Searching with Structured Keys
Objectives

To look at search structures for variable length and multidimensional keys.

To understand several basic radix search techniques, their advantages and disadvantages

To look at range queries and partial match searches in multidimensional structures.

Digital Search Trees - Costs

The number of nodes inspected in a search is, at most, the number of leading bits needed to distinguish it from other keys

For “randomly distributed” keys this should be Θ(log n), but if keys agree on lead bits problems arise

But, visiting each node involves a key comparison, this may be the bulk of the cost

Radix Search Tries
“trie” as in “retrieval” is pronounced “try”

Radix Search Trie Updates

If you “fall out”, hang a new node on there

If you come to a leaf continue
the path in the tree with
the two elements
until the bits differ

"Only n nodes of storage"

Only n nodes of storage

Just one full key compare

No “useless” tests

Patricia: Step 1
Leave out 1 way branch nodes

Add to each node the position of the bit to be tested (we will still move left to right)

"Patricia:"

Patricia: Step 2

Add the “dummy value” 00000 (which is either not stored or stored in the header).

Store keys in internal nodes that are ancestors of their leaf positions in the tree

Patricia Insertions

Insertions can be made in the natural way.

Follow the search path of new leaf.

Determine the highest number bit where new key and leaf key differ.

Retrace search path to the point where a test on this bit could occur.

Insert new node there with the node on the path to leaf node as one child and a new leaf containing the new key

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Radix Techniques

Insensitive to order of insertions

Sensitive to particular value

Very useful for long or variable length keys


	To look at search structures for variable length and multidimensional keys.
	To understand several basic radix search techniques, their advantages and disadvantages
	To look at range queries and partial match searches in multidimensional structures.


	The number of nodes inspected in a search is, at most, the number of leading bits needed to distinguish it from other keys
	For “randomly distributed” keys this should be Θ(log n), but if keys agree on lead bits problems arise
	But, visiting each node involves a key comparison, this may be the bulk of the cost


	If you “fall out”, hang a new node on there
	If you come to a leaf continue the path in the tree with the two elements until the bits differ


	Only n nodes of storage
	Just one full key compare
	No “useless” tests


	Add to each node the position of the bit to be tested (we will still move left to right)


	Patricia: Step 2
	Add the “dummy value” 00000 (which is either not stored or stored in the header).
	Store keys in internal nodes that are ancestors of their leaf positions in the tree


	Insertions can be made in the natural way.
	Follow the search path of new leaf.
	Determine the highest number bit where new key and leaf key differ.
	Retrace search path to the point where a test on this bit could occur.
	Insert new node there with the node on the path to leaf node as one child and a new leaf containing the new key


	Insensitive to order of insertions
	Sensitive to particular value
	Very useful for long or variable length keys