Showing papers on "Decision tree model published in 1984"

Optimal decision trees and one-time-only branching programs for symmetric Boolean functions

Abstract: Combinational complexity and depth are the most important complexity measures for Boolean functions. It has turned out to be very hard to prove good lower bounds on the combinational complexity or the depth of explicitly defined Boolean functions. Therefore one has restricted oneself to models where nontrivial lower bounds are easier to prove. Here decision trees, branching programs, and one-time-only branching programs are considered, where each variable may be tested on each path of computation only once. Efficient algorithms for the construction of optimal decision trees and optimal one-time-only branching programs for symmetric Boolean functions are presented. Furthermore, the following trade-off results are proved. An exponential lower bound on the decision tree complexity of some Boolean function is shown having linear formula size and linear one-time-only branching program complexity. Furthermore, a quadratic lower bound on the one-time-only branching program complexity of some Boolean function is shown having linear combinational complexity.

On the complexity of decentralized decision making and detection problems

Abstract: We study the computational complexity of the discrete versions of some simple but basic decentralized decision problems. These problems are variations of the classical "team decision problem" and include the problem of decentralized detection, whereby a central processor is to select one of two hypotheses, based on 1-bit messages from two noncommunicating sensors. Our results point to the inherent difficulty of decentralized decision making and suggest that optimality may be an elusive goal.

A Concise Model for the Management of Possible Appendicitis

Abstract: A conventional decision tree model for the management of patients with possible appendicitis requires at least 15 pieces of information. It makes three assumptions. A more concise decision tree model of the problem is presented here. It requires six pieces of information and prevents unintended mathematical shifts between disease states. It makes an additional five assumptions. The concise decision tree model was compared with the larger original model. Expert surgeons gave opinions on the management of a theoretical patient with possible appendicitis and the information necessary to solve the problem by decision analysis. Information from 25 surgeons whose opinions were consistent with analyses of their information using the original large decision tree was used to reanalyze the problem using the concise decision tree. The concise decision tree agreed with the original analysis in 24 of 25 cases (96%), despite violation of the five additional assumptions in 103 of 125 possible instances. The surgeon's decision can be shown by this model to be a function of the probability of appendicitis, the probability of perforation during observation, and the relative differences in outcomes between avoidable perforation and unnecessary operation.

Applications of Ramsey's Theorem to Decision Trees Complexity (Preliminary Version)

Abstract: 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 dornain 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 /spl Omega/(n log n) lower bound for the element uniqueness problem and several other problems for any k-bounded decision tree, such that k - )(n/sup c/) and c < 1/2. This lower bound is tight since that there exist n/sup 1/2/-bounded decision trees of complexity 0(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.