Algorithms - Sequential, Parallel - A Unified Appr. by R. Miller, L. Boxer

By R. Miller, L. Boxer

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In the covering phase, sensor i is assigned a sensing radius ri and covers the interval [yi − ri , yi + ri ]; let r = (r1 , . . , rn ) be the radii vector. We call this interval the covering interval of sensor i. An example of movement and coverage by one sensor is given in Fig. 1. e. [0, 1] ⊆ ∪i [yi − ri , yi + ri ]. A pair (y, r) is called feasible if it covers [0, 1]. Sensor i expends energy both in moving and sensing. Given a deployment point yi , the energy sensor i spends in movement is proportional to the distance i has traveled, and given by a|xi − yi |, where a is the constant of proportionality, also referred to as the cost of friction.

Chazelle By our assumption that xB can vary by at most σ0 in each coordinate and x = (xA , xB ), the cell Vv is enclosed within a cube of side-length σ1 , where σ1 ≤ σ0 + 21−γ s + n21−(tv −ν−p)γ/p . (24) We update (21) to estimate the new reproduction rate on the assumption that s > s0 and σ1 is sufficiently smaller than 1/νnO(n) : μ ≤ 1 + σ1 νnO(n) . If Vv intersects the discontinuity uTA xA + uTB xB = 1, then, by (23) and u nO(1) , (uTA Cs +uTB B≤w Bw )xB −1 ≤ 2−γ s+O(log n) (25) 2 = +2−(tv −ν−p)γ/p+O(log n) ≤ δ, (26) with the last inequality ensuring that the constraints fit within δ-slabs.

If there are swaps, then there must exist at least one swap due to a pair of adjacent sensors. Let i and j be such sensors. , with yi = yj and ri = rj , yj = yi and rj = ri , and yk = yk and rk = rk , for every k = i, j. Clearly, the barrier [0, 1] remains covered. We show that the energy sum does not increase, since the total distance traveled by the sensors does not increase. If both sensors move to the right in y, then we have that xi < xj ≤ yj < yi . 38 A. Bar-Noy et al. In this case (yi − xi ) + (yj − xj ) = (yi − xi ) + (yj − xj ), and we are done.

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