1. 15.082 and 6.855J February 25, 2003 Radix Heap Animation

2. 2  An Example from AMO (with a small change) 1 0 1 2323 4747 8 15 16 31 32 63 1 24 53 6 13 5 2 8 15 20 9 0 Initialize distance labels Insert nodes into buckets. Initialize buckets and their ranges.    0 2 3 4 5 6

3. 3  Select 1 0 1 2323 4747 8 15 16 31 32 63 1 24 53 6 13 5 2 8 15 20 9 0 Select the node with minimum distance label    0 2 3 4 5 6 1

4. 4  Update 3 0 1 2323 4747 8 15 16 31 32 63 1 24 53 6 13 5 2 8 15 20 9 0 Scan arcs out of node 1 and update distance labels.    0 2 3 4 5 6 1 13 2 0 15 4 20 5

5. 5  Select 3 0 1 2323 4747 8 15 16 31 32 63 1 24 53 6 13 5 2 8 15 20 9 0 Select the node with minimum distance label  0 6 1 2 4 5 Node 3 has label 0, which is minimum. 13 0 15 20 3

6. 6  Update 3 0 1 2323 4747 8 15 16 31 32 63 1 24 53 6 13 5 2 8 15 20 9 0 Scan arcs out of node 3 and update distance labels.  0 6 1 2 4 5 13 0 15 20 3 9 5

7. 7  Select: part 1 0 1 2323 4747 8 15 16 31 32 63 1 24 53 6 13 5 2 8 15 20 9 0 Find the first non-empty bucket, by scanning buckets from left to right.  0 6 1 2 4 13 0 15 20 3 9 5

8. 8  Select: part 2 0 1 2323 4747 8 15 16 31 32 63 1 24 53 6 13 5 2 8 15 20 9 0 Determine the minimum distance value in the bucket, by scanning all nodes in the bucket.  0 6 1 2 4 13 0 15 20 3 9 5 d(5) = 9, which is minimum.

9. 9  Select: part 3 0 1 2323 8 15 16 31 32 63 1 24 53 6 13 5 2 8 15 20 9 0 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.  0 6 1 2 4 13 0 15 20 3 9 5 910 11 12  4747 13 15

10. 10  Select: part 4 0 1 2323 13 15 8 15 16 31 32 63 1 24 53 6 13 5 2 8 15 20 9 0 Reinsert nodes in the correct bucket. Determine the bucket by scanning left.  0 6 1 2 4 13 0 15 20 3 9 5 910 11 12  4 2 5 At this point the leftmost bucket is non-empty

11. 11  Select: part 5 0 1 2323 13 15 8 15 16 31 32 63 1 24 53 6 13 5 2 8 15 20 9 0 Select a node in the leftmost bucket.  0 6 1 13 0 15 20 3 9 910 11 12  4 2 5 5

12. 12  Update 0 1 2323 13 15 8 15 16 31 32 63 1 24 53 6 13 5 2 8 15 20 9 0 Scan arcs out of node 5 and update distance labels.  0 6 1 13 0 15 20 3 9 910 11 12  4 2 5 17 6 Reinsert nodes in correct buckets by scanning left.

13. 13 Select: parts 1 and 2 0 1 2323 13 15 8 15 16 31 32 63 1 24 53 6 13 5 2 8 15 20 9 0 Find the minimum non-empty bucket.  0 1 13 0 15 20 3 9 910 11 12  4 2 5 17 6 Find the minimum distance label in the bucket

14. 14 Select: parts 3 and 4 0 1 2323 13 15 8 15 16 31 32 63 1 24 53 6 13 5 2 8 15 20 9 0 Redistribute bucket ranges in the minimum bucket  0 1 13 0 15 20 3 9 910 11 12  4 2 5 17 6 Reinsert nodes in correct buckets.  1314 15 24

15. Select: part 5 0 1 2323 13 15 8 15 16 31 32 63 1 24 53 6 13 5 2 8 15 20 9 0 Select a node from the leftmost bucket.  0 1 13 0 15 20 3 9 910 11 12  5 17 6  1314 15 24 2

16. 16 Update 0 1 2323 13 15 8 15 16 31 32 63 1 24 53 6 13 5 2 8 15 20 9 0 Scan the arc out of node 2.  0 1 13 0 15 20 3 9 910 11 12  5 17 6  1314 15 4 2

17. 17 Select, Modified rule 1 0 1 2323 13 15 8 15 16 31 32 63 1 24 53 6 13 5 2 8 15 20 9 0 Find the minimum non-empty bucket  0 1 13 0 15 20 3 9 910 11 12  5 17 6  1314 15 4 2 If the bucket has a width of 1, select any node in the bucket. 4

18. 18 Update 0 1 2323 13 15 8 15 16 31 32 63 1 24 53 6 13 5 2 8 15 20 9 0 Scan the arc out of node 4  0 1 13 0 15 20 3 9 910 11 12  5 17 6  1314 15 24

19. 19 Select: modified rule 2 0 1 2323 13 15 8 15 16 31 32 63 1 24 53 6 13 5 2 8 15 20 9 0 Find the minimum non-empty bucket  0 1 13 0 15 20 3 9 910 11 12  5 17 6  1314 15 24 If the bucket has a single node, then select the node. 6 Modified rules and heuristics often help in practice, but must be used carefully.

20. 20 The algorithm ends 0 1 2323 13 15 8 15 16 31 32 63 1 24 53 6 13 5 2 8 15 20 9 0 There are no arcs to update.  0 1 13 0 15 20 3 9 910 11 12  5 17  1314 15 24 There are no nodes that need to be permanently labeled. 6