From Fibonacci Numbers to Central Limit Type Theorems

A beautiful theorem of Zeckendorf states that every integer can be written uniquely as a sum of non-consecutive Fibonacci numbers $\{F_n\}_{n=1}^{\infty}$. Lekkerkerker proved that the average number of summands for integers in $[F_n, F_{n+1})$ is $n/(\phi^2 + 1)$, with $\phi$ the golden mean. This has been generalized to the following: given nonnegative integers $c_1,c_2,...,c_L$ with $c_1,c_L>0$ and recursive sequence $\{H_n\}_{n=1}^{\infty}$ with $H_1=1$, $H_{n+1} =c_1H_n+c_2H_{n-1}+...+c_nH_1+1$ $(1\le n< L)$ and $H_{n+1}=c_1H_n+c_2H_{n-1}+...+c_LH_{n+1-L}$ $(n\geq L)$, every positive integer can be written uniquely as $\sum a_iH_i$ under natural constraints on the $a_i$'s, the mean and the variance of the numbers of summands for integers in $[H_{n}, H_{n+1})$ are of size $n$, and the distribution of the numbers of summands converges to a Gaussian as $n$ goes to the infinity. Previous approaches used number theory or ergodic theory. We convert the problem to a combinatorial one. In addition to re-deriving these results, our method generalizes to a multitude of other problems (in the sequel paper \cite{BM} we show how this perspective allows us to determine the distribution of gaps between summands in decompositions). For example, it is known that every integer can be written uniquely as a sum of the $\pm F_n$'s, such that every two terms of the same (opposite) sign differ in index by at least 4 (3). The presence of negative summands introduces complications and features not seen in previous problems. We prove that the distribution of the numbers of positive and negative summands converges to a bivariate normal with computable, negative correlation, namely $-(21-2\phi)/(29+2\phi) \approx -0.551058$.Comment: This is a companion paper to Kologlu, Kopp, Miller and Wang's On the number of summands in Zeckendorf decompositions. Version 2.0 (mostly correcting missing references to previous literature

Performance Guarantees for Distributed Reachability Queries

In the real world a graph is often fragmented and distributed across different sites. This highlights the need for evaluating queries on distributed graphs. This paper proposes distributed evaluation algorithms for three classes of queries: reachability for determining whether one node can reach another, bounded reachability for deciding whether there exists a path of a bounded length between a pair of nodes, and regular reachability for checking whether there exists a path connecting two nodes such that the node labels on the path form a string in a given regular expression. We develop these algorithms based on partial evaluation, to explore parallel computation. When evaluating a query Q on a distributed graph G, we show that these algorithms possess the following performance guarantees, no matter how G is fragmented and distributed: (1) each site is visited only once; (2) the total network traffic is determined by the size of Q and the fragmentation of G, independent of the size of G; and (3) the response time is decided by the largest fragment of G rather than the entire G. In addition, we show that these algorithms can be readily implemented in the MapReduce framework. Using synthetic and real-life data, we experimentally verify that these algorithms are scalable on large graphs, regardless of how the graphs are distributed.Comment: VLDB201