SLIDE 1
Outline of Talk
- Cache-oblivious model
- Basic cache-oblivious techniques
- Cache-oblivious string algorithms
- Cache-oblivious string dictionaries
Cache-Oblivious String Dictionaries Gerth Stlting Brodal University - - PowerPoint PPT Presentation
Cache-Oblivious String Dictionaries Gerth Stlting Brodal University - - PowerPoint PPT Presentation
SLIDE 2
SLIDE 3
Hierarchical Memory Models
✁✄✂ ☎ ✆✞✝ ✟ ✠ ✡☛ ☞✍✌ ☞✞✎ ✏ ✑ ✒ ✓ ✔ ☞✖✕ ✗ ✘ ☞ ☎ ✓ ☞✞✎ ✕ ✂ ✔ ☞ ✝ ✑ SLIDE 4
Hierarchical Memory
✁- ✁
SLIDE 5
I/O Model
Aggarwal and Vitter 1988
- ✁
= problem size = memory size
✔= I/O block size
- One I/O moves
consecutive records from/to disk
- Complexity measure = number of I/Os
SLIDE 6
Ideal Cache Model — no parameters!?
Frigo, Leiserson, Prokop, Ramachandran 1999
- Program with only one memory
- Analyze in the I/O model for
- ✁
- ✁
- Optimal off-line cache replacement
strategy arbitrary
✔and
✁✄✂ ☎ ✆✞✝ ✟ ✠ ✡☛ ☞✍✌ ☞✞✎ ✏ ✑ ✒ ✓ ✔ ☞✖✕ ✗ ✘ ☞ ☎ ✓ ☞✞✎ ✕ ✂ ✔ ☞ ✝ ✑ ✠ SLIDE 7
Ideal Cache Model — no parameters!?
Frigo, Leiserson, Prokop, Ramachandran 1999
- Program with only one memory
- Analyze in the I/O model for
- ✁
- ✁
- Optimal off-line cache replacement
strategy arbitrary
✔and Advantages
- Optimal on arbitrary level
- ptimal on all levels
- Portability,
and not hard-wired into algorithm
- Dynamic changing
(and
✔)
✁✄✂ ☎ ✆✞✝ ✟ ✠ ✡☛ ☞✍✌ ☞✞✎ ✏ ✑ ✒ ✓ ✔ ☞✖✕ ✗ ✘ ☞ ☎ ✓ ☞✞✎ ✕ ✂ ✔ ☞ ✝ ✑ ✠ SLIDE 8
Cache-Oblivious Preliminaries
✁✄✂ ☎ ✆✞✝ ✟ ✠ ✡☛ ☞✍✌ ☞✞✎ ✏ ✑ ✒ ✓ ✔ ☞✖✕ ✗ ✘ ☞ ☎ ✓ ☞✞✎ ✕ ✂ ✔ ☞ ✝ ✑ SLIDE 9
Cache-Oblivious Scanning
✓ ✔- ✏
I/Os
✁✄✂ ☎ ✆✞✝ ✟ ✠ ✡☛ ☞✍✌ ☞✞✎ ✏ ✑ ✒ ✓ ✔ ☞✖✕ ✗ ✘ ☞ ☎ ✓ ☞✞✎ ✕ ✂ ✔ ☞ ✝ ✑ ✄ SLIDE 10
Cache-Oblivious Scanning
✓ ✔- ✏
I/Os Corollary Cache-oblivious selection requires
✂- ✓
I/Os
Hoare 1961 / Blum et al. 1973
✁✄✂ ☎ ✆✞✝ ✟ ✠ ✡☛ ☞✍✌ ☞✞✎ ✏ ✑ ✒ ✓ ✔ ☞✖✕ ✗ ✘ ☞ ☎ ✓ ☞✞✎ ✕ ✂ ✔ ☞ ✝ ✑ ✄ SLIDE 11
Cache-Aware B-trees
- ✁✂✁✄✁☎✁✂✁✄✁☎✁✂✁✄✁☎✁✂✁✄✁☎✁✂✁✄✁☎✁✂✁✄✁☎✁✂✁✄✁☎✁✂✁✄✆
SLIDE 12
Static Cache-Oblivious B-Tree
✁ ✂ ☎✄ ✂ ☎✄ ✁ ✌ ✌ ✌ ✌ ✌ ✌ ✆ ✝ ✆ ✞ ✟ ✠ ✡ ✆ ✞ ✟ ☛Recursive layout of binary tree
☞van Emde Boas layout
✁✄✂ ☎ ✆✞✝ ✟ ✠ ✡☛ ☞✍✌ ☞✞✎ ✏ ✑ ✒ ✓ ✔ ☞✖✕ ✗ ✘ ☞ ☎ ✓ ☞✞✎ ✕ ✂ ✔ ☞ ✝ ✑ ✩✌ SLIDE 13
Static Cache-Oblivious B-Tree
✁✄✂ ☎ ✆✞✝ ✟ ✠ ✡☛ ☞✍✌ ☞✞✎ ✏ ✑ ✒ ✓ ✔ ☞✖✕ ✗ ✘ ☞ ☎ ✓ ☞✞✎ ✕ ✂ ✔ ☞ ✝ ✑ ✩ ✩ SLIDE 14
Static Cache-Oblivious B-Tree
- ✁✄✂
SLIDE 15
Static Cache-Oblivious B-Tree
- ✁✄✂
SLIDE 16
Static Cache-Oblivious B-Tree
- ✁✄✂
SLIDE 17
Static Cache-Oblivious B-Tree
- Each green tree has height between
- ✁
and
- ✁
- Searches visit between
- ✁
- ✓
and
✄- ✁
- ✓
green trees, i.e. perform at most
☎- ✁
- ✓
I/Os (misalignment)
✁✄✂ ☎ ✆✞✝ ✟ ✠ ✡☛ ☞✍✌ ☞✞✎ ✏ ✑ ✒ ✓ ✔ ☞✖✕ ✗ ✘ ☞ ☎ ✓ ☞✞✎ ✕ ✂ ✔ ☞ ✝ ✑ ✩ ✩ SLIDE 18
Summary Cache-Oblivious Tools
Scanning :
✂- ✓
B-tree searching :
✂- ✁
- ✓
Sorting
- :
- ✁
- ✓
- requires a tall cache assumption
Frigo, Leiserson, Prokop, Ramachandran 1999 Brodal and Fagerberg 2002, 2003
✁✄✂ ☎ ✆✞✝ ✟ ✠ ✡☛ ☞✍✌ ☞✞✎ ✏ ✑ ✒ ✓ ✔ ☞✖✕ ✗ ✘ ☞ ☎ ✓ ☞✞✎ ✕ ✂ ✔ ☞ ✝ ✑ ✩ ✙ SLIDE 19
Cache-Oblivious String Algorithms
✁✄✂ ☎ ✆✞✝ ✟ ✠ ✡☛ ☞✍✌ ☞✞✎ ✏ ✑ ✒ ✓ ✔ ☞✖✕ ✗ ✘ ☞ ☎ ✓ ☞✞✎ ✕ ✂ ✔ ☞ ✝ ✑ ✩ SLIDE 20
Knuth-Morris-Pratt String Matching
Knuth, Morris, Pratt 1977
✍- ☛
- ☛
- ☛
- ☛
- ☛
- ☛
- ✂
- ☛
- ☛
- ✍
- ☛
- ☛
- ✍
- ☛
- ☛
- Time
- Scans text left-to-right
- Accesses the pattern (and failure function) like a stack
SLIDE 21
Knuth-Morris-Pratt String Matching
Knuth, Morris, Pratt 1977
✍- ☛
- ☛
- ☛
- ☛
- ☛
- ☛
- ✂
- ☛
- ☛
- ✍
- ☛
- ☛
- ✍
- ☛
- ☛
- Time
- Scans text left-to-right
- Accesses the pattern (and failure function) like a stack
- KMP is cache-oblivious and uses
I/Os
✁✄✂ ☎ ✆✞✝ ✟ ✠ ✡☛ ☞✍✌ ☞✞✎ ✏ ✑ ✒ ✓ ✔ ☞✖✕ ✗ ✘ ☞ ☎ ✓ ☞✞✎ ✕ ✂ ✔ ☞ ✝ ✑ ✩ ✤ SLIDE 22
Suffix Tree/Suffix Array Construction
Farach et al. 2000
b $ a abacdacabab$ a b$ cdacabab$ abab$ dacabab$ c a b $ $ dacabab$ cdacabab$ b$ c abab$ dacabab$
aabacdacabab$
- Reduces to sorting, i.e.
- ✁
- ✓
I/Os
✁✄✂ ☎ ✆✞✝ ✟ ✠ ✡☛ ☞✍✌ ☞✞✎ ✏ ✑ ✒ ✓ ✔ ☞✖✕ ✗ ✘ ☞ ☎ ✓ ☞✞✎ ✕ ✂ ✔ ☞ ✝ ✑ ✩ ✕ SLIDE 23 ✁✄✂ ☎ ✆✞✝ ✟ ✠ ✡☛ ☞✍✌ ☞✞✎ ✏ ✑ ✒ ✓ ✔ ☞✖✕ ✗ ✘ ☞ ☎ ✓ ☞✞✎ ✕ ✂ ✔ ☞ ✝ ✑ ✩ ✠
SLIDE 24
String Dictionaries
✁✄✂ ☎ ✆✞✝ ✟ ✠ ✡☛ ☞✍✌ ☞✞✎ ✏ ✑ ✒ ✓ ✔ ☞✖✕ ✗ ✘ ☞ ☎ ✓ ☞✞✎ ✕ ✂ ✔ ☞ ✝ ✑ ✩ SLIDE 25
Tries vs Blind Tries
- ✁
- ✁
- ✁
- ✂
Trie
- ✁
- ✂
- ☎
- ☎
Blind trie Searches take
✂ ✄ ✂ ✄ ✂time in internal memory for constant sized alphabets and
✂- ✁
time for comparison based alphabets
✁✄✂ ☎ ✆✞✝ ✟ ✠ ✡☛ ☞✍✌ ☞✞✎ ✏ ✑ ✒ ✓ ✔ ☞✖✕ ✗ ✘ ☞ ☎ ✓ ☞✞✎ ✕ ✂ ✔ ☞ ✝ ✑ ✩ ✄ SLIDE 26
The Trouble Starts...
– Tries cannot be stored cache-aware to support top-down searches in
✂- ✁
- ✓
I/Os
Demaine et al 2004
– Can construct suffix trees cache-obliviously using
✂- ✁
- ✓
I/Os, but cannot search in it efficiently... + Cache-aware string B trees support searches in a set of strings in
✂- ✁
- ✡
I/Os
Ferragina and Grossi 1999
✁✄✂ ☎ ✆✞✝ ✟ ✠ ✡☛ ☞✍✌ ☞✞✎ ✏ ✑ ✒ ✓ ✔ ☞✖✕ ✗ ✘ ☞ ☎ ✓ ☞✞✎ ✕ ✂ ✔ ☞ ✝ ✑ ✩ ✓ SLIDE 27
String Dictionary
✍- ✍
- ✔
- ✂
- ✟
- ☛
- ✍
- ✍
- ✔
- ✂
- ✟
- ☛
- ✍
- ✄
- ✄
Queries: Search blind trie + Verify one string
✁✄✂ ☎ ✆✞✝ ✟ ✠ ✡☛ ☞✍✌ ☞✞✎ ✏ ✑ ✒ ✓ ✔ ☞✖✕ ✗ ✘ ☞ ☎ ✓ ☞✞✎ ✕ ✂ ✔ ☞ ✝ ✑ ✙ ✌ SLIDE 28
String Dictionary
✍- ✍
- ✔
- ✂
- ✟
- ☛
- ✍
- ✍
- ✔
- ✂
- ✟
- ☛
- ✍
- ✄
- ✄
Queries: Search blind trie + Verify one string
✁✄✂ ☎ ✆✞✝ ✟ ✠ ✡☛ ☞✍✌ ☞✞✎ ✏ ✑ ✒ ✓ ✔ ☞✖✕ ✗ ✘ ☞ ☎ ✓ ☞✞✎ ✕ ✂ ✔ ☞ ✝ ✑ ✙ ✌ SLIDE 29
Suffix Tree
✂✁ ✄ ☎ ✆ ✁ ✝ ✞ ✁ ✄ ✆ ✁ ✝- ✁
- ✁
- ✁
- ✁
- ✁
- ✠
- ✞
- ✞
- ✠
- ✠
- ✌
- ✆
- ✄
Queries: Search blind trie + Verify one suffix
✁✄✂ ☎ ✆✞✝ ✟ ✠ ✡☛ ☞✍✌ ☞✞✎ ✏ ✑ ✒ ✓ ✔ ☞✖✕ ✗ ✘ ☞ ☎ ✓ ☞✞✎ ✕ ✂ ✔ ☞ ✝ ✑ ✙ ✩ SLIDE 30
Suffix Tree
✂✁ ✄ ☎ ✆ ✁ ✝ ✞ ✁ ✄ ✆ ✁ ✝- ✁
- ✁
- ✁
- ✁
- ✁
- ✠
- ✞
- ✞
- ✠
- ✠
- ✌
- ✆
- ✄
Queries: Search blind trie + Verify one suffix
✁✄✂ ☎ ✆✞✝ ✟ ✠ ✡☛ ☞✍✌ ☞✞✎ ✏ ✑ ✒ ✓ ✔ ☞✖✕ ✗ ✘ ☞ ☎ ✓ ☞✞✎ ✕ ✂ ✔ ☞ ✝ ✑ ✙ ✩ SLIDE 31
Tries
1 2 3 3 2 1
- ✁
- ✁
- ✁
- ✁
- ✁
- ✂
- ☎
- ☎
Queries: Search blind trie + Verify prefix of one path
✁✄✂ ☎ ✆✞✝ ✟ ✠ ✡☛ ☞✍✌ ☞✞✎ ✏ ✑ ✒ ✓ ✔ ☞✖✕ ✗ ✘ ☞ ☎ ✓ ☞✞✎ ✕ ✂ ✔ ☞ ✝ ✑ ✙ ✙ SLIDE 32
Tries
1 2 3 3 2 1
- ✁
- ✁
- ✁
- ✁
- ✁
- ✂
- ☎
- ☎
Queries: Search blind trie + Verify prefix of one path
✁✄✂ ☎ ✆✞✝ ✟ ✠ ✡☛ ☞✍✌ ☞✞✎ ✏ ✑ ✒ ✓ ✔ ☞✖✕ ✗ ✘ ☞ ☎ ✓ ☞✞✎ ✕ ✂ ✔ ☞ ✝ ✑ ✙ ✙ SLIDE 33
Verifying a Prefix of a Path in a Tree
✁✄✂ ☎ ✆✞✝ ✟ ✠ ✡☛ ☞✍✌ ☞✞✎ ✏ ✑ ✒ ✓ ✔ ☞✖✕ ✗ ✘ ☞ ☎ ✓ ☞✞✎ ✕ ✂ ✔ ☞ ✝ ✑ ✙ SLIDE 34
Verifying Paths in Giraffe Trees is Easy
Definition A tree is a giraffe tree if all root-to-leaf paths share at least half
- f the nodes of the tree (long neck)
SLIDE 35
Verifying Paths in Giraffe Trees is Easy
Definition A tree is a giraffe tree if all root-to-leaf paths share at least half
- f the nodes of the tree (long neck)
- A prefix of length
- f a path in a giraffe tree using a BFS
layout can be traversed in
✂- ✁
I/Os
✁✄✂ ☎ ✆✞✝ ✟ ✠ ✡☛ ☞✍✌ ☞✞✎ ✏ ✑ ✒ ✓ ✔ ☞✖✕ ✗ ✘ ☞ ☎ ✓ ☞✞✎ ✕ ✂ ✔ ☞ ✝ ✑ ✙ ✤ SLIDE 36
Giraffe Cover of a Tree
✁✄✂ ☎ ✆✞✝ ✟ ✠ ✡☛ ☞✍✌ ☞✞✎ ✏ ✑ ✒ ✓ ✔ ☞✖✕ ✗ ✘ ☞ ☎ ✓ ☞✞✎ ✕ ✂ ✔ ☞ ✝ ✑ ✙ ✕ SLIDE 37
Giraffe Cover of a Tree
- Uses space
- ✓
and can be constructed greedily from left-to-right using
✂- ✓
I/Os by an Euler traversal of
✁- BFS layout of each giraffe
- A prefix of length
- f a path in a known giraffe can be
traversed in
✂- ✁
I/Os
✁✄✂ ☎ ✆✞✝ ✟ ✠ ✡☛ ☞✍✌ ☞✞✎ ✏ ✑ ✒ ✓ ✔ ☞✖✕ ✗ ✘ ☞ ☎ ✓ ☞✞✎ ✕ ✂ ✔ ☞ ✝ ✑ ✙ ✕ SLIDE 38
Summary so far...
String dictionary search Suffix tree search Trie search
- ✁✄✁✂✁
reduce to blind trie search Query : Blind trie search +
✂ ✆ ☛ ✄ ✂ ✄ ✔I/Os
✁✄✂ ☎ ✆✞✝ ✟ ✠ ✡☛ ☞✍✌ ☞✞✎ ✏ ✑ ✒ ✓ ✔ ☞✖✕ ✗ ✘ ☞ ☎ ✓ ☞✞✎ ✕ ✂ ✔ ☞ ✝ ✑ ✙ ✠ SLIDE 39
Cache-Oblivious (Blind) Tries
✁✄✂ ☎ ✆✞✝ ✟ ✠ ✡☛ ☞✍✌ ☞✞✎ ✏ ✑ ✒ ✓ ✔ ☞✖✕ ✗ ✘ ☞ ☎ ✓ ☞✞✎ ✕ ✂ ✔ ☞ ✝ ✑ ✙ SLIDE 40
Cache-Oblivious (Blind) Tries
- ✁
- ✁
- Partition input trie
into components (generalization of heavy paths)
- ✁
= collapse components in
✁into high degree nodes and replace by weight balanced trees
- Apply van Emde Boas layout out to
SLIDE 41
Cache-Oblivious (Blind) Tries
- ✁
- ✁
- Partition input trie
into components (generalization of heavy paths)
- ✁
= collapse components in
✁into high degree nodes and replace by weight balanced trees
- Apply van Emde Boas layout out to
Search:
✂- ✁
- ✡
I/O
✁✄✂ ☎ ✆✞✝ ✟ ✠ ✡☛ ☞✍✌ ☞✞✎ ✏ ✑ ✒ ✓ ✔ ☞✖✕ ✗ ✘ ☞ ☎ ✓ ☞✞✎ ✕ ✂ ✔ ☞ ✝ ✑ ✙ ✄ SLIDE 42
Cache-Oblivious (Blind) Tries
- ✁
- ✁
- Partition input trie
into components (generalization of heavy paths)
- ✁
= collapse components in
✁into high degree nodes and replace by weight balanced trees
- Apply van Emde Boas layout out to
Search:
✂- ✁
- ✡
I/O — ignoring searching inside components
✁✄✂ ☎ ✆✞✝ ✟ ✠ ✡☛ ☞✍✌ ☞✞✎ ✏ ✑ ✒ ✓ ✔ ☞✖✕ ✗ ✘ ☞ ☎ ✓ ☞✞✎ ✕ ✂ ✔ ☞ ✝ ✑ ✙ ✄ SLIDE 43
Decomposition into Components
✩ ✩ ✙- ✩
- ✙
- ✌
- ✤
- ✄
- ✩
- ✩
- ✩
- ✙
- ✩
- ✠
- ✩
- ✩
- ✩
- ✙
- ✩
- ✞
- ✞
- ✡
- ☛
- ✤
- ✤
- ✪
- ✤
- ✤
SLIDE 44
Storing and Searching Components
✩ ✩ ✙- ✩
- ✙
- ✌
- ✤
- ✄
- ✩
- ✩
- ✩✌
- ✙
- ✩
- ✠
- ✩
- ✩
- ✩
- ✙
- ✩
- ✞
- ✞
- ✞
- Store each layer
separately
- Make a giraf-decompostition of
- For
have a blind trie
- f size
- ✄
(using BFS layout) to select the right giraffe-tree
- Search:
search the blind trie + search in one giraffe-tree
- Distribute
- ✚
- ✚
- ✁
in the van Emde Boas layout of
✁ ✛- Analysis:
– Search in blind trie for
✚ ✘ ✄ ✂ ✜dominated by the matched characters in
✚ ✘ ✜– Space in van Emde Boas layout for a subtree of size
✂becomes
✂- ✂
- ✌
SLIDE 45
Cache-Oblivious Tries
There exists a cache-oblivious trie supporting prefix queries in
✂- ✁
- ✄
I/Os
- where
is the query string, and
✡is the number of leaves in the trie. It can be constructed in
✂- ✁
- ✓
time, where
✓is the total number of characters in the input. The space required is
✂- ✓
. The structure assumes
✁ ✔ ✟ ✄- .
- ✩
SLIDE 46
Conclusion
- A string dictionary (trie data structure) was presented that
supports queries in
✂- ✁
- ✡
I/Os. The data structure uses
✂- ✓
space and can be constructed using
✂- ✁
- ✓
I/Os.
- Lookahead in the query string is crucial
(both cache-aware and cache-oblivious)
- A giraffe cover is a simple construction allowing topdown
path traversals in a tree using
✂ ✄ ✂ ✄ ✁ ✔ ✂I/Os
✁✄✂ ☎ ✆✞✝ ✟ ✠ ✡☛ ☞✍✌ ☞✞✎ ✏ ✑ ✒ ✓ ✔ ☞✖✕ ✗ ✘ ☞ ☎ ✓ ☞✞✎ ✕ ✂ ✔ ☞ ✝ ✑- ✙
SLIDE 47
Open problems
- Prove a lower bound trade-off between the number of I/Os
required for a query and the lookahead used
- Implementation: compare with string B-trees, tries, ternary
trees, different trie layouts, ...
✁✄✂ ☎ ✆✞✝ ✟ ✠ ✡☛ ☞✍✌ ☞✞✎ ✏ ✑ ✒ ✓ ✔ ☞✖✕ ✗ ✘ ☞ ☎ ✓ ☞✞✎ ✕ ✂ ✔ ☞ ✝ ✑ SLIDE 48
The End
✁✄✂ ☎ ✆✞✝ ✟ ✠ ✡☛ ☞✍✌ ☞✞✎ ✏ ✑ ✒ ✓ ✔ ☞✖✕ ✗ ✘ ☞ ☎ ✓ ☞✞✎ ✕ ✂ ✔ ☞ ✝ ✑- ✤