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Randomization and the Computational Power of Analytic and Algebraic Decision Trees
, 1997
"... We introduce a new powerful method for proving lower bounds on randomized and deterministic analytic decision trees, and give direct applications of our results towards some concrete geometric problems. We design also randomized algebraic decision trees for recognizing the positive octant in R n ..."
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Cited by 9 (7 self)
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We introduce a new powerful method for proving lower bounds on randomized and deterministic analytic decision trees, and give direct applications of our results towards some concrete geometric problems. We design also randomized algebraic decision trees for recognizing the positive octant in R n or computing MAX in R n+1 in depth log 0(1) n. Both problems are known to have linear lower lower bounds for the depth of any deterministic analytic decision tree recognizing them. The main new (and unifying) proof idea of the paper is in the reduction technique of the signs of testing functions in a decision tree to the signs of their leading terms at the specially chosen points. This allows us to reduce the complexity of a decision tree to the complexity of a certain boolean circuit.
Complexity Lower Bounds for Approximation Algebraic Computation Trees
, 1999
"... We prove lower bounds for approximate computations of piecewise polynomial functions which, in particular, apply for roundoff computations of such functions. The goal of this paper is to prove lower bounds for approximated computations. As it is customary for lower bounds, we consider some form of ..."
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Cited by 5 (0 self)
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We prove lower bounds for approximate computations of piecewise polynomial functions which, in particular, apply for roundoff computations of such functions. The goal of this paper is to prove lower bounds for approximated computations. As it is customary for lower bounds, we consider some form of algebraic tree as our computational model (cf. [Burgisser, Clausen, and Shokrollahi 1996] or [Blum, Cucker, Shub, and Smale 1998] for algebraic trees). But, unlike the usual proofs of lower bounds, which deal with decision problems, we will consider computations of real functions. That is, we consider trees computing functions f : IR n ! IR and, also unlike the usual results on lower bounds, we will allow for approximate computations. To understand the nature of our results let us look first at an example. Example 1 Given a compact polygon P ae IR 2 consider the function f : IR 2 ! IR defined by f(c) = max x2P hc; xi 2 : Obviously, there is a partition of IR 2 into a finite n...
Randomized OBDDs and the Model Checking
 Proc. Probabilistic Methods in Verification, PROBMIV'98
, 1998
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Randomized Ω(n²) Lower Bound For Knapsack
, 1996
"... We prove Ω(n²) complexity lower bound for the general model of randomized computation trees solving the Knapsack Problem, and more generally Restricted Integer Programming. This is the first nontrivial lower bound proven for this model of computation. The method of the proof depends crucially on t ..."
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We prove Ω(n²) complexity lower bound for the general model of randomized computation trees solving the Knapsack Problem, and more generally Restricted Integer Programming. This is the first nontrivial lower bound proven for this model of computation. The method of the proof depends crucially on the new technique for proving lower bounds on the border complexity of a polynomial which could be of independent interest.
Randomized Complexity of Linear Arrangements and Polyhedra
, 1999
"... We survey some of the recent results on the complexity of recognizing ndimensional linear arrangements and convex polyhedra by randomized algebraic decision trees. We give also a number of concrete applications of these results. In particular, we derive first nontrivial, in fact quadratic, randomi ..."
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We survey some of the recent results on the complexity of recognizing ndimensional linear arrangements and convex polyhedra by randomized algebraic decision trees. We give also a number of concrete applications of these results. In particular, we derive first nontrivial, in fact quadratic, randomized lower bounds on the problems like Knapsack and Bounded Integer Programming. We formulate further several open problems and possible directions for future research. 1 Introduction. Linear search algorithms, algebraic decision trees, and computation trees were introduced early to simulate random access machines (RAM) model. They are also a very useful and simplified abstraction of various other RAMrelated computations cf. [AHU74], [DL78], [Y81], [SP82], [M84], [M85a], [KM90], and a useful tool in computational geometry. The same applies for the randomized models of computation. Starting with the papers of Manber and Tompa [MT85], Snir [S85], Meyer auf der Heide [M85a], [M85c] there wa...
On the Decisional Complexity of Problems Over the Reals
"... We consider the role of randomness for the decisional complexity in algebraic decision (or computation) trees, i.e. the number of comparisons ignoring all other computation. Recently Ting and Yao showed that the problem of finding the maximum of n elements has decisional complexity O(log 2 n). In ..."
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We consider the role of randomness for the decisional complexity in algebraic decision (or computation) trees, i.e. the number of comparisons ignoring all other computation. Recently Ting and Yao showed that the problem of finding the maximum of n elements has decisional complexity O(log 2 n). In contrast, Rabin showed in 1972 an W(n) bound for the deterministic case. We point out that their technique is applicable to several problems for which corresponding W(n) lower bounds hold. We show that in general the randomized decisional complexity is logarithmic in the size of the decision tree. We then turn to the question of the number of random bits needed to obtain the Ting and Yao result. We provide deterministic algorithms for finding the k largest elements, given the k + 1th element that have complexity O(k 2 log n) (constructive) and O(k log n) (nonconstructive). We use them to obtain an O(log 2 n) random bits and O(log 2 n) queries algorithm for finding the maximum. 1 I...