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Skewed Binary Search Trees 6 2 10 1 3 7 12 4 8 11 13 5 9 14 15 Gerth Stlting Brodal University of Aarhus Joint work with Gabriel Moruz Dagstuhl seminar on Data Structures, February 26-March 3, 2006. 1 Skewed Binary Search

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Skewed Binary Search Trees

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Gerth Stølting Brodal University of Aarhus Joint work with Gabriel Moruz

Dagstuhl seminar on Data Structures, February 26-March 3, 2006.

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Skewed Binary Search Trees

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Gerth Stølting Brodal University of Aarhus Joint work in progress with Gabriel Moruz

Dagstuhl seminar on Data Structures, February 26-March 3, 2006.

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Perfectly Balanced Search Trees

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Skewed Binary Search Trees

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Skewed Binary Search Trees

x

1 − α

⌊α(n − 1)⌋ ⌈(1 − α)(n − 1)⌉

α

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Skewed Binary Search Trees

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α = 0.5

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Skewed Binary Search Trees

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α = 0.4

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Skewed Binary Search Trees

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α = 0.2

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Skewed Binary Search Trees

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α = 0.05

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Skewed Binary Search Trees

— Average Node Depth

≤ 1 −α log2 α − (1 − α) log2(1 − α)

  • H(α)

· n + 1 n · log2(n + 1) − 2

Nievergelt and E. M. Reingold, 1972

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H(α)

α H(α) 1 0.8 0.6 0.4 0.2 10 8 6 4 2

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Comparisons

α Comparisons 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 300 250 200 150 100 50

n = 20.000

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Running Time

α Running time, 10−6 sec 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.15 0.145 0.14 0.135 0.13 0.125 0.12

Best running time achieved for α ≈ 0.3 !?

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Conclusion

Skewed binary search trees beat Perfectly balanced binary search trees !

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Conclusion

Skewed binary search trees c a n beat Perfectly balanced binary search trees !

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Why ?

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Why ?

The costs going left and right are different ! Possible reasons

  • Number of instructions
  • Branch mispredictions
  • Cache faults (what is a good memory layout?)
  • ...

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Expected Cost

cost(α) = (α · {left cost} + (1 − α) · {right cost}) · H(α)

α cost(α) 1 0.8 0.6 0.4 0.2 8 7 6 5 4 3 2 1

left cost = 1 and right cost = 3

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Expected Cost

cost(α) = (α · {left cost} + (1 − α) · {right cost}) · H(α)

α cost(α) 1 0.8 0.6 0.4 0.2 8 7 6 5 4 3 2 1

left cost = 1 and right cost = 0 .. 28

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Search Code

int search(int root, int key) { if (root==NULLV) return NULLV; if (key==t[root].key) return root; if(key>t[root].key) return search(t[root].right, key); else return search(t[root].left, key); }

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Branch Mispredictions

α Branch mispredictions 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 22 20 18 16 14 12 10 8 6 4 2

Static branch prediction: left cost = 1 and right cost = 0

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Simple Layouts

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Random

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Inorder

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BFS

3 5 6 7 4 1 2

DFS

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Running Time for Simple Layouts

Inorder Random BFS DFSr DFSl α Running time, 10−6 sec 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.35 0.3 0.25 0.2 0.15 0.1

DFS < Inorder < BFS < Random DFS achieves the best performance for α ≈ 0.2 !

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Cache Faults for Simple Layouts

Inorder Random BFS DFSr DFSl α Cache faults 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 14 12 10 8 6 4 2

DFS ≈ expected left cost = 1 and right cost = 1 cache-line size

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Blocked Layouts — pqDFSk

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k = 3

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  • layout the k heavest nodes in order of decreasing size
  • recurse on subtrees in order of decreasing size

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Blocked Layouts — veb

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9 2 1 1 15 5 2 3 1 1 2 1 5 3 3

top

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  • top = ⌈√n⌉ heavest nodes
  • recurse on top and bottom trees in order of decreasing size

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Running Time for Blocked Layouts

vEB pqDFS8 pqDFS4 pqDFS2 DFSr α Running time, 10−6 sec 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.2 0.19 0.18 0.17 0.16 0.15 0.14 0.13 0.12

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Cache Faults for Blocked Layouts

vEB pqDFS8 pqDFS4 pqDFS2 DFSr α Cache faults 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 5 4.5 4 3.5 3 2.5 2 1.5

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Experimental Summary

α Comparisons 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 300 250 200 150 100 50 vEB pqDFS8 pqDFS4 pqDFS2 DFSr α Running time, 10−6 sec 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.2 0.19 0.18 0.17 0.16 0.15 0.14 0.13 0.12 α Branch mispredictions 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 22 20 18 16 14 12 10 8 6 4 2 vEB pqDFS8 pqDFS4 pqDFS2 DFSr α Cache faults 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 5 4.5 4 3.5 3 2.5 2 1.5

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Conclusion

Skewed binary search trees c a n beat Perfectly balanced binary search trees because The costs going left and right are different !

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Questions

  • Can we give a precise theoretical explanation of the

experimental results ?

  • Different values of n ?
  • Different computer architectures ?
  • What is the best layout of a tree on a given machine ?
  • What is the best tree ?
  • ...

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Experimental setup

  • AMD Athlon XP 2400+
  • 2.0 GHz
  • 256 Kb L2 cache
  • 64 Kb L1 data cache
  • 64 Kb L1 instruction cache
  • 1Gb RAM
  • Linux 2.6.8.1
  • GCC 3.3.2
  • Tree nodes = 12 bytes

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