Complexity measures and decision tree complexity: a survey
Theoretical Computer Science - Complexity and logic
Improved Lower Bounds on the Randomized Complexity of Graph Properties
ICALP '01 Proceedings of the 28th International Colloquium on Automata, Languages and Programming,
Bounds for Small-Error and Zero-Error Quantum Algorithms
FOCS '99 Proceedings of the 40th Annual Symposium on Foundations of Computer Science
Every decision tree has an in.uential variable
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
Improved lower bounds on the randomized complexity of graph properties
Random Structures & Algorithms
A note on the query complexity of the Condorcet winner problem
Information Processing Letters
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In this simple model, a decision tree algorithm must determine whether an unknown digraph on nodes {1, 2, …, n} has a given property by asking questions of the form “Is edge in the graph?”. The complexity of a property is the number of questions which must be asked in the worst case.Aanderaa and Rosenberg conjectured that any monotone, nontrivial, (isomorphism-invariant) n-node digraph property has complexity &OHgr;(n2). This bound was proved by Rivest and Vuillemin and subsequently improved to n2/4+&ogr;(n2). In Part I, we give a bound of n2/2+&ogr;(n2). Whether these properties are evasive remains open.In Part II, we investigate the power of randomness in recognizing these properties by considering randomized decision tree algorithms in which coins may be flipped to determine the next edge to be queried. Yao's lower bound on the randomized complexity of any monotone nontrivial graph property is improved from &OHgr;(nlog1/12n) to &OHgr;(n5/4), and improved bounds for the complexity of monotone, nontrivial bipartite graph properties are shown.