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I/O-algorithmsI/O-Model

- 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
- I/O: Movement of block between memory and disk

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Block I/O

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I/O-AlgorithmsFundamental Bounds

Internal External

- Scanning: N
- Sorting: N log N
- Permuting
- Searching:
- Note:
- Linear I/O: O(N/B)
- Permuting not linear
- Permuting and sorting bounds are equal in all practical cases
- B factor VERY important:
- Cannot sort optimally with search tree

I/O-AlgorithmsScalability Problems: Block Access Matters

- Example: Traversing linked list (List ranking)
- Array size N = 10 elements
- Disk block size B = 2 elements
- Main memory size M = 4 elements (2 blocks)
- Large difference between N and N/B large since block size is large
- Example: N = 256 x 106, B = 8000 , 1ms disk access time

N I/Os take 256 x 103 sec = 4266 min = 71 hr

N/B I/Os take 256/8 sec = 32 sec

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Algorithm 1: N=10 I/Os

Algorithm 2: N/B=5 I/Os

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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
- One of the simplest graph problem one can think of
- Straightforward O(N) internal algorithm
- Also uses O(N) I/Os in external memory
- Much harder to get external algorithm

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I/O-algorithmsList 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
- Note: Weight stored in array entry together with edge (next vertex)

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I/O-algorithms

List Ranking- Algorithm:
- Find and mark independent set of vertices
- “Bridge-out” independent set: Add new edges
- Recursively rank resulting list
- “Bridge-in” independent set: Compute rank of independent set
- Step 1, 2 and 4 in I/Os
- Independent set of size αN for 0 < α≤ 1

I/Os

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I/O-algorithms

List Ranking: Bridge-out/in- 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

I/Os

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I/O-algorithms

List Ranking: Independent Set- Easy to design 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

- 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

- 3-coloring I/O algorithm

I/O-algorithmsList 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

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
- Scan of list
- O(N) priority-queue operations

I/Os

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I/O-algorithms

Summary: List Ranking- Simplest graph problem: Traverse linked list
- Very easy O(N) algorithm in internal memory
- Much more difficult external memory
- Finding independent set via 3-coloring
- Bridging vertices in/out
- Permuting bound 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

I/O-algorithmsReferences

- 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
- 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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