Christian Ortolf

Online multi-robot exploration of grid graphs with rectangular obstacles

We consider the multi-robot exploration problem of an unknown n x n grid graph with oriented disjoint rectangular obstacles. All robots start at a given node and have to visit all nodes of the graph. The robots are unrestricted in their computational power and storage. In the local communication model the robots can exchange any information if they meet at the same node. In the global communication model all robots share the same knowledge. In this paper we present the first nontrivial upper and lower bounds. We show that k robots can explore the graph using only local communication in time O( n log2(n) + (f log n)/k), where f is the number of free nodes in the graph. This establishes a competitive upper bound of O(log2 n). For the lower bound we show a competitive factor of Ω((log k)/(log log k)) for deterministic exploration and Ω(√(log k)/(log log k)) for randomized exploration strategies using global communication.

The wake up dominating set problem

Recently developed wake-up receivers pose a viable alternative for duty-cycling in wireless sensor networks. Here, a special radio signal can wake up close-by nodes. We model the wake-up range by the unit-disk graph. Such wake-up radio signals are very energy expensive and limited in range. Therefore, their number must be minimized.We revisit the Connected Dominating Set (CDS) problem for unit-disk graphs and consider an online variant, where starting from an initial node all nodes need to be woken up, while the online algorithm knows only the nodes woken up so far and has no information about the number and location of the sleeping nodes.We show that in general this problem cannot be solved effectively, since a worst-case setting exists where the competitive ratio, i.e. the number of wake-up signals divided by the size of the minimum CDS, is (n) for n nodes. For dense random uniform placements, this problem can be solved within a constant factor competitive ratio with high probability, i.e. 1nc, when the nodes positions are known or at least some rough distance estimator.For a restricted adversary with a reduced wake-up range of 1 we present a deterministic wake-up algorithm with a competitive ratio of O(12) for the general problem in two dimensions.In the case of random placement without any explicit position information we present an O(logn)-competitive epidemic algorithm to wake up all nodes with high probability. Simulations show that a simplified version of this oblivious online algorithm already produces reasonable results, that allow its application in the real world.

A Recursive Approach to Multi-robot Exploration of Trees

The multi-robot exploration problem is to explore an unknown graph of size n and depth d with k robots starting from the same node. For known graphs a traversal of all nodes takes at most \({\cal O}(d + n/k)\) steps. The ratio between the time until cooperating robots explore an unknown graph and the optimal traversal of a known graph is called the competitive exploration time ratio.

Strategies for parallel unaware cleaners

We investigate the parallel traversal of a graph with multiple robots unaware of each other. All robots traverse the graph in parallel forever and the goal is to minimize the time needed until the last node is visited (first visit time) and the time between revisits of a node (revisit time). We also want to minimize the visit time, i.e. the maximum of the first visit time and the time between revisits of a node. We present randomized algorithms for uncoordinated robots, which can compete with the optimal coordinated traversal by a small factor, the so-called competitive ratio.For any number of robots ring and path graph simple traversal strategies allow constant competitive factors even in the worst case. For grid and torus graphs with n nodes and any number of robots there is an O(logn)-competitive algorithm for both visit problems succeeding with high probability, i.e. with probability 1nO(1). For general graphs we present an O(log2n)-competitive algorithm for the first visit problem, while for the visit problem we show an O(log3n)-competitive algorithm both succeeding with high probability.

Classifying peer-to-peer network coding schemes

Modern peer-to-peer file sharing systems distribute large files among peers using block partitioning. Blocks can be redistributed by a peer even before the whole file is available which highly decreases the distribution time. All peer-to-peer networks face the problem of dynamic participation of the peers and dynamic bandwidth in the network. A leaving peer can cause an unrecoverable loss of blocks and obstruct further downloads of the file. Furthermore, the choice which block needs to be sent to which peer is a hard question. A random choice leads to the coupon collector problem which decreases the transmission rate. Filesharing networks like BitTorrent or Splitstream face such problems. Network Coding overcomes this problem by using error redundant codes of all blocks of the file. An efficient randomized variant of it, Practical Network Coding, transmits and recombines random linear combinations of the blocks of the partitioned file. As soon as enough linear combinations have been gathered, the original file can be decoded by a matrix operation, optimizing the network flow in any peer-to-peer network. All known Network Coding schemes, however, suffer from a quadratic cost of read/write disk operations for both encoding and decoding. Since there is an increasing gap between the speed of mass storage devices and the main memory, this poses an obstacle to a wider use of Network Coding schemes. In this paper we present and investigate new network coding schemes, which form a compromise between Network Coding and uncoded block transfer schemes like BitTorrent. These schemes, called Paircoding and Treecoding have smaller read/write costs for encoding and decoding than Practical Network Coding and higher throughput than BitTorrent. We develop a new framework for comparing the throughput of data (performance) of such peer-to-peer file sharing systems and classify these systems, as well as a BitTorrent variant which uses forward error correction. The dynamics of peer-to-peer networks are described by a round model where the set of participating peers and their link quality changes after each round. The framework compares two schemes for all possible dynamic scenarios. If the transmission rate of scheme A is at least as well as scheme B, then we say A performs as well as B. If this is the case and there is a scenario where A is better than B, we say A outperforms B. We show that all of our proposed coding schemes outperform BitTorrent, while being outperformed by Network Coding. This leads to a hierarchy, where BitTorrent is the worst performer and Network Coding is the best performer regarding throughput. Regarding computation (disk read/write) complexity for decoding, BitTorrent and Foward Error Correction have linear time behavior, for Paircoding it is almost linear, Treecoding with one coding tree needs time O(n) and Network Coding has time O(n2).

Strategies for Parallel Unaware Cleaners

We investigate the parallel traversal of a graph with multiple robots unaware of each other. All robots traverse the graph in parallel forever and the goal is to minimize the time needed until the last node is visited (first visit time) and the time between revisits of a node (revisit time). We also want to minimize the visit time, i.e. the maximum of the first visit time and the time between revisits of a node. We present randomized algorithms for uncoordinated robots, which can compete with the optimal coordinated traversal by a small factor, the so-called competitive ratio.

The Wake Up Dominating Set Problem

Recently developed wake-up receivers pose a viable alternative for duty-cycling in wireless sensor networks. Here, a special radio signal can wake up close-by nodes. We model the wake-up range by the unit-disk graph. Such wake-up radio signals are very energy expensive and limited in range. Therefore, the number of signals must be minimized. So, we revisit the Connected Dominating Set (CDS) problem for unit-disk graphs and consider an online variant, where starting from an initial node all nodes need to be woken up, while the online algorithm knows only the nodes woken up so far and has no information about the number and location of the sleeping nodes.

Tree network coding for peer-to-peer networks

Partitioning is the dominant technique to transmit large files in peer-to-peer networks. A peer can redistribute each part immediately after its download. BitTorrent combines this approach with incentives for uploads and has thereby become the most successful peer-to-peer network. However, BitTorrent fails if files are unpopular and are distributed by irregularly participating peers. It is known that Network Coding always provides the optimal data distribution, referred as optimal performance. Yet, for encoding or decoding a single code block the whole file must be read and users are not willing to read <i>O</i>(<i>n</i>²) data blocks from hard disk for sending <i>n</i> message blocks. We call this the disk read/write complexity of an encoding. It is an open question whether fast network coding schemes exist. In this paper we present a solution for simple communication patterns. Here, in a round model each peer can send a limited amount of messages to other peers. We define the depth of this directed acyclic communication graph as the maximum path length (not counting the rounds). In our online model each peer knows the bandwidth of its communication links for the current round, but neither the existence nor the weight of links in future rounds. In this paper we analyze BitTorrent, Network Coding, Tree Coding, and Tree Network Coding. We show that the average encoding and decoding complexity of Tree Coding is bounded by <i>O</i>(<i>kn</i> log² <i>n</i>) disk read/write-operations where <i>k</i> is the number of trees and <i>n</i> the number of data blocks. Tree Coding has perfect performance in communication networks of depth two with a disk read/write complexity of <i>O</i>(<i>pnt</i> log³ <i>n</i>) where <i>p</i> is the number of peers, <i>t</i> is the number of rounds, and <i>n</i> is the number of data blocks. For arbitrary networks Tree Coding performs optimally using 2(δ+1) ^t-1 <i>p</i> log² <i>n</i> trees which results in a read/write complexity of O((δ+1)^t-1 <i>n</i> log³ <i>n</i>) for <i>t</i> rounds and in-degree δ