Balanced Allocations: A Simple Proof for the Heavily Loaded Case

We provide a relatively simple proof that the expected gap between the maximum load and the average load in the two choice process is bounded by $(1+o(1))\log \log n$, irrespective of the number of balls thrown. The theorem was first proven by Berenbrink et al. Their proof uses heavy machinery from Markov-Chain theory and some of the calculations are done using computers. In this manuscript we provide a significantly simpler proof that is not aided by computers and is self contained. The simplification comes at a cost of weaker bounds on the low order terms and a weaker tail bound for the probability of deviating from the expectation

Balanced Allocation on Graphs: A Random Walk Approach

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 $d=\omega(\log n)$. Then we establish an upper bound for the maximum number of balls at any bin after allocating $n$ balls by the algorithm, called {\it maximum load}, in terms of $l$ with high probability. We also show that the upper bound is at most an $O(\log\log n)$ factor above the lower bound that is proved for the algorithm. In particular, we show that if we set $l=\lfloor(\log n)^{\frac{1+\epsilon}{2}}\rfloor$, for every constant $\epsilon\in (0, 1)$, and $G$ has girth at least $\omega(l)$, then the maximum load attained by the algorithm is bounded by $O(1/\epsilon)$ with high probability.Finally, we slightly modify the algorithm to have similar results for balanced allocation on $d$-regular graph with $d\in[3, O(\log n)]$ and sufficiently large girth

Locally Optimal Load Balancing

This work studies distributed algorithms for locally optimal load-balancing: We are given a graph of maximum degree $\Delta$, and each node has up to $L$ units of load. The task is to distribute the load more evenly so that the loads of adjacent nodes differ by at most $1$. If the graph is a path ($\Delta = 2$), it is easy to solve the fractional version of the problem in $O(L)$ communication rounds, independently of the number of nodes. We show that this is tight, and we show that it is possible to solve also the discrete version of the problem in $O(L)$ rounds in paths. For the general case ($\Delta > 2$), we show that fractional load balancing can be solved in $\operatorname{poly}(L,\Delta)$ rounds and discrete load balancing in $f(L,\Delta)$ rounds for some function $f$, independently of the number of nodes.Comment: 19 pages, 11 figure

Worst case and probabilistic analysis of the 2-Opt algorithm for the TSP

2-Opt is probably the most basic local search heuristic for the TSP. This heuristic achieves amazingly good results on “real world” Euclidean instances both with respect to running time and approximation ratio. There are numerous experimental studies on the performance of 2-Opt. However, the theoretical knowledge about this heuristic is still very limited. Not even its worst case running time on 2-dimensional Euclidean instances was known so far. We clarify this issue by presenting, for every p∈N , a family of L p instances on which 2-Opt can take an exponential number of steps. Previous probabilistic analyses were restricted to instances in which n points are placed uniformly at random in the unit square [0,1]2, where it was shown that the expected number of steps is bounded by O~(n10) for Euclidean instances. We consider a more advanced model of probabilistic instances in which the points can be placed independently according to general distributions on [0,1] d , for an arbitrary d≥2. In particular, we allow different distributions for different points. We study the expected number of local improvements in terms of the number n of points and the maximal density ϕ of the probability distributions. We show an upper bound on the expected length of any 2-Opt improvement path of O~(n4+1/3⋅ϕ8/3) . When starting with an initial tour computed by an insertion heuristic, the upper bound on the expected number of steps improves even to O~(n4+1/3−1/d⋅ϕ8/3) . If the distances are measured according to the Manhattan metric, then the expected number of steps is bounded by O~(n4−1/d⋅ϕ) . In addition, we prove an upper bound of O(ϕ√d) on the expected approximation factor with respect to all L p metrics. Let us remark that our probabilistic analysis covers as special cases the uniform input model with ϕ=1 and a smoothed analysis with Gaussian perturbations of standard deviation σ with ϕ∼1/σ d

Scalable Multi-Party Private Set-Intersection

In this work we study the problem of private set-intersection in the multi-party setting and design two protocols with the following improvements compared to prior work. First, our protocols are designed in the so-called star network topology, where a designated party communicates with everyone else, and take a new approach of leveraging the 2PC protocol of [FreedmanNP04]. This approach minimizes the usage of a broadcast channel, where our semi-honest protocol does not make any use of such a channel and all communication is via point-to-point channels. In addition, the communication complexity of our protocols scales with the number of parties. More concretely, (1) our first semi-honest secure protocol implies communication complexity that is linear in the input sizes, namely $O((\sum_{i=1}^n m_i)\cdot\kappa)$ bits of communication where $\kappa$ is the security parameter and $m_i$ is the size of $P_i$\u27s input set, whereas overall computational overhead is quadratic in the input sizes only for a designated party, and linear for the rest. We further reduce this overhead by employing two types of hashing schemes. (2) Our second protocol is proven secure in the malicious setting. This protocol induces communication complexity O((n^2 + nm_\maxx + nm_\minn\log m_\maxx)\kappa) bits of communication where m_\minn (resp. m_\maxx) is the minimum (resp. maximum) over all input sets sizes and $n$ is the number of parties

Almost Optimal Permutation Routing on Hypercubes

This paper deals with permutation routing on hypercube networks in the store-and-forward model. We introduce the first (on-line and off-line) algorithms routing any permutation on the $d$-dimensional hypercube in $d+o(d)$ steps. The best previously known results were $2d+o(d)$ (oblivious on-line) and $2d-3$ (off-line). In particular, we present \begin{itemize} \item a randomized, oblivious on-line algorithm with routing time $d + O(d / \log d)$, \item a matching lower bound of $d + \Omega(d / \log d)$ for (randomized) oblivious on-line routing, and \item a deterministic, off-line algorithm with routing time $d+O(\sqrt{d \log d})$. \end{itemize} Previous algorithms lose a factor of two mainly because packets are first sent to intermediate destinations in order to resolve congestion. As a consequence, the maximum path length becomes $2d - o(d)$. Our algorithms use intermediate destinations as well, but we introduce a simple, elegant trick ensuring that the routing paths are not stretched too much. In fact, we achieve small congestion using paths of length at most $d$. The main focus of our work, however, lies on the scheduling aspect. On one hand, we investigate well-known and practical scheduling policies for on-line routing, namely {\em Farthest-to-Go\/} and {\em Nearest-to-Origin}. On the other hand, we present a new off-line scheduling scheme that is based on frugal colorings of multigraphs. This scheme might be of interest for other sparse scheduling problems, too

Random Knapsack in Expected Polynomial Tme

We present the first average-case analysis proving a polynomial upper bound on the expected running time of an exact algorithm for the 0/1 knapsack problem. In particular, we prove for various input distributions, that the number of Pareto-optimal knapsack fillings is polynomially bounded in the number of availa ble items. An algorithm by Nemhauser and Ullmann can enumerate these solutions very efficiently so that a polynomial upper bound on the number of Pareto-optimal sol utions implies an algorithm with expected polynomial running time. The random input model underlying our analysis is quite general and not restricted to a particular input distribution. We assume adversarial weights and randomly drawn profits (or vice versa). Our analysis covers general probability distributions with finite mean and, in its most general form, can even handle different probability distributions for the profits of different items. This feature enables us to study the effects of correlations between profits and weights. Our analysis confirms and explains practical studies showing that so-called \em strongly correlated\/} instances are harder to solve than {\em weakly correlated\/ ones