Optimization of Discrete-parameter Multiprocessor Systems using a Novel Ergodic Interpolation Technique

Modern multi-core systems have a large number of design parameters, most of which are discrete-valued, and this number is likely to keep increasing as chip complexity rises. Further, the accurate evaluation of a potential design choice is computationally expensive because it requires detailed cycle-accurate system simulation. If the discrete parameter space can be embedded into a larger continuous parameter space, then continuous space techniques can, in principle, be applied to the system optimization problem. Such continuous space techniques often scale well with the number of parameters. We propose a novel technique for embedding the discrete parameter space into an extended continuous space so that continuous space techniques can be applied to the embedded problem using cycle accurate simulation for evaluating the objective function. This embedding is implemented using simulation-based ergodic interpolation, which, unlike spatial interpolation, produces the interpolated value within a single simulation run irrespective of the number of parameters. We have implemented this interpolation scheme in a cycle-based system simulator. In a characterization study, we observe that the interpolated performance curves are continuous, piece-wise smooth, and have low statistical error. We use the ergodic interpolation-based approach to solve a large multi-core design optimization problem with 31 design parameters. Our results indicate that continuous space optimization using ergodic interpolation-based embedding can be a viable approach for large multi-core design optimization problems.Comment: A short version of this paper will be published in the proceedings of IEEE MASCOTS 2015 conferenc

Hamiltonian cycles and subsets of discounted occupational measures

We study a certain polytope arising from embedding the Hamiltonian cycle problem in a discounted Markov decision process. The Hamiltonian cycle problem can be reduced to finding particular extreme points of a certain polytope associated with the input graph. This polytope is a subset of the space of discounted occupational measures. We characterize the feasible bases of the polytope for a general input graph $G$, and determine the expected numbers of different types of feasible bases when the underlying graph is random. We utilize these results to demonstrate that augmenting certain additional constraints to reduce the polyhedral domain can eliminate a large number of feasible bases that do not correspond to Hamiltonian cycles. Finally, we develop a random walk algorithm on the feasible bases of the reduced polytope and present some numerical results. We conclude with a conjecture on the feasible bases of the reduced polytope.Comment: revised based on referees comment

An SPQR-Tree Approach to Decide Special Cases of Simultaneous Embedding with Fixed Edges

We present a linear-time algorithm for solving the simulta- neous embedding problem with ?xed edges (SEFE) for a planar graph and a pseudoforest (a graph with at most one cycle) by reducing it to the following embedding problem: Given a planar graph G, a cycle C of G, and a partitioning of the remaining vertices of G, does there exist a planar embedding in which the induced subgraph on each vertex partite of G C is contained entirely inside or outside C ? For the latter prob- lem, we present an algorithm that is based on SPQR-trees and has linear running time. We also show how we can employ SPQR-trees to decide SEFE for two planar graphs where one graph has at most two cycles and the intersection is a pseudoforest in linear time. These results give rise to our hope that our SPQR-tree approach might eventually lead to a polynomial-time algorithm for deciding the general SEFE problem for two planar graphs

Map Matching with Simplicity Constraints

We study a map matching problem, the task of finding in an embedded graph a path that has low distance to a given curve in R^2. The Fr\'echet distance is a common measure for this problem. Efficient methods exist to compute the best path according to this measure. However, these methods cannot guarantee that the result is simple (i.e. it does not intersect itself) even if the given curve is simple. In this paper, we prove that it is in fact NP-complete to determine the existence a simple cycle in a planar straight-line embedding of a graph that has at most a given Fr\'echet distance to a given simple closed curve. We also consider the implications of our proof on some variants of the problem

Cycle factors and renewal theory

For which values of $k$ does a uniformly chosen $3$-regular graph $G$ on $n$ vertices typically contain $n/k$ vertex-disjoint $k$-cycles (a $k$-cycle factor)? To date, this has been answered for $k=n$ and for $k \ll \log n$; the former, the Hamiltonicity problem, was finally answered in the affirmative by Robinson and Wormald in 1992, while the answer in the latter case is negative since with high probability most vertices do not lie on $k$-cycles. Here we settle the problem completely: the threshold for a $k$-cycle factor in $G$ as above is $\kappa_0 \log_2 n$ with $\kappa_0=[1-\frac12\log_2 3]^{-1}\approx 4.82$. Precisely, we prove a 2-point concentration result: if $k \geq \kappa_0 \log_2(2n/e)$ divides $n$ then $G$ contains a $k$-cycle factor w.h.p., whereas if $k<\kappa_0\log_2(2n/e)-\frac{\log^2 n}n$ then w.h.p. it does not. As a byproduct, we confirm the "Comb Conjecture," an old problem concerning the embedding of certain spanning trees in the random graph $G(n,p)$. The proof follows the small subgraph conditioning framework, but the associated second moment analysis here is far more delicate than in any earlier use of this method and involves several novel features, among them a sharp estimate for tail probabilities in renewal processes without replacement which may be of independent interest.Comment: 45 page