By Vangelis Th. Paschos, Peter Widmayer
This publication constitutes the refereed convention complaints of the ninth foreign convention on Algorithms and Complexity, CIAC 2015, held in Paris, France, in might 2015.
The 30 revised complete papers offered have been rigorously reviewed and chosen from ninety three submissions and are offered including 2 invited papers. The papers current unique study within the idea and functions of algorithms and computational complexity.
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In addition, the radius of sensor i may increase due to its movement from yi to yi and due to covering 2 . up to grid points. Hence, ri ≤ ri + m 1 2 ε ε If ri ≥ 2μ , then ri ≤ ri + m ≤ ri + 4αμ . Hence, ≤ ri 1 + 2α Ei (y , r ) = a|yi − xi | + t(ri )α ≤ a|yi − xi | + triα 1 + ε α 2α ≤ 1+ ε/2 ≤e Otherwise, if ri < 1 2μ , then ri ≤ ri + 2 m ≤ 1 2μ + ε 4αμ ≤ 1 μ. ε α 2α Ei (y, r) Ei (y, r) . Hence, Ei (y , r ) = a|yi − xi | + t(ri )α ≤ a|yi − xi | + t μ1α = a|yi − xi | + t 2αεnα “Green” Barrier Coverage with Mobile Sensors 39 Putting it all together we get i Ei (y , r ) ≤ eε/2 i Ei (y, r) + nt 2αεnα ≤ (1 + ε) i Ei (y, r) + ε · opt (1) where the second inequality follows from (i) eε/2 ≤ 1 + ε, for any ε ∈ (0, 1), and 1 (ii) opt ≥ nt (2n) α , as observed after Theorem 1.
Let (x, t) be a SumVar instance, and let (y, r) be an optimal solution for (x, t) that minimizes the number of swaps. If there are no swaps, then we are done. Otherwise, we show that the number of swaps may be decreased. 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.
Our FPTAS for the non-zero friction case is obtained by the following approach. We ﬁrst show that any SumVar instance has a non-swapping optimal solution. Then, we show that we pay an approximation factor of (1 + ε) for only considering a certain family of solutions. Finally, we design a dynamic programming algorithm that computes an optimal solution within this family. Lemma 1. Any SumVar instance has a non-swapping optimal solution. Proof. Let (x, t) be a SumVar instance, and let (y, r) be an optimal solution for (x, t) that minimizes the number of swaps.