1. The TRIE
Amihood Amir

2. Labeled Trees Edge Labeled Tree: T=(V,E,ℓ) Where ℓ:VΣ,
Σ is the alphabet.
Example: Σ={A,B,C} A A B C B

3. Path String
A path v0,…,vi in an edge labeled tree defines the path string ℓ(v0),…,ℓ(vi) of the labels of the vertices on the path.
Example:
Path: A A B C B
Path string: AAB

4. Root Paths
A root path v0,…,vi in an edge labeled tree is a path that starts at the root, i.e. v0 is the root of the tree.
Example:
Root Path: A
Not Root Path: A B C B

5. Longest Common Prefix
Let S=S[1],…,S[m] and T=T[1],…,T[n] be two strings over alphabet Σ.
The Longest Common Prefix (LCP) of S and T is the string a[1],…,a[k] such that a[i]=S[i]=T[i], i=1,…,k and such that S[k+1]≠T[k+1].
Example: The LCP of
ABCAABCDABCCC and
ABCAABCDACACC is:
ABCAABCDA

6. reTRIEval We define a Trie T of n strings S1 = S1[1],…,S1[m1]
Sn = Sn[1],…,Sn[mn]
over alphabet Σ by induction on n as follows:
Let Λ,$ є Σ.

7. reTRIEval – base case For n=1: S1 = S1[1],…,S1[m1] The trie is: . . .Λ
For n=1:
S1 = S1[1],…,S1[m1]
The trie is:
S1[1] . . .
S1[m1] $

8. reTRIEval – inductive case (1)
Assume we have defined he trie Tn of n strings.
The trie Tn+1 of the n+1 strings:
S1 = S1[1],…,S1[m1]
S2 = S2[1],…,S2[m2] …
Sn = Sn[1],…,Sn[mn]
Sn+1 = Sn+1[1],…,Sn+1[mn+1]
Is defined as follows:

9. reTRIEval - inductive case (2)
Let Tn be the trie of the n strings
S1 = S1[1],…,S1[m1]
S2 = S2[1],…,S2[m2] …
Sn = Sn[1],…,Sn[mn]
And let a[1],…a[k] be the longest LCP(Sn+1,Si), i=1,…,n.

10. reTRIEval – inductive case (3)
Concatenate the path:
To the node where the
root path of string
a[1],…,a[k]
ends.
The resulting tree is
Tn+1.
Sn+1[k+1] . . .
Sn+1[mn+1] $

11. Trie construction Example
ABCABC
ABB
ABBA
ABCB
BBAB
BABC

12. Trie construction Time
For a Trie T of n strings:
S1 = S1[1],…,S1[m1]
S2 = S2[1],…,S2[m2] …
Sn = Sn[1],…,Sn[mn]
Over fixed finite alphabet Σ:

13. Trie Insertion, Lookup, Deletion Time
For string: S = S[1],…,S[m]
Over fixed finite alphabet Σ: O(m)
Over ubounded alphabet Σ: O(m log n)

14. How do we deal with numbers?
An n-digit number is the string composed of the digits.
Insertion/deletion/lookup time of number m: O(log m)
Compare with AVL:
O(log n)