Digital Search Trees & Binary Tries

Digital Search Trees & Binary Tries • Analog of radix sort to searching. • Keys are binary bit strings. • Fixed length – 0110, 0010, 1010, 1011. • Variable length – 01, 00, 101, 1011. • 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. • 123.1.44.161 • 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 224~ 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 214 = 16,384 Class B nets. • Each Class B net may have up to 216= 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. • (0111011*, 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.

0110 0110 0010 Example • Start with an empty digital search tree and insert a pair whose key is 0110. • Now, insert a pair whose key is 0010.

0110 0010 1001 0110 0010 Example • Now, insert a pair whose key is 1001.

1001 0110 0010 1001 0110 0010 1011 Example • Now, insert a pair whose key is 1011.

1001 0110 0010 1011 1001 0110 0010 1011 0000 Example • Now, insert a pair whose key is 0000.

1001 0110 0010 1011 0000 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.

0 1 0 0 1 0 1 0 0 1 1000 1100 0011 0001 1001 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.

0 null 1 0 00 01100 10 11111 0 0 0000 001 1000 101 1 00100 001100 Example At most one key comparison for a search.

1 0 1 0 0 1 0 1 0 0 1 1000 1100 0011 0001 0111 1001 Fixed Length Insert Insert 0111. Zero compares.

1 0 1 0 0 1 0 1 0 0 1 1000 1100 0111 0011 0001 1001 Fixed Length Insert Insert 1101.

1 0 1 0 0 1 0 0 0 1 0 1001 0011 0001 1100 0111 1000 Fixed Length Insert 0 1 Insert 1101.

1 0 1 0 0 1 0 0 0 1 0 1 1000 0011 0001 0111 1100 1101 1001 Fixed Length Insert 0 1 Insert 1101. One compare.

1 0 1 0 0 1 0 1 0 0 0 1 0 1 1000 0011 0111 0001 1100 1101 1001 Fixed Length Delete Delete 0111.

0 1 0 0 1 0 1 0 0 0 1 0 1 1000 0011 0001 1100 1101 1001 Fixed Length Delete Delete 0111. One compare.

0 1 0 0 1 0 1 0 0 0 1 0 1 1000 0011 0001 1100 1101 1001 Fixed Length Delete Delete 1100.

0 1 1101 1001 0001 0011 1000 Fixed Length Delete 0 1 0 0 1 0 1 0 0 1 Delete 1100.

0 1101 1001 1000 0001 0011 Fixed Length Delete 0 1 0 0 1 0 1 0 0 1 Delete 1100.

1101 1001 1000 0011 0001 Fixed Length Delete 0 1 0 0 1 0 1 0 0 1 Delete 1100.

1001 1000 0011 0001 1101 Fixed Length Delete 0 1 0 0 1 0 1 0 0 1 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;

S B’ J(S,B’) a b c c J(a,b) Fixed Length Join(S,m,B) • S has empty right subtree. + = J(X,Y)ajoin X and Y, all keys in X< all in Y.

S B’ J(S,B’) a b c a J(b,c) 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. + = Complexity = O(height).

Digital Search Trees & Binary Tries