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6.3.4 Indexing Operations on Weight-Balanced Trees

Weight balanced trees support operations that view the tree as sorted sequence of associations. Elements of the sequence can be accessed by position, and the position of an element in the sequence can be determined, both in logarthmic time.

— procedure+: wt-tree/index wt-tree index
— procedure+: wt-tree/index-datum wt-tree index
— procedure+: wt-tree/index-pair wt-tree index

Returns the 0-based indexth association of wt-tree in the sorted sequence under the tree's ordering relation on the keys. wt-tree/index returns the indexth key, wt-tree/index-datum returns the datum associated with the indexth key and wt-tree/index-pair returns a new pair (key . datum) which is the cons of the indexth key and its datum. The average and worst-case times required by this operation are proportional to the logarithm of the number of associations in the tree.

These operations signal an error if the tree is empty, if index<0, or if index is greater than or equal to the number of associations in the tree.

Indexing can be used to find the median and maximum keys in the tree as follows:

     median:  (wt-tree/index wt-tree (quotient (wt-tree/size wt-tree) 2))
     
     maximum: (wt-tree/index wt-tree (-1+ (wt-tree/size wt-tree)))
— procedure+: wt-tree/rank wt-tree key

Determines the 0-based position of key in the sorted sequence of the keys under the tree's ordering relation, or #f if the tree has no association with for key. This procedure returns either an exact non-negative integer or #f. The average and worst-case times required by this operation are proportional to the logarithm of the number of associations in the tree.

— procedure+: wt-tree/min wt-tree
— procedure+: wt-tree/min-datum wt-tree
— procedure+: wt-tree/min-pair wt-tree

Returns the association of wt-tree that has the least key under the tree's ordering relation. wt-tree/min returns the least key, wt-tree/min-datum returns the datum associated with the least key and wt-tree/min-pair returns a new pair (key . datum) which is the cons of the minimum key and its datum. The average and worst-case times required by this operation are proportional to the logarithm of the number of associations in the tree.

These operations signal an error if the tree is empty. They could be written

          (define (wt-tree/min tree)        (wt-tree/index tree 0))
          (define (wt-tree/min-datum tree)  (wt-tree/index-datum tree 0))
          (define (wt-tree/min-pair tree)   (wt-tree/index-pair tree 0))
— procedure+: wt-tree/delete-min wt-tree

Returns a new tree containing all of the associations in wt-tree except the association with the least key under the wt-tree's ordering relation. An error is signalled if the tree is empty. The average and worst-case times required by this operation are proportional to the logarithm of the number of associations in the tree. This operation is equivalent to

          (wt-tree/delete wt-tree (wt-tree/min wt-tree))
— procedure+: wt-tree/delete-min! wt-tree

Removes the association with the least key under the wt-tree's ordering relation. An error is signalled if the tree is empty. The average and worst-case times required by this operation are proportional to the logarithm of the number of associations in the tree. This operation is equivalent to

          (wt-tree/delete! wt-tree (wt-tree/min wt-tree))