Succinct Indexable Dictionaries with Applications to Encoding kk-ary Trees, Prefix Sums and Multisets

We consider the {\it indexable dictionary} problem, which consists of storing a set $S \subseteq \{0,...,m-1\}$ for some integer $m$, while supporting the operations of \Rank(x), which returns the number of elements in $S$ that are less than $x$ if $x \in S$, and -1 otherwise; and \Select(i) which returns the $i$-th smallest element in $S$. We give a data structure that supports both operations in O(1) time on the RAM model and requires ${\cal B}(n,m) + o(n) + O(\lg \lg m)$ bits to store a set of size $n$, where {\cal B}(n,m) = \ceil{\lg {m \choose n}} is the minimum number of bits required to store any $n$-element subset from a universe of size $m$. Previous dictionaries taking this space only supported (yes/no) membership queries in O(1) time. In the cell probe model we can remove the $O(\lg \lg m)$ additive term in the space bound, answering a question raised by Fich and Miltersen, and Pagh. We present extensions and applications of our indexable dictionary data structure, including: An information-theoretically optimal representation of a $k$-ary cardinal tree that supports standard operations in constant time, A representation of a multiset of size $n$ from $\{0,...,m-1\}$ in ${\cal B}(n,m+n) + o(n)$ bits that supports (appropriate generalizations of) \Rank and \Select operations in constant time, and A representation of a sequence of $n$ non-negative integers summing up to $m$ in ${\cal B}(n,m+n) + o(n)$ bits that supports prefix sum queries in constant time.Comment: Final version of SODA 2002 paper; supersedes Leicester Tech report 2002/1

Residual Minimizing Model Interpolation for Parameterized Nonlinear Dynamical Systems

We present a method for approximating the solution of a parameterized, nonlinear dynamical system using an affine combination of solutions computed at other points in the input parameter space. The coefficients of the affine combination are computed with a nonlinear least squares procedure that minimizes the residual of the governing equations. The approximation properties of this residual minimizing scheme are comparable to existing reduced basis and POD-Galerkin model reduction methods, but its implementation requires only independent evaluations of the nonlinear forcing function. It is particularly appropriate when one wishes to approximate the states at a few points in time without time marching from the initial conditions. We prove some interesting characteristics of the scheme including an interpolatory property, and we present heuristics for mitigating the effects of the ill-conditioning and reducing the overall cost of the method. We apply the method to representative numerical examples from kinetics - a three state system with one parameter controlling the stiffness - and conductive heat transfer - a nonlinear parabolic PDE with a random field model for the thermal conductivity.Comment: 28 pages, 8 figures, 2 table

Control efficacy of complex networks

Acknowledgements W.-X.W. was supported by CNNSF under Grant No. 61573064, and No. 61074116 the Fundamental Research Funds for the Central Universities and Beijing Nova Programme, China. Y.-C.L. was supported by ARO under Grant W911NF-14-1-0504.Peer reviewedPublisher PD

Robust Rotation Synchronization via Low-rank and Sparse Matrix Decomposition

This paper deals with the rotation synchronization problem, which arises in global registration of 3D point-sets and in structure from motion. The problem is formulated in an unprecedented way as a "low-rank and sparse" matrix decomposition that handles both outliers and missing data. A minimization strategy, dubbed R-GoDec, is also proposed and evaluated experimentally against state-of-the-art algorithms on simulated and real data. The results show that R-GoDec is the fastest among the robust algorithms.Comment: The material contained in this paper is part of a manuscript submitted to CVI

Line Polar Grassmann Codes of Orthogonal Type

Polar Grassmann codes of orthogonal type have been introduced in I. Cardinali and L. Giuzzi, \emph{Codes and caps from orthogonal Grassmannians}, {Finite Fields Appl.} {\bf 24} (2013), 148-169. They are subcodes of the Grassmann code arising from the projective system defined by the Pl\"ucker embedding of a polar Grassmannian of orthogonal type. In the present paper we fully determine the minimum distance of line polar Grassmann Codes of orthogonal type for $q$ odd