Tight Cell Probe Bounds for Succinct Boolean Matrix-Vector Multiplication

The conjectured hardness of Boolean matrix-vector multiplication has been used with great success to prove conditional lower bounds for numerous important data structure problems, see Henzinger et al. [STOC'15]. In recent work, Larsen and Williams [SODA'17] attacked the problem from the upper bound side and gave a surprising cell probe data structure (that is, we only charge for memory accesses, while computation is free). Their cell probe data structure answers queries in $\tilde{O}(n^{7/4})$ time and is succinct in the sense that it stores the input matrix in read-only memory, plus an additional $\tilde{O}(n^{7/4})$ bits on the side. In this paper, we essentially settle the cell probe complexity of succinct Boolean matrix-vector multiplication. We present a new cell probe data structure with query time $\tilde{O}(n^{3/2})$ storing just $\tilde{O}(n^{3/2})$ bits on the side. We then complement our data structure with a lower bound showing that any data structure storing $r$ bits on the side, with $n < r < n^2$ must have query time $t$ satisfying $t r = \tilde{\Omega}(n^3)$. For $r \leq n$, any data structure must have $t = \tilde{\Omega}(n^2)$. Since lower bounds in the cell probe model also apply to classic word-RAM data structures, the lower bounds naturally carry over. We also prove similar lower bounds for matrix-vector multiplication over $\mathbb{F}_2$

On the Distribution of Random Geometric Graphs

Random geometric graphs (RGGs) are commonly used to model networked systems that depend on the underlying spatial embedding. We concern ourselves with the probability distribution of an RGG, which is crucial for studying its random topology, properties (e.g., connectedness), or Shannon entropy as a measure of the graph's topological uncertainty (or information content). Moreover, the distribution is also relevant for determining average network performance or designing protocols. However, a major impediment in deducing the graph distribution is that it requires the joint probability distribution of the $n(n-1)/2$ distances between $n$ nodes randomly distributed in a bounded domain. As no such result exists in the literature, we make progress by obtaining the joint distribution of the distances between three nodes confined in a disk in $\mathbb{R}^2$. This enables the calculation of the probability distribution and entropy of a three-node graph. For arbitrary $n$, we derive a series of upper bounds on the graph entropy; in particular, the bound involving the entropy of a three-node graph is tighter than the existing bound which assumes distances are independent. Finally, we provide numerical results on graph connectedness and the tightness of the derived entropy bounds.Comment: submitted to the IEEE International Symposium on Information Theory 201

Fast hashing with Strong Concentration Bounds

Previous work on tabulation hashing by Patrascu and Thorup from STOC'11 on simple tabulation and from SODA'13 on twisted tabulation offered Chernoff-style concentration bounds on hash based sums, e.g., the number of balls/keys hashing to a given bin, but under some quite severe restrictions on the expected values of these sums. The basic idea in tabulation hashing is to view a key as consisting of $c=O(1)$ characters, e.g., a 64-bit key as $c=8$ characters of 8-bits. The character domain $\Sigma$ should be small enough that character tables of size $|\Sigma|$ fit in fast cache. The schemes then use $O(1)$ tables of this size, so the space of tabulation hashing is $O(|\Sigma|)$. However, the concentration bounds by Patrascu and Thorup only apply if the expected sums are $\ll |\Sigma|$. To see the problem, consider the very simple case where we use tabulation hashing to throw $n$ balls into $m$ bins and want to analyse the number of balls in a given bin. With their concentration bounds, we are fine if $n=m$, for then the expected value is $1$. However, if $m=2$, as when tossing $n$ unbiased coins, the expected value $n/2$ is $\gg |\Sigma|$ for large data sets, e.g., data sets that do not fit in fast cache. To handle expectations that go beyond the limits of our small space, we need a much more advanced analysis of simple tabulation, plus a new tabulation technique that we call \emph{tabulation-permutation} hashing which is at most twice as slow as simple tabulation. No other hashing scheme of comparable speed offers similar Chernoff-style concentration bounds.Comment: 54 pages, 3 figures. An extended abstract appeared at the 52nd Annual ACM Symposium on Theory of Computing (STOC20

Quantifying Link Stability in Ad Hoc Wireless Networks Subject to Ornstein-Uhlenbeck Mobility

The performance of mobile ad hoc networks in general and that of the routing algorithm, in particular, can be heavily affected by the intrinsic dynamic nature of the underlying topology. In this paper, we build a new analytical/numerical framework that characterizes nodes' mobility and the evolution of links between them. This formulation is based on a stationary Markov chain representation of link connectivity. The existence of a link between two nodes depends on their distance, which is governed by the mobility model. In our analysis, nodes move randomly according to an Ornstein-Uhlenbeck process using one tuning parameter to obtain different levels of randomness in the mobility pattern. Finally, we propose an entropy-rate-based metric that quantifies link uncertainty and evaluates its stability. Numerical results show that the proposed approach can accurately reflect the random mobility in the network and fully captures the link dynamics. It may thus be considered a valuable performance metric for the evaluation of the link stability and connectivity in these networks.Comment: 6 pages, 4 figures, Submitted to IEEE International Conference on Communications 201

Nearly Optimal Static Las Vegas Succinct Dictionary

Given a set $S$ of $n$ (distinct) keys from key space $[U]$, each associated with a value from $\Sigma$, the \emph{static dictionary} problem asks to preprocess these (key, value) pairs into a data structure, supporting value-retrieval queries: for any given $x\in [U]$, $\mathtt{valRet}(x)$ must return the value associated with $x$ if $x\in S$, or return $\bot$ if $x\notin S$. The special case where $|\Sigma|=1$ is called the \emph{membership} problem. The "textbook" solution is to use a hash table, which occupies linear space and answers each query in constant time. On the other hand, the minimum possible space to encode all (key, value) pairs is only $\mathtt{OPT}:= \lceil\lg_2\binom{U}{n}+n\lg_2|\Sigma|\rceil$ bits, which could be much less. In this paper, we design a randomized dictionary data structure using $\mathtt{OPT}+\mathrm{poly}\lg n+O(\lg\lg\lg\lg\lg U)$ bits of space, and it has \emph{expected constant} query time, assuming the query algorithm can access an external lookup table of size $n^{0.001}$. The lookup table depends only on $U$, $n$ and $|\Sigma|$, and not the input. Previously, even for membership queries and $U\leq n^{O(1)}$, the best known data structure with constant query time requires $\mathtt{OPT}+n/\mathrm{poly}\lg n$ bits of space (Pagh [Pag01] and P\v{a}tra\c{s}cu [Pat08]); the best-known using $\mathtt{OPT}+n^{0.999}$ space has query time $O(\lg n)$; the only known non-trivial data structure with $\mathtt{OPT}+n^{0.001}$ space has $O(\lg n)$ query time and requires a lookup table of size $\geq n^{2.99}$ (!). Our new data structure answers open questions by P\v{a}tra\c{s}cu and Thorup [Pat08,Tho13]. We also present a scheme that compresses a sequence $X\in\Sigma^n$ to its zeroth order (empirical) entropy up to $|\Sigma|\cdot\mathrm{poly}\lg n$ extra bits, supporting decoding each $X_i$ in $O(\lg |\Sigma|)$ expected time.Comment: preliminary version appeared in STOC'2

Emotional Qualities of VR Space

The emotional response a person has to a living space is predominantly affected by light, color and texture as space-making elements. In order to verify whether this phenomenon could be replicated in a simulated environment, we conducted a user study in a six-sided projected immersive display that utilized equivalent design attributes of brightness, color and texture in order to assess to which extent the emotional response in a simulated environment is affected by the same parameters affecting real environments. Since emotional response depends upon the context, we evaluated the emotional responses of two groups of users: inactive (passive) and active (performing a typical daily activity). The results from the perceptual study generated data from which design principles for a virtual living space are articulated. Such a space, as an alternative to expensive built dwellings, could potentially support new, minimalist lifestyles of occupants, defined as the neo-nomads, aligned with their work experience in the digital domain through the generation of emotional experiences of spaces. Data from the experiments confirmed the hypothesis that perceivable emotional aspects of real-world spaces could be successfully generated through simulation of design attributes in the virtual space. The subjective response to the virtual space was consistent with corresponding responses from real-world color and brightness emotional perception. Our data could serve the virtual reality (VR) community in its attempt to conceive of further applications of virtual spaces for well-defined activities.Comment: 12 figure

SystemC Model Generation for Realistic Simulation of Networked Embedded Systems

Verification and design-space exploration of today's embedded systems require the simulation of heterogeneous aspects of the system, i.e., software, hardware, communications. This work shows the use of SystemC to simulate a model-driven specification of the behavior of a networked embedded system together with a complete network scenario consisting of the radio channel, the IEEE 802.15.4 protocol for wireless personal area networks and concurrent traffic sharing the medium. The paper describes the main issues addressed to generate SystemC modules from Matlab/Stateflow descriptions and to integrate them in a complete network scenario. Simulation results on a healthcare wireless sensor network show the validity of the approach