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Dijkstra’s Algorithm with two levels of buckets

15.082J & 6.855J & ESD.78J Visualizations. Dijkstra’s Algorithm with two levels of buckets. ∞. ∞. 0. ∞. ∞. ∞. ∞. 2. 3. 4. 5. 6. Initialize. Initialize distance labels. 46. 2. 4. 8. 18. Initialize buckets. 22. 25. 26. 1. 6.

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Dijkstra’s Algorithm with two levels of buckets

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  1. 15.082J & 6.855J & ESD.78JVisualizations Dijkstra’s Algorithm with two levels of buckets

  2. ∞ 0 ∞ ∞ ∞ ∞ 2 3 4 5 6 Initialize Initialize distance labels 46 2 4 8 18 Initialize buckets. 22 25 26 1 6 Select the first nonempty lower bucket. Then select an element of the bucket. 1 41 25 37 3 5 10-19 20-29 30-39 40-49 50-59 60-69 70-79 80-89 90-99 0 1 2 3 4 5 6 7 8 9 1

  3. 1 ∞ Update ∞ ∞ 8 Scan arcs out of node 1 and update data structures 46 2 4 8 18 0 22 25 26 ∞ 1 6 41 25 37 3 5 41 ∞ ∞ 10-19 20-29 30-39 40-49 50-59 60-69 70-79 80-89 90-99 2 3 4 0 1 2 3 4 5 6 7 8 9 5 1 6

  4. 1 ∞ Find Min Lower Bucket ∞ 8 Find min non-empty lower bucket. Select a node in the bucket. 46 2 4 2 8 18 0 22 25 26 ∞ 1 6 41 25 37 3 5 41 ∞ 10-19 20-29 30-39 40-49 50-59 60-69 70-79 80-89 90-99 3 4 0 1 2 3 4 5 6 7 8 9 5 2 6

  5. 1 ∞ Update ∞ 8 54 Scan arcs out of node 2 and update distance labels and buckets. 46 2 4 2 8 18 0 22 25 26 ∞ 1 6 41 25 37 3 5 33 30 41 ∞ 10-19 20-29 30-39 40-49 50-59 60-69 70-79 80-89 90-99 3 4 0 1 2 3 4 5 6 7 8 9 5 2 6

  6. 1 ∞ Find Min Lower Bucket 54 8 Scan lower bucket for first non-empty bucket. If none exist, then go to scanning upper buckets. 46 2 4 2 8 18 0 22 25 26 ∞ 1 6 41 25 37 3 5 33 30 10-19 20-29 30-39 40-49 50-59 60-69 70-79 80-89 90-99 3 4 5 0 1 2 3 4 5 6 7 8 9 6

  7. 1 ∞ Find Min Upper Bucket 54 8 Scan upper buckets to find the first non-empty bucket. 46 2 4 2 8 18 0 22 25 26 ∞ 1 6 41 25 37 3 5 33 30 10-19 20-29 30-39 40-49 50-59 60-69 70-79 80-89 90-99 3 4 5 0 1 2 3 4 5 6 7 8 9 6

  8. 1 ∞ Move contents down 54 8 Adjust bucket numbers of lower buckets and move contents of upper bucket to lower buckets 46 2 4 2 8 18 0 22 25 26 ∞ 1 6 41 25 37 3 5 33 30 30-39 40-49 50-59 60-69 70-79 80-89 90-99 3 4 5 30 0 1 31 32 2 3 33 4 34 5 35 6 36 7 37 8 38 9 39 6

  9. 1 ∞ Find Min Lower Bucket 54 8 Find the first nonempty lower bucket and select a node in the bucket 46 2 4 2 8 18 0 22 25 26 ∞ 1 6 41 25 37 3 5 5 33 30 40-49 50-59 60-69 70-79 80-89 90-99 4 30 31 32 33 34 35 36 37 38 39 5 3 6

  10. 1 ∞ Update 54 8 Scan arcs out of node 5 and update distance labels and buckets. 46 2 4 2 8 18 0 22 25 26 ∞ 1 6 41 55 25 37 3 5 5 33 30 40-49 50-59 60-69 70-79 80-89 90-99 4 30 31 32 33 34 35 36 37 38 39 5 3 6

  11. 1 Find Min Lower Bucket 54 8 Find the first nonempty lower bucket and select a node in the bucket 46 2 4 2 8 18 0 22 25 26 1 6 41 55 25 37 3 5 3 5 33 30 40-49 50-59 60-69 70-79 80-89 90-99 4 6 30 31 32 33 34 35 36 37 38 39 3

  12. 1 Update 54 8 Scan arcs out of node 3 and perform updates. 46 2 4 2 8 18 0 22 25 26 1 6 41 55 25 37 3 5 3 5 33 30 40-49 50-59 60-69 70-79 80-89 90-99 4 6 30 31 32 33 34 35 36 37 38 39 3

  13. 1 Find Min Lower Bucket 54 8 Scan lower buckets looking for a non-empty bucket. 46 2 4 2 8 18 0 22 25 26 1 6 41 55 25 37 3 5 5 33 30 40-49 50-59 60-69 70-79 80-89 90-99 4 6 30 31 32 33 34 35 36 37 38 39

  14. 1 Find Min Upper Bucket 54 8 Scan upper buckets looking for a non-empty bucket. 46 2 4 2 8 18 0 22 25 26 1 6 41 55 25 37 3 5 5 33 30 40-49 50-59 60-69 70-79 80-89 90-99 4 6 30 31 32 33 34 35 36 37 38 39

  15. 1 Move contents down 54 8 Adjust bucket numbers of lower buckets and move contents of upper bucket to lower buckets. 46 2 4 2 8 18 0 22 25 26 1 6 41 55 25 37 3 5 5 33 30 50-59 60-69 70-79 80-89 90-99 4 6 30 50 51 31 32 52 33 53 34 54 35 55 36 56 37 57 58 38 59 39

  16. 1 Find Min Lower Bucket 54 8 Select the min non-empty lower bucket. Select an element from the bucket. 46 2 4 2 4 8 18 0 22 25 26 1 6 41 55 25 37 3 5 5 33 30 60-69 70-79 80-89 90-99 50 51 52 53 54 55 56 57 58 59 4 6

  17. 1 Update 54 8 Scan arcs out of node 4 and perform all updates. 46 2 4 2 4 8 18 0 22 25 26 1 6 41 55 25 37 3 5 5 33 30 60-69 70-79 80-89 90-99 50 51 52 53 54 55 56 57 58 59 4 6

  18. 1 Find Min Lower Bucket 54 8 Scan lower buckets looking for a non-empty bucket. Select a node of the bucket 46 2 4 2 6 8 18 0 22 25 26 1 6 41 55 25 37 3 5 5 33 30 60-69 70-79 80-89 90-99 50 51 52 53 54 55 56 57 58 59 6

  19. 1 Update 54 8 Scan arcs out of node 6, and perform updates 46 2 4 2 6 8 18 0 22 25 26 1 6 41 55 25 37 3 5 5 33 30 60-69 70-79 80-89 90-99 50 51 52 53 54 55 56 57 58 59 6

  20. 1 Termination 54 8 All nodes are permanent. The algorithm terminates with the optimal shortest path tree. 46 2 4 2 8 18 0 22 25 26 1 6 41 55 25 37 3 5 5 33 30 60-69 70-79 80-89 90-99 50 51 52 53 54 55 56 57 58 59

  21. MITOpenCourseWare http://ocw.mit.edu 15.082J / 6.855J / ESD.78J Network Optimization Fall 2010 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.

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