Van Emde Boaz Tries (The presentation follows Willard

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Van Emde Boaz Tries(The presentation follows
Willards fast tries)
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RAM model

• We have integers in the input
• Each integer U
• A computer work can store log(U) bits

Suppose we want to solve the dictionary
problem Insert(x), delete(x), find(x)
Hashing
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Perfect hashing
Build an O(m) table in (randomized) O(m) time
Table stores m keys so that find(x) takes O(1)
time
Dynamic perfect hashing Can also insert(x) and
delete(x) in O(1) time (amortized expected)
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The successor problem
In addition to find(x), insert(x), delete(x)
We have successor(x) and predecessor(x)
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Trie
0
1
1101
0000
0011
1000
1001
Depth is log(U) and we have m leaves
Can do all operations in O(log(U))
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Willards trie
0, 00, 000, 0000, 001, 0011, 1, 10, 100, 1000, .
0
1
1101
0000
0011
1000
1001
Store all prefixes of the keys in a hash table
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0, 00, 000, 0000, 001, 0011, 1, 10, 100, 1000, .
0
1
1101
0000
0011
1000
1001
1100
Find(x) We can find the longest prefix of x in
the trie by a binary search
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0, 00, 000, 0000, 001, 0011, 1, 10, 100, 1000, .
0
1
1101
0000
0011
1000
1001
1100
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0, 00, 000, 0000, 001, 0011, 1, 10, 100, 1000, .
0
1
1101
0000
0011
1000
1001
1100
Now if each node knows the minimum and the
maximum in its subtree, we have either the
successor or the predecessor
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0, 00, 000, 0000, 001, 0011, 1, 10, 100, 1000, .
0
1
1101
0000
0011
1000
1001
1100
To have them both we doubly link the items
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Summary
So we can search in O(loglogU) time
Remaining problems
Space is not linear its O(nlog(U)) !
Updates take O(log(U)) time
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Partition the items into groups of size O(logU)
each
Maintain the minimum of each group in a trie as
before
0100
0000
1001
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0
1
0100
0000
1001
Use a binary search tree to represent each group
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0
1
0100
0000
1001
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Summary
Now space is linear
Search still takes O(loglogU) time
To insert or delete work within subtree -- This
takes O(loglogU) time
If a subtree get too large split it and insert a
new item into the trie. This would take O(logU)
but we peform it once for O(logU) insertions
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Summary (Cont)
Delete is similar We work within a set, when a
set gets too small we merge it with its neighbor