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module7

# module7 - Radix Trees(Tries Suppose we have data items...

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Unformatted text preview: Radix Trees (Tries) Suppose we have data items whose keys are binary strings, not necessarily of the same length. We can organize our data into a dictionary using a binary tree based on bitwise comparisons of strings. This data structure is called a radix tree or trie (pronounced “try”). Its main features are as follows: 1. a left child corresponds to a 0 bit, 2. a right child corresponds to a 1 bit, 3. items (or, alternatively, links to items) are stored only in leaf nodes of the tree. Note: property 3 implies that no string should be a prefix of another string (e.g., 011 is a prefix of 01101). We will always make this assumption unless otherwise stated. 156/ 193 Operations on Tries Search: Suppose X = x 1 x 2 ...x ‘ . We can use a binary search technique to locate an item with key X . There are four possible outcomes: 1. A successful search which ends at the correct leaf node. 2. an unsuccessful search that ends at a leaf node storing a key Y where Y is a prefix of X . 3. an unsuccessful search that ends at a non-leaf node at level ‘ of the trie (in this case, X is a prefix of a string Y in the trie). 4. an unsuccessful search that ends at a non-leaf node at some level k < ‘ of the trie. The complexity of Search is O ( | X | ), where | X | denotes the length of the string X . 157/ 193 Trie Search Algorithm: TrieSearch ( T,X ) N ← Root ( T ); ‘ ← Length ( X ); i ← 1; case ← while i ≤ ‘ and case = 0 do if N is a leaf node then case ← 2 else if x i = 0 then N ← LeftChild ( N ) else N ← RightChild ( N ) if N = nil then case ← 4 else i ← i + 1 if i = ‘ + 1 then if N is a leaf node then case ← 1 else case ← 3 return ( case ,N ) 158/ 193 Operations on Tries (cont.) Insert: Suppose X = x 1 x 2 ...x ‘ . To insert a node containing the (new) key X , we first do an unsuccessful search for X . If we terminate in case 2 or 3, then we cannot insert X into the tree because it will violate the “prefix-free” property. So assume we end up in case 4. Then the path from the root node to N corresponds to a prefix of X , say x 1 x 2 ...x k , where k < ‘ . Now create ‘- k new nodes in the trie to “complete” this prefix to X . The complexity of Insert is O ( | X | ). 159/ 193 Operations on Tries (cont.) Delete: Suppose X = x 1 x 2 ...x ‘ . To delete the node containing the key X , we first do a successful search for X that terminates at a leaf node N . We delete N and all ancestors of N until we reach an ancestor that has two children ; then stop. The complexity is O ( | X | ). 160/ 193 Compressed Tries A compressed trie eliminates nodes with only one child, which decreases the storage requirement of the data structure. Roughly speaking, we compress every path of one-child nodes to a single edge . Each node stores an index indicating the next bit to be tested during a search....
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module7 - Radix Trees(Tries Suppose we have data items...

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