To reduce computational complexity and delay in randomized network coded content distribution, and for some other practical reasons, coding is not performed simultaneously over all content blocks, but over much smaller, possibly overlapping subsets of these blocks, known as generations. A penalty of this strategy is throughput reduction. To analyze the throughput loss, we model coding over generations with random generation scheduling as a coupon collector's brotherhood problem. This model enables us to derive the expected number of coded packets needed for successful decoding of the entire content as well as the probability of decoding failure (the latter only when generations do not overlap) and further, to quantify the tradeoff between computational complexity and throughput. Interestingly, with a moderate increase in the generation size, throughput quickly approaches link capacity. Overlaps between generations can further improve throughput substantially for relatively small generation sizes.
The performance of codes with random scheduling of disjoint generations was first theoretically analyzed in @cite_14 by , who referred to them as chunked codes . Chunked codes allow convenient encoding at intermediate nodes, and are readily suitable for peer-to-peer file dissemination. In @cite_14 , the authors used an adversarial schedule as the network model and characterized the code performance under certain restrictions on the chunk(generation) size when the length of the information to be encoded tends to infinity. Coding with overlapping generations was first studied in @cite_3 and @cite_9 with the goal to improve throughput. Reference @cite_9 studied a head-to-toe'' overlapping scheme in which only contiguous generations overlap for a given number of information packets, and analyzed its asymptotic performance over a line network when the length of information goes to infinity. Another overlapping scheme with a grid structure was proposed in @cite_3 , analyzed for short lengths (e.g., @math generations) and simulated for practical lengths. When properly designed, these codes show improved performance over codes with disjoint generations. In our work, we offer an analysis of coding over disjoint and overlapping generations for finite but practically long information lengths.
@cite_9 @cite_14 @cite_3
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Traditionally, the idea of overlapping generations in network coding research has focused on reducing the complexity of decoding large data files while maintaining the delay performance expected of a system that combines all data packets. However, the effort for encoding and decoding individual generations can still be quite high compared to other sparse coding approaches. This paper focuses on an inherently different approach that combines (i) sparsely coded generations configured on-the- fly based on (ii) controllable and infrequent feedback that allows the system to remove some original packets from the pool of packets to be mixed in the linear combinations. The latter is key to maintain a high impact of the coded packets received during the entire process while maintaining very sparsely coded generations. Interestingly, our proposed approach naturally bridges the idea of overlapping generations with that of tunable sparse network coding, thus providing the system with a seamless and adaptive strategy to balance complexity and delay performance. We analyze two families of strategies focused on these ideas. We also compare them to other standard approaches both in terms of delay performance and complexity as well as providing measurements in commercial devices to support our conclusions. Our results show that a judicious choice of the overlapping of the generations provides close-to-optimal delay performance, while reducing the decoding complexity by up to an order of magnitude with respect to other schemes. To reduce computational complexity and delay in randomized network coded content distribution, and for some other practical reasons, coding is not performed simultaneously over all content blocks, but over much smaller, possibly overlapping subsets of these blocks, known as generations. A penalty of this strategy is throughput reduction. To analyze the throughput loss, we model coding over generations with random generation scheduling as a coupon collector's brotherhood problem. This model enables us to derive the expected number of coded packets needed for successful decoding of the entire content as well as the probability of decoding failure (the latter only when generations do not overlap) and further, to quantify the tradeoff between computational complexity and throughput. Interestingly, with a moderate increase in the generation size, throughput quickly approaches link capacity. Overlaps between generations can further improve throughput substantially for relatively small generation sizes. This paper presents a novel approach to network coding for distribution of large files. Instead of the usual approach of splitting packets into disjoint classes (also known as generations) we propose the use of overlapping classes. The overlapping allows the decoder to alternate between Gaussian elimination and back substitution, simultaneously boosting the performance and reducing the decoding complexity. Our approach can be seen as a combination of fountain coding and network coding. Simulation results are presented that demonstrate the promise of our approach. Selective families, a weaker variant of superimposed codes [KS64, F92, 197, CR96], have been recently used to design Deterministic Distributed Broadcast (DDB) protocols for unknown radio networks (a radio network is said to be unknown when the nodes know nothing about the network but their own label) [CGGPR00, CGOR00]. We first provide a general almost tight lower bound on the size of selective families. Then, by reverting the selective families - DDB protocols connection, we exploit our lower bound to construct a family of “hard” radio networks (i.e. directed graphs). These networks yield an O(n log D) lower bound on the completion time of DDB protocols that is superlinear (in the size n of the network) even for very small maximum eccentricity D of the network, while all the previous lower bounds (e.g. O(D log n) [CGGPR00]) are superlinear only when D is almost linear. On the other hand, the previous upper bounds are all superlinear in n independently of the eccentricity D and the maximum in-degree d of the network. We introduce a broadcast technique that exploits selective families in a new way. Then, by combining selective families of almost optimal size with our new broadcast technique, we obtain an O(Dd log3 n) upper bound that we prove to be almost optimal when d = O(n D). This exponentially improves over the best known upper bound [CGR00) when D, d = O(polylogn). Furthermore, by comparing our deterministic upper bound with the best known randomized one [BGI87] we obtain a new, rather surprising insight into the real gap between deterministic and randomized protocols. It turns out that this gap is exponential (as discovered in [BGI87]), but only when the network has large maximum in-degree (i.e. d = O(na), for some constant a > O). We then look at the multibroadcast problem on unknown radio networks. A similar connection to that between selective families and (single) broadcast also holds between superimposed codes and multibroadcast. We in fact combine a variant of our (single) broadcast technique with superimposed codes of almost optimal size available in literature [EFF85, HS87, I97, CHI99]. This yields a multibroadcast protocol having completion time O(Dd2 log3 n). Finally, in order to determine the limits of our multibroadcast technique, we generalize (and improve) the best known lower bound [CR96] on the size of superimposed codes.