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Reading Material

Reading Material. Shortest Paths 2: R-Heaps. Radix Heap Animation. . . 5. 2. 4. 15. 2. 13. 0. . 1. 6. 20. 0. 8. 9. 3. 5. . . . An Example from AMO (with a small change). Initialize distance labels. Initialize buckets and their ranges. Insert nodes into buckets.

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Reading Material

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  1. Reading Material Shortest Paths 2: R-Heaps

  2. Radix Heap Animation

  3.  5 2 4 15 2 13 0  1 6 20 0 8 9 3 5    An Example from AMO (with a small change) Initialize distance labels Initialize buckets and their ranges. Insert nodes into buckets. 2 3 47 8 15 16 31 3263 2 0 1 3 4 5 1 6

  4. Select   5 Select the node with minimum distance label 2 4 15 2 13 0  1 1 6 20 0 8 9 3 5   2 3 47 8 15 16 31 3263 2 0 1 3 4 5 1 6

  5.  0     Update 13 15 5 Scan arcs out of node 1 and update distance labels. 2 4 15 2 13 1 1 6 20 0 8 9 3 5 0 20 2 3 47 8 15 16 31 3263 2 0 1 3 4 2 5 5 3 4 6

  6. Select 13 15 5 Select the node with minimum distance label 2 4 15 2 13 0  1 1 6 Node 3 has label 0, which is minimum. 20 0 8 9 3 3 5 0 20 2 3 47 8 15 16 31 3263 0 1 2 5 6 3 4

  7. Update 13 15 5 Scan arcs out of node 3 and update distance labels. 2 4 15 2 13 0  1 1 6 20 0 8 9 3 3 5 0 20 9 2 3 47 8 15 16 31 3263 0 1 2 5 6 3 4 5

  8. Select: part 1 13 15 5 Find the first non-empty bucket, by scanning buckets from left to right. 2 4 15 2 13 0  1 1 6 20 0 8 9 3 3 5 0 20 9 2 3 47 8 15 16 31 3263 0 1 2 6 4 5

  9. Select: part 2 13 15 5 Determine the minimum distance value in the bucket, by scanning all nodes in the bucket. 2 4 15 2 13 0  1 1 6 20 0 8 9 3 3 5 d(5) = 9, which is minimum. 0 20 9 2 3 47 8 15 16 31 3263 0 1 2 6 4 5

  10. Select: part 3 13 15 5 Redistribute the range of bucket 5 into the first 4 buckets, starting with value 9. Bucket widths stay the same, except that some may be smaller. 2 4 15 2 13 0  1 1 6 20 0 8 9 3 3 5 0 20 9 11 12 1315 2 3 47  8 15 16 31 3263 0 9 1 10 2 6 4 5

  11. Select: part 4 13 15 5 Reinsert nodes in the correct bucket. Determine the bucket by scanning left. 2 4 15 2 13 0  1 1 6 20 0 8 9 At this point the leftmost bucket is non-empty 3 3 5 0 20 9 2 3 11 12 1315  8 15 16 31 3263 0 9 10 1 2 6 2 5 4 4 5

  12. Select: part 5 13 15 5 Select a node in the leftmost bucket. 2 4 15 2 13 0  1 1 6 20 0 8 9 3 3 5 5 0 20 9 2 3 11 12 1315 8 15  16 31 3263 0 9 10 1 6 2 5 4

  13. Update 13 15 5 Scan arcs out of node 5 and update distance labels. 2 4 15 2 13 0  17 1 1 6 Reinsert nodes in correct buckets by scanning left. 20 0 8 9 3 3 5 5 0 20 9 2 3 11 12 1315  8 15 16 31 3263 0 9 1 10 6 2 6 4

  14. Select: parts 1 and 2 13 15 5 Find the minimum non-empty bucket. 2 4 15 2 13 0 17  1 1 6 Find the minimum distance label in the bucket 20 0 8 9 3 3 5 5 0 20 9 11 12 2 3 1315  8 15 16 31 3263 0 9 1 10 2 6 4

  15. Select: parts 3 and 4 13 15 5 Redistribute bucket ranges in the minimum bucket 2 4 15 2 13 0  17 1 1 6 Reinsert nodes in correct buckets. 20 0 8 9 3 3 5 5 0 20 9 2 3 11 12 15 1315   8 15 16 31 3263 0 13 9 14 1 10 2 2 4 6 4

  16. Select: part 5 13 15 5 Select a node from the leftmost bucket. 2 2 4 15 2 13 0 17  1 1 6 20 0 8 9 3 3 5 5 0 20 9 15 11 12 2 3  1315 8 15  16 31 3263 0 9 13 1 10 14 2 4 6

  17. Update 13 15 5 Scan the arc out of node 2. 2 2 4 15 2 13 0 17  1 1 6 20 0 8 9 3 3 5 5 0 20 9 15 2 3 11 12 1315  8 15  16 31 3263 0 9 13 1 14 10 4 6

  18. Select, Modified rule 1 13 15 5 Find the minimum non-empty bucket 2 2 4 4 15 2 13 0  17 If the bucket has a width of 1, select any node in the bucket. 1 1 6 20 0 8 9 3 3 5 5 0 20 9 15 11 12 2 3 1315  8 15  16 31 3263 0 13 9 10 14 1 4 6

  19. Update 13 15 5 Scan the arc out of node 4 2 2 4 4 15 2 13 0  17 1 1 6 20 0 8 9 3 3 5 5 0 20 9 15 2 3 11 12 1315  8 15  16 31 3263 0 9 13 1 14 10 6

  20. Select: modified rule 2 13 15 5 Find the minimum non-empty bucket 2 2 4 4 15 2 13 If the bucket has a single node, then select the node. 0 17  1 1 6 6 20 0 Modified rules and heuristics often help in practice, but must be used carefully. 8 9 3 3 5 5 0 20 9 11 12 15 2 3 1315   8 15 16 31 3263 0 9 13 10 14 1 6

  21. The algorithm ends 13 15 5 There are no arcs to update. 2 2 4 4 15 2 13 0 17  1 1 6 6 There are no nodes that need to be permanently labeled. 20 0 8 9 3 3 5 5 0 20 9 15 11 12 2 3 1315   8 15 16 31 3263 0 9 13 1 14 10

  22. Exercise • Solve the shortest path problem shown in Figure 4.15(a) (page127 in textbook) using the Radix-heap algorithm

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