To make the
analysis simple, let us assume the boundary is a Lipschitz function [3], which
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Ren-Shiou Liu, Lifeng Sang, and Prasun Sinha
Fig. 15. A sensor network in the square cell field.
includes linear boundaries, parametric curves and some boundaries that cannot
be described parametrically as well. Based on these assumptions, there
will be O(pn) nodes lying on the boundary within a resolution of 1/pn. Also,
[3] infers that the mean-square error (MSE) can not decay faster than O(pn)
given the above conditions.
In order to quantify the total energy that is required to send messages
along the boundary, we assume that each node on the boundary just sends
one message to inform the desired destination that it detects the boundary. If
each message costs a roughly equal unit of energy, the least total energy that
is needed would be O(pn). According to the description above, we have the
following relation between energy cost and boundary estimation [9]:
MSE ?ยป
1
Energy
. (25)
Note that the energy here only contains the basic requirements for transmission.
It does not include other necessary energy (e.g., energy needed to
identify if a node belongs to the set of boundary nodes). In addition, this
formula does not imply that if we provide more energy for a fixed number
of nodes, the MSE would be decreased. Actually, both the MSE and energy
depend on the number of sensor nodes. This formula just gives us a general
idea about how the MSE and energy would behave once the density of nodes
increases.
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