorithms pp 314-325

# Chains-into-Bins Processes

• Tuğkan Batu
• Petra Berenbrink
• Colin Cooper
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6460)

## Abstract

The study of balls-into-bins processes or occupancy problems has a long history. These processes can be used to translate realistic problems into mathematical ones in a natural way. In general, the goal of a balls-into-bins process is to allocate a set of independent objects (tasks, jobs, balls) to a set of resources (servers, bins, urns) and, thereby, to minimize the maximum load. In this paper, we analyze the maximum load for the chains-into-bins problem, which is defined as follows. There are n bins, and m objects to be allocated. Each object consists of balls connected into a chain of length ℓ, so that there are m ℓ balls in total. We assume the chains cannot be broken, and that the balls in one chain have to be allocated to ℓ consecutive bins. We allow each chain d independent and uniformly random bin choices for its starting position. The chain is allocated using the rule that the maximum load of any bin receiving a ball of that chain is minimized. We show that, for d ≥ 2 and m·ℓ= O(n), the maximum load is ((ln ln m)/ln d) + O(1) with probability $$1-\tilde O(1/m^{d-1})$$.

## Keywords

Balls-into-bins processes chains-into-bins processes random processes offline assignment

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## Authors and Affiliations

• Tuğkan Batu
• 1
• Petra Berenbrink
• 2
• Colin Cooper
• 3
1. 1.Department of MathematicsLondon School of EconomicsLondonUK
2. 2.School of Computing ScienceSimon Fraser UniversityBurnabyCanada
3. 3.Department of Computer ScienceKing’s College LondonLondonUK