1. Data Structures and Algorithms for Information Processing
Lecture 10: Searching II
Lecture 10: Searching

2. Outline Analysis of hashing algorithms Some practical considerations
Radix searching
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3. Open vs. Chained Hashing
How big should the table be?
Open addressing can be inconvenient when the number of insertions and deletions is unpredictable - overflow.
Simple solution to overflow: Resize (double) table, rehashing everything into the new table
Use Knuth’s approach and double hashing to avoid clustering.
Lecture 10: Searching

4. Variant: Ordered Hashing
In linear probing, we stop search when we find an empty cell or a record with a key equal to the search key
In ordered hashing we stop when we find a key less than or equal to the search key (tough schoolboy hashing)
Lecture 10: Searching

5. Tough Schoolboy hashing
13 chairs in the classroom
Each boy has a preferred seat
Each boy has a jump value
Boys later in the alphabet are bigger
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6. Class in the morning Inserts Don prefers 3 jumps 2
Bill prefers 5 jumps 4
Al prefers 3 jumps 6
Joe prefers 3 jumps 4
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7. 1 2
3 DON 4 5 6 7 8 9 10 11 12
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8. 1 2
3 DON 4
5 BILL 6 7 8 9 10 11 12
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9. 1 2 DON Al can’t sit here!! 4 5 BILL 6 7 8 9 10 11 121 2
DON
Al can’t sit here!! 4
5 BILL 6 7 8 9 10 11 12
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10. 1 2
DON 4
5 BILL 6 7 8
9 Al 10 11 12
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11. 1 2 DON Joe kicks out Don 4 5 BILL 6 7 8 9 Al 10 11 121 2
DON
Joe kicks out Don 4
5 BILL 6 7 8
9 Al 10 11 12
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12. 1 2
Joe 4
Don kicks out Bill 6 7 8
9 Al 10 11 12
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13. 9 Al and Bill argue and Al gets kicked out 10 111 2
Joe 4
Don 6 7 8
9 Al and Bill argue and Al gets kicked out 10 11 12
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14. 1
2 AL
Joe 4
Don 6 7 8
9 Bill 10 11 12
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15. Searching the classroom
Search for Don, Bill, Al, and Joe
Search for Ken who prefers 3 and jumps 1
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16. Variant: Ordered Hashing
This reduces the time of unsuccessful search to about the same as successful search
Useful for applications where we expect to have a large number of unsuccessful searches
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17. Summary of Basic Searching
Hashing is preferred to binary tree methods in general, since it is faster.
But binary search trees are truly dynamic (no advance info on size needed).
BSTs also give worst case guarantees (hash function could be lousy).
BSTs support more operations — sorting.
Lecture 10: Searching

18. Time Analysis
Open address hashing methods store N records in a table of size M. M > N
The performance of the operations depends on the load factor
alpha = N/M
For chained hashing, alpha may be greater than 1.
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19. Linear Probing
Open address hashing with linear probing requires, on average:
1/2 ( 1 + 1/(1-alpha)^2) operations for an unsuccessful search
1/2 ( 1 + 1/(1-alpha)) operations for a successful search
E.g., for alpha = 2/3 we’ll make 5 probes for an average unsuccessful search, and 2 for a successful search
Lecture 10: Searching

20. Double Hashing
Open address hashing with double hashing requires, on average:
1/(1-alpha) operations for an unsuccessful search
-log(1-alpha)/alpha operations for a successful search
E.g., for alpha = 2/3 we’ll make 3 probes for an average unsuccessful search, and 1.65 for a successful search
Lecture 10: Searching

21. Chained Hashing Chained hashing requires, on average:
1+alpha operations for an unsuccessful search
1+alpha/2 operations for a successful search
E.g., for alpha = 2/3 we’ll make 1.66 probes for an average unsuccessful search, and 1.33 for a successful search
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22. Time Analysis
These formulas require significant mathematical analysis, which we won’t go into.
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23. Average Number of Probes
Successful Search
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24. Radix Searching
For many applications, keys can be thought of as numbers
Searching methods that take advantage of digital properties of these keys are called radix searches
Radix searches treat keys as numbers in base M (the radix) and work with individual digits
Lecture 10: Searching

25. Radix Searching
Provide reasonable worst-case performance without complication of balanced trees.
Provide way to handle variable length keys.
Biased data can lead to degenerate data structures with bad performance.
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26. The Simplest Radix Search
Digital Search Trees — like BSTs but branch according to the key’s bits.
Key comparison replaced by function that accesses the key’s next bit.
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27. Digital Search Example
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28. Digital Search Requires O(log N) comparisons on average
Requires b comparisons in the worst case for a tree built with N random b-bit keys
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29. Digital Search
Problem: At each node we make a full key comparison — this may be expensive, e.g. very long keys
Solution: store keys only at the leaves, use radix expansion to do intermediate key comparisons
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30. Radix Tries Used for Retrieval [sic]
Internal nodes used for branching, external nodes used for final key comparison, and to store data
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31. Radix Trie Example H E A C S R A 00001 S 10011 E 00101 R 10010 C 00011
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32. Radix Tries
Left subtree has all keys which have 0 for the leading bit, right subtree has all keys which have 1 for the leading bit
An insert or search requires O(log N) bit comparisons in the average case, and b bit comparisons in the worst case
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33. Radix Tries
Problem: lots of extra nodes for keys that differ only in low order bits (See R and S nodes in example above)
This is addressed by Patricia trees, which allow “lookahead” to the next relevant bit
Practical Algorithm To Retrieve Information Coded In Alphanumeric (Patricia)
In the slides that follow the entire alphabet would be included in the indexes
Lecture 10: Searching

34. // Insert word K (see Drozdek and Simon – needs work) i=0; p=root;
While not inserted
if (K[i] == ‘\0’) set end-of-word marker in p to true
else if (p.ptrs[K[i]] == null) create leaf containing K and put its
address in p.ptrs[K[i]]
else if (refernce p.ptrs[k[i]] refers to a leaf)
K_L = key in leaf p.ptrs[K[i]];
do create a non-leaf and put its address in p.ptrs[K[i]]
p = the new non-leaf;
i++;
while (K[i] == K_L[i]);
create a leaf containing K and put its address in p.ptrs[K[--i]]
if (end-of-word K reached) set end-of-word marker in p to true
else create leaf containing K_L and put address in p.ptrs[K_L[i]]
else p = p.ptrs[K[i++]]
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35. Empty Radix Trie
Insert “ARA” # A E I P R
ARA
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36. # A E I P R ARA # A E I P R ARA K_L AREA K Insert “AREA” # A E I P R P
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37. P # A E I P R
ARA
AREA
Insert “A” P A
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38. # A E I P R # A E I P R # A E I P R # A E I P R # A E I P R # A E I P
PIER
EIRE
IPA
IRE
EERIE A # A E I P R # A E I P R
ARA # A E I P R
ERA
ERIE
ERE
PEER
ARE
PEAR
PER
AREA
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39. A L Radix Trie O G G E I A N D R ADAM LOGGIA LOGGING LOGGED LOGGERHEAD
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40. A L Patricia Tree 4 E 5 I 5 D R A N ADAM LOGGIA LOGGING LOGGERHEAD4
ADAM E 5 I 5 D R A N
LOGGIA
LOGGING
LOGGERHEAD
LOGGED
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