Mobility of robots is utilized to save energy in long-term communication tasks (e.g., video monitoring) in Liu *et al*. (2007). In such long-term communication tasks, the traffic will be regular and large enough in volume to warrant nodes expending energy moving in order to forward traffic in a more energy-efficient manner. Given a communication request between a source-destination pair, the problem is to find an initial route between them if possible (using current robot locations), and move each node on the route to its desired final location (while maintaining the route if possible). The objective is to minimize the total transmission power of nodes for this long-term communication, while keeping low movement distances from the initial to the final positions of intermediate nodes (robots).

Suppose a source *S* needs to find a route to a destination *D*. According to Goldenberg *et al*. (2004), the optimal locations of relay nodes must lie evenly on the line between *S* and *D*. Theorem 9.1 shows the optimal number of hops and optimal distance of adjacent nodes on the line. Assume that the energy needed for transmitting and receiving between two nodes at distance *d* is proportional to *d*^{α} + *c* for some constant *α* (between 2 and 6) and *c* > 0. Let *d*(*S*, *D*) be the Euclidean distance between *S* and *D*.

**Theorem 9.1.** Stojmenovic and Lin (2001) Total transmission power of route from S to D is minimal when the optimal number of ...

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