I/O-Algorithms
Lars Arge
Aarhus University
April 16, 2008
I/O-algorithms
I/O-Model
D
Block I/O
M
• Parameters
N = # elements in problem instance
B = # elements that fits in disk block
M = # elements that fits in main memory
T = # output size in searching problem
• We often assume that M>B2
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• I/O: Movement of block between memory
and disk
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I/O-algorithms
Until Now
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I/O-algorithms
Graph Algorithms
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I/O-algorithms
List Ranking
• Problem:
– Given N-vertex linked list stored in array
– Compute rank (number in list) of each vertex
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• One of the simplest graph problem one can think of
• Straightforward O(N) internal algorithm
– Also use O(N) I/Os in external memory
• Much harder to get O( NB log M B NB ) external algorithm
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I/O-algorithms
List Ranking
• We will solve more general problem:
– Given N-vertex linked list with edge-weights stored in array
– Compute sum of weights (rank) from start for each vertex
• List ranking: All edge weights one
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• Note: Weight stored in array entry together with edge (next vertex)
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List Ranking
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Algorithm:
1. Find and mark independent set of vertices
2. “Bridge-out” independent set: Add new edges
3. Recursively rank resulting list
4. “Bridge-in” independent set: Compute rank of independent set
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Step 1, 2 and 4 in O( NB log M B NB ) I/Os
Independent set of size αN for 0 < α ≤ 1
 T ( N )  T ((1   ) N )  O( NB log M B NB )  O( NB log M
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N
B B)
I/Os
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I/O-algorithms
List Ranking: Bridge-out/in
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• Obtain information (edge or rang) of successor
– Make copy of original list
– Sort original list by successor id
– Scan original and copy together to obtain successor information
– Sort modified original list by id
 O( NB log M B NB ) I/Os
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I/O-algorithms
List Ranking: Independent Set
• Easy to design O( NB log M B NB ) randomized algorithm:
– Scan list and flip a coin for each vertex
– Independent set is vertices with head and successor with tails
 Independent set of expected size N/4
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• Deterministic algorithm:
– 3-color vertices (no vertex same color as predecessor/successor)
– Independent set is vertices with most popular color
 Independent set of size at least N/3
• O( NB log M
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N
B B ) 3-coloring
 O( NB log M
N
B B)
I/O algorithm
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I/O-algorithms
List Ranking: 3-coloring
• Algorithm:
– Consider forward and backward lists (heads/tails in two lists)
– Color forward lists (except tail) alternately red and blue
– Color backward lists (except tail) alternately green and blue
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3-coloring
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I/O-algorithms
List Ranking: Forward List Coloring
• Identify heads and tails
• For each head, insert red element in priority-queue (priority=position)
• Repeatedly:
– Extract minimal element from queue
– Access and color corresponding element in list
– Insert opposite color element corresponding to successor in queue
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• Scan of list
• O(N) priority-queue operations
 O( NB log M B NB ) I/Os
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I/O-algorithms
Summary: List Ranking
• Simplest graph problem: Traverse linked list
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• Very easy O(N) algorithm in internal memory
• Much more difficult O( NB log M B NB ) external memory
– Finding independent set via 3-coloring
– Bridging vertices in/out
• Permuting bound O(min{ N , NB log M B NB }) best possible
– Also true for other graph problems
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I/O-algorithms
Summary: List Ranking
• External list ranking algorithm similar to PRAM algorithm
– Sometimes external algorithms by “PRAM algorithm simulation”
• Forward list coloring algorithm example of “time forward processing”
– Use external priority-queue to send information “forward in time”
to vertices to be processed later
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References
• External-Memory Graph Algorithms
Y-J. Chiang, M. T. Goodrich, E.F. Grove, R. Tamassia. D. E.
Vengroff, and J. S. Vitter. Proc. SODA'95
– Section 3-6
• The Buffer Tree: A Technique for Designing Batched External
Data Structures
Lars Arge, Algorithmica 37:1-24, 2003
– Section 4.1
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I/O-algorithms
References
• I/O-Efficient Graph Algorithms
Norbert Zeh. Lecture notes
– Section 2-4
• Cache-Oblivious Priority Queue and Graph Algorithm
Applications
L. Arge, M. Bender, E. Demaine, B. Holland-Minkley and I. Munro.
SICOMP, 36(6), 2007
– Section 3.1-3-2
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I/O-algorithms
Algorithms on Trees
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