A. C. Yao. Probabilistic computations: Towards a unied measure of complexity. In Proceedings of the 17th Annual Symposium on Foundations of Computer Science, pages 222227, 1977.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... the following theorem of Karp et al. Theorem 3 (Karp et al. No randomized algorithm can achieve competitive ratio better than ln( c ) 1 in the xed range model with range [c; d] For completeness, we restate the proof of Theorem 3 from [6] Proof (Karp et al. Apply Yao s minimax technique [10]. Instead of trying to prove a lower bound for randomized algorithms against an adversary, we can consider deterministic algorithms against a speci c randomized adversary. In this case, Karp et al. have the adversary pick u t = y with the probability density function g(y) 2 for y d. This ....

A.C.C. Yao. Probabilistic computations: Towards a uni ed measure of complexity. In 18th Symposium on Foundations of Computer Science, pages 222-227, 1977.

....ratio for this problem. Moreover, we show that the algorithm we give in Section 2. 2 is optimal in the sense that no randomized algorithm can achieve a competitive ratio of o(log B) To do this we derive lower bounds, based on novel applications of Yao s randomness shifting technique [93] , that show the competitive ratios for our algorithm is worst case optimal. In order to show that our algorithm performs well in practice we undertook a number of experiments. The results, detailed in Section 2.4, demonstrate that our algorithm handles di erent types of input sequences with ease ....

....to any sequence that is either of length n or n 1. This bleak outlook for deterministic algorithms is not improved much by knowing the number of bids to expect, however, as shown in Theorem 2.3.2. Furthermore we show that even randomization does not help us too much. We can use Yao s principle [93] to show that no randomized algorithm can be more competitive against an oblivious adversary than Price and Pack. Theorem 2.3.3 Any randomized algorithm for the single item B bounded online auctioning problem is 24 B) competitive in the worst case. Proof. We use Yao s Principle [93, 61] to ....

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A. C.-C. Yao. Probabilistic computations: Towards a uni ed measure of complexity. In Proceedings of the 17th Annual Symposium on Foundations of Computer Science, pages 222-227, 1977.

....instances of size k and k 1 untill all agents have been found. This proves that a non adaptive algorithm will make at least k 1 calls on an instance of size k 1. ut We now proceed to prove a lower bound on linear mutual search algorithms. In the proof, we use the following result of Yao [Yao77] Theorem 3.6 ( Yao77] Let t be the expected running time of a randomized algorithm solving problem P over all possible inputs, where the expected time is taken over the random choice made by the algorithm. Let t be the average running time over the distribution of all inputs, minimized ....

....k and k 1 untill all agents have been found. This proves that a non adaptive algorithm will make at least k 1 calls on an instance of size k 1. ut We now proceed to prove a lower bound on linear mutual search algorithms. In the proof, we use the following result of Yao [Yao77] Theorem 3. 6 ( Yao77] Let t be the expected running time of a randomized algorithm solving problem P over all possible inputs, where the expected time is taken over the random choice made by the algorithm. Let t be the average running time over the distribution of all inputs, minimized over all determinisitc ....

Yao, A. Probabilistic computations: toward a uni ed measure of complexity. In 18th FOCS (Long Beach, CA, USA, 1977), IEEE Comp. Soc. Press, pp. 222-227.

.... theorem of Karp et al. Theorem 3 (Karp et al. No randomized algorithm can achieve competitive ratio better than ln( d c ) 1 in the xed range model with range [c; d] For completeness, we restate the proof of Theorem 3 from [7] 4 Proof (Karp et al. Apply Yao s minimax technique [12]. Instead of trying to prove a lower bound for randomized algorithms against an adversary, we can consider deterministic algorithms against a speci c randomized adversary. In this case, Karp et al. have the adversary pick b i = y with the probability density function g(y) c y 2 for y d. ....

A.C.C. Yao. Probabilistic computations: Towards a unied measure of complexity. In 18th Symposium on Foundations of Computer Science, pages 222-227, 1977. 9

....[KN97] for de nitions) We denote the one round randomized communication complexity of a function f : X Y Z with error by R 1 (f ) For a distribution over the inputs of f , we denote by 13 D 1; f) the one round ( distributional communication complexity of f . Yao s theorem [Yao77] implies that R 1 (f) max D 1; f ) The family G we de ne is indexed by (S 1 ; S n ; i; j) where S 1 ; S n are n subsets of [n] of size t = n=10 and i; j 2 [n] G S1 ; S n ;i;j is an undirected graph (U V W;E) where U = fu 1 ; u n g, V = fv 1 ; v ....

A.C-C. Yao. Probabilistic computations: toward a unied measure of complexity. In Proceedings of the 18th IEEE Annual Symposium on Foundations of C mputer Science (FOCS), pages 222-227, 1977. 17

....If the adversary can observe those choices, it can generate requests as if the algorithm was deterministic, which is then at best 2 competitive. We therefore consider only the interesting situation of the oblivious adversary. Lower bounds for the competitive ratio can be proved using Yao s theorem [18]: If there is a probability distribution on request sequences so that the resulting expected competitive ratio for any deterministic online algorithm is d or higher, then every deterministic or randomized online algorithm has competitive ratio d or higher [8] Teia [16] described a simple ....

....competitive ratio [8] resulting from a distribution on request sequences. This distribution is a mixed strategy of the adversary with probabilities q for in S so that for all online strategies j = 1; N X 2S q ON j ( OFF ( d : 9) The minimax theorem for zero sum games [18] asserts that there are mixed strategies for both players and reals c and d so that (8) and (9) hold with d = c. Then c is the value of the game and the optimal strict competitive ratio for the chosen nite approximation of the list update problem. Note that it depends on the admitted length of ....

A. C. Yao (1977), Probabilistic computations: Towards a unied measure of complexity, Proc. 18th FOCS, 222-227. 14

....codewords are of Hamming distance n=2. Thus, in this case the second part of the tester succeeds with high probability in nding y 0 and rejects because y 0 62 A. We note also that this algorithm has one sided error. 2 Proof of Lemma 3.4. The lower bound makes use of the Yao principle [Yao77] let D be an arbitrary probability distribution on positive and negative inputs (i.e. inputs that either belong to PA or are n far from PA ) Then if every deterministic algorithm that makes at most q queries, errs with probability at least 1=8 (with respect to input chosen according to D, ....

....for L must make p N) queries (even when allowing twosided error. Theorem 4.3 There is a quantum property tester for L making O(log N log log N) queries. Moreover, this quantum property tester makes all its queries nonadaptively. Proof of Theorem 4.2. We again apply the Yao principle [Yao77] as in the proof of Lemma 3.4: we construct two distributions, P and U , on positive and negative (at least N=8 far) inputs, respectively, such that any deterministic (adaptive) decision tree T has error 1=2 o(1) when trying to distinguish whether an input is chosen from U or P . Indeed, we will ....

A. C-C. Yao. Probabilistic computations: Toward a unied measure of complexity. In Proceedings of 18th IEEE FOCS, pages 222-227, 1977. A Appendix Proof of Lemma 4.10. We follow the steps of subroutine Q when passed k linearly independent vectors z 1 ; : : : ; z k so that all i j := minfi : z j [i] = 1g are distinct for 1  j  k.

....case we show that the relation between models holds also for d = 2 by giving an algorithm of ratio larger than 0:4. All upper bounds (negative results) in this paper hold both for deterministic and for randomized algorithms. In order to prove upper bounds we use an adaptation of Yao s theorem [12] which states that an upper bound of a deterministic algorithm against a xed distribution on the input is also an upper bound for randomized algorithms. For maximization problems the bound is given by E (V on =V opt ) or by E(V on ) E(V opt ) 4, 5] All upper bounds are proved for algorithms ....

A.C.C. Yao. Probabilistic computations: Towards a unied measure of complexity. In Proceedings of the 18th ACM Symposium on Theory of Computing, pages 222{ 227, 1977.

....methods, that are used in more than one proof, in Section 2. Section 3 contains our results on total (weighted) completion time, and Section 4 discusses the total (weighted) ow time measure. 2 Methods To prove lower bounds for randomized algorithms we use the adaptation of Yao s theorem [17]. It states that a lower bound for the competitive ratio of deterministic algorithms on a xed distribution on the input is also a lower bound for randomized algorithms and is given by E(TON =TOPT ) where TON is the cost of the on line algorithm (see [2] A useful method for weighted problems ....

....whether it restarts or not on arrival of the last 2 jobs of size 0. Theorem 2 Any randomized algorithm for minimizing the total completion time on a single machine which is allowed to restart jobs, has a competitive ratio of at least R 3 = 114=103 1:1068. Proof. We use Yao s minimax principle [17] and consider a randomized adversary against a deterministic algorithm. Assume there exists an on line algorithm A with a competitive ratio of R 3 . At time 0, a job of size 1 arrives. A will certainly start this job immediately since it is allowed to restart. At time 1=3, two jobs of size 0 ....

A. C. Yao. Probabilistic computations: Towards a unied measure of complexity. In Proc. 18th Annual Symposium on Foundations of Computer Science, pages 222-227. IEEE, 1977. 11

....of T , S e (T ) and S w (T ) are the maximum of S e (T ; x) and S w (T ; x) respectively, over all inputs x 2 A n . Notice that this model can simulate, with the same eciency, various models of decision trees, including Boolean, comparison, and algebraic decision trees. Yao s Theorem [Yao77] gives an equivalent characterization of a randomized decision tree as a distribution over deterministic decision trees. The expected query complexity of the tree on input x is the expected length (over ) of the paths corresponding to x in these trees (and similarly for the worst case query ....

A.C-C. Yao. Probabilistic computations: toward a unied measure of complexity. In Proceedings of the 18th IEEE Annual Symposium on Foundations of Computer Science (FOCS), pages 222-227, 1977.

....competitive ratio for this problem. Moreover, we show that the algorithm we give in Section 2 is optimal in the sense that no randomized algorithm can achieve a competitive ratio of o(log B) To do this we derive lower bounds, based on novel applications of Yao s randomness shifting technique [20] , that show the competitive ratios for our algorithm is worst case optimal. In order to show that our algorithm performs well in practice we undertook a number of experiments. The results, detailed in Section 4, demonstrate that our algorithm handles di erent types of input sequences with ease ....

....to any sequence that is either of length n or n 1. This bleak outlook for deterministic algorithm is not improved much by knowing the number of bids to expect, however, as shown in Theorem 3. Furthermore we show that even randomization does not help us too much. We can use Yao s principle [20] to show that no randomized algorithm can be more competitive against an oblivious adversary than Sell One. Theorem 4. Any randomized algorithm for the single item B bounded online auctioning problem is (log B) competitive in the worst case. The proof is in Appendix B. 2 Bids are not always ....

A. C-C. Yao. Probabilistic computations: Towards a unied measure of complexity. In Proceedings of the 17th Annual Symposium on Foundations of Computer Science, pages 222-227, 1977. A Proof of Theorem 1 Let the optimal oine algorithm OPT achieve prot density p on a given input sequence I .

....can dynamically adjust its service rate (allocated bandwidth) according to queue size. Our main goal is to understand the limitations of such adaptive schemes, i.e. to provide lower bounds for a whole class of allocation algorithms. To do this, we carefully employ Yao s min max principle (see e.g. [9, 10]) which relates lower bounds of any (randomized) scheme for worst case inputs with (lower bounds to) the performance of an optimal deterministic algorithm for suitably chosen random inputs. We chose the Poisson stream as the random input (whose rate, will be suitably selected to give the worst ....

A. Yao, \Probabilistic Computations: Towards a Unied Measure of Complexity", in Proc. of the 17th Symp. on Foundations of Computer Science (FOCS), pp. 222-227, 1977. 10

....supports one frequency with (G) 2 and 1=2 p 1, the algorithm p random has (strictly) better competitive ratio than any deterministic algorithm. Obviously, if (G) 1 or (G) 0, the greedy deterministic algorithm is optimal. 3. 2 Lower bounds Using the minimax principle proposed by Yao [12] (see also [8] we prove two lower bounds on the competitive ratio against oblivious adversaries of any randomized algorithm on cellular and arbitrary planar networks. We consider networks that support one frequency; our lower bounds can be trivially extended for networks that support multiple ....

A. C. Yao. Probabilistic Computations: Towards a Unied Measure of Complexity. In Proceedings of the 17th Annual Symposium on Foundations of Computer Science (FOCS '77),pp. 222-227, 1977.

....0 , choosing N large enough, and setting N 0 = 2N 2k 3, the result follows. 25 2.3 Randomized Lower Bounds Theorem 2.6. For the total distance cost function, no randomized on line algorithm R has a competitive ratio lower than 1 (lg lg k) on the K k 1 metric space. Proof. We use Yao s [32] method of deriving a lower bound on the performance of any randomized algorithm on a worst case input instance by deriving a lower bound on the expected performance of any deterministic algorithm against a speci c probability distribution of input instances. Let the input distribution be the ....

A.C.-C. Yao. Probabilistic computations: Towards a unied measure of complexity. In Proc. of 18th IEEE Symp. on Foundations of Computer Science, pages 222-227, 1977. A Appendix Denition A.1. B(n) is the set of non-negative integers such that

....r (it is easy to check that the probabilities sum to 1) When the adversary selects u t = y, the online gain is equal to a 1 r R y a f(x)x dx = y=r. The optimal gain is y and the competitive ratio is r, independently of the choice of y. To show that this is optimal we employ Yao s Lemma [5] (the classical minimax theorem of Game Theory adapted to on line algorithms) It suces to consider a randomized adversary against deterministic on line algorithms. In particular, let the adversary select y with probability density function g(y) a=y 2 ; in a similar manner with the upper ....

A. C. C. Yao. Probabilistic computations: Towards a unied measure of complexity. In Proc. 18th Symp. Foundations of Computer Science, pages 222-227, 1977.

....to the value of the game. More formally, max i iMy = x My = min j x Mj ; 5) where i and j range over all pure strategies for player 1 and player 2, respectively. This result is known as Loomis s lemma [9] This implies a simple but very useful conclusion: Yao s principle [19]. For any mixed strategy x for player 1 min j xM j x My = max i iMy : 6) Equation (6) is not especially surprising from a game theoretic perspective, but when one thinks of the game modeling an online problem it leads to a surprising conclusion. Recall that, the row player is ....

A. C. Yao. Probabilistic computations: Towards a unied measure of complexity. In Proceedings of the 18th Annual Symposium on Foundations of Computer Science (FOCS'77), pages 222-227, 1977.

....possible: We show that no randomized algorithm can be better than max 0 w 1 4e w 1 e 4e w 1 2e w 2ew 1:75037 (1) 5 s t v u 1 1 1 1 Figure 2: The ring C 4 . competitive, even on a 4 node ring. The construction makes use of Yao s corollary to the von Neumann minimax principle [24, 25]. This principle states that, for a given problem, one can show a lower bound cost of c for any randomized algorithm by showing a distribution over inputs such that the expected cost to any deterministic algorithm is c. We use the graph C 4 illustrated in Figure 2 and a distribution over request ....

A. C. Yao. Probabilistic computations: Toward a unied measure of complexity. In Proceedings of the 18th Symposium on Foundations of Computer Science (FOCS'77), pages 222-227, 1977. 10

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A. C. Yao. Probabilistic computations: Towards a unied measure of complexity. In Proceedings of the 17th Annual Symposium on Foundations of Computer Science, pages 222227, 1977.

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A. C. C. Yao. Probabilistic computations: Towards a uni ed measure of complexity. In Proc. 18th Symp. Foundations of Computer Science, pages 222-227, 1977. 11 A: Appendix A1. Optimality We de ne: maxcost c (i; j) = minA

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A. C. Yao. Probabilistic computations: Toward a uni ed measure of complexity. In 18th IEEE Symposium on Foundations of Computer Science, FOCS'77, pp. 222-227. 11 Output 1 Output n Crossbar Switch Scheduler Q 1,1 Q 1,n Q m,1 Q m,n Input 1 Input m

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A.C.-C. Yao. Probabilistic computations: Towards a uni ed measure of complexity. In Proc. of 18th IEEE Symp. on Foundations of Computer Science, pages 222-227, 1977. A Appendix De nition A.1. B(n) is the set of non-negative integers such that

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A.C.-C. Yao. Probabilistic computations: Towards a uni ed measure of complexity. Proc. 18th Annual Symposium on Foundations of Computer Science, 222{ 227, 1977.

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A. C-C. Yao. Probabilistic computations: Toward a uni ed measure of complexity. In Proceedings of 18th IEEE FOCS, pages 222-227, 1977.

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