TY - GEN

T1 - Applications of ramsey's theorem to decision trees complexity

AU - Moran, Shlomo

AU - Snir, Marc

AU - Manber, Udi

N1 - Funding Information:
* Supported in part by the National Science Foundation under Grant
Funding Information:
Supported in part by the National Science Foundation under Grant MCS83-03134.
Publisher Copyright:
© 1984 IEEE.

PY - 1984

Y1 - 1984

N2 - Combinatorial techniques for extending lower bounds results for decision trees to general types of queries are presented. We consider problems, which we call order invariant, that are defined by simple inequalities between inputs. A decision tree is called k - bounded if each query depends on at most k variables. We make no further assumptions on the type of queries. We prove that we can replace the queries of any k-bounded decision tree that solves an order invariant problem over a large enough input domain with k-bounded queries whose outcome depends only on the relative order of the inputs. As a consequence, all existing lower bounds for comparison based algorithms are valid for general k-bounded decision trees, where k is a constant. We also prove an Ωl(n logn) lower bound for the element uniqueness problem and several other problems for any k-bounded decision tree, such that k = O(nc) and c<1/2. This lower bound is tight since that there exist n1/2-bounded decision trees of complexity O(n) that solve the element uniqueness problem. All the lower bounds mentioned above are shown to hold for nondeterministic and probabilistic decision trees as well.

AB - Combinatorial techniques for extending lower bounds results for decision trees to general types of queries are presented. We consider problems, which we call order invariant, that are defined by simple inequalities between inputs. A decision tree is called k - bounded if each query depends on at most k variables. We make no further assumptions on the type of queries. We prove that we can replace the queries of any k-bounded decision tree that solves an order invariant problem over a large enough input domain with k-bounded queries whose outcome depends only on the relative order of the inputs. As a consequence, all existing lower bounds for comparison based algorithms are valid for general k-bounded decision trees, where k is a constant. We also prove an Ωl(n logn) lower bound for the element uniqueness problem and several other problems for any k-bounded decision tree, such that k = O(nc) and c<1/2. This lower bound is tight since that there exist n1/2-bounded decision trees of complexity O(n) that solve the element uniqueness problem. All the lower bounds mentioned above are shown to hold for nondeterministic and probabilistic decision trees as well.

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M3 - Conference contribution

AN - SCOPUS:85115235100

T3 - Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS

SP - 332

EP - 337

BT - 25th Annual Symposium on Foundations of Computer Science, FOCS 1984

PB - IEEE Computer Society

T2 - 25th Annual Symposium on Foundations of Computer Science, FOCS 1984

Y2 - 24 October 1984 through 26 October 1984

ER -