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Thus, there is a link labeled ji+1 connecting vi to vi+1 , and clearly vt = y. But this means that v0 , v1 , v2 , . . , vt is a path from x to y, and the sequence of labels on this path is j1 , j2 , . . , jt , which is decreasing. , it contains all the nodes of) Hk , regardless of x. In other words, within ﬁnite time, every entity will have the information. Let us now concentrate on the cost of HyperFlood. First of all observe that M[HyperFlood/Hk ] = n − 1. 2) BROADCAST 35 To prove that only n − 1 messages will be sent during the broadcast, we just need to show that every entity will receive the information only once.

In other words, the ideal time complexity will be at most the diameter of G: T[Flooding] ≤ d(G). 6 STATES AND EVENTS Once we have deﬁned the behavior of the entities, their communication topology, and the set of restrictions under which they operate, we must describe the initial conditions of our environment. This is done ﬁrst of all by specifying the initial condition of all the entities. The initial content of all the registers of entity x and the initial value of its alarm clock cx at time t constitute the initial internal state σ (x, 0) of x.

Otherwise, xd−1 received the message from a different neighbor, and it then sends the message to all its neighbors, including x. Hence x will eventually receive the message. 2 The total number of messages sent without the improvement was x∈E |N (x)| = 2|E| = 2m; in Flooding, every entity (except the initiator) will send one message less. Hence the total number of messages is 2m − (|V | − 1) = 2m − n + 1. 6 (Basis of Induction only) Consider ﬁrst the case k = 1: Only one child, say z, has a dirty forehead.