Energy latency tradeoffs for medium access and sleep scheduling in wireless sensor networks.  Page 64 
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4.4.1.2 Optimal Assignment on a Ring We first show the optimal assignment for the case where the number of nodes n on a ring is a multiple of the number of slots k i.e. n = mk. We then present a lower bound for the case when the number of nodes is not an exact multiple. Theorem 2: Consider n = mk nodes 0, 1, · · ·mk−1 arranged on a ring in the clockwise direction. The optimal slot assignment function f is specified as follows: f(0) = 0. 8i : 1 i mk−1 : f(i) = (f(i − 1) + 1) mod k. Proof: We will refer to such an f as the sequential slot assignment as it assigns a sequentially increasing slot (modulo k) to the nodes around the ring (see figure 4.7 (a)). We prove theorem 2 by contradiction. For k = 2, it is easy to show that assigning 2 adjacent nodes the same slot incurs a delay of 2 in both directions on that link, while a sequential assignment will yield a delay of 1 in either direction. Hence, we focus on the case where k 3. For a sequential slot assignment f, it is easy to show that the delay diameter is given by: Df = m(k − 1) (4.7) Assume that there exists a slot assignment function f0, such that Df0 < Df . In the rest of the proof, we will focus on the delay in the ring due to f0. Consider a block of m links on the ring from node 0 to node m as shown in figure 4.2. Since we assumed that Df0 < m(k − 1), the shortest delay path from node 0 to node m (and vice versa) must lie completely within the block. The alternative path has m(k − 1) links each incurring a delay of at least 1 (If this alternative path is the shortest delay path, it contradicts our assumption that Df0 < m(k −1)). This is true for every block of m links on the ring. Figure 4.3 shows the shortest delay path for nodes within each of k such blocks. 8i : i 2 [1, k], 8j : j 2 [1, 2], let di1 be the delay in block i from node (i − 1)m to im, while di2 be the delay in block i from node im to (i − 1)m as shown in the figure 4.3. We claim that dmin = mini,j{dij} < 2m. This can again be proved by contradiction as follows: Consider a path from node 0 to node k−1 2 m. There are two possibilities as shown in figure 4.3: 1. 0 ! m ! 2m· · · ! k−1 2 m. The delay along this path is at least k−1 2 dmin. 2. 0 ! mk − m· · · ! k−1 2 m. The delay along this path is at least k+1 2 dmin 51
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Title  Energy latency tradeoffs for medium access and sleep scheduling in wireless sensor networks.  Page 64 
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Full text  4.4.1.2 Optimal Assignment on a Ring We first show the optimal assignment for the case where the number of nodes n on a ring is a multiple of the number of slots k i.e. n = mk. We then present a lower bound for the case when the number of nodes is not an exact multiple. Theorem 2: Consider n = mk nodes 0, 1, · · ·mk−1 arranged on a ring in the clockwise direction. The optimal slot assignment function f is specified as follows: f(0) = 0. 8i : 1 i mk−1 : f(i) = (f(i − 1) + 1) mod k. Proof: We will refer to such an f as the sequential slot assignment as it assigns a sequentially increasing slot (modulo k) to the nodes around the ring (see figure 4.7 (a)). We prove theorem 2 by contradiction. For k = 2, it is easy to show that assigning 2 adjacent nodes the same slot incurs a delay of 2 in both directions on that link, while a sequential assignment will yield a delay of 1 in either direction. Hence, we focus on the case where k 3. For a sequential slot assignment f, it is easy to show that the delay diameter is given by: Df = m(k − 1) (4.7) Assume that there exists a slot assignment function f0, such that Df0 < Df . In the rest of the proof, we will focus on the delay in the ring due to f0. Consider a block of m links on the ring from node 0 to node m as shown in figure 4.2. Since we assumed that Df0 < m(k − 1), the shortest delay path from node 0 to node m (and vice versa) must lie completely within the block. The alternative path has m(k − 1) links each incurring a delay of at least 1 (If this alternative path is the shortest delay path, it contradicts our assumption that Df0 < m(k −1)). This is true for every block of m links on the ring. Figure 4.3 shows the shortest delay path for nodes within each of k such blocks. 8i : i 2 [1, k], 8j : j 2 [1, 2], let di1 be the delay in block i from node (i − 1)m to im, while di2 be the delay in block i from node im to (i − 1)m as shown in the figure 4.3. We claim that dmin = mini,j{dij} < 2m. This can again be proved by contradiction as follows: Consider a path from node 0 to node k−1 2 m. There are two possibilities as shown in figure 4.3: 1. 0 ! m ! 2m· · · ! k−1 2 m. The delay along this path is at least k−1 2 dmin. 2. 0 ! mk − m· · · ! k−1 2 m. The delay along this path is at least k+1 2 dmin 51 