With double hashing, for a key $x$, one generates two hash values $f(x)$ and $g(x)$, and then uses combinations $(f(x) +i g(x)) \bmod n$ for $i=0,1,2,...$ to generate multiple hash values in the range $[0,n-1]$ from the initial two. For balanced allocations, keys are hashed into a hash table where each bucket can hold multiple keys, and each key is placed in the least loaded of $d$ choices. It has been shown previously that asymptotically the performance of double hashing and fully random hashing is the same in the balanced allocation paradigm using fluid limit methods. Here we extend a coupling argument used by Lueker and Molodowitch to show that double hashing and ideal uniform hashing are asymptotically equivalent in the setting of open address hash tables to the balanced allocation setting, providing further insight into this phenomenon. We also discuss the potential for and bottlenecks limiting the use this approach for other multiple choice hashing schemes.
Consistent Hashing functions are widely used for load balancing across a variety of applications. However, the original presentation and typical implementations of Consistent Hashing rely on randomised allocation of hash codes to keys which results in a flawed and approximately-uniform allocation of keys to hash codes. We analyse the desired properties and present an algorithm that perfectly achieves them without resorting to any random distributions. The algorithm is simple and adds to our understanding of what is necessary to create a consistent hash function.
Designing algorithms for balanced allocation of clients to servers in dynamic settings is a challenging problem for a variety of reasons. Both servers and clients may be added and/or removed from the system periodically, and the main objectives of allocation algorithms are: the uniformity of the allocation, and the number of moves after adding or removing a server or a client. The most popular solution for our dynamic settings is Consistent Hashing. However, the load balancing of consistent hashing is no better than a random assignment of clients to servers, so with $n$ of each, we expect many…
We consider a variation of balls-into-bins which randomly allocates $m$ balls into $n$ bins. Following Godfrey's model (SODA, 2008), we assume that each ball $t$, $1\le t\le m$, comes with a hypergraph $\mathcal{H}^{(t)}=\{B_1,B_2,\ldots,B_{s_t}\}$, and each edge $B\in\mathcal{H}^{(t)}$ contains at least a logarithmic number of bins. Given $d\ge 2$, our $d$-choice algorithm chooses an edge $B\in \mathcal{H}^{(t)}$, uniformly at random, and then chooses a set $D$ of $d$ random bins from the selected edge $B$. The ball is allocated to a least-loaded bin from $D$, with ties are broken randomly. W…
We consider the allocation problem in which $m \leq (1-ε) dn $ items are to be allocated to $n$ bins with capacity $d$. The items $x_1,x_2,\ldots,x_m$ arrive sequentially and when item $x_i$ arrives it is given two possible bin locations $p_i=h_1(x_i),q_i=h_2(x_i)$ via hash functions $h_1,h_2$. We consider a random walk procedure for inserting items and show that the expected time insertion time is constant provided $ε= Ω\left(\sqrt{ \frac{ \log d}{d}} \right).$
In this paper, we study the maximum loads of explicit hash families in the $d$-choice schemes when allocating sequentially $n$ balls into $n$ bins. We consider the \emph{Uniform-Greedy} scheme, which provides $d$ independent bins for each ball and places the ball into the bin with the least load, and its non-uniform variant --- the \emph{Always-Go-Left} scheme introduced by Vöcking. We construct a hash family with $O(\log n \log \log n)$ random bits based on the previous work of Celis et al. and show the following results. 1. With high probability, this hash family has a maximum load of $\frac…
In the 2-choice allocation problem, $m$ balls are placed into $n$ bins, and each ball must choose between two random bins $i, j \in [n]$ that it has been assigned to. It has been known for more than two decades, that if each ball follows the Greedy strategy (i.e., always pick the less-full bin), then the maximum load will be $m/n + O(\log \log n)$ with high probability in $n$ (and $m / n + O(\log m)$ with high probability in $m$). It has remained open whether the same bounds hold in the dynamic version of the same game, where balls are inserted/deleted with up to $m$ balls present at a time. W…
To overcome incompatibility issues, kidney patients may swap their donors. In international kidney exchange programmes (IKEPs), countries merge their national patient-donor pools. We consider a recently introduced credit system. In each round, countries are given an initial ``fair'' allocation of the total number of kidney transplants. This allocation is adjusted by a credit function yielding a target allocation. The goal is to find a solution that approaches the target allocation as closely as possible, to ensure long-term stability of the international pool. As solutions, we use maximum matc…
In this paper we propose algorithms for allocating $n$ sequential balls into $n$ bins that are interconnected as a $d$-regular $n$-vertex graph $G$, where $d\ge3$ can be any integer.Let $l$ be a given positive integer. In each round $t$, $1\le t\le n$, ball $t$ picks a node of $G$ uniformly at random and performs a non-backtracking random walk of length $l$ from the chosen node.Then it allocates itself on one of the visited nodes with minimum load (ties are broken uniformly at random). Suppose that $G$ has a sufficiently large girth and $d=ω(\log n)$. Then we establish an upper bound for the m…
In the standard ball-in-bins experiment, a well-known scheme is to sample $d$ bins independently and uniformly at random and put the ball into the least loaded bin. It can be shown that this scheme yields a maximum load of $\log\log n/\log d+O(1)$ with high probability. Subsequent work analyzed the model when at each time, $d$ bins are sampled through some correlated or non-uniform way. However, the case when the sampling for different balls are correlated are rarely investigated. In this paper we propose three schemes for the ball-in-bins allocation problem. We assume that there is an underly…
We consider load balancing in a network of caching servers delivering contents to end users. Randomized load balancing via the so-called power of two choices is a well-known approach in parallel and distributed systems that reduces network imbalance. In this paper, we propose a randomized load balancing scheme which simultaneously considers cache size limitation and proximity in the server redirection process. Since the memory limitation and the proximity constraint cause correlation in the server selection process, we may not benefit from the power of two choices in general. However, we prove…
Load balancing is a well-studied problem, with balls-in-bins being the primary framework. The greedy algorithm $\mathsf{Greedy}[d]$ of Azar et al. places each ball by probing $d > 1$ random bins and placing the ball in the least loaded of them. With high probability, the maximum load under $\mathsf{Greedy}[d]$ is exponentially lower than the result when balls are placed uniformly randomly. Vöcking showed that a slightly asymmetric variant, $\mathsf{Left}[d]$, provides a further significant improvement. However, this improvement comes at an additional computational cost of imposing structure on…
We consider the allocation of $m$ balls into $n$ bins with incomplete information. In the classical Two-Choice process a ball first queries the load of two randomly chosen bins and is then placed in the least loaded bin. In our setting, each ball also samples two random bins but can only estimate a bin's load by sending binary queries of the form "Is the load at least the median?" or "Is the load at least 100?". For the lightly loaded case $m=O(n)$, Feldheim and Gurel-Gurevich (2021) showed that with one query it is possible to achieve a maximum load of $O(\sqrt{\log n/\log \log n})$, and pose…
We consider the allocation of $m$ balls (jobs) into $n$ bins (servers). In the Two-Choice process, for each of $m$ sequentially arriving balls, two randomly chosen bins are sampled and the ball is placed in the least loaded bin. It is well-known that the maximum load is $m/n+\log_2 \log n + O(1)$ w.h.p. Berenbrink, Czumaj, Englert, Friedetzky and Nagel (2012) introduced a parallel version of this process, where $m$ balls arrive in consecutive batches of size $b=n$ each. Balls within the same batch are allocated in parallel, using the load information of the bins at the beginning of the batch. …
We consider the allocation of $m$ balls (jobs) into $n$ bins (servers). In the standard Two-Choice process, at each step $t=1,2,\ldots,m$ we first sample two bins uniformly at random and place a ball in the least loaded bin. It is well-known that for any $m \geq n$, this results in a gap (difference between the maximum and average load) of $\log_2 \log n + Θ(1)$ (with high probability). In this work, we consider the Memory process [Mitzenmacher, Prabhakar and Shah 2002] where instead of two choices, we only sample one bin per step but we have access to a cache which can store the location of o…
