Energy latency tradeoffs for medium access and sleep scheduling in wireless sensor networks.  Page 63 
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4.4.1 Optimal Assignment on Specific Topologies In this section, we formally characterize the optimal assignment function f (that minimizes the delay diameter Df ) for 2 specific topologies: tree and ring. Using results from simulated annealing on a grid, we also show how an optimal assignment for a ring might form a basic building block of a good assignment on cyclic graphs. 4.4.1.1 Optimal Assignment on a Tree Theorem 1: Consider a tree T = (V,E). Let the number of slots be k. Let the diameter of T (in hops) be h (from node a to b, say). Then for every slot assignment f : V ! [0, · · · k − 1], Df hk 2 . Proof: Consider a path between two nodes p to q having x hops. Since T is a tree, this is the only path between p and q. Consider an arbitrary slot assignment function f : V ! [0, · · · k − 1]. Now, df (p ! q) = Xx j=1 df (ij , ij+1) df (q ! p) = Xx j=1 (k − df (ij , ij+1)) Thus, df (p ! q) + df (q ! p) = kx. max {df (p ! q), df (q ! p)} kx 2 (4.6) This is true for each pair of nodes including a and b. Thus, for every slot assignment function f, Df hk 2 , where h is the diameter of T. Based on theorem 1, the following assignment function f will minimize the delay diameter of the tree T = (V,E) whose hop diameter is h (from a to b): Just use 2 slot values, 0 and d k 2 e. Let df (a) = 0. Adjacent vertices are assigned different slots (similar to a chess board pattern). In this case 8i, j : (i, j) 2 E : max {df (i, j) = df (j, i)} = d k 2 e. Hence max {df (a ! b), df (b ! a)} = d hk 2 e, which tightly matches the lower bound on the delay diameter of T. Thus, an optimal slot assignment for a tree balances the delay in each direction along a path as shown in figure 4.1(a). 50
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Title  Energy latency tradeoffs for medium access and sleep scheduling in wireless sensor networks.  Page 63 
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Full text  4.4.1 Optimal Assignment on Specific Topologies In this section, we formally characterize the optimal assignment function f (that minimizes the delay diameter Df ) for 2 specific topologies: tree and ring. Using results from simulated annealing on a grid, we also show how an optimal assignment for a ring might form a basic building block of a good assignment on cyclic graphs. 4.4.1.1 Optimal Assignment on a Tree Theorem 1: Consider a tree T = (V,E). Let the number of slots be k. Let the diameter of T (in hops) be h (from node a to b, say). Then for every slot assignment f : V ! [0, · · · k − 1], Df hk 2 . Proof: Consider a path between two nodes p to q having x hops. Since T is a tree, this is the only path between p and q. Consider an arbitrary slot assignment function f : V ! [0, · · · k − 1]. Now, df (p ! q) = Xx j=1 df (ij , ij+1) df (q ! p) = Xx j=1 (k − df (ij , ij+1)) Thus, df (p ! q) + df (q ! p) = kx. max {df (p ! q), df (q ! p)} kx 2 (4.6) This is true for each pair of nodes including a and b. Thus, for every slot assignment function f, Df hk 2 , where h is the diameter of T. Based on theorem 1, the following assignment function f will minimize the delay diameter of the tree T = (V,E) whose hop diameter is h (from a to b): Just use 2 slot values, 0 and d k 2 e. Let df (a) = 0. Adjacent vertices are assigned different slots (similar to a chess board pattern). In this case 8i, j : (i, j) 2 E : max {df (i, j) = df (j, i)} = d k 2 e. Hence max {df (a ! b), df (b ! a)} = d hk 2 e, which tightly matches the lower bound on the delay diameter of T. Thus, an optimal slot assignment for a tree balances the delay in each direction along a path as shown in figure 4.1(a). 50 