Showing papers on "Decision tree model published in 1995"

On rank vs. communication complexity

Abstract: This paper concerns the open problem of Lovasz and Saks regarding the relationship between the communication complexity of a boolean function and the rank of the associated matrix. We first give an example exhibiting the largest gap known. We then prove two related theorems.

Fractional Covers and Communication Complexity

Abstract: It is possible to view communication complexity as the minimum solution of an integer programming problem. This integer programming problem is relaxed to a linear programming problem and from it information regarding the original communication complexity question is deduced. A particularly appealing avenue this opens is the possibility of proving lower bounds on the communication complexity (which is a minimization problem) by exhibiting upper bounds on the maximization problem defined by the dual of the linear program. This approach works very neatly in the case of nondeterministic communication complexity. In this case a special case of Lovasz's fractional cover measure is obtained. Through it the amortized nondeterministic communication complexity is completely characterized. The power of the approach is also illustrated by proving lower and upper bounds on the nondeterministic communication complexity of various functions. In the case of deterministic complexity the situation is more complicated. Two attempts are discussed and some results using each of them are obtained. The main result regarding the first attempt is negative: one cannot use this method for proving superpolynomial lower bounds for formula size. The main result regarding the second attempt is a "direct-sum" theorem for two-round communication complexity.

On the Monte Carlo Boolean decision tree complexity of read-once formulae

Abstract: In the boolean decision tree model there is at least a linear gap between the Monte Carlo and the Las Vegas complexity of a function depending on the error probability. We prove for a large class of read‐once formulae that this trivial speed‐up is the best that a Monte Carlo algorithm can achieve. For every formula F belonging to that class we show that the Monte Carlo complexity of F with two‐sided error p is (1 − 2p)R(F), and with one‐sided error p is (1 − p)R(F), where R(F) denotes the Las Vegas complexity of F. The result follows from a general lower bound that we derive on the Monte Carlo complexity of these formulae. This bound is analogous to the lower bound due to Saks and Wigderson on their Las Vegas complexity. © 1995 Wiley Periodicals, Inc.

Decision tree model describing alternate health care choices made by oncology patients

Abstract: What common sense decision strategies do patients with cancer use when they are making health care choices that include alternate therapies? Existing research indicates that oncology patients are making alternate choices while associated with biomedicine. Often patients' decision strategies are exploited by the alternate system to promote and market alternate products. Although some of these practices are benign, others are dangerous or may interfere or delay successful treatment in biomedicine. Therefore, it seems important for biomedical professionals to understand patients' common sense decision patterns. A decision tree model, outlining patients' decision strategies, has recently been developed through intensive interviews with 300 patients who were diagnosed with cancer of the respiratory and digestive systems. The two-phase methodology included, first, a context sensitive approach to develop the model, followed by a predictive approach testing the model developed in the first phase on a separate yet similar random sample of patients. The discussion in this article focuses on the research, the patterns of the decision tree model, and the implications and adaptability of this model to nursing practice.

Search Problems in the Decision Tree Model

Abstract: The relative power of determinism, randomness, and nondeterminism for search problems in the Boolean decision tree model is studied. It is shown that the gaps between the nondeterministic, the randomized, and the deterministic complexities can be arbitrarily large for search problems. An interesting connection of this model to the complexity of resolution proofs is also mentioned.

Complexity measures for object-oriented program based on the entropy

Abstract: Software metrics are widely advocated as fundamental elements of an engineering approach to planning and controlling software development. They are especially important in object-oriented programming. We propose a new software complexity model which consists of class complexity, inter-object complexity, and the total complexity for the object-oriented program. We measure the complexities based on the entropy concept. The class complexity for a class measures the information flows in a class based on the information passing relationship among member data and member functions. The inter-object complexity for a program measures the information flows between objects. The total complexity for a program is measured by the class complexity and the inter-object complexity. We evaluate the proposed metrics using the complexity properties proposed by Weyuker (1988). Experimental results of C++ classes show the effectiveness of the proposed metrics. We assert that the class complexity and the inter-object complexity are correlated to other metrics for object-oriented concepts.

Decision trees with Boolean threshold queries

Abstract: We investigate decision trees in which one is allowed to query threshold functions of subsets of variables. We are mainly interested in the case where only queries of AND and OR are allowed. This model is a generalization of the classical descision tree model. Its complexity (depth) is related to the parallel time that is required to compute Boolean functions in certain CRCW PRAM machines with only one cell of constant size. It is also related to the computation using Ethernet channel. We prove a tight lower bound of θ(k log(n/k)) for the required depth of a decision tree for the threshold-k function. As a corollary of the method we also prove a tight lower bound for the "direct sum" problem of computing simultaneously k copies of threshold-2 in this model. Next, the size complexity is considered. A relation to depth-three circuits is established and a lower bound is proven. Finally the relation between randomized, nondeterminism, and determinism is also investigated, we show separation results between these models.

Decision tree classifier designed using hidden markov models

Abstract: A decision tree classifier (20) is designed using the hidden Markov model to yield a final classifier that can be implemented without using any mathematical operations. The apparatus of this invention moves all of the mathematical calculations to the construction of the decision tree. Once this is completed, the decision tree can be implemented using only logical operations such as memory addressing and binary comparison operations. This simplicity allows the decision tree to be implemented in a simple hardware form using conventional gates.

Stochastic complexity in learning

Abstract: This is an expository paper on the latest results in the theory of stochastic complexity and the associated MDL principle with special interest in modeling problems arising in machine learning. As an illustration we discuss the problem of designing MDL decision trees, which are meant to improve the earlier designs in two ways: First, by use of the sharper formula for the stochastic complexity at the nodes the earlier found tendency of getting too small trees appears to be overcome. Secondly, a dynamic programming based pruning algorithm is described for finding the optimal trees, which generalizes an algorithm described in Nohre (1994).

Decision trees with AND, OR queries

Abstract: We investigate decision trees in which one is allowed to query threshold functions of subsets of variables. We are mainly interested in the case where only queries of AND and OR are allowed. This model is a generalization of the classical decision tree model. Its complexity (depth) is related to the parallel time that is required to compute Boolean functions in certain CRCW PRAM machines with only one cell of constant size. It is also related to the computation using the Ethernet channel. We prove a tight lower bound of /spl theta/(k log(n/k)) for the required depth of a decision tree for the threshold-k function. As a corollary of the method we also prove a tight lower bound for the "direct sum" problem of computing simultaneously k copies of threshold-2 in this model. Next, the size complexity is considered. A relation to depth-three circuits is established and a lower bound is proven. Finally the relation between randomization, nondeterminism and determinism is also investigated, we show separation results between these models.

Communication complexity of matrix computation over finite fields

Abstract: We investigate the communication complexity of singularity testing in a finite field, where the problem is to determine whether a given square matrixM is singular. We show that, forn×n matrices whose entries are elements of a finite field of sizep, the communication complexity of this problem is Θ(n2 logp). Our results imply tight bounds for several other problems likedetermining the rank andcomputing the determinant.

Average Circuit Depth and Average Communication Complexity

Abstract: We use the techniques of Karchmer and Widgerson [KW90] to derive strong lower bounds on the expected parallel time to compute boolean functions by circuits. By average time, we mean the time needed on a self-timed circuit, a model introduced recently by Jakoby, Reischuk, and Schindelhauer, [JRS94] in which gates compute their output as soon as it is determined (possibly by a subset of the inputs to the gate).

A VLSI architecture for 2D object classification based on tree matching

Abstract: This paper presents a real-time classification algorithm for 2D object contours using a multi-resolution tree model which is implemented in a modular VLSI architecture. The hardware implementation takes advantage of pipelining, parallelism, and the speed of VLSI technology to perform real-time object classification. Using the multi-resolution tree model, the classification algorithm is invariant under 2D similarity transformations and recognizes the visible portions of occluded objects. The VLSI classification system is implemented in 0.8 /spl mu/m CMOS and is capable of performing 34000 matchings per second.

Computer models for computation and verification of bronchial morphology

Abstract: Accurate physiological measurements of the parameters like branching angles, branch lengths, and diameters of bronchial tree structures help in addressing the mechanistic and diagnostic questions related to obstructive lung disease. In order to facilitate these measurements, bronchial trees are reduced to a central axis tree. The approach we take employs first setting up a theoretical computerized tree structure, and then applying a 3D analysis to obtain the required anatomical data. A stick model was set up in 3D, with segment endpoints and diameters as input parameters to the model generator. By fixing the direction in which the slices are taken, a stack of 2D images of the generated 3D tree model is obtained, thereby simulating bronchial data sets. We design a two pass algorithm to compute the central axis tree and apply it on our models. In the first pass, the topological tree T is obtained by implementing a top-down seeded region growing algorithm of the 3D tree model. In the second pass, T is used to region growth along the axes of the branches. As the 3D tree model is traversed bottom-up, the centroid values of the cross sections of the branches are stored in the corresponding branch of T. At each bifurcation, the branch point and the three direction vectors along the branches are computed, by formulating it as a nonlinear optimization problem that minimizes the sum of least squares error of the centroid points of the corresponding branches. By connecting the branch points with straight lines, we obtain a reconstructed central axis tree which closely corresponds to the input stick model. We also studied the effect of adding external noise to out tree models and evaluating the physiological parameters. We conclude with the results of our algorithm on real airway trees.

Lower Bounds for the Modular Communication Complexity of Various Graph Accessibility Problems

Abstract: We investigate the modular communication complexity of the graph accessibility problem GAP and its modular counting versions MODk-GAP, k≥2 Due to arguments concerning variation ranks and certain projection reductions, we prove that, for any partition of the input variables and for any moduls k and m, GAP and MODk-GAP have MOD m -communication complexity Ω(n), where n denotes the number of nodes of the graphs under consideration

Read-once Projections and Formal Circuit Verification with Binary Decision Diagrams

Abstract: Computational complexity is concerned with the complexity of solving problems and computing functions and not with the complexity of verifying circuit designs. The importance of formal verification is evident. Therefore, the framework of a complexity theory for formal verification with binary decision diagrams is developed. This theory is based on read-once projections. For many problems it is determined whether and how they are related with respect to read-once projections. The result that circuits for squaring may be harder to verify than circuits for multiplication is derived and discussed. It is shown that the class of functions with polynomial-size free binary decision diagrams has no complete problem while for the corresponding classes for the other considered types of binary decision diagrams complete problems are presented.

Transforming Comparison Model Lower Bounds to the PRAM

Abstract: This note provides general transformations of lower bounds in Valiant's parallel comparison decision tree model to lower bounds in the priority concurrent-read concurrent-write parallel-random-access-machine model. The proofs rely on standard Ramsey-theoretic arguments that simplify the structure of the computation by restricting the input domain. The transformation of comparison model lower bounds, which are usually easier to obtain, to the parallel-random-access-machine, unifies some known lower bounds and gives new lower bounds for several problems.

Feature Construction during Tree Learning

Abstract: This paper addresses the “problem of new terms” in the context of learning decision trees using the approach based on ID3. We discuss an algorithm for efficiently constructing new features from given primitive features and relate it to constructive induction. In our approach, feature construction is integrated with selecting a (new) feature for building the decision tree in one process. Hence, appropriate features are constructed during tree generation. The representation of constructed features is based on sets. While the search space of possible features is exponential, we use a geometric interpretation to show that this algorithm provides linear time and space complexity. Moreover, we show that it finds features with optimal value for the tree construction procedure of ID3. Results of experiments are reported, and besides of considerations related to the size of the generated trees we also discuss the important issue of how comprehensible these trees are. In particular, we are interested in the intelligibility of the discovered features.

Decision Trees with AND, OR Queries (Extended Abstract)

Abstract: We investigate decision trees in which one is allowed to query threshold functions of subsets of variables. We are mainly interested in the case where only queries of AND and OR are allowed. This model is a generalization of the classical decision tree model. Its complexity (depth) is related to the parallel time that as required to compute Boolean functions in certain CRCW PRAM machines with only one cell of constant size. It is also related to the computation using Ethernet eh ann el. We prove a tight lower bound of b'(klog(n/k)) for the required depth of a decision tree for the threshold-k function. As a corollary of the method we also prove a tight lower bound for the 'direct sum' problem of computing simultaneously k copies of threshold-2 in this model. Next, the size complexity is considered. A relation to depth-three circuits is established and a lower bound is proven. Finally the relation between randomized, nondeterminism and determinism is also investigated, we show separation results between these models.