Cache-oblivious priority queue and graph algorithm applications

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1. What contributions have the authors mentioned in the paper "Cache-oblivious priority queue and graph algorithm applications" ?

2. What is the standard approach to obtaining good locality?

3. What is the key difficulty in designing a cache-oblivious priority queue?

4. How can directed DFS be solved in memory?

Accepted Answer

In this paper the authors develop an optimal cache-oblivious priority queue data structure, supporting insertion, deletion, and deletemin operations in O ( 1 B logM/B N B ) amortized memory transfers, where M and B are the memory and block transfer sizes of any two consecutive levels of a multilevel memory hierarchy.. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for pro£t or commercial advantage and that copies bear this notice and the full citation on the £rst page.

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The standard approach to obtaining good locality is to design algorithms parameterized by several aspects of the memory hierarchy, such as the size of each memory level, and the speed and block sizes of memory transfers between levels.

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One key difficulty in designing a cache-oblivious priority queue is that in order to be optimal the structure has to adapt to arbitrary values of both B and M , in contrast to existing cache-oblivious data structures, which only adapt to B [13, 14, 16].

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In the RAM model, directed DFS can be solved in linear time using a stack containing vertices v that have not yet been visited but have an edge (w, v) incident to a visited vertex w.

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The idea in the cache-oblivious model is to design and analyze algorithms in the I/O model but without using the parameters M and B in the algorithm description.

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MSF algorithm can be obtained by running the phase 1 algorithm until the number of vertices have been reduced to V ′ = E/B using O((sort(E)·log2 log2( V B E )) memory transfers and then finishing the MSF in O(V ′ + sort(E)) = O(sort(E)) memory transfers using the phase 2 algorithm.

Accepted Answer

To select the leaders, the authors use of the following three properties: the selected edges of a connected component form a tree, exactly one edge in each tree (the minimal weight edge in the tree) is selected twice, and the weights of the edges along a simple path from this edge to a leaf of the tree appear in increasing order.

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Using their cache-oblivious BRT structure and cache-oblivious priority queue in the algorithm by Buchsbaum et al. [18] the authors can obtain a cache-oblivious DFS algorithm.

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Using ideas utilized in efficient PRAM algorithms, an Euler Tour of such a modified tree can easily be computed using a scan of the graph followed by a list ranking step.

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Since only forward list are colored, the nodes will be colored (accessed) in the order they are stored on disk, that is, in a single scan of the list.

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The authors can easily perform the rebuilding in a sorting step using O(NB logM/B N B )memory transfers, or O( 1 B logM/B N B ) transfers per operation.

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The main advantage of the cache-oblivious model is that it allows us to reason about a simple two-level memory model, but prove results about an unknown, multilevel memory model.

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Because the authors fill up the down buffers after a recursive pull using one sort and one scan of X3/2 element, this cost is dominated by the cost of the recursive pull operation on the next level up.

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Because these elements do not necessarily have smaller keys than the, say U , elements in the up buffer uX 3/2, the authors first sort this up buffer and merge the two sorted lists.

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Next the authors distribute the elements in the sorted list into the X1/2 down buffers of level X3/2 by scanning through the list and simultaneously visiting the down buffers in (linked) order.

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The number ofmemory transfers needed to search for an element among a set of N elements is Ω(logB N) (the searching bound) and this bound is matched by the B-tree, which also supports updates in O(logB N) memory transfers [12, 23, 27, 26].

Accepted Answer

Next the nodes in the forward lists are colored red or blue by coloring the head nodes red and the other nodes alternatingly red and blue.

Accepted Answer

In this paper the authors presented an optimal cache-oblivious priority queue and used it to develop efficient cache-oblivious algorithms for several graph problems.

Accepted Answer

Theorem 3. The Euler Tour, BFS, DFS, and centroid decomposition problems on a tree can be solved cache-obliviously in O(sort(V )) memory accesses.

Cache-oblivious priority queue and graph algorithm applications

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Most frequently asked questions

1. What contributions have the authors mentioned in the paper "Cache-oblivious priority queue and graph algorithm applications" ?

2. What is the standard approach to obtaining good locality?

3. What is the key difficulty in designing a cache-oblivious priority queue?

4. How can directed DFS be solved in memory?

5. What is the idea of the cache-oblivious model?

6. How can the authors obtain an O(sort(E) log2 log2( V?

7. How do the authors select the leaders of a graph?

8. How do the authors obtain a cache-oblivious DFS algorithm?

9. How can an Euler Tour be computed?

10. How is the color of the head nodes in the forward list?

11. How can the authors perform the rebuilding in a sorting step?

12. What is the main advantage of the cache-oblivious model?

13. What is the cost of a recursive pull?

14. Why do the authors first sort the up buffer and merge the two sorted lists?

15. How do the authors distribute the elements in the sorted list?

16. What is the number of memory transfers needed to search for an element among a set of N?

17. How do you color the head nodes in the forward lists?

18. What is the way to make the cache-oblivious priority queue?

19. How can a tree be cache-oblivious to the underlying memory?

Figures

Citations

Cache-oblivious B-trees

Cache-oblivious streaming B-trees

Algorithms for Memory Hierarchies

High-Performance Streaming Dictionary

Cache-oblivious algorithms and data structures

References

The Art in Computer Programming

Shortest connection networks and some generalizations

Ubiquitous B-Tree

The input/output complexity of sorting and related problems

Organization and maintenance of large ordered indexes

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