1. PAT Trees Index for arbitrary character sequence in text
Gonnet(1983) – based on Patricia Tree
Used for indexing OED (Morrison 68)
SISTRINGS — Semi-Infinite-Strings
pos
A I rise on a point of order which …
B I rise on a point of objection to ….
B < A in sistring order
What if we encountered all sistrings and sorted then?

2. STEP 1 : SISTRINGS .can .a.can.can.cans? –
STRING: . c a n a c a n c a n c a n s ?
SISTRING OFFSET
.can.a.can.can.cans?
can.a.can.can.cans?
an.a.can.can.cans?
n.a.can.can.cans?
.a.can.can.cans?
a.can.can.cans?
.can.can.cans?
can.can.cans?
an.can.cans?
n.can.cans?
.can.cans?
can.cans?
an.cans?
n.cans?
.cans?
cans?
ans?
ns? s? ? –

3. STEP 2 : Sort and find minimal distinguishing prefixes
.can .a.can.can.cans?
STRING : c a n a c a n c a n c a n s ?
SISTRING OFFSET MINIMAL DISTINGUISHING PREFIX
.a.can.can.cans? a
.can.a.can.can.cans? can.a
.can.can.cans? can.can.
.can.cans? can.cans
.cans? cans
? ?
a.can.can.cans? a.
an.a.can.can.cans? an.a
an.can.cans? an.cans
ans? ans
can.a.can.can.cans? can.a
can.can.cans? can.can.
can.cans? can.cans
cans? cans
n.a.can.can.cans? n.a
n.can.cans? n.can.
n.cans? n.cans
ns? ns
s? s

4. STEP 4 : Create a Digital Trie from Prefixes
ⓐ ⓒ ⓝ ⓢ
ⓝ ⓐ ⓢ
ⓐ ⓒ ⓢ ⓝ ⓐ ⓒ
ⓐ ⓢ
ⓝ ⓐ ⓒ ⓐ
ⓢ ⓐ
ⓝ ⓝ
ⓐ ⓒ ⓢ ⓢ ⓐ ⓝ
ⓢ ( Label with substring beginning ) • ? 18 19 • • 5 12 • 16 3 4 • 15 1 14 • • • 11 9 13 7 • 10 16

5. STEP 5 : Simplify tree with use of skipped bits
ⓐ ⓒ ⓝ ⓢ
x x
x x
ⓐ ⓒ ⓝ ⓢ ⓢ
x x
x x
ⓢ ⓢ ⓐ ⓒ ⓐ ⓒ
x x x
x x x
ⓐ ⓒ ⓐ ⓒ ⓢ ⓐ ⓢ
x x x x
x x x x
ⓢ ⓢ
# of missing letters c a n s ? • ? 19 18 • • • 4 5 15 • • 16 1 3 • 2 7 12 9 13 • • 16 10 8 12

6. Convert character to ASCII bits and make trie
Patricia Tree
Binary digital trie
Convert character to ASCII bits and make trie
a c n

7. O(height)  O(return set)
Applications
Prefix Searching
If no branch for next character,
then fail. • c a n
Enumerate all leaves sharing prefix
O(height)  O(return set)
O(log n)

8. Applications
• Longest Repetition Search
.can.can farthest internal node from root
— Simplify on-line calculation by storing a bit to show which
direction longest subtree goes
• Most Frequent N-gram

9. Problem with Pat Tree
Enumerating subtree costly

10. Create Suffix Arrays