1. Selection Trees

2. What are selection trees? Complete binary tree Each node represents a “match” Winner Trees Loser Trees

3. Diagram – max winner tree 9585548 988 5 89 9 4 7 0 … 3 6 9 … 1 5 2 … 8 6 7 … 5 3 0 … 5 4 1 … 2 8 3 … 4 1 3 … 7

4. Diagram – max loser tree 9585548 754 5 55 4 7 4 0 … 9 6 3 … 5 2 1 … 8 7 6 … 5 3 0 … 5 4 1 … 8 3 2 … 4 3 1 … 7

5. Runtimes Complexities Initialize  r = the number of runs = number of leaf nodes  r-1 comparison nodes  O(r) to initialize r-runs Compare  O(1) Remove winner and Replay  O(logr) Total Time for n elements  O(nlogr)

6. What’s this good for? External Sorting Package Loading Problem

7. Truck Loading Problem n parcels from r sources each parcel has a weight and a size minimize the number of trucks

8. First Fit Packing with Max Winner Tree Use a max winner tree in which n parcels are t trucks  One at a time in left to right order  Current item is packed in the left-most into which it fits  If there is no truck into which it fits, start a new truck

9. Best Fit Object is packed into the bin with the least unused capacity that is at least the size of the object  If conditions are not met, start a new truck