Degree-constrained Minimum Latency Trees are APX-Hard

When transmitting data from a single source to many recipients, it is often desirable to use some recipients of the stream to re-broadcast the stream to other users. In such multicast systems each client may be used as a server, serving up to B other clients. Formally, we will take a set X of clients, along with a distance function d that specifies the latency (in the host network) between each pair of clients in X. Our goal will be to produce a directed spanning tree of X, rooted at some specified root 2 X, with out-degree bounded by B, and minimizing the sum of the latencies from root to every point in X. In addition to being motivated by current experimental algorithms work, the problem also interpolates naturally between the traveling repairman problem (when B = 1) and single source shortest paths (when B = n − 1). The former problem is APX-complete (in metric spaces) and the latter is in P. We explore the hardness of the problem for other values of B. In particular, we show that the problem remains APX-Hard at least up to B = Cpn for some universal constant C when the host space is a general semi-metric

A note on the data-driven capacity of P2P networks

We consider two capacity problems in P2P networks. In the first one, the nodes have an infinite amount of data to send and the goal is to optimally allocate their uplink bandwidths such that the demands of every peer in terms of receiving data rate are met. We solve this problem through a mapping from a node-weighted graph featuring two labels per node to a max flow problem on an edge-weighted bipartite graph. In the second problem under consideration, the resource allocation is driven by the availability of the data resource that the peers are interested in sharing. That is a node cannot allocate its uplink resources unless it has data to transmit first. The problem of uplink bandwidth allocation is then equivalent to constructing a set of directed trees in the overlay such that the number of nodes receiving the data is maximized while the uplink capacities of the peers are not exceeded. We show that the problem is NP-complete, and provide a linear programming decomposition decoupling it into a master problem and multiple slave subproblems that can be resolved in polynomial time. We also design a heuristic algorithm in order to compute a suboptimal solution in a reasonable time. This algorithm requires only a local knowledge from nodes, so it should support distributed implementations. We analyze both problems through a series of simulation experiments featuring different network sizes and network densities. On large networks, we compare our heuristic and its variants with a genetic algorithm and show that our heuristic computes the better resource allocation. On smaller networks, we contrast these performances to that of the exact algorithm and show that resource allocation fulfilling a large part of the peer can be found, even for hard configuration where no resources are in excess.Comment: 10 pages, technical report assisting a submissio

Networked Computing in Wireless Sensor Networks for Structural Health Monitoring

This paper studies the problem of distributed computation over a network of wireless sensors. While this problem applies to many emerging applications, to keep our discussion concrete we will focus on sensor networks used for structural health monitoring. Within this context, the heaviest computation is to determine the singular value decomposition (SVD) to extract mode shapes (eigenvectors) of a structure. Compared to collecting raw vibration data and performing SVD at a central location, computing SVD within the network can result in significantly lower energy consumption and delay. Using recent results on decomposing SVD, a well-known centralized operation, into components, we seek to determine a near-optimal communication structure that enables the distribution of this computation and the reassembly of the final results, with the objective of minimizing energy consumption subject to a computational delay constraint. We show that this reduces to a generalized clustering problem; a cluster forms a unit on which a component of the overall computation is performed. We establish that this problem is NP-hard. By relaxing the delay constraint, we derive a lower bound to this problem. We then propose an integer linear program (ILP) to solve the constrained problem exactly as well as an approximate algorithm with a proven approximation ratio. We further present a distributed version of the approximate algorithm. We present both simulation and experimentation results to demonstrate the effectiveness of these algorithms

On the Stability of Distribution Topologies in Peer-to-Peer Live Streaming Systems

Peer-to-Peer Live-Streaming-Systeme sind ständigen Störungen ausgesetzt.Insbesondere ermöglichen unzuverlässige Teilnehmer Ausfälle und Angriffe, welche überraschend Peers aus dem System entfernen. Die Folgen solcher Vorfälle werden großteils von der Verteilungstopologie bestimmt, d.h. der Kommunikationsstruktur zwischen den Peers.In dieser Arbeit analysieren wir Optimierungsprobleme welche bei der Betrachtung von Stabilitätsbegriffen für solche Verteilungstopologien auftreten. Dabei werden sowohl Angriffe als auch unkoordinierte Ausfälle berücksichtigt.Zunächst untersuchen wir die Berechnungskomplexität und Approximierbarkeit des Problems resourcen-effiziente Angriffe zu bestimmen. Dies demonstriert Beschränkungen in den Planungsmöglichkeiten von Angreifern und zeigt inwieweit die Topologieparameter die Schwierigkeit solcher Angriffsrobleme beeinflussen. Anschließend studieren wir Topologieformationsprobleme. Dabei sind Topologieparameter vorgegeben und es muss eine passende Verteilungstopologie gefunden werden. Ziel ist es Topologien zu erzeugen, welche den durch Angriffe mit beliebigen Parametern erzeugbaren maximalen Schaden minimieren.Wir identifizieren notwendige und hinreichende Eigenschaften solcher Verteilungstopologien. Dies führt zu mathematisch fundierten Zielstellungen für das Topologie-Management von Peer-to-Peer Live-Streaming-Systemen.Wir zeigen zwei große Klassen effizient konstruierbarer Verteilungstopologien, welche den maximal möglichen, durch Angriffe verursachten Paketverlust minimieren. Zusätzlich beweisen wir, dass die Bestimmung dieser Eigenschaft für beliebige Topologien coNP-vollständig ist.Soll die maximale Anzahl von Peers minimiert werden, bei denen ein Angriff zu ungenügender Stream-Qualität führt, ändern sich die Anforderungen an Verteilungstopologien. Wir zeigen, dass dieses Topologieformationsproblem eng mit offenen Problemen aus Design- und Kodierungstheorie verwandt ist.Schließlich analysieren wir Verteilungstopologien die den durch unkoordinierte Ausfälle zu erwartetenden Paketverlust minimieren. Wir zeigen Eigenschaften und Existenzbedingungen. Außerdem bestimmen wir die Berechnungskomplexität des Auffindens solcher Topologien. Unsere Ergebnisse liefern Richtlinien für das Topologie-Management von Peer-to-Peer Live-Streaming-Systemen und zeigen auf, welche Stabilitätsziele effizient erreicht werden können.The stability of peer-to-peer live streaming systems is constantly challenged. Especially, the unreliability and vulnerability of their participants allows for failures and attacks suddenly disabling certain sets of peers. The consequences of such events are largely determined by the distribution topology, i.e., the pattern of communication between the peers.In this thesis, we analyze a broad range of optimization problems concerning the stability of distribution topologies. For this, we discuss notions of stability against both attacks and failures.At first, we investigate the computational complexity and approximability of finding resource-efficient attacks. This allows to point out limitations of an attacker's planning capabilities and demonstrates the influence of the chosen system parameters on the hardness of such attack problems.Then, we turn to study topology formation problems. Here, a set of topology parameters is given and the task consists in finding an eligible distribution topology. In particular, it has to minimize the maximum damage achievable by attacks with arbitrary attack parameters.We identify necessary and sufficient conditions on attack-stable distribution topologies. Thereby, we give mathematically sound guidelines for the topology management of peer-to-peer live streaming systems.We find large classes of efficiently-constructable topologies minimizing the system-wide packet loss under attacks. Additionally, we show that determining this feature for arbitrary topologies is coNP-complete.Considering topologies minimizing the maximum number of peers for which an attack leads to a heavy decrease in perceived streaming quality, the requirements change. Here, we show that the corresponding topology formation problem is closely related to long-standing open problems of Design and Coding Theory.Finally, we study topologies minimizing the expected packet loss due to uncoordinated peer failures. We investigate properties and existence conditions of such topologies. Furthermore, we determine the computational complexity of constructing them.Our results provide guidelines for the topology management of peer-to-peer live streaming systems and mathematically determine which goals can be achieved efficiently

Abstract Depth-Latency Tradeoffs in Multicast Tree Algorithms

The construction of multicast trees is complicated by the need to balance a number of important objectives, including: minimizing latencies, minimizing depth/hops, and bounding the degree. In this paper we study the problem of determining a degree-bounded directed spanning tree of minimum average-latency in a complete graph where the inter-node latencies determine a metric. In particular, we focus on measuring the effects on average latency when imposing depth constraints (i.e., bounds on hop count) on degree-bounded spanning trees. The general problem is a well known NP-hard problem, and several recent works have proposed approximate solutions which aim at minimizing either depth or latency. In this work, we present a new heuristic algorithm which improves upon previous solutions by considering both depth and latency and the tradeoffs between them. Our algorithms are shown to improve the theoretical worst-case approximation factors, and we show improvements under empirical evaluation. Our experiments examine and analyze several different topologies, including, low-dimensional random geometric networks, random transit-stub networks, and high-dimensional hypercube networks. We show how our solutions can be applied in the context of enabling multicasting support in locality aware peer-to-peer overlay networks.