Efficient symbolic computation of approximated small-signal characteristics of analog integrated circuits

A symbolic analysis tool is presented that generates simplified symbolic expressions for the small-signal characteristics of large analog integrated circuits. The expressions are approximated while they are computed, so that only those terms are generated which remain in the final expression. This principle causes drastic savings in CPU time and memory, compared with previous symbolic analysis tools. In this way, the maximum size of circuits that can be analyzed, is largely increased. By taking into account a range for the value of a circuit parameter rather than one single number, the generated expressions are also more generally valid. Mismatch handling is explicitly taken into account in the algorithm. The capabilities of the new tool are illustrated with several experimental result

The scaling limits of the Minimal Spanning Tree and Invasion Percolation in the plane

We prove that the Minimal Spanning Tree and the Invasion Percolation Tree on a version of the triangular lattice in the complex plane have unique scaling limits, which are invariant under rotations, scalings, and, in the case of the MST, also under translations. However, they are not expected to be conformally invariant. We also prove some geometric properties of the limiting MST. The topology of convergence is the space of spanning trees introduced by Aizenman, Burchard, Newman & Wilson (1999), and the proof relies on the existence and conformal covariance of the scaling limit of the near-critical percolation ensemble, established in our earlier works.Comment: 56 pages, 21 figures. A thoroughly revised versio

Sampling Random Spanning Trees Faster than Matrix Multiplication

We present an algorithm that, with high probability, generates a random spanning tree from an edge-weighted undirected graph in $\tilde{O}(n^{4/3}m^{1/2}+n^{2})$ time (The $\tilde{O}(\cdot)$ notation hides $\operatorname{polylog}(n)$ factors). The tree is sampled from a distribution where the probability of each tree is proportional to the product of its edge weights. This improves upon the previous best algorithm due to Colbourn et al. that runs in matrix multiplication time, $O(n^\omega)$. For the special case of unweighted graphs, this improves upon the best previously known running time of $\tilde{O}(\min\{n^{\omega},m\sqrt{n},m^{4/3}\})$ for $m \gg n^{5/3}$ (Colbourn et al. '96, Kelner-Madry '09, Madry et al. '15). The effective resistance metric is essential to our algorithm, as in the work of Madry et al., but we eschew determinant-based and random walk-based techniques used by previous algorithms. Instead, our algorithm is based on Gaussian elimination, and the fact that effective resistance is preserved in the graph resulting from eliminating a subset of vertices (called a Schur complement). As part of our algorithm, we show how to compute $\epsilon$-approximate effective resistances for a set $S$ of vertex pairs via approximate Schur complements in $\tilde{O}(m+(n + |S|)\epsilon^{-2})$ time, without using the Johnson-Lindenstrauss lemma which requires $\tilde{O}( \min\{(m + |S|)\epsilon^{-2}, m+n\epsilon^{-4} +|S|\epsilon^{-2}\})$ time. We combine this approximation procedure with an error correction procedure for handing edges where our estimate isn't sufficiently accurate

Multi-Channel Scheduling for Fast Convergecast in Wireless Sensor Networks

We explore the following fundamental question - how fast can information be collected from a wireless sensor network? We consider a number of design parameters such as, power control, time and frequency scheduling, and routing. There are essentially two factors that hinder efficient data collection - interference and the half-duplex single-transceiver radios. We show that while power control helps in reducing the number of transmission slots to complete a convergecast under a single frequency channel, scheduling transmissions on different frequency channels is more efficient in mitigating the effects of interference (empirically, 6 channels suffice for most 100-node networks). With these observations, we define a receiver-based channel assignment problem, and prove it to be NP-complete on general graphs. We then introduce a greedy channel assignment algorithm that efficiently eliminates interference, and compare its performance with other existing schemes via simulations. Once the interference is completely eliminated, we show that with half-duplex single-transceiver radios the achievable schedule length is lower-bounded by max(2nk − 1,N), where nk is the maximum number of nodes on any subtree and N is the number of nodes in the network. We modify an existing distributed time slot assignment algorithm to achieve this bound when a suitable balanced routing scheme is employed. Through extensive simulations, we demonstrate that convergecast can be completed within up to 50% less time slots, in 100-node networks, using multiple channels as compared to that with single-channel communication. Finally, we also demonstrate further improvements that are possible when the sink is equipped with multiple transceivers or when there are multiple sinks to collect data