In this article we present a protocol for building dense numeration in network with unknown topology. Additionally to a unique number each node as result of the protocol will get information about a spanning tree. This spanning tree is constructed during BFS search from the leader node. This property of the numeration can be useful in other tasks, as example we present a protocol for searching bridges in network. The time of numeration building in our protocol is linear in network size, simple informational lower bounds also linear (it is required at least linear number of bits for code tree structure). In bridges searching problem our protocol also heats lower linear bound: in result each node knows about all bridges.
Distributed computations are hardly investigated last time, some basics can be found in [1], [2]. The main concept is a network a set of nodes and bound between them, nodes a participants of an interactive protocol, bounds are used for information transmitting. The range of problems and available methods depends on used network model, and there are many different models. Therefore, in the first chapter we focus on formal definition of used in this work model. The main points of used model are following:
- synchronous timetransition through all bounds happen in the same moment; 2. limited bandwidth of boundsduring one communication round only constant number of bits can be transmitted through each bound; 3. lack of information about network topologyprotocols we study must be correct in some family of networks, at the beginning of protocol execution nodes do not known in which exactly network from considered family they are; 4. uniformity of protocolseach node execute the same program, which must solve a considered problem in each network from family.
The same model of network are investigated in, for example, in works [3], [4], but in more specialized way, we will use definitions that are more generic. Other popular models with synchronized time use either full information about topology either prior dense numeration, or both. Also, popular property for networks models is “signed” messages: bandwidth of bounds is 𝑂(log 𝑉), there 𝑉 is number of nodes, so each message can contain the number of the author.
Mainly we use family of networks with a leader: exactly one node at the beginning of execution knows that it is leader, and other nodes know, that we are not. This model can be considered as basic, because networks with unique numbers (they form sparse numeration) is reducible to the model with a leader [4], meanwhile in networks without unique identifiers most problems are not solvable [4].
The main result of the work (section 2, Theorem 1) is protocol for building dense numeration (each node will get unique number from segment[1, 𝑉]) in network with a leader, the protocol finishes in linear time. In other words, we reduce network with a leader into more researched model. Note, that linear time of work good enough, because protocol in distribute information about a spanning tree: each node will know structure of some tree and its place inside this tree. Therefore, each node will receive sufficient part of information about network’s topology. Note that building spanning tree is a basic task for solving many problems. Additionally, as information about tree is linear, we get linear lower bound (a node with constrained degree cannot receive more than constant bits of information per round).
To sum up, the model we use from one side is more basic, from other side it allow to solve some problems more effective. For example in work [6] presented a protocol for searching bridges in model with signed messages, which solves the problems in linear time. If we reinterpret that protocol in basic model, the time bound will be 𝑂(𝑉log 𝑉), that worse than linear bound for protocol we build in section 3. In addition the protocol we build is better by total traffic through network.
Note, that definition of time of protocol execution is not trivial: it is required either synchronized stop of all nodes either lack of conflicts if some nodes already stopped. Formal definitions we .give in section 3.
Main stages of protocol we build in section 3 is following:
- Finding out “height” of networkmaximal distance between leader and other node. This step partly technical and required for synchronize starting each next stage 2. Tree numeration (protocol described in section 2). In result each node get unique number from segment [1, 𝑉] (there V number of nodes) and also it get structure of some network’s spanning tree, more other number it got is number of the node in this tree. 3. Nodes send their number to each neighbor. As result, at first each edge get number (as pair of numbers of adjacent nodes); at second nodes find out place of neighbors in the tree build in second stage. 4. Nodes transmit to tree-parents information based on previous stage. As result each node find out adjacent bridges. 5. Last optional stage allow share information between all nodes in linear time.
Note that idea of this protocol is very similar to protocol from [6], but using other model and evicting unnecessary data from messages allow to improve protocol time. I would like to emphasize that the Pritchard’s protocol reaches lower bound but only in model with signed messages, reinterpreting the protocol in more generic model do not save optimality.
For make definition of protocol easier, we define network with help of index function. Note, in this definition we do not prohibit loops and multiple edges. In problems we investigate, they do not play any significant role. For some simplification, we consider only connected graphs without loops and mul
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