Kernkonzepte
Zip-zip trees are a simple variant of zip trees that provide improved balance and bias properties while maintaining strong history independence and compact metadata.
Zusammenfassung
The paper introduces zip-zip trees, a variant of the zip tree data structure that improves upon the balance and bias properties of the original zip tree design.
Key highlights:
Zip-zip trees define each node's rank as a pair (r1, r2), where r1 is drawn from a geometric distribution as in the original zip tree, and r2 is drawn uniformly from a suitable range. Rank comparisons are done lexicographically.
This simple modification results in zip-zip trees having an expected node depth of at most 1.3863 log n - 1 + o(1), matching the expected depth of treaps and binary search trees built by random insertions, while using only O(log log n) bits of metadata per node w.h.p.
The expected depths of the smallest and largest keys in a zip-zip tree are the same, at most 0.6932 log n + γ + o(1), where γ is the Euler-Mascheroni constant.
The authors also introduce biased zip-zip trees, which support searches with expected performance logarithmic in the weight of the search key relative to the total weight of all keys.
A just-in-time (JIT) variant of zip-zip trees is presented that uses only an expected O(1) bits of metadata per node, though it lacks history independence.
Experimental results confirm the theoretical analysis and demonstrate the practical advantages of zip-zip trees over the original zip tree design.
Statistiken
The expected depth of the smallest key in an original zip tree is 0.5 log n + O(1).
The expected depth of the largest key in an original zip tree is log n + O(1).
The expected depth of any node in an original zip tree is at most 1.5 log n + O(1).
The expected depth of any node in a zip-zip tree is at most 1.3863 log n - 1 + o(1).
The expected depth of the smallest and largest keys in a zip-zip tree is at most 0.6932 log n + γ + o(1), where γ is the Euler-Mascheroni constant.
Zitate
"The expected depth of the smallest key in an original zip tree is 0.5 log n + O(1) whereas the expected depth of the largest key is log n + O(1)."
"The expected depth of any node in a zip-zip tree is at most 1.3863 log n - 1 + o(1)."
"The expected depth of the smallest and largest keys in a zip-zip tree is at most 0.6932 log n + γ + o(1), where γ is the Euler-Mascheroni constant."