A.-C. Yao. Probabilistic computations: Toward a unified measure of complexity. In Proceedings of the 18th Annual Symposium on Foundations of Computer Science, pages 222--227, 1977.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....path coloring problem on meshes. The lower bound also applies to the load balancing problem on meshes, where we seek for minimizing the maximum number of paths crossing a link, irrespective of the color ( AAFPW93] The lower bound is based on an application of Yao s Lemma to on line algorithms [Y77] We construct a distribution over request sequences, such that the number of colors used by an optimal algorithm is always bounded by a constant, while the expected on line load (i.e. the maximum number of paths crossing an edge) of a deterministic algorithm is Omega Gamma 44 n) We recall that ....

A.C. Yao. Probabilistic Computations: Towards a Unified Measure of Complexity. In Proc. of the 17th Annual Symposium on Foundations of Computer Science, pp. 222-227, 1977. 19

....number of colors used by the algorithm and the chromatic number of the interval graph, i.e. the maximum number of intervals overlapping at the same point of the line. A lower bound for randomized algorithms against an oblivious adversary is established using the application of Yao s Lemma [Yao77] to online algorithms [BEY98,BFL96] A lower bound over the competitive ratio of randomized algorithms is obtained proving a lower bound on the competitive ratio of deterministic online algorithms for a specific probability distribution over the input sequences for the problem. We first give ....

.... , thus implying the claim of Lemma 1. 3 A lower bound for online path coloring on trees We prove that any randomized algorithm for online path coloring on trees of diameter Delta = O(log n) has competitive ratio Omega (log Delta) We establish the lower bound using Yao s Lemma [Yao77] We prove a lower bound on the competitive ratio of any deterministic algorithm for a given probability distribution on the input sequences for the problem. The tree network we use for generating the input sequence is a complete binary tree of L 4 levels. The root of the tree is at level 0, ....

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

....label is at least n=2 Gamma 1. This holds even if the out degree of nodes is bounded by some Delta 2. The proof constructs a probability distribution on request sequences that causes every deterministic labeling scheme to perform bad on this probability distribution. Then from Yao s lemma [16] it follows that this bad performance holds also for randomized labeling. The details are omitted. Furthermore notice that the Omega Gamma d log Delta) lower bound for trees with bounded depth and degree applies to randomized schemes as well. 4. LABELING WITH A CLUE We have seen that for ....

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

....one frequency with jff(G)j 2 and 1=2 p 1, algorithm p Random has (strictly) better competitive ratio than any deterministic algorithm. Obviously, if jff(G)j = 1 or jff(G)j = 0, the greedy deterministic algorithm is optimal. 3. 2 Lower bounds Using the Minimax Principle proposed by Yao [13] (see also [8] we prove two lower bounds on the competitive ratio of any randomized algorithm in cellular and arbitrary planar networks against oblivious adversaries. 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 Unified Measure of Complexity. In Proceedings of the 17th Annual Symposium on Foundations of Computer Science (FOCS '77), pp. 222--227, 1977. 22

....factor relative to the desired approximation ratio. In our proofs, we assume an oracle model of computation in which the algorithm is charged only for asking the oracle the distance between a pair of points. 42 We refer to each call to the oracle as a probe. By a generalization of Yao s technique [46] due to Mackenzie [34] we can establish a lower bound of p on the success probability of a randomized algorithm by exhibiting an input distribution for which every deterministic algorithm has a success probability of at most p. The intuition underlying this reduction is that the success ....

A. Yao. Probabilistic computations: Toward a unified measure of complexity. In Proceedings of the 18th IEEE Symposium on Foundations of Computer Science, pages 222--227, 1977.

.... following theorem of Karp et al. Theorem 3 (Karp et al. No randomized algorithm can achieve competitive ratio better than ln( 1 in the fixed 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 specific randomized adversary. In this case, Karp et al. have the adversary pick u t = y with the probability density function g(y) y 2 for y d. This ....

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

....ON correspond to deterministic algorithms. We omit the formal proof of this fact from this abstract. We can also express the on line performance of an on line algorithm with respect to an input sequence s as e(A; s) M2M;M s jM j p(s; M ) 2. 1) Yao s Lemma: The often quoted Lemma of Yao [15] relates the performance of randomized algorithms to the performance of deterministic algorithms on random inputs. In order to state Yao s lemma for our model we extend some functions defined in Section 1 on input sequences to probability distributions on input sequences. Let D be an arbitrary ....

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

....to choose a t . A black box algorithm is very powerful, since all calculations are free, only sampling is charged. However, it is limited, since the parameters of the instance are not known. In order to prove lower bounds for randomized algorithms, one can apply Yao s minimax principle [81]. If the number of instances is finite, a lower bound for the average case optimization time of deterministic algorithms with respect to a given probability distribution on the inputs is a lower bound for the worst case expected optimization time for all randomized algorithms. Droste, Jansen, ....

A. C. Yao. Probabilistics computations: Towards a unified measure of complexity. In Proc. of 17th IEEE Symp. on Foundations of Computer Science (FOCS), pages 222--227, 1977.

....any probabilistic approximation of the square grid by tree metrics. Bartal s lower bound proof used sparse graphs of logarithmic girth which are known not to have small excluded minors due to the work of Robertson and Seymour. Our lower bound proof uses the easy direction of Yao s minimax theorem [17] and bounds the average distortion of any edge of the square grid in any tree metric approximation by extending a result of Alon et al. 2] Finally, we turn to applications of probabilistic approximation by tree metrics to network design problems [3, 6, 16] In these applications, it is ....

.... 2) cannot be c probabilistically approximated by a tree for any c = o(log n) The result is an extension of a theorem of Alon, Karp, Peleg and West [2] An often used method of proving lower bounds for (either the running time or the performance guarantee of) randomized algorithms is that of Yao [17, 13], based on the minimax principle of linear programming. Consider a randomized algorithm as a distribution p on some set of deterministic algorithms A C .4. If q is any distribution on the set of inputs I C I, then (4) Eq[cost( I , A ) Ep[ cost( I , A ) Definition 4.1. Let T be a ....

A. C.-C. Yao. Probabilistic computations: Towards a unified measure of complexity. In Proceedings of the 18thAnnual IEEE Symposium on Foundations of Computer Science, pages 222-227, 1977.

....traversing graphs drawn from G(w) there exists a layered graph K of width at most w such that the ratio of the expected distance traversed by B to the length of the shortest root Gammatarget path in K is at least r w . Proof. The proof follows Yao s observation regarding the minimax principle [Yao]. Choose a randomized algorithm B and a width w. Lemma 11 implies that the expected cost incurred by every deterministic algorithm A on a graph drawn randomly from G(w) is at least r wLw . However, B is nothing more than a probability distribution on deterministic algorithms. It follows that the ....

A.C.C. Yao. Probabilistic Computations: Towards a Unified Measure of Complexity. In Proc. of the 18th IEEE Annual Symp. on Foundations of Computer Science, pages 222--227, October 1977. 15

....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) flow 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 fixed 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 unified measure of complexity. In Proc. 18th Annual Symposium on Foundations of Computer Science, pages 222--227. IEEE, 1977.

....least X #2 so that the density of 0 entries in M is at most = the constants in the big O s depending on : and = only) Proof. Given a randomized protocol, repeat it O(1) times to get the error probability lower than where 2 #( By Yao s version of the von Neuman minmax theorem [Yao77], we can find a deterministic protocol with the same parameters which errs on a fraction of at most of X Y. Let f be the function computed by this protocol. The possible histories of the communication protocol induces a partition of the matrix M f into disjoint submatrices. Consider the ....

.... . For any distribution D on X Y we will construct a deterministic t 1 round algorithm for f that errs on at most =#2 of the inputs weighted according to the distribution D. A randomized algorithm for f with error probability = follows from Yao s version of the von Neuman minmax theorem [Yao77]. Let I= 1, m] Define a distribution D onX I Y as follows: For each 1# j#m we choose (independently) x j , y j ) according to distribution D, and we choose i uniformly at random in I. We set y= y (and throw away all other y j s) Let A be a deterministic algorithm for P m ( f ) that ....

A. C. Yao, Probabilistic computations: Toward a unified measure of complexity, in Proc. 18th IEEE Symposium on Foundations of Computer Science (FOCS)," 1977, pp. 222# 227.

....in Example 6.3, there is an algorithm that makes only 2 random accesses and no sorted accesses. Therefore, the optimality ratio can be arbitrarily large. The theorem follows (in the deterministic case) For probabilistic algorithms that never make a mistake, we appeal to Yao s Minimax Principle [Yao77] see also [MR95, Section 2.2] and see [FMRW85, Lemma 4] for a simple proof) which says that the expected cost of the optimal deterministic algorithm for an arbitrary input distribution is a lower bound on the expected cost of the optimal probabilistic algorithm that never makes a mistake. ....

A. C-C. Yao. Probabilistic computations: Towards a unified measure of complexity. In Proc. 17th IEEE Symp. on Foundations of Computer Science, 1977.

....away from 1. This is a contradiction, and hence an on line algorithm A that fulfills (12) cannot exist. Theorem 5.1 For any on line algorithm A for d dimensional vector covering with d 2, the inequality RA 2= 2d 1) holds. A simple extension of the above argument combined with Yao s theorem [15] yields an analogous result for randomized on line algorithms. We omit the straightforward details. Theorem 5.2 For any randomized on line algorithm A for d dimensional vector covering with d 2, the inequality RA 2= 2d 1) holds. 6 Off line Results for Vector Covering In this section, we ....

A.C.C. Yao, Probabilistic Computations: Towards a unified measure of complexity, in Proceedings of the 18th ACM Symposium on Theory of Computing, 1977, 222-227.

....of routing in switchless optical networks benefit version, and that graph coloring is a subproblem of routing in optical switchless networks coloring version. 1. 4 Structure of the paper In Section 2 we give formal definitions of competitive analysis and explain the use of Yao s Lemma ( Y77] for lower bounds for on line randomized algorithms. Section 3 deals with the on line induced subgraph problem. In Section 3.1 we present the lower bound for randomized non preemptive algorithms for the on line induced subgraph problem. In Section 3.2 we present the lower bound for randomized ....

....oe such that OPT(oe) a, OPT(oe) ae Delta E(ON(oe) 6 The value ae is called the competitive ratio of the algorithm. 2. 1 Lower bounds using the minimax principle Lower bounds for on line randomized algorithms can be proved using Yao s Lemma based on the von Neumann minimax principle (see [Y77] applied to on line algorithms. Yao s Lemma can be stated in this context in the following way: Let f(ON; oe) be some function of the algorithm ON and the sequence oe to the reals. Let D be the set of deterministic on line algorithms, R be the set of randomized on line algorithms, S be the set ....

A.C. Yao. Probabilistic Computations: Towards a Unified Measure of Complexity. In Proc. of the 17th Annual Symposium on Foundations of Computer Science, pp. 222-227, 1977. 35

....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 [24] , 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 different 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 [24] 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 sketched in Appendix B. 4 Experimental ....

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

....points is some constant ffl 0, the competitive ratio ae(P ) is 2. Proof: By Lemma III.1 the optimal policy cannot have miss rate which is less than an AUTH source, and thus, using Lemma V.2, we get that ae(P ) 2. Let ffi 0, and we prove that ae(P ) 2 Gamma ffi. We use Yao s lemma [8], and construct a distribution on sequences of requests r 1 ; r 2 ; at times t 1 ; t 2 ; such that the miss rate of the optimal algorithm is always 1 2 , while the expected miss rate of P is 1 Gamma ffi=2. Let t 1 be chosen uniformly from an arbitrary interval of length ffl. For ....

....algorithm. Thus, when using P , r 2k Gamma1 is always a miss, while r 2k is a miss with probability at least 1 Gamma ffi, so that the expected miss rate is (2 Gamma ffi) 2. The optimal policy can rejuvenate in the times of the requests, hence get a miss rate of 1=2. Therefore, using Yao s lemma [8], ae(P ) 2 Gamma ffi, and the theorem holds. Notice that Theorem V.2 implies that rejuvenating in fixed intervals, an EXCLUSIVE v source) has a competitive ratio 2. Another related problem is the worst case performance of an INDEPENDENT v source with respect to an AUTH source. Since an ....

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

....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 suffices 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 unified measure of complexity. In Proc. 18th Symp. Foundations of Computer Science, pages 222--227, 1977.

....using Lemma 1, a lower bound on E(jD(XY; Y X)j) is obtained by setting jF 1 j = jF 2 j = 2 n Gamma1 in (13) Xi 10.3. 3 Average Case Bound for Probabilistic Algorithms The lower bound for probabilistic algorithms follows from the average case lower bound of Theorem 10.2, using a method of Yao [27]. To use Theorem 10.2, which was proved only for deterministic algorithms, a probabilistic algorithm is viewed as random choice of a deterministic algorithm. Lemma 2 Let n; m be positive integers, and let P be a probabilistic single pass differencing algorithm for n bit strings with m bits of ....

A. C. Yao. Probabilistic computation: towards a unified measure of complexity. In Proc. 18th IEEE Symposium on Foundations of Computer Science, pages 222--227, 1977.

....in its own right, and this is by now widely realized. As observed by Megiddo [Mg2] the relationship between randomized and deterministic algorithms resembles that between mixed and pure strategies, though some interesting differences are pointed out as well. An easy but important result of Yao [Yao] is based on viewing the run time of a randomized algorithm as the value of a game between the algorithm and an adversary designing a hard instance for it to run on. The min max theorem implies that it is possible to replace the analysis of a randomized algorithm running on a hard instance by a ....

A. C. Yao, Probabilistic computation: Towards a unified measure of complexity, Proc. 18th Annual IEEE Symposium on Foundations of Computer Science (1977) 222 - 227.

....Theorem 5.2 If G is a branch tree then c EE;k;k Gamma1 (G) 1 o(1) c k;k Gamma1 (G) where the o(1) term refers to a function of k. We use in the proof the next lemma. Lemma 5.3 If G is a branch tree then c k;k Gamma1 (G) k= lg k. Proof sketch: We derive a lower bound using Yao s method [33]. Given a tree, T , rooted at r 1 , the adversary generates the sequence oe by selecting at each node the left child with probability 1=2. Now, we allow the online algorithm, A, to start fetching the first k blocks at time t = 0 (rather than one block per time unit) then, A waits for k steps and ....

A. C.-C. Yao, "Probabilistic Computations: towards a unified measure of complexity", FOCS 1977.

....might ask if randomization helps the online algorithm do better. In the next theorem, we prove otherwise. Theorem 2 Any randomized non clairvoyant online scheduling algorithm for mean slowdown has Omega Gamma n) competitive ratio. Proof: To prove this theorem, we use Yao s Minimax principle [15]. This states that the expected running time of the optimal deterministic algorithm for an arbitrarily chosen input distribution is a lower bound on the expected running time of the optimal randomized algorithm. Hence we only need to prove a lower bound on the expected performance ratio of any ....

A. C-C. Yao. Probabilistic computations: Toward a unified measure of complexity (extended abstract). In IEEE Symposium on Foundations of Computer Science (FOCS), 1977. 18

....the distance from B i to B j is ffl for i 6= j. We refer to the set B 1 ; B 2 ; B 3 ; as the Bcluster. Let the initial positions of the servers be A and B 1 . We define a randomized oblivious adversary against which no deterministic algorithm is better than (1 p 2=2) competitive. By [9], this proves the lower bound. The adversary request sequence consists of a sufficiently large number of phases. At the beginning of each phase, the optimal servers are located at A and at some B k , and the algorithm s two servers are located at A and B for some . We now describe one phase. ....

A. C. C. Yao. Probabilistic computations: Towards a unified measure of complexity. In Proc. 11th Symp. Theory of Computing, 1980. 10

....at best 2 competitive. We therefore consider only the interesting situation of the oblivious adversary. In this case, lower bounds for the competitive ratio are harder to find; the first nontrivial bounds are due to Karp and Raghavan, see the remark in [12] A general technique is Yao s theorem [15]: 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 [9] In the partial cost model, a lower ....

....M 3. Let us call an algorithm satisfying (i) and (ii) regular. Given any 0 and b, we will show that there is a probability distribution on a finite set of request sequences so that X 2 ( A( OFF( b 1:6 Gamma ; 8) for any deterministic regular algorithm A. Then Yao s theorem [15] asserts that also any randomized regular algorithm has competitive ratio 1:6 Gamma or larger. This holds for any fixed p and M . Hence the competitive ratio is at least 1.6. This is achieved by COMB and therefore a tight bound for projective algorithms. The same holds for general projective ....

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

....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 ....

....ratio [8] resulting from a distribution on request sequences. This distribution is a mixed strategy of the adversary with probabilities q oe for oe in S so that for all online strategies j = 1; N X oe2S q oe ON j (oe) OFF (oe) 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 finite approximation of the list update problem. Note that it depends on the admitted length of ....

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

....between elements. Our lower bounds are based on decision trees in which each node corresponds to either a binary comparison ( between two elements, or a question on the equality ( of the elements. The Yao von Neumann Minimax principle. We recall the Yao von Neumann Minimax principle [5, 6] used in our lower bound. The expected number of comparisons of the best deterministic algorithm on some input distribution is a lower bound to the expected number of comparisons of the best randomized algorithm on an arbitrary input. Formally: minA E[C(I p ; A) max I E[C(I; A q ) where ....

....of the form A i [p] a i 0 [p 0 ] such that if all are satisfied then the intersection is restricted to the elements given in I . For instance, the k tuple (p 1 ; p k ) means that A 1 [p 1 ] A 2 [p 2 ] A k [p k ] belongs to I . And the inequality A 1 [1] A 2 [5] implies that no element of rank strictly less than 6 in set A 2 is in the intersection. One instance can have many different certificates, with very different sizes. For example, if A = f0g, B = f1; 3; 5; 7g and C = f2; 4; 6; 8g, the intersection I is empty, and a possible set of inequalities ....

[Article contains additional citation context not shown here]

A. C. Yao, Probabilistic computations: Toward a unified measure of complexity, in Proc. 18th IEEE Symposium on Foundations of Computer Science (FOCS), 1977, pp. 222-227.

....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 efficiency, various models of decision trees, including Boolean, comparison, and algebraic decision trees. Yao s Theorem [30] 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. Yao. Probabilistic computations: toward a unified measure of complexity. In Proceedings of the 18th IEEE Annual Symposium on Foundations of Computer Science (FOCS), pages 222--227, 1977.

....#(log d n # log n log log n) We remark that the limiting condition d # n log 3 n occurs in Lemma 4, and in fact any d that is o(n log 2 n) will suffice for its proof. It is plausible that the theorem holds for any d asymptotically smaller than n log n . By the von Neumann minimax principle [Yao 1977], we immediately have: COROLLARY 2.2.2. For any n node network of degree d # n log 3 n, and any ORS scheme, there is a permutation for which the expected routing time is #(log d n # log n log log n) PROOF OF THEOREM 2.2.1. The log d n term is the diameter lower bound. The second term is due ....

....cannot 733 How Much Can Hardware Help Routing decrease for the good nodes w. Thus, the probability that the good node w has congestion log n 4 log log n is at least 1 #n. e To conclude this section, we prove a high probability lower bound using a variation of the von Neumann minimax principle [Yao 1977]. THEOREM 2.2.6. For any n node network of degree d # n log 3 n, and any ORS scheme, there is a permutation for which with probability at least 1 # n #c ( for any constant c) the routing time is #(log d n # log n log log n) PROOF. Let # be the set of all ODS schemes, and let S n be the ....

YAO, A. C-C. 1977. Probabilistic computations: Towards a unified measure of complexity. In Proceedings of the 17th Annual Symposium on Foundations of Computer Science. IEEE, New York, pp. 222--227.

....in Example 4.5, there is an algorithm that makes only 2 random accesses and no sorted accesses. Therefore, the optimality ratio can be arbitrarily large. The theorem follows (in the deterministic case) For probabilistic algorithms that never make a mistake, we appeal to Yao s Minimax Principle [Yao77] see also [MR95, Section 2.2] and see [FMRW85, Lemma 4] for a simple proof) which says that the expected cost of the optimal deterministic algorithm for an arbitrary input distribution is a lower bound on the expected cost of the optimal probabilistic algorithm that never makes a mistake. 13 ....

A. C-C. Yao. Probabilistic computations: Towards a unified measure of complexity. In Proc. 17th IEEE Symp. on Foundations of Computer Science, 1977.

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A.-C. Yao. Probabilistic computations: Toward a unified measure of complexity. In Proceedings of the 18th Annual Symposium on Foundations of Computer Science, pages 222--227, 1977.

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Yao, A. (1977), Probabilistic computations: Towards a unified measure of complexity, in "Proceedings, 18th IEEE Symposium on Foundations of Computer Science," pp. 222--227. 21

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A.Yao. "Probabilistic computations: Toward a unified measure of complexity." Proceedings of the 18th IEEE FOCS, pp. 222--227, 1977. 16

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

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Yao, A. C. (1977). Probabilistic computations: Towards a unified measure of complexity. Proc. of 17th IEEE Symp. on Foundations of Computer Science (FOCS), 222--227.

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A. C. Yao. Probabilistic computations: Towards a unified measure of complexity. In Proc. of the 1977.

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A. C. Yao. Probabilistic Computation : Towards a unified measure of complexity, In Proc. 18th FOCS, pages 222-227, 1997

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

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

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Yao, A. C. (1977). Probabilistic computations: Towards a unified measure of complexity. Proc. of 17th IEEE Symp. on Foundations of Computer Science (FOCS), 222--227.

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A. C. Yao. Probabilistic computations: Toward a unified measure of complexity. In Proceedings of the 18th Annual Symposium on Foundations of Computer Science, pages 222--227, 1977. 13

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A. C-C. Yao. Probabilistic computations: Toward a unified measure of complexity. In Proceedings of 18th IEEE FOCS, pages 222--227, 1977. 15

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A. Yao. Probabilistic computations: Toward a unified measure of complexity. In Proceedings of 18th IEEE Symposium on Foundations of Computer Science, pages 222--227, 1977.

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A. C. Yao. Probabilistic computations: Toward a unified measure of complexity. In Proceedings of the 18th Symposium on Foundations of Computer Science (FOCS'77), pages 222--227, 1977. A Calculations for the Proof of Theorem 10 To show the desired result, we split the analysis into three cases, based on the the relative values of #, i and i + j. We use the following two facts:

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Yao, A.C. (1977). Probabilistic computations: Towards a unified measure of complexity. Proc. of 17th IEEE Symp. on Foundations of Computer Science (FOCS), 222--227. 27

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A. C. Yao. Probabilistic computations: Towards a unified measure of complexity. In Proc. 18th Annu. IEEE Symp. Found. Comp. Sci., pages 222--227, 1977.

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A. Yao. Probabilistic computations: Towards a unified measure of complexity. In Proceedings of the 18th Annual Symposium on Foundations of Computer Science, pages 222--227, 1977.

No context found.

A.C. Yao. Probabilistic computations: Towards a unified measure of complexity. In Proc. 12th ACM Symposium on Theory of Computing, 1980. 51

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Yao, A.C. (1977). Probabilistic computations: Towards a unified measure of complexity. Proc. of 17th IEEE Symp. on Foundations of Computer Science (FOCS), 222--227.

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A.C. Yao. Probabilistic computations: Toward a unified measure of complexity. In Proc. 18th IEEE Symposium on Foundations of Computer Science (FOCS) (1977) 222--227.

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A. C. Yao. Probabilistic computations: Towards a unified measure of complexity. In Proc. 12th ACM Symposium on Theory of Computing, 1980.

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