Statistical Pruning for Near-Maximum Likelihood Decoding

In many communications problems, maximum-likelihood (ML) decoding reduces to finding the closest (skewed) lattice point in N-dimensions to a given point xisin CN. In its full generality, this problem is known to be NP-complete. Recently, the expected complexity of the sphere decoder, a particular algorithm that solves the ML problem exactly, has been computed. An asymptotic analysis of this complexity has also been done where it is shown that the required computations grow exponentially in N for any fixed SNR. At the same time, numerical computations of the expected complexity show that there are certain ranges of rates, SNRs and dimensions N for which the expected computation (counted as the number of scalar multiplications) involves no more than N3 computations. However, when the dimension of the problem grows too large, the required computations become prohibitively large, as expected from the asymptotic exponential complexity. In this paper, we propose an algorithm that, for large N, offers substantial computational savings over the sphere decoder, while maintaining performance arbitrarily close to ML. We statistically prune the search space to a subset that, with high probability, contains the optimal solution, thereby reducing the complexity of the search. Bounds on the error performance of the new method are proposed. The complexity of the new algorithm is analyzed through an upper bound. The asymptotic behavior of the upper bound for large N is also analyzed which shows that the upper bound is also exponential but much lower than the sphere decoder. Simulation results show that the algorithm is much more efficient than the original sphere decoder for smaller dimensions as well, and does not sacrifice much in terms of performance

Efficient Construction of Probabilistic Tree Embeddings

In this paper we describe an algorithm that embeds a graph metric $(V,d_G)$ on an undirected weighted graph $G=(V,E)$ into a distribution of tree metrics $(T,D_T)$ such that for every pair $u,v\in V$, $d_G(u,v)\leq d_T(u,v)$ and ${\bf{E}}_{T}[d_T(u,v)]\leq O(\log n)\cdot d_G(u,v)$. Such embeddings have proved highly useful in designing fast approximation algorithms, as many hard problems on graphs are easy to solve on tree instances. For a graph with $n$ vertices and $m$ edges, our algorithm runs in $O(m\log n)$ time with high probability, which improves the previous upper bound of $O(m\log^3 n)$ shown by Mendel et al.\,in 2009. The key component of our algorithm is a new approximate single-source shortest-path algorithm, which implements the priority queue with a new data structure, the "bucket-tree structure". The algorithm has three properties: it only requires linear time in the number of edges in the input graph; the computed distances have a distance preserving property; and when computing the shortest-paths to the $k$-nearest vertices from the source, it only requires to visit these vertices and their edge lists. These properties are essential to guarantee the correctness and the stated time bound. Using this shortest-path algorithm, we show how to generate an intermediate structure, the approximate dominance sequences of the input graph, in $O(m \log n)$ time, and further propose a simple yet efficient algorithm to converted this sequence to a tree embedding in $O(n\log n)$ time, both with high probability. Combining the three subroutines gives the stated time bound of the algorithm. Then we show that this efficient construction can facilitate some applications. We proved that FRT trees (the generated tree embedding) are Ramsey partitions with asymptotically tight bound, so the construction of a series of distance oracles can be accelerated

Four Soviets Walk the Dog-Improved Bounds for Computing the Fr\'echet Distance

Given two polygonal curves in the plane, there are many ways to define a notion of similarity between them. One popular measure is the Fr\'echet distance. Since it was proposed by Alt and Godau in 1992, many variants and extensions have been studied. Nonetheless, even more than 20 years later, the original $O(n^2 \log n)$ algorithm by Alt and Godau for computing the Fr\'echet distance remains the state of the art (here, $n$ denotes the number of edges on each curve). This has led Helmut Alt to conjecture that the associated decision problem is 3SUM-hard. In recent work, Agarwal et al. show how to break the quadratic barrier for the discrete version of the Fr\'echet distance, where one considers sequences of points instead of polygonal curves. Building on their work, we give a randomized algorithm to compute the Fr\'echet distance between two polygonal curves in time $O(n^2 \sqrt{\log n}(\log\log n)^{3/2})$ on a pointer machine and in time $O(n^2(\log\log n)^2)$ on a word RAM. Furthermore, we show that there exists an algebraic decision tree for the decision problem of depth $O(n^{2-\varepsilon})$, for some $\varepsilon > 0$. We believe that this reveals an intriguing new aspect of this well-studied problem. Finally, we show how to obtain the first subquadratic algorithm for computing the weak Fr\'echet distance on a word RAM.Comment: 34 pages, 15 figures. A preliminary version appeared in SODA 201

Cyclic best first search in branch-and-bound algorithms

In this dissertation, we study the application of a search strategy called cyclic best first search (CBFS) in branch-and-bound (B&B) algorithms. First, we solve a one machine scheduling problem with release and delivery times with the minimum makespan objective with a B&B algorithm using a variant of CBFS called CBFS-depth and a modified heuristic for finding feasible schedules. Second, we investigate the conditions of the search trees that may lead to CBFS-depth outperforming BFS in terms of the average number of nodes explored to prove optimality. Finally, we present a B&B algorithm using CBFS for a close-enough traveling salesman problem that demonstrates the benefit of using CBFS even if it does not improve the number of nodes explored to prove optimality. Overall, we show that using CBFS has a number of advantages to the performance of a B&B algorithm in comparison to the other search strategies given the right problems