Skewed Binary Search Trees
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Skewed Binary Search Trees 13 14 15 11 12 7 8 9 10 3 4 5 1 2 6 Gerth Stølting Brodal University of Aarhus Joint work with Gabriel Moruz presented at ESA’06 CPH STL Workshop, University of Copenhagen, October 30, 2006. 1
Transcript of Skewed Binary Search Trees
Skewed Binary Search Trees
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Gerth Stølting BrodalUniversity of Aarhus
Joint work with Gabriel Moruz presented at ESA’06
CPH STL Workshop, University of Copenhagen, October 30, 2006. 1
Perfectly Balanced Search Trees
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Skewed Binary Search Trees
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⌊α(n − 1)⌋ 1 − α
⌈(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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α = 0.2
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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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1/H(α)
α
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Comparisons
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3e+08
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2e+08
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1e+08
5e+07
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n = 50.000
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Running Time
α
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time
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0.275
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0.255
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0.235
Best running time achieved for α ≈ 0.3 !?
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Conclusion
Skewed binary search trees
can 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 Costcost(α) = (α · {left cost} + (1 − α) · {right cost})/H(α)
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cost
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left cost = 1 and right cost = 3
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Expected Costcost(α) = (α · {left cost} + (1 − α) · {right cost})/H(α)
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left cost = 1 and right cost = 0 .. 28
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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
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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;
}
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Branch Mispredictions
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n = 50.000
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Simple Layouts
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Random – O( log nH(α)
) I/Os
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Inorder – O( log nH(α)
− log B) I/Os
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BFS – O( log nH(α)
− log B) I/Os
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DFSr – O(α+(1−α)/BH(α)
· log n) I/Os.
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Running Time for Simple Layouts
RandInordBFS
DFSrDFSl
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DFS < Inorder < BFS < Random
DFS achieves the best performance for α ≈ 0.2 !
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Cache Faults for Simple Layouts
RandInordBFS
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DFS ≈ expected left cost = 1 and right cost = 1/B.
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Blocked Layouts — k-level blocking
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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
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Blocked Layouts — pqDFS k
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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
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Blocked Layouts — veb
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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
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Running Time for Blocked Layouts
pqDFSbDFS
vEBDFSr
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vEB achieves the fastest running time for α ≈ .25
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Cache Faults for Blocked Layouts
pqDFSbDFS
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vEB achieves the smallest number of cache faults
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Experimental Summary
α
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Conclusion
Skewed binary search trees
beat
Perfectly balanced binary search trees
because
The costs going left and right are different !
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