We present an iterative algorithm for solving a class of Laplacian system of equations in @math iterations, where @math is a measure of nonlinearity, @math is the number of variables, @math is the number of nonzero entries in the graph Laplacian @math , @math is the solution accuracy and @math neglects (non-leading) logarithmic terms. This algorithm is a natural nonlinear extension of the one by of Kelner et. al., which solves a linear Laplacian system of equations in nearly linear time. Unlike the linear case, in the nonlinear case each iteration takes @math time so the total running time is @math . For sparse graphs where @math and fixed @math this nonlinear algorithm is @math which is slightly faster than standard methods for solving linear equations, which require approximately @math time. Our analysis relies on the construction of a nonlinear "energy function" and a nonlinear extension of the duality analysis of Kelner et. al to the nonlinear case without any explicit references to spectral analysis or electrical flows. These new insights and results provide tools for more general extensions to spectral theory and nonlinear applications.
Our algorithm is based on Kelner et. al's algorithm @cite_3 , which dramatically reduces the amount of machinery required. Its origins are clearly implicitly based on the more sophisticated machinery in the earlier papers, such as low-stretch spanning trees (e.g., @cite_6 ) and ultra-sparsifiers (e.g., @cite_10 ), but does not require them directly.
@cite_10 @cite_6 @cite_3
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We present a Las Vegas algorithm for dynamically maintaining a minimum spanning forest of an n-node graph undergoing edge insertions and deletions. Our algorithm guarantees an O(n^ o(1) ) worst-case update time with high probability. This significantly improves the two recent Las Vegas algorithms by Wulff-Nilsen Wulff-Nilsen16a with update time O(n^ 0.5-&#x2265;ilon ) for some constant &#x2265;ilon 0 and, independently, by Nanongkai and Saranurak NanongkaiS16 with update time O(n^ 0.494 ) (the latter works only for maintaining a spanning forest).Our result is obtained by identifying the common framework that both two previous algorithms rely on, and then improve and combine the ideas from both works. There are two main algorithmic components of the framework that are newly improved and critical for obtaining our result. First, we improve the update time from O(n^ 0.5-&#x2265;ilon ) in Wulff-Nilsen16a to O(n^ o(1) ) for decrementally removing all low-conductance cuts in an expander undergoing edge deletions. Second, by revisiting the contraction technique by Henzinger and King HenzingerK97b and HolmLT01, we show a new approach for maintaining a minimum spanning forest in connected graphs with very few (at most (1+o(1))n) edges. This significantly improves the previous approach in Wulff-Nilsen16a, NanongkaiS16 which is based on Fredericksons 2-dimensional topology tree Frederickson85 and illustrates a new application to this old technique. Spanners, emulators, and approximate distance oracles can be viewed as lossy compression schemes that represent an unweighted graph metric in small space, say O(n1+Δ) bits. There is an inherent tradeoff between the sparsity parameter Δ and the stretch function f of the compression scheme, but the qualitative nature of this tradeoff has remained a persistent open problem. It has been known for some time that when Δ ≥ 1 3 there are schemes with constant additive stretch (distance d is stretched to at most f(d) = d + O(1)), and recent results of Abboud and Bodwin show that when Δ In this paper we show that the lower bound of Abboud and Bodwin is just the first step in a hierarchy of lower bounds that characterize the asymptotic behavior of the optimal stretch function f for sparsity parameter Δ ∈ (0,1 3). Specifically, for any integer k ≥ 2, any compression scheme with size [EQUATION] has a sublinear additive stretch function f: f(d) = d + Ω(d1−1 k). This lower bound matches Thorup and Zwick's (2006) construction of sublinear additive emulators. It also shows that Elkin and Peleg's (1 + ϵ, β)-spanners have an essentially optimal tradeoff between Δ, ϵ, and β, and that the sublinear additive spanners of Pettie (2009) and Chechik (2013) are not too far from optimal. To complement these lower bounds we present a new construction of (1 + ϵ, O(k ϵ)k−1)-spanners with size [EQUATION], where hk Our lower bound technique exhibits several interesting degrees of freedom in the framework of Abboud and Bodwin. By carefully exploiting these freedoms, we are able to obtain lower bounds for several related combinatorial objects. We get lower bounds on the size of (β, ϵ)-hopsets, matching Elkin and Neiman's construction (2016), and lower bounds on shortcutting sets for digraphs that preserve the transitive closure. Our lower bound simplifies Hesse's (2003) refutation of Thorup's conjecture (1992), which stated that adding a linear number of shortcuts suffices to reduce the diameter to polylogarithmic. Finally, we show matching upper and lower bounds for graph compression schemes that work for graph metrics with girth at least 2γ + 1. One consequence is that 's (2010) additive O(γ)-spanners with size [EQUATION] cannot be improved in the exponent. We propose a compact and efficient index structure for massive data sets. Several indexing techniques are widely-used and well-known such as binary trees and B+trees. Unfortunately, we find that these techniques suffer major two shortcomings when applied to massive sets; first, their indices are so large they could overflow regular main memory, and, second, they suffer from a variety of penalties (e.g., conditional branches, low cache hits, and TLB misses), which restricts the number of instructions executed per processor cycle. Our state-of-the-art index structure, called VAST-Tree, classifies branch nodes into multiple layers. It applies existing techniques such as cache-conscious, aligned, and branch-free structures to the top layers of branch nodes in trees. Next, it applies the adaptive compression technique to save space and harness data parallelism with SIMD instructions to the middle and bottom layers of branch nodes. Moreover, a processor-friendly compression technique is applied to leaf nodes. The end result is that trees are much more compact and traversal efficiency is high. We implement a prototype and show its resulting index size and performance as compared to binary trees, and the hardware-conscious technique called FAST which currently offers the highest performance. Compared to current alternatives, VAST-Tree compacts the branch nodes by more than 95 , and the overall index size by 47-84 given that there are 230 keys. With 228 keys, it has roughly 6.0-times and 1.24-times throughput and saves the memory consumption by more than 94.7 and 40.5 as compared to binary trees and FAST, respectively. We describe a novel method for efficient reconstruction of phylogenetic trees, based on sequences of whole genomes or proteomes. The core of our method is a new measure of pairwise distances between sequences, whose lengths may greatly vary. This measure is based on information theoretic tools (Kullback-Leibler relative entropy). We present an algorithm for efficiently computing these distances. The algorithm uses suffix arrays to compute the distance of two l long sequences in O(l) time. It is fast enough to enable the construction of the phylogenomic tree for hundreds of species, and the phylogenomic forest for almost two thousand viruses. An initial analysis of the results exhibits a remarkable agreement with “acceptable phylogenetic truth”. To assess our approach, it was implemented together with a number of alternative approaches, including two that were previously published in the literature. Comparing their outcome to ours, using a “traditional” tree and a standard tree comparison method, our algorithm improved upon the “competition” by a substantial margin.