We use network coding to improve the speed of distributed computation in the dynamic network model of Kuhn, Lynch and Oshman [STOC '10]. In this model an adversary adaptively chooses a new network topology in every round, making even basic distributed computations challenging. show that n nodes, each starting with a d-bit token, can broadcast them to all nodes in time O(n^2) using b-bit messages, where b > d + log n. Their algorithms take the natural approach of token forwarding : in every round each node broadcasts some particular token it knows. They prove matching Omega(n^2) lower bounds for a natural class of token forwarding algorithms and an Omega(n log n) lower bound that applies to all token-forwarding algorithms. We use network coding, transmitting random linear combinations of tokens, to break both lower bounds. Our algorithm's performance is quadratic in the message size b, broadcasting the n tokens in roughly d b^2 * n^2 rounds. For b = d = O(log n) our algorithms use O(n^2 log n) rounds, breaking the first lower bound, while for larger message sizes we obtain linear-time algorithms. We also consider networks that change only every T rounds, and achieve an additional factor T^2 speedup. This contrasts with related lower and upper bounds of implying that for natural token-forwarding algorithms a speedup of T, but not more, can be obtained. Lastly, we give a general way to derandomize random linear network coding, that also leads to new deterministic information dissemination algorithms.
Secondly, in all prior literature including @cite_1 , it is also assumed that tokens are uniquely numbered indexed and that this index is known to any node that starts with a token. This is needed to allow nodes to specify in the coding header which packets are coded together in a message. In this paper such an assumption would be unacceptable. For example, for the task of counting the number of nodes in a dynamic network @cite_7 having the IDs consecutively indexed would essentially amount to assuming that a solution to the counting problem is already part of the input.
@cite_1 @cite_7
1512160561 2130507035 2089001139 2145438521
Index coding studies multiterminal source-coding problems where a set of receivers are required to decode multiple (possibly different) messages from a common broadcast, and they each know some messages a priori . In this paper, at the receiver end, we consider a special setting where each receiver knows only one message a priori , and each message is known to only one receiver. At the broadcasting end, we consider a generalized setting where there could be multiple senders, and each sender knows a subset of the messages. The senders collaborate to transmit an index code. This paper looks at minimizing the number of total coded bits the senders are required to transmit. When there is only one sender, we propose a pruning algorithm to find a lower bound on the optimal (i.e., the shortest) index codelength, and show that it is achievable by linear index codes. When there are two or more senders, we propose an appending technique to be used in conjunction with the pruning technique to give a lower bound on the optimal index codelength; we also derive an upper bound based on cyclic codes. While the two bounds do not match in general, for the special case where no two distinct senders know any message in common, the bounds match, giving the optimal index codelength. The results are expressed in terms of strongly connected components in directed graphs that represent the index-coding problems. Motivated by a problem of transmitting supplemental data over broadcast channels (Birk and Kol, INFOCOM 1998), we study the following coding problem: a sender communicates with n receivers R1,..., Rn. He holds an input x ∈ 0,01l n and wishes to broadcast a single message so that each receiver Ri can recover the bit xi. Each Ri has prior side information about x, induced by a directed graph Grain nodes; Ri knows the bits of a; in the positions j | (i,j) is an edge of G .G is known to the sender and to the receivers. We call encoding schemes that achieve this goal INDEXcodes for 0,1 n with side information graph G. In this paper we identify a measure on graphs, the minrank, which exactly characterizes the minimum length of linear and certain types of nonlinear INDEX codes. We show that for natural classes of side information graphs, including directed acyclic graphs, perfect graphs, odd holes, and odd anti-holes, minrank is the optimal length of arbitrary INDEX codes. For arbitrary INDEX codes and arbitrary graphs, we obtain a lower bound in terms of the size of the maximum acyclic induced subgraph. This bound holds even for randomized codes, but has been shown not to be tight. We consider a group of n wireless clients and a set of k messages. Each client initially holds a subset of messages and is interested in an arbitrary subset of messages. Each client cooperates with other clients to obtain the set of messages it wants by exchanging instantly decodable network coded (IDNC) packets. This problem setting is known as the cooperative index coding problem. Clients are assumed to be connected through an arbitrary topology. In the absence of any known algorithm to complete the exchange of packets for general network topologies, we propose a greedy algorithm to satisfy the demands of all the clients with the aim of reducing the mean completion time. Our algorithm, in a completely distributed fashion, decides which subset of clients should transmit at each round of transmission and which messages should be coded together by each transmitting client to generate an IDNC packet. The algorithm encourages transmissions which are decodable for a larger number of clients and attempts to avoid collisions. We evaluate the performance of our algorithm via numerical experiments. This letter investigates a new class of index coding problems. One sender broadcasts packets to multiple users, each desiring a subset, by exploiting prior knowledge of linear combinations of packets. We refer to this class of problems as index coding with coded side-information. Our aim is to characterize the minimum index code length that the sender needs to transmit to simultaneously satisfy all user requests. We show that the optimal binary vector index code length is equal to the minimum rank (minrank) of a matrix whose elements consist of the sets of desired packet indices and side-information encoding matrices. This is the natural extension of matrix minrank in the presence of coded side information. Using the derived expression, we propose a greedy randomized algorithm to minimize the rank of the derived matrix.