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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 presented at ESA06 CPH STL Workshop, University of Copenhagen, October 30, 2006. 1

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Skewed Binary Search Trees

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Gerth Stølting Brodal University of Aarhus Joint work with Gabriel Moruz presented at ESA’06

CPH STL Workshop, University of Copenhagen, October 30, 2006.

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Perfectly Balanced Search Trees

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Skewed Binary Search Trees

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Skewed Binary Search Trees

⌊α(n − 1)⌋

1 − α

⌈(1 − α)(n − 1)⌉

α

Skewed Binary Search Trees

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Skewed Binary Search Trees

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

α = 0.5

Skewed Binary Search Trees

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Skewed Binary Search Trees

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α = 0.4

Skewed Binary Search Trees

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Skewed Binary Search Trees

9 7 4 5 8 6 1 2 3 10 11 12 13 14 15

α = 0.2

Skewed Binary Search Trees

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Skewed Binary Search Trees

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

α = 0.05

Skewed Binary Search Trees

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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

Skewed Binary Search Trees

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1/H(α)

α 1/H(α) 1 0.8 0.6 0.4 0.2 10 8 6 4 2

Skewed Binary Search Trees

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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 3.5e+08 3e+08 2.5e+08 2e+08 1.5e+08 1e+08 5e+07

n = 50.000

Skewed Binary Search Trees

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Running Time

α Running time 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.28 0.275 0.27 0.265 0.26 0.255 0.25 0.245 0.24 0.235

Best running time achieved for α ≈ 0.3 !?

Skewed Binary Search Trees

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Conclusion

Skewed binary search trees can beat Perfectly balanced binary search trees !

Skewed Binary Search Trees

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Why ?

Skewed Binary Search Trees

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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?)
...

Skewed Binary Search Trees

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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

Skewed Binary Search Trees

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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

Skewed Binary Search Trees

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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
No unsuccesful searches

Skewed Binary Search Trees

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Search Code

while(root!=NULLV) { if(key==t[root].key) return root; if(key>t[root].key) root=t[root].right; else root=t[root].left; }

Skewed Binary Search Trees

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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 8e+06 7e+06 6e+06 5e+06 4e+06 3e+06 2e+06

n = 50.000

Skewed Binary Search Trees

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Simple Layouts

7 1 3 4 5 6 2

4 5 1 7 6 2 3

Random – O( log n

H(α)) I/Os

1 2 3 4 5 6 7

Inorder – O( log n

H(α) − log B) I/Os

3 1 5 2 4 6 7

BFS – O( log n

H(α) − log B) I/Os

3 5 6 7 4 1 2

DFSr – O( α+(1−α)/B

H(α)

· log n) I/Os.

Skewed Binary Search Trees

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Running Time for Simple Layouts

Rand Inord BFS DFSr DFSl α Running time 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.4 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 !

Skewed Binary Search Trees

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Cache Faults for Simple Layouts

Rand Inord BFS DFSr DFSl α Cache misses 1 0.8 0.6 0.4 0.2 3e+07 2.5e+07 2e+07 1.5e+07 1e+07 5e+06

DFS ≈ expected left cost = 1 and right cost = 1/B.

Skewed Binary Search Trees

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Blocked Layouts — k-level blocking

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k = 2

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layout the nodes of the first k levels
recurse on subtrees
a search uses O(logB n/H(α)) I/Os

Skewed Binary Search Trees

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Blocked Layouts — pqDFSk

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

k = 3

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layout the k heavest nodes in order of decreasing size
recurse on subtrees in order of decreasing size
a search uses O(logBα+1 n) I/Os

Skewed Binary Search Trees

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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
a search uses O(logBα+1 n) I/Os

Skewed Binary Search Trees

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Running Time for Blocked Layouts

pqDFS bDFS vEB DFSr α Running time 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.22 0.21 0.2 0.19 0.18 0.17 0.16 0.15 0.14 0.13

vEB achieves the fastest running time for α ≈ .25

Skewed Binary Search Trees

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Cache Faults for Blocked Layouts

pqDFS bDFS vEB DFSr α Cache misses 1 0.8 0.6 0.4 0.2 1e+07 8e+06 6e+06 4e+06 2e+06

vEB achieves the smallest number of cache faults

Skewed Binary Search Trees

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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 3.5e+08 3e+08 2.5e+08 2e+08 1.5e+08 1e+08 5e+07 pqDFS bDFS vEB DFSr α Running time 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.22 0.21 0.2 0.19 0.18 0.17 0.16 0.15 0.14 0.13 α Branch mispredictions 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 8e+06 7e+06 6e+06 5e+06 4e+06 3e+06 2e+06 pqDFS bDFS vEB DFSr α Cache misses 1 0.8 0.6 0.4 0.2 1e+07 8e+06 6e+06 4e+06 2e+06

Skewed Binary Search Trees

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Conclusion

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

Skewed Binary Search Trees