1 Funnel Heap Introduced by Brodal et al. [1]. Based on merging - - PDF document

▶
1 funnel heap
SMART_READER_LITE
LIVE PREVIEW

1 Funnel Heap Introduced by Brodal et al. [1]. Based on merging - - PDF document

Oct 17, 2022 119 likes •225 views

1 Funnel Heap Introduced by Brodal et al. [1]. Based on merging instead of distribution. References [1] G. S. Brodal and R. Fagerberg, Funnel heapa cache oblivious priority queue, Proceedings of the 13th Annual International

slide-1
SLIDE 1

1 Funnel Heap

  • Introduced by Brodal et al. [1].
  • Based on merging instead of distribution.

References

[1] G. S. Brodal and R. Fagerberg, Funnel heap—a cache oblivious priority queue, Proceedings of the 13th Annual International Symposium on Algorithms and Computation, Lecture Notes in Computer Science 2380, Springer-Verlag (2002), 426–438.

1

slide-2
SLIDE 2

2 Funnel

❞ ✲ ✂ ✂ ✂ ✍ ❇ ❇ ❇ ◆ ❞ ✲ ✡ ✡ ✣ ❏ ❏ ❫ ❞ ✲ ✸ s ❞ ✲ ✶ q ❞ ✲ ✶ q ❞ ✲ ✸ s ❞ ✲ ✶ q ❞ ✲ ✶ q ❞ ✲ ✡ ✡ ✣ ❏ ❏ ❫ ❞ ✲ ✸ s ❞ ✲ ✶ q ❞ ✲ ✶ q ❞ ✲ ✸ s ❞ ✲ ✶ q ❞ ✲ ✶ q

2

slide-3
SLIDE 3

3 Structure

(k1, s1) = (2, 8), (1) si = si−1(ki + 1), and (2) ki = ⌈⌈si

1/3⌉⌉,

(3) where ⌈⌈x⌉⌉ is “x rounded to the nearest power of 2”.

3

slide-4
SLIDE 4

B2 v 2 Bi v i A2 K2 S 2,1 S 2,k2 S 2,2 Ai B1 v 1 A1 S 1,1 S 1,2 K1 Ki S i,1 S i,2 Link i

...

...... Link 2 I Link 1 S i,ki

...

4

slide-5
SLIDE 5

4 Extract

  • Fill A1 if it is empty.
  • Extract from A1

5 Insert

  • Insert element in I.
  • If I is full perform sweep.

5

slide-6
SLIDE 6

6 Sweep

Idea: Merge elements, and move them to a higher link.

  • 1. Find first empty S buffer.
  • 2. Create σ1 and σ2, and merge them to form σ.
  • 3. Insert elements on path from the root to the

empty S buffer.

B2 v 2 B1 v 1 A1 A2 K2 K1 B3 v 3 A3 K3

σ

2

σ

1

I S

...

S S

3,1 3,2 2,k 2

S S S S

1,1 1,2 2,1 2,2

*

6

slide-7
SLIDE 7

7 Space Complexity

  • In a link i there are ki S buffers of size si.
  • By the definition of k and s we know that:

ki = O(s1/3

i

).

  • The space consumption of the S buffers are

thus O(k4

i ) = O(s4/3 i

).

  • Since Ki, Ai and Bi are all of size O(si), the

space in the S buffers dominates the space used for a link i.

  • Since si and ki is increasing, the space

consumption of a Funnel Heap with i links is dominated by the ith link.

  • Since the space consumption of a Funnel Heap

with i links is O(k4

i ), and ki+1 = O(k4/3 i

), the worst case space complexity is O(N 4/3).

7

slide-8
SLIDE 8

8 I/O Complexity

They prove that the amortized I/O complexity of an extract is O 1 B logM/B N B

  • .

Which is optimal for cache-oblivious priority queues.

8