Consistent Hashing with Bounded Loads

Designing algorithms for balanced allocation of clients to servers in dynamic settings is a challenging problem for a variety of reasons. We want a hashing-style solution where we given the ID of a client can efficiently find its server in a distributed dynamic environment where both servers and clients may be added and/or removed from the system in any order. The main objectives of allocation algorithms are as follows: the uniformity of the allocation, and the number of moves after adding or removing a server or a client. More specifically, while one main goal is to achieve a proper load balancing in the allocation of clients to servers, after adding/removing clients or servers, re-computing allocations should not result in moving too many clients from their current server. The most popular solutions for our dynamic settings is Consistent Hashing [KLL97, SML03] or Rendezvous Hashing [TR98]. However, the load balancing of these schemes is no better than a random assignment of clients to servers, so with n of each, we expect many servers to be overloaded with Θ(log n/ log log n) clients. In this paper, with n clients and n servers, we get a guaranteed max-load of 2 while only moving an expected constant number of clients each time a client or server is added or removed. Our general result is described below. In this paper, we consider the following problem. We take an arbitrary user specified balancing parameter c = 1 + ε > 1. With m balls and n bins1 in the system, we want no load above dcm/ne. Meanwhile we want to bound the expected number of balls that have to be moved when a ball or server is added or removed. Our algorithmic starting point is consistent hashing where current balls and bins are hashed to a unit cycle, and a ball is placed in the first bin succeeding it in clock-wise order. In this paper, we suggest to let bins have capacities, and forward the ball to the first non-full bin. For balancing, we use maximum bin capacity dcm/ne. Compared with general lower bounds without capacity constraints, we show that when a ball or bin is inserted or deleted, the expected number of balls that have to be moved is increased only by a multiplicative factor O( 1 ε2 ) for ε ≤ 1 (Theorem 4) and by a factor 1 + O( log c c ) for ε ≥ 1 (Theorem 3). Technically, the latter bound is the most challenging to prove. It implies that we for superconstant c only pay a negligible cost in extra moves. We also get the same bounds for the simpler problem where we instead of a user specified balancing parameter have a fixed bin capacity C for all bins, and define c = 1 + ε = Cn/m. ∗Supported in part by Advanced Grant DFF-0602-02499B from the Danish Council for Independent Research under the Sapere Aude research career programme. Throughout this paper, we use balls as clients, and also bins as servers, interchangeably. ar X iv :1 60 8. 01 35 0v 2 [ cs .D S] 1 2 M ay 2 01 7