Efficient motion planning for problems lacking optimal substructure

We consider the motion-planning problem of planning a collision-free path of a robot in the presence of risk zones. The robot is allowed to travel in these zones but is penalized in a super-linear fashion for consecutive accumulative time spent there. We suggest a natural cost function that balances path length and risk-exposure time. Specifically, we consider the discrete setting where we are given a graph, or a roadmap, and we wish to compute the minimal-cost path under this cost function. Interestingly, paths defined using our cost function do not have an optimal substructure. Namely, subpaths of an optimal path are not necessarily optimal. Thus, the Bellman condition is not satisfied and standard graph-search algorithms such as Dijkstra cannot be used. We present a path-finding algorithm, which can be seen as a natural generalization of Dijkstra's algorithm. Our algorithm runs in $O\left((n_B\cdot n) \log( n_B\cdot n) + n_B\cdot m\right)$ time, where~$n$ and $m$ are the number of vertices and edges of the graph, respectively, and $n_B$ is the number of intersections between edges and the boundary of the risk zone. We present simulations on robotic platforms demonstrating both the natural paths produced by our cost function and the computational efficiency of our algorithm

Approximation algorithm for QoS routing with multiple additive constraints

In this paper, we study the problem of computing the supported QoS from a source to a destination with multiple additive constraints. The problem has been shown to be NP-complete and many approximation algorithms have been developed. We propose a new approximation algorithm called multi-dimensional relaxation algorithm. We formally prove that our algorithm produces smaller approximation error than the existing algorithms. We further verify the performance by extensive simulations. ©2009 IEEE.published_or_final_versio

Towards Internet QoS Provisioning Based on Generic Distributed QoS Adaptive Routing Engine

Increasing efficiency and quality demands of modern Internet technologies drive today’s network engineers to seek to provide quality of service (QoS). Internet QoS provisioning gives rise to several challenging issues. This paper introduces a generic distributed QoS adaptive routing engine (DQARE) architecture based on OSPFxQoS. The innovation of the proposed work in this paper is its undependability on the used QoS architectures and, moreover, splitting of the control strategy from data forwarding mechanisms, so we guarantee a set of absolute stable mechanisms on top of which Internet QoS can be built. DQARE architecture is furnished with three relevant traffic control schemes, namely, service differentiation, QoS routing, and traffic engineering. The main objective of this paper is to (i) provide a general configuration guideline for service differentiation, (ii) formalize the theoretical properties of different QoS routing algorithms and then introduce a QoS routing algorithm (QOPRA) based on dynamic programming technique, and (iii) propose QoS multipath forwarding (QMPF) model for paths diversity exploitation. NS2-based simulations proved the DQARE superiority in terms of delay, packet delivery ratio, throughput, and control overhead. Moreover, extensive simulations are used to compare the proposed QOPRA algorithm and QMPF model with their counterparts in the literature

Computing Shortest Paths for Any Number of Hops

In this paper, we introduce and investigate a “new” path optimization problem that we denote the all hops optimal path (AHOP) problem. The problem involves identifying, for all hop counts, the optimal, i.e., minimum weight, path(s) between a given source and destination(s). The AHOP problem arises naturally in the context of quality-of-service (QoS) routing in networks, where routes (paths) need to be computed that provide services guarantees, e.g., delay or bandwidth, at the minimum possible “cost” (amount of resources required) to the network. Because service guarantees are typically provided through some form of resource allocation on the path (links) computed for a new request, the hop count, which captures the number of links over which resources are allocated, is a commonly used cost measure. As a result, a standard approach for determining the cheapest path available that meets a desired level of service guarantees is to compute a minimum hop shortest (optimal) path. Furthermore, for efficiency purposes, it is desirable to precompute such optimal minimum hop paths for all possible service requests. Providing this information gives rise to solving the AHOP problem. The paper’s contributions are to investigate the computational complexity of solving the AHOP problem for two of the most prevalent cost functions (path weights) in networks, namely, additive and bottleneck weights. In particular, we establish that a solution based on the Bellman–Ford algorithm is optimal for additive weights, but show that this does not hold for bottleneck weights for which a lower complexity solution exists

Fast Monotone Summation over Disjoint Sets

We study the problem of computing an ensemble of multiple sums where the summands in each sum are indexed by subsets of size $p$ of an $n$-element ground set. More precisely, the task is to compute, for each subset of size $q$ of the ground set, the sum over the values of all subsets of size $p$ that are disjoint from the subset of size $q$. We present an arithmetic circuit that, without subtraction, solves the problem using $O((n^p+n^q)\log n)$ arithmetic gates, all monotone; for constant $p$, $q$ this is within the factor $\log n$ of the optimal. The circuit design is based on viewing the summation as a "set nucleation" task and using a tree-projection approach to implement the nucleation. Applications include improved algorithms for counting heaviest $k$-paths in a weighted graph, computing permanents of rectangular matrices, and dynamic feature selection in machine learning