Digital Search Trees & Binary Tries Analog of radix sort to searching. Keys are binary bit strings.  Fixed length – 0110, 0010, 1010,  Variable length – 01, 00, 101, Application – IP routing.  IPv4 – 32 bit IP address.  IPv6 – 128 bit IP address.

IPv4 Addresses 32-bit address divided into four 8-bit segments.  a1.a2.a3.a4  Each 8-bit segment is written as a decimal number in the range 0 – 255.   Class A Internet address.  a1 is net address.  Most significant bit of a1 is 0.  a2.a3.a4 gives host address within network.  <= 128 Class A nets.  Each Class A net may have up to 2 24 ~ 17 million hosts.

IPv4 Addresses  Class B Internet address.  a1.a2 is net address.  Two most significant bits are 10.  a3.a4 gives host address within network.  Up to 2 14 = 16,384 Class B nets.  Each Class B net may have up to 2 16 = 65,536 hosts.  IP addresses in a small company/department that has up to 256 hosts could have the format a1.a2.a3.*.  IP addresses of hosts in the same network/subnetwork have the same prefix.

IPv4 Routers Routing table entry … (prefix, nextMachine)  Prefix is a binary string whose length is <= 32 bits.  Packets whose destination address matches prefix may be routed to nextMachine.  (011*, M1) => all packets whose destination address begins with 011 may be routed next to machine M1.  ( *, M2)  Packet routing requires us to find the longest prefix (in the routing table) that matches the packet destination.

IP Router Table Collection of (prefix, nextMachine) pairs. Operations.  Add a pair.  Remove a pair.  Find pair with longest matching prefix.

Digital Search Tree Assume fixed number of bits. Not empty =>  Root contains one dictionary pair (any pair).  All remaining pairs whose key begins with a 0 are in the left subtree.  All remaining pairs whose key begins with a 1 are in the right subtree.  Left and right subtrees are digital subtrees on remaining bits.

Example Start with an empty digital search tree and insert a pair whose key is Now, insert a pair whose key is

Example Now, insert a pair whose key is

Example Now, insert a pair whose key is

Example Now, insert a pair whose key is

Search/Insert/Delete Complexity of each operation is O(#bits in a key). #key comparisons = O(height). Expensive when keys are very long

Binary Trie Information Retrieval. At most one key comparison per operation. Fixed length keys.  Branch nodes.  Left and right child pointers.  No data field(s).  Element nodes.  No child pointers.  Data field to hold dictionary pair.

Example At most one key comparison for a search

Variable Key Length Left and right child fields. Left and right pair fields.  Left pair is pair whose key terminates at root of left subtree or the single pair that might otherwise be in the left subtree.  Right pair is pair whose key terminates at root of right subtree or the single pair that might otherwise be in the right subtree.  Field is null otherwise.

Example At most one key comparison for a search null

Fixed Length Insert Insert Zero compares.

Fixed Length Insert Insert

Fixed Length Insert Insert

Fixed Length Insert Insert One compare

Fixed Length Delete Delete

Fixed Length Delete Delete 0111.One compare

Fixed Length Delete Delete

Fixed Length Delete Delete

Fixed Length Delete Delete

Fixed Length Delete Delete

Fixed Length Delete Delete 1100.One compare

Fixed Length Join(S,m,B) Insert m into B to get B’. S empty => B’ is answer; done. If you get to this step, the root of both S and B’ is a branch node. S is element node => insert S element into B’; done; B’ is element node => insert B’ element into S; done;

Fixed Length Join(S,m,B) S has empty right subtree. a S bc B’ c J(S,B’) J(a,b) J(X,Y)  join X and Y, all keys in X < all in Y.

Fixed Length Join(S,m,B) S has nonempty right subtree. Left subtree of B’ must be empty, because all keys in B’ > all keys in S. c B’ a J(S,B’) J(b,c) a S b Complexity = O(height).