# Probabilistic Boolean decision trees and the complexity of evaluating game trees

@article{Saks1986ProbabilisticBD, title={Probabilistic Boolean decision trees and the complexity of evaluating game trees}, author={Michael E. Saks and Avi Wigderson}, journal={27th Annual Symposium on Foundations of Computer Science (sfcs 1986)}, year={1986}, pages={29-38} }

The Boolean Decision tree model is perhaps the simplest model that computes Boolean functions; it charges only for reading an input variable. We study the power of randomness (vs. both determinism and non-determinism) in this model, and prove separation results between the three complexity measures. These results are obtained via general and efficient methods for computing upper and lower bounds on the probabilistic complexity of evaluating Boolean formulae in which every variable appears…

## 204 Citations

### Randomized Boolean Decision Trees: Several Remarks

- Computer Science, MathematicsTheor. Comput. Sci.
- 1998

### On the power of randomness in the decision tree model

- Mathematics, Computer ScienceProceedings Fifth Annual Structure in Complexity Theory Conference
- 1990

Results suggest that there are relations between the decision tree complexity of a Boolean function and its symmetry and consideration is given to the question of what distinguishes graph properties from other highly symmetric Boolean functions, where randomization can help significantly.

### Principles of Optimal Probabilistic Decision Tree Construction

- Computer ScienceFCS
- 2006

A method is provided, which can be used to construct the optimal probabilistic decision tree for a given Boolean function, which takes into account the symmetries of given function.

### Lower bounds for uniform read-once threshold formulae in the randomized decision tree model

- Computer Science, MathematicsArXiv
- 2022

This work investigates the randomized decision tree complexity of a class of read-once threshold functions and proves lower bounds of the form c ( k, n ) d , where d is the depth of the tree.

### On P versus NP $ \cap $ co-NP for decision trees and read-once branching programs

- Mathematics, Computer Sciencecomputational complexity
- 1999

It is shown that this simulation cannot be made polynomial: explicit Boolean functions f that require deterministic trees of size exp (\Omega({\rm log^2} N) $ where N is the total number of monomials in minimal DNFs for f and ¬f are exhibited.

### Lower Bounds on the Randomized Communication Complexity of Read-Once Functions

- Computer Science, Mathematics2009 24th Annual IEEE Conference on Computational Complexity
- 2009

Information theory methods are used to prove lower bounds on the randomized two-party communication complexity of functions that arise from read-once boolean formulae, which are optimal up to the constant in the base of the denominator.

### Randomized vs. deterministic decision tree complexity for read-once Boolean functions

- Computer Science, Mathematics[1991] Proceedings of the Sixth Annual Structure in Complexity Theory Conference
- 1991

This work generalizes an existing lower bound technique and combines it with restriction arguments to provide a lower bound ofn0.51 on the number of positions that have to be evaluated by any randomized α-β pruning algorithm computing the value of any two-person zero-sum game tree withn final positions.

### On parallel evaluation of game trees

- Computer ScienceSPAA '89
- 1989

These algorithms parallelize the "left-to-right" sequential algori thm for evaluating A N D ] O R trees and the a-fl pruning procedure for evaluating MIN/MAX trees, and it is shown that, on every instance of a uniform tree, these parallel algorithms achieve a linear speed-up over their corresponding sequential algorithms.

### On Directional vs. Undirectional Randomized Decision Tree Complexity for Read-Once Formulas

- MathematicsCATS
- 2010

A systematic search for a certain class of functions is conducted and an explicit construction of a read-once Boolean formula f on n variables is provided such that the cost of the optimal directional randomized decision tree for f is Ω(nα) and the cost for the optimal randomized undirectional decision tree is O(nβ) with α -- β > 0.0101.

## References

SHOWING 1-10 OF 14 REFERENCES

### The complexity of problems on probabilistic, nondeterministic, and alternating decision trees

- Computer ScienceJACM
- 1985

This work generalizes decision trees in order to study lower bounds on the running times of algorithms that allow probabilistic, nondeterministic, or alternating control. It is shown that decision…

### Asymptotic Properties of Minimax Trees and Game-Searching Procedures

- Computer ScienceArtif. Intell.
- 1980

It is shown that a game with WIN-LOSS terminals can be solved by examining, on the average, O [(d) h 2 ] terminal positions if positions if P 0 ≠ P∗ and O [(P∗ (1 − P ∗) ) h ] positionsif P 0 = P∷, the former performance being optimal for all search algorithms.

### Nondeterministic versus probabilistic linear search algorithms

- Computer Science26th Annual Symposium on Foundations of Computer Science (sfcs 1985)
- 1985

The proof of the lower bound differs fundamentally from all known lower bounds for LSA's or PLSA's, because it does not reduce the problem to a combinatorial one but argues extensively about e.g. a non-discrete measure for similarity of sets in Rn.

### On the time required to recognize properties of graphs: a problem

- MathematicsSIGA
- 1973

In a recent paper [i], Holt and Reingold have proved the following results: any algorithm which, given an n-node graph, detects whether or not the graph enjoys property P must in the worst case probe 0(n 2) entries of the incidence matrix.

### Probabilistic computations: Toward a unified measure of complexity

- Mathematics18th Annual Symposium on Foundations of Computer Science (sfcs 1977)
- 1977

Two approaches to the study of expected running time of algoritruns lead naturally to two different definitions of intrinsic complexity of a problem, which are the distributional complexity and the randomized complexity, respectively.

### The solution for the branching factor of the alpha-beta pruning algorithm and its optimality

- Computer ScienceCACM
- 1982

When k > top > 2, one can show that the probability for a specific value of top that a[top] = stop(top) is a(top + l) /a( top) , which reduces to (k top + l ) / (n top) so it is very unlikely that the next combination is generated by using the theoretical maxim of operations.

### Optimal Search on Some Game Trees

- Computer ScienceJACM
- 1983

It is proved that the dlrecUonal algorithm for solving a game tree is optimal, in the sense of average run trine, for balanced trees (a family containing all uniform trees). This result implies that…

### A topological approach to evasiveness

- Mathematics24th Annual Symposium on Foundations of Computer Science (sfcs 1983)
- 1983

It is shown that Karp's conjecture follows from another conjecture concerning group actions on topological spaces as follows: every graph property that is monotone (preserved by addition of edges) and nontrivial has complexity θ(v2) where v is the number of vertices.