Energy latency tradeoffs for medium access and sleep scheduling in wireless sensor networks.  Page 89 
Save page Remove page  Previous  89 of 140  Next 

small (250x250 max)
medium (500x500 max)
Large (1000x1000 max)
Extra Large
large ( > 500x500)
Full Resolution
All (PDF)

This page
All

Figure 5.6: An example of finding minimum length 2 node disjoint paths. briefly describe the algorithm. Figure 5.6 shows an example. We can simply use any shortest path algorithm to find the first minimum weight node disjoint path from a to z, where a is the super pseudo source node, and z is the pseudo sink node. We assume PM is a given optimal set of M nodedisjoint paths in delay graph DG. Now we need to find optimal (M + 1)th nodedisjoint paths PM+1, as follows: 1. Reverse the direction of each edge on PM, and make its length negative. These edges are called negative arcs. Other edges are called positive arcs. 2. Split each vertex v on PM into two nodes v1 and v2, joined by an arc of length zero, directed towards a. Assign output links on v2 and input links on v1. 3. Find a shortest path from a to z on this transformed delay graph. We call this path an interlacing S, which may contain both positive arcs and negative arcs. 4. let PM + S represent the graph obtained by adding to PM the positive arcs of S, and removing from PM the negative arcs of S. This is called Augmentation, which results in the optimal PM+1 paths, the minimum weight M + 1 nodedisjoint paths. We present a lemma that will be used in the following proofs. Lemma 3: The output of the augmentation of PM + S is a set of minimum weight M + 1 noddisjoint paths: PM+1 = PM + S. We omit proofs here. Interested readers could see [81, 84] for details. The authors [81, 84] only proposed algorithm to compute PM+1 from PM. This can be extended to compute M − 1 nodedisjoint paths from M nodedisjoint path as follows: 1. Reverse the direction of each edge on PM, and make its length negative. These edges are called negative arcs. Other edges are called positive arcs. 2. Split each vertex v on PM into two nodes v1 and v2, joined by an arc of length zero, directed towards a. Assign output links on v2 and input links on v1. 76
Object Description
Description
Title  Energy latency tradeoffs for medium access and sleep scheduling in wireless sensor networks.  Page 89 
Repository email  cisadmin@lib.usc.edu 
Full text  Figure 5.6: An example of finding minimum length 2 node disjoint paths. briefly describe the algorithm. Figure 5.6 shows an example. We can simply use any shortest path algorithm to find the first minimum weight node disjoint path from a to z, where a is the super pseudo source node, and z is the pseudo sink node. We assume PM is a given optimal set of M nodedisjoint paths in delay graph DG. Now we need to find optimal (M + 1)th nodedisjoint paths PM+1, as follows: 1. Reverse the direction of each edge on PM, and make its length negative. These edges are called negative arcs. Other edges are called positive arcs. 2. Split each vertex v on PM into two nodes v1 and v2, joined by an arc of length zero, directed towards a. Assign output links on v2 and input links on v1. 3. Find a shortest path from a to z on this transformed delay graph. We call this path an interlacing S, which may contain both positive arcs and negative arcs. 4. let PM + S represent the graph obtained by adding to PM the positive arcs of S, and removing from PM the negative arcs of S. This is called Augmentation, which results in the optimal PM+1 paths, the minimum weight M + 1 nodedisjoint paths. We present a lemma that will be used in the following proofs. Lemma 3: The output of the augmentation of PM + S is a set of minimum weight M + 1 noddisjoint paths: PM+1 = PM + S. We omit proofs here. Interested readers could see [81, 84] for details. The authors [81, 84] only proposed algorithm to compute PM+1 from PM. This can be extended to compute M − 1 nodedisjoint paths from M nodedisjoint path as follows: 1. Reverse the direction of each edge on PM, and make its length negative. These edges are called negative arcs. Other edges are called positive arcs. 2. Split each vertex v on PM into two nodes v1 and v2, joined by an arc of length zero, directed towards a. Assign output links on v2 and input links on v1. 76 