Citations: Fast probabilistic algorithms for hamiltonian circuits and matchings

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D. Angluin and L. Valiant. Fast probabilistic algorithms for hamiltonian circuits and matchings. J. Comput. Syst. Sci., 18:155--193, 1979.

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On The Probable Performance Of Heuristics - For Bandwidth Minimization (Correct)

....1 and V 2 . As we shall see, these cases differ from the rest and will be handled in Lemma 3.5. First however, we need a proposition concerning the binomial distribution, B(n; p) By definition if x 2 B(n; p) then P(x = k) n Gammak . The following proposition is from Angluin and Valiant [1]. Proposition 3.1: If x 2 B(n; p) then for any ffl, 0 ffl 1; P(x (1 Gamma ffl)np) e and P(x (1 ffl)np) e . Lemma 3.5: Let ffl 0; 0 p 1 be fixed, c = Gamma(1 ffl) ln(1 Gamma p ) ff = c ln n n. For almost all G 2 Psi n ( p) 1 Gamma ffl)p jV 1 j ....

Angluin, D., L. G. Valiant. "Fast Probabilistic Algorithms for Hamiltonian Circuits and Matchings". In Journal of Computer and System Sciences 18 , 155-193, 1979.

Randomized Selection in n + C + o(n) Comparisons - Gerbessiotis, Siniolakis (Correct)

....of x in this expression is the following one. # #1 1 i=0 # # i=2c#s . The sum in the term above gives the probability of having at least 2c#s t 2 successes in B 1 independent Bernoulli trials with probability of success q. Cherno# s upper bound [1] on the probability of having at least (1 #) B 1)q successes in such an experiment gives an upper bound of e (# (B 1)q) 3 . We set 2c#s t 2 = 1 #) B 1)q in this bound and get e (2c#s t 2 (B 1)q) We choose f(N) so that f(N) N t) 2 #s) A lower bound for 2c#s (B 1)q and an ....

D. Angluin and L. G. Valiant. Fast probabilistic algorithms for hamiltonian circuits and matchings. Journal of Computer and System Sciences, 18:155-193, 1979.

Space-Efficient Routing Tables For Almost All - Networks And The (Correct)

....e 2k 3(n 1) 2.2) possibilities. The last inequality follows from a general estimate of the tail probability of the binomial distribution with s n the number of successful outcomes in n experiments with probability of success p = Namely, by Cherno# s bounds, in the form used in [1, 9], Pr( s n pn k) 2e k 3pn . 2.3) To describe G, it then su#ces to modify the old code of G by prefixing it with (i) a description of this discussion in O(1) bits; ii) the identity of node i in 1)# bits; iii) the value of k in 1)# bits, possibly adding nonsignificant ....

L.G. Valiant and D. Angluin, Fast probabilistic algorithms for Hamiltonian circuits and matchings, J. Comput. System Sci., 18 (1979), pp. 155--193.

Noise Tolerant Algorithms for Learning and Searching - Aslam (1995) (3 citations) (Correct)

....be the probability of at least n successes in m Bernoulli trials, where each trial has probability of success p. Similarly, let LE(p, m, n) be the probability of at most n successes in m Bernoulli trials, where each trial has probability of success p. Chernoffs bounds may then be stated as follows [3]: GE(p, m, mp(1 ( e mpa2 3 LE(p,m, mp(1 ( e mpa2 2 Furthermore, we often make use of the following properties of GE and LE: p p LE(p,m,n) LE(p ,m,n) p p GE(p,m,n) GE(p ,m,n) 4.2) 4.3) We may now prove following theorem. from query space Q with worst case relative ....

Dana Angluin and Leslie G. Valiant. Fast probabilistic algorithms for Hamiltonian circuits and matchings. Journal of Computer and System Sciences, 18(2):155-193, April 1979.

Improved Sampling with Applications to Dynamic Graph Algorithms. - Henzinger, Thorup (1996) (5 citations) (Correct)

.... Gamma 1= 4r) 4r ln ln s Gamma4r ln ln s= 4r) 1= ln s: Case 2: jRj=jSj 1= 4r) The algorithm did return an element in Step A.6, so jR 2 j 4 ln s. However, the expected value of jR 2 j is at most 2 ln s. Note that p = P r(jR 2 j (1 ffi ) with ffi = 1: Using a Chernoff bound from [2,9], P r(jR 2 j (1 ffi ) e Gamma2 ln s=3 Gamma2(e ln ln s) 3 1= ln s: Above, it was used that x= ln x e for any real x 0. We are now ready to show that Algorithm B satisfies the conditions of Lemma 1. i) First we find the expected number of samples if the algorithm ....

....we are returning in Step B.2.3. Then jR i j is at least x = n i . However, the expected value of jR i j is at most n i (1 Gamma n i Gamma1 =2:1) n i (1 Gamma 1= 2:1 lnn i ) Note that p i P r(jR i Gamma1 j (1 ffi ) with ffi = x Gamma ) Using the Chernoff bound (according to [2,9]) P r(jR i Gamma1 j (1 ffi ) e Gamma(x Gamma) 3) For n i (1 Gamma 1= 2:1 lnn i ) this function is maximized for = n i (1 Gamma 1= 2:1 lnn i ) Thus, Gamma(n i = 2:1 ln n i ) 3n i (1 Gamma 1= 2:1 lnn i ) Gamman i 13(ln n i ) 2 ) n For the last ....

D. Angluin, L. G. Valiant. Fast probabilistic algorithms for Hamiltonian circuits and matchings. J. Comput. System Sci. 18 (2), 1979, 155--193.

The efficiency of resolution and Davis-Putnam procedures - Beame, Karp, Pitassi (1999) (8 citations) (Correct)

.... are monotone (or anti monotone) with respect to sets of clauses, the almost certain properties under both distributions are the same up to a change from m to m o(m) This is just a natural extension of the similar (and more precise) equivalences for the random graph model as shown for example in [AV79]. We generally assume the distribution F m . We write F F to mean F is a random formula selected according to distribution F . We make frequent use of two well known tail bounds for the binomial distribution (see [ASE92] Appendix A) Proposition 6: If Y is a random variable distributed ....

D. Angluin and L. Valiant. Fast probabilistic algorithms for Hamiltonian circuits and matchings. Journal of Computer and System Sciences, 18:155--193, 1979.

Selection on the Bulk-Synchronous Parallel Model with.. - Gerbessiotis, Siniolakis (1996) (12 citations) (Correct)

.... The random distribution of items to processors (line 8) requires at most max fL; grg time with probability 1 Gamma O(n ) for any constant ae 0, where r = d(1 ) n=p)e, and = O( 1= n ) This claim can be derived by way of Chernoff bounds on the right tail of the binomial distribution [1]. For the remainder of this proof, we therefore assume that each processor has at most r = d(1 ) n=p)e items from any node in Q. The median selection process (line 10) takes at most T sel (2pr; n; p) time. We note that we use the expression 2pr to compensate for the uneven distribution of items ....

D. Angluin and L. G. Valiant. Fast probabilistic algorithms for Hamiltonian circuits and matchings. Journal of Computer and System Sciences, 18:155-193, 1979.

Concurrent Heaps on the BSP Model - Gerbessiotis, Siniolakis (1996) (Correct)

....q(1 Gamma x) The coefficient of x in this expression is the following one. The sum in the term above gives the probability of having at least 2cs Gamma t=2 successes in B 1 independent Bernoulli trials with probability of success q. Chernoff s upper bound [1] on the probability of having at least (1 fi) B 1)q successes in such an experiment gives an upper bound of e Gamma(fi (B 1)q) 3 . We set 2cs Gamma t=2 = 1 fi) B 1)q in this bound and get e . We choose the function f(N) so that f(N) N t) 2 s) A lower bound for 2cs Gamma ....

....1: In step 1, at most max fL; d2=peg time is spent on each processor for random processor identifier generation. We distinguish four cases in the analysis of step 2. Case 1: If d2=pe 3ae log n and since processor identifiers are selected uniformly at random, then the the probability bound in [1], Prob[S n;P (1 ) nP ] e , 0 1, where S n;P is the number of successes in n independent Bernoulli trials with individual probability of success P , for n = 2 Gamma 1, P = 1=p and = 1 Gamma o(1) shows that with probability 1 Gamma O(n ) for any ae 1, no processor gets ....

[Article contains additional citation context not shown here]

D. Angluin and L. G. Valiant. Fast probabilistic algorithms for Hamiltonian circuits and matchings. Journal of Computer and System Sciences, 18:155-193, 1979.

Noise Tolerant Algortihms for Learning and Searching - Aslam (1995) (Correct)

....the probability of at least n successes in m Bernoulli trials, where each trial has probability of success p. Similarly, let LE(p; m; n) be the probability of at most n successes in m Bernoulli trials, where each trial has probability of success p. Chernoff s bounds may then be stated as follows [3]: GE(p; m;mp(1 ff) e LE(p; m; mp(1 Gamma ff) e Furthermore, we often make use of the following properties of GE and LE: LE(p; m; n) LE(p = GE(p; m; n) GE(p We may now prove following theorem. Theorem 10 If F is learnable by a statistical query algorithm which makes ....

Dana Angluin and Leslie G. Valiant. Fast probabilistic algorithms for Hamiltonian circuits and matchings. Journal of Computer and System Sciences, 18(2):155--193, April 1979.

Approximation Algorithms Via Randomized Rounding: A Survey - Srinivasan (Correct)

.... (7) For ffi 2 [0; 1] Hoeffding s approach also gives Pr[X (1 Gamma ffi) Pr[n Gamma X n Gamma (1 Gamma ffi) F (n; Gammaffi ) 1 Gamma ffi (n Gamma(1 Gammaffi) n Gamma(1 Gammaffi) 1 Gamma ffi) 1 Gammaffi) 8) The following related results are useful (see, e.g. [33, 5, 4, 56]) If ffi 0, Pr[X (1 ffi) Q i2[n] E[ 1 ffi) X i ] 1 ffi) 1 ffi) G( ffi) e ffi (1 ffi) 1 ffi) 9) If ffi 2 [0; 1] Pr[X (1 Gamma ffi) Q i2[n] E[ 1 Gamma ffi) X i ] 1 Gamma ffi) 1 Gammaffi) H( ffi) e Gammaffi 2 =2 : 10) It can ....

D. Angluin and L.G. Valiant. Fast probabilistic algorithms for Hamiltonian circuits and matchings. Journal of Computer and System Sciences, 18:155--193, 1979.

On Restricted-Focus-of-Attention Learnability of Boolean Functions - Birkendorf (1998) (6 citations) (Correct)

....M( t) d max(3;2 2 ) 1) 2 log t e. For a sample of size m M( jAj) the following holds with con dence 1 : For every A 2 A, p(A) p(A) p(A) p(A) p(A) p(A) 2 Proof: For a sample of size m, and for every A 2 A, the Cherno bounds imply [2]: Pr[ p(A) p(A) e mp(A) 1) 2 =3 Pr[ p(A) p(A) e mp(A) 1) 2 = 2 2 ) If we upper bound all inequalities (for every A 2 A) by jAj , then all the estimates are within of their true values. Solving for m yields m = M( jAj) 4.1. n 1) RFA learnability of (n ....

Angluin, D. and Valiant, L. G. (1979). Fast probabilistic algorithms for hamiltonian circuits and matchings. Journal of Computer Systems and Sciences, 18:155-193.

Specification and Simulation of Statistical Query Algorithms.. - Aslam, Decatur (1995) (16 citations) (Correct)

....sample which satisfy , and P fi to be the fraction of examples in the corrupted sample which satisfy . The sample size m is chosen sufficiently large to ensure, with high probability, that fi 2fi and that for all 2 Q, d =8 ( P ; P ) 8. Applying standard Chernoff bounds [1], the former condition can be guaranteed with probability at least 1 Gamma ffi=2 using a sample of size 72 ln 2 ffi . Applying Theorem 2, the latter condition can be guaranteed with probability at least 1 Gamma ffi=2 using a sample of size 512 2 ln 4jQj ffi if Q is ....

Dana Angluin and Leslie G. Valiant. Fast probabilistic algorithms for Hamiltonian circuits and matchings. Journal of Computer and System Sciences, 18(2):155--193, April 1979.

On Key Storage In Secure Networks - Dyer, Fenner, Frieze (1995) (22 citations) (Correct)

....it is sucient to ensure that, for any set C, jC A Bj jA Bj=w: 5) Clearly this implies (4) for any collection C 1 ; Cw (i = 1; w) We bound the probability of (5) as follows, using a version of Cherno s bound for the tails of the binomial distribution. See, for example, [1]. Pr(jA Bj 0:68p 2 k) exp( 1 2 (0:32) 2 (0:2) 2 500w ln n) n 2:048 (w 2) Hence, Pr(Any such A; B exist) n 2 n 2:048 1 2 : 6) 14 Now, for xed C, Pr(jA B Cj 0:68p 2 k=w) Pr(jA B Cj 3:4p 3 k) e 3:4 3:4p 3 k (7) n 3:04 : ....

D. Angluin and L. G. Valiant, \Fast probabilistic algorithms for Hamiltonian circuits and matchings", Journal of Computer and System Sciences 18 (1979), 155-193.

Learning with Restricted Focus of Attention - Ben-David, Dichterman (1997) (Correct)

....of functions computable by an AC 0 circuit of depth d is denoted by AC 0 [d] Using a result by Linial,Mansour and Nisan [23] we show in Section 4.2 that this class is 1 wRFA learnable by a quasi polynomial algorithm. 3. 2 Chernoff Bounds The following form of Chernoff bounds originate in [3], and is used throughout the work. Theorem 3.5 Let X 1 ; Xm be independent random boolean variables, with Pr[X i = 1] p for every 1 i m. Then, for every 0 fl 1, the following are bounds on the empirical average p 4 = 1 m P m i=1 X i : Pr [p p fl] e Gamma2mfl 2 Pr [p ....

Dana Angluin and L. G. Valiant. Fast probabilistic algorithms for hamiltonian circuits and matchings. Journal of Computer Systems and Sciences, 18:155--193, 1979.

A new randomized sorting algorithm on the BSP model - Gerbessiotis, Siniolakis (2001) (3 citations) (Correct)

....is 1 q 1 q s 1 # i=0 # B 1 i # q i (1 q) B 1 i = 1 q B 1 # i=s # B 1 i # q i (1 q) B 1 i . The sum in the term above gives the probability of having at least s successes in B 1 independent Bernoulli trials with probability of success q. Cherno# bounds [5] on the probability of having at least (1 #) B 1)q successes gives the upper bound of e (# 2 (B 1)q) 3 . By setting s = 1 #) B 1)q in this bound we obtain e (s (B 1)q) 2 (3(B 1)q) We first derive a lower bound for the term s (B 1)q, for 0 # 1 and ks # N 2. s (B ....

D. Angluin and L. G. Valiant. Fast probabilistic algorithms for Hamiltonian circuits and matchings. Journal of Computer and System Sciences, 18:155-193, 1979. 32

On the b-partite Random Asymmetric Traveling Salesman.. - Baltz, Schoen, Srivastav (Correct)

....a tour via light edges. While Frieze, Karp and Reed may rely on known facts about random permutations, we have to establish similar results for unions of b random bijections. 2 Preliminaries As a tool from probability theory the following two large deviation inequalities of Angluin and Valiant [1] will be useful. Theorem 3 (Angluin Valiant) Let X 1 ; X n be independent random variables with 0 X i 1 for all i and set X : P n i=1 X i . Then for every 0 1: i) P[X (1 )E[X] e 2 E[X] 3 ; ii) P[X (1 )E[X] e 2 E[X] 2 : Moreover we will need the ....

D. Angluin, L.G. Valiant, Fast probabilistic algorithms for Hamiltonian circuits and matchings, J. Comp. Sys. Sci. 18: 155-194, 1979.

Learning Binary Relations and Total Orders - Goldman, Rivest, Schapire (1989) (14 citations) (Correct)

....d(i) Gamma 1 rows of the same type as i are in I ij . In other words, this probability is bounded by the chance of fewer than y successes in a sequence of d(i) Gamma 1 Bernoulli trials, each succeeding with probability p. We use the following form of Chernoff bounds, due to Angluin and Valiant [3], to bound this probability: Lemma 5 Consider a sequence of m independent Bernoulli trials, each succeeding with probability p. Let S be the random variable describing the total number of successes. Then for 0 fl 1, the following hold: ffl Pr[S (1 Gamma fl)mp] e Gammafl 2 mp=2 , and ....

Dana Angluin and Leslie G. Valiant. Fast probabilistic algorithms for hamiltonian circuits and matchings. Journal of Computer and System Sciences, 18(2):155--193, April 1979.

Valid Generalisation from Approximate Interpolation - Anthony, Bartlett (1996) (3 citations) (Correct)

....for t. Now, the probability of drawing a sample of length m that is not in Q is no more than the probability of k=2 successes in a sequence of m Bernoulli trials, where the probability of success at each trial is 2ffl. From standard Chernoff bounds on the tails of the binomial distribution (see [3]) this probability is no more than exp Gamma 2mffl 3 k 4mffl Gamma 1 2 and for 0 ffi 1=6 and k 1, this is no more than 1 Gamma ffi when m k= 12ffl) ut In what follows, it will be convenient to define d H;C (j) to be infinite if the set fV Cdim(H [j;t] t 2 Cg is ....

Angluin, D. and Valiant, L. (1979). Fast probabilistic algorithms for Hamiltonian circuits and matchings. Journal of Computer and System Sciences, 18: 155-193.

Dominated Error Correcting Codes with Distance Two - Sagols, Riccio, Colbourn (2001) (Correct)

.... 659 146 179 191 182 162 174 1007 929 1001 905 925 661 949 901 919 790 276 342 886 895 369 375 372 112 376 381 361 379 314 319 307 273 309 52 181 439 433 176 241 128 234 254 236 239 206 494 454 386 406 471 451 463 398 140 158 148 244 247 230 4 397 Such a cycle can be found using the heuristic in [1], for example. The graph D 11;0 is hamiltonian: By Corollary 2.2 there exists a hamiltonian cycle H 10;0 of 2D 10;0 containing the edge (0000000000; 0000000011) Apply an appropriate permutation to the hamiltonian cycle H 10;1 to obtain another hamiltonian cycle, say H 0 10;1 , containing the ....

D. Angluin and L. G. Valiant, Fast probabilistic algorithms for Hamiltonian circuits and matchings, Journal of Computer and System Sciences 18 (1979), 155-193.

Probabilistic Integer Sorting - Gerbessiotis, Siniolakis (Correct)

....Then, the random variable X n,P = # n i=1 X i follows a binomial distribution. Let We therefore obtain the following bound for the probability on the tail of the binomial distribution [3] For any u 1, Prob[X n,P # unP ] # # e u # unP e nP . 1) The following bounds also hold ([1]) Let 0 # 1. Then Prob[X n,P # (1 #)nP ] e (1 3)# 2 nP , 2) and Prob[X n,P # (1 #)nP ] e (1 2)# 2 nP . 3) We also use the following result that derives from Stirling s formula [2] for all integers n, m such that n # m. # n m # # # ne m # m , 4) The main ....

D. Angluin and L. G. Valiant. Fast probabilistic algorithms for Hamiltonian circuits and matchings. Journal Computer and System Sciences,18:155-193, 1979.

Stochastic Graphs Have Short Memory: Fully Dynamic.. - Nikoletseas Reif.. (1995) (Correct)

.... may not be always known) For extensive surveys motivating the exploration of efficient average case algorithms and their probabilistic analysis see ( 17] Basic good average case graph algorithms were presented in various settings: sequential, parallel, distributed, NP hard and so on (e.g. [2], 5] 18] 19] 20] 21] 23] 26] 29] We are not aware of any investigation prior to ours concerning fully dynamic graph theoretic problems. 1.3 Average Case Analysis of Dynamic Graph Algorithms The investigation of the average case of fully dynamic graph suggests random graph ....

D. Angluin and L. Valiant, "Fast Probabilistic Algorithms for Hamiltonian Circuits and Matchings", JCSS, vol. 18, pp. 155--193, 1979.

On Packing Bipartite Graphs - Hajnal, Szegedy (Correct)

....random variables. Lemma 1.4. Chernoff [Ch52] Let X 1 ; X 2 ; XN be independent 0 1 random variables such that P rob(X i = 1) p. If m Np is an integer then P rob( XN i=1 X i m) Np m m exp(m Gamma Np) An easy consequence of this is the following. Lemma 1.5. ES74] [AV79]) Let X 1 ; X 2 ; XN be independent 0 1 random variables such that P rob(X i = 1) p. Then for every 0 fi 1, i) P rob( P N i=1 X i b(1 Gamma fi)Npc) exp i Gamma fi 2 Np 2 j : ii) P rob( P N i=1 X i b(1 fi)Npc) exp i Gamma fi 2 Np 3 j : And now let us see ....

D. Angluin and L. G. Valiant, Fast probabilistic algorithms for Hamiltonian circuits and matchings, Journal of Computer and System Sciences, 19 (1979), 155--193.

Lower Bound on the Randomized Complexity of Graph Properties - Hajnal (1991) (19 citations) (Correct)

....that we use a well known inequality for Bernoulli random variable. Theorem 5.2. Let X 1 ; X 2 ; XN be independent 0 1 random variables such that P rob(X i = 1) p. i) Chernoff s inequality [Ch] If m Np is an integer then P rob( XN i=1 X i m) Np m m exp(m Gamma Np) ii) [AV,Ch] For every 0 fi 1, P rob( XN i=1 X i b(1 Gamma fi)Npc) exp Gamma fi 2 Np 2 : Lemma 5.3. Let 0 d 1 ; d 2 ; d n L = n 10 log n integers and let us define d average by P n i=1 d i = d average n. Let X 1 ; X 2 ; X n be independent 0 Gamma 1 random variables ....

D. Angluin and L. G. Valiant, Fast probabilistic algorithms for Hamiltonian circuits and matchings, Journal of Computer and System Sciences, 19, 155-193.

Learning From a Consistently Ignorant Teacher - Frazier, Goldman, al. (1994) (13 citations) (Correct)

.... statement of Part 3 is bounded above by: LE ffl 2 d 1 ; mLBA 2 ; mBPQ ffl 2 d 1 ; ffi 3 Delta 2 d (1) since each example is independently drawn and falls in Q with probability p ffl 2 d 1 (because Q is significant) By applying a version of Chernoff bounds presented in [AV79], we know: LE(p; m; pm=2) e Gammamp=8 (2) It is easily verified that mBPQ ( ffl 2 d 1 ; ffi 3 Delta2 d ) p Deltam LBA 2 =2. Thus by substituting p = ffl 2 d 1 , m = mLBA 2 into Equation (2) and using this observation, we can apply it to Equation (1) to obtain that for a particular ....

D. Angluin and L. G. Valiant. Fast probabilistic algorithms for hamiltonian circuits and matchings. Journal of Computer and System Sciences, 18(2):155--193, 1979.

On the parallel complexity of Hamiltonian Cycle and.. - Dahlhaus, Hajnal.. (1993) (1 citation) (Correct)

....instead of maximal matching. If we don t worry about the running time then our algorithm can be simplified. 6. Conclusion and open problems The sequential deterministic algorithm computing a Hamiltonian cycle for any dense graph seems related to the probabilistic solution of Angluin and Valiant [AV] which computes a Hamiltonian cycle with high probability for any graph if it has one. One would hope to be able to turn Frieze s [Fr] probabilistic parallel algorithm into a deterministic algorithm. But we were not successful in dividing any dense graph into two dense graphs of nearly equal size ....

D.Angluin and L. Valiant, Fast Probabilistic Algorithms for Hamiltonian Circuits and Matchings,J. Computer Syst., 18 (1979), 155-193.

Function Learning from Interpolation - Anthony, Bartlett (1994) (1 citation) (Correct)

....Now, the probability of drawing a sample of length m that is not in Q is at most the probability of at least r=2 successes in a sequence of m Bernoulli trials, where the probability of success at each trial is 2ffl. From standard Chernoff bounds on the tails of the binomial distribution (see [2, 13]) this probability is no more than exp Gamma 2mffl 3 r 4mffl Gamma 1 2 and this is less than 1 Gamma ffi when m 3 2ffl r 6 Gamma ln 1 1 Gamma ffi : For ffi 1=13, m r= 8ffl) will suffice, from which the result follows. ut 2 Definitions and the Main Result A ....

Angluin, D. and Valiant, L. (1979). Fast probabilistic algorithms for Hamiltonian circuits and matchings. Journal of Computer and System Sciences, 18: 155-193.

Computational Sample Complexity and Attribute-Efficient Learning - Servedio (2000) (Correct)

....this distribution, each bit of each A (v i ) is equally likely to occur as the label of a useful example. Let S be a sample of qff =18ffl examples which are drawn from EX(c;D n ) Since the expected number of useful examples in S is q=6; a simple application of Chernoff bounds (see, e.g. [1, 20]) shows that with overwhelmingly high probability the sample S will contain at least one useful example. Since each useful example contains f(v 1 ) ffi Delta Delta Delta f(v q ) as its mq bit prefix, it follows that with overwhelmingly high probability a polynomial time learning algorithm ....

D. Angluin and L. G. Valiant, Fast probabilistic algorithms for Hamiltonian circuits and matchings, J. Comput. System Sci. 48 (1979), 155--193. 28

On PAC Learning Using Winnow, Perceptron, and a Perceptron-Like.. - Servedio (Correct)

....7=12 of the updates must have been on examples in B: As noted earlier, though, if the Perceptron s hypothesis has never been ffl accurate, then at each update step the probability of that update occurring on a point in B is at most 1=2. By a straightforward application of Chernoff bounds (see [3], 23] it follows that the probability that more than 7=12 of m = maxf144 ln ffi 2 ; m 1 g updates occur on points in B is at most ffi=2. Consequently, if m updates have been made, then with probability at least 1 Gamma ffi=2 the Perceptron algorithm will have found an ffl accurate hypothesis. ....

D. Angluin and L. G. Valiant. Fast probabilistic algorithms for Hamiltonian circuits and matchings. J. of Computer and System Sciences, 18(2):155-193, 1979.

Toward a usable theory of Chernoff Bounds for heterogeneous and.. - Siegel (1992) (2 citations) (Correct)

.... is sharper, for almost all ranges of a and p than the more expressive approximations commonly used, and implies, via trivial maximization, the simpler but weaker estimates P rfX Gamma E[X] ang e Gamma2a 2 n [Ho 63] P rfX Gamma E[X] fflE[X]g e Gammaffl 2 E[X] 3 , for ffl 1 [AV 79]; P rfX Gamma E[X] GammafflE[X ]g e Gammaffl 2 E[X] 2 [ASE 91] 3 2.3)Let X = X 1 Delta Delta Delta X k , where E[X i ] n i = p i , and X i is the sum of n i mutually independent random variables, each belonging to [0; 1] The X i s may exhibit arbitrary mutual dependencies. ....

....gives a bound with exponential decay, P rfX Gamma E[X] ang e Gammanf (a) the error in the exponent is always o(n) c.f. SW 92] Unfortunately, even the Chernoff Hoeffding estimate (2) is rather inconvenient to use, and the standard approach in the Computer Science literature (c.f. [AV 79], ASE 92] often uses approximations that, while based on the Chernoff Hoeffding estimation procedure, are exponentially worse but considerably more expressive. In summary, it is fair to say that the contributions of Chernoff Hoeffding bounds are likely endure, and the value of Chernoff tight ....

[Article contains additional citation context not shown here]

D. Angluin and L. G. Valiant. Fast probabilistic algorithms for Hamiltonian circuits and matchings. JCSS 18, 1979, 155--193.

Algorithmic and Complexity Issues Concerning.. - Dunne, Gibbons, Zito (Correct)

....for tackling questions such as these have been outlined in Dunne, Gibbons, and Zito [13] Although speci c examples are few, rather stronger support for the presence of extreme changes in expected run time has emerged from studies of some NP hard search problems. Thus, Angluin and Valiant [3] describe O(n log 2 n) average time methods for nding Hamiltonian cycles in graphs whose edge density (this can be viewed as related to p in the G p model of random graphs introduced above) is large enough to make it almost certain that such cycles exist. Similarly, 80 using a particular model ....

Angluin, D., Valiant, L.G.: Fast probabilistic algorithms for Hamiltonian Circuits and Matchings. Jnl. of Comp. and System Sci., 18 (1979), 82-93

The efficiency of resolution and Davis-Putnam procedures - Beame, Karp (1999) (8 citations) (Correct)

.... are monotone (or anti monotone) with respect to sets of clauses, the almost certain properties under both distributions are the same up to a change from m to m o(m) This is just a natural extension of the similar (and more precise) equivalences for the random graph model as shown for example in [AV79]. We generally assume the distribution F k,n m . We write F # F to mean F is a random formula selected according to distribution F . We make frequent use of two well known tail bounds for the binomial distribution (see [ASE92] Appendix A) Proposition 6: If Y is a random variable ....

D. Angluin and L. Valiant. Fast probabilistic algorithms for Hamiltonian circuits and matchings. Journal of Computer and System Sciences, 18:155--193, 1979.

A Hamilton Path Heuristic with Applications to the Middle Two .. - Shields, Savage (Correct)

....reversing at v a path with endpoint u. Even if the edge uv were not present, using PosaSearch, the path shown would never be extended to a Hamilton path as it can never include the vertex x. Nevertheless, the path reversal approach has proven useful in studying random graphs. Angluin and Valiant [1] devised a fast variant that was always terminating by removing edges from the graph once they had been included in a path. This was shown to almost surely find a Hamilton cycle in random graphs with n vertices and cn log n edges. The middle two levels graph, although certainly not random, has a ....

D. Angluin and L. G. Valiant. Fast probabilistic algorithms for Hamiltonian circuits and matchings. J. Comput. System Sci., 18(2):155--193, 1979.

A Hamilton Path Heuristic with Applications to the Middle Two .. - Shields, Savage (Correct)

On the Occurrence of Null Clauses in Random Instances of.. - John Franco Department (1989) (4 citations) (Correct)

....(x) 1) where is the mean number of serious variables in an instance. First, we obtain a bound on the second sum in (1) Since variables are placed independently in clauses, the number of serious variables in an instance is binomially distributed. By the Chernoff bound for binomial distributions [1], Pr(I ( 1 fi) e Gammafi 2 =3 , fi 0. Thus, r X x=b3:82c 2 2 x Gamma1 Delta Pr(I (x) r X x=b3:82c 2 2 x e Gamma(x= Gamma1) 2 =3 4 = r X x=d3:82e 1 e x ln(2) Gammax 2 = 3) 2x=3 Gamma=3 r X x=d3:82e 1 1 r: Next, we obtain an upper bound on the first sum ....

Angluin, D., and Valiant, L. G., "Fast probabilistic algorithms for Hamiltonian circuits and matchings ", Journal of Computer and System Sciences, Vol. 18, (1979) pp.155-193.

On the Learnability of Boolean Formulae - Michael Kearns Harvard (1987) (56 citations) Self-citation (Valiant) (Correct)

No context found.

D. Angluin and L.G. Valiant. Fast Probabilistic Algorithms for Hamiltonian Circuits and Matchings. JCSS, 18(2):155-193, 1979.

A General Lower Bound on the Number - Of Examples Needed Self-citation (Valiant) (Correct)

No context found.

Angluin, D., L.G. Valiant, "Fast probabilistic algorithms for Hamiltonian circuits and matchings", Journal of Computer and Systems Sciences, 18, 1979, pp. 155-193.

Cryptographic Limitations on Learning - Boolean Formulae And Self-citation (Valiant) (Correct)

No context found.

D. Angluin, L.G. Valiant. Fast probabilistic algorithms for Hamiltonian circuits and matchings. Journal of Computer and Systems Sciences, 18, 1979, pp. 155-193.

Online Performance-Improvement - Prasad Chalasani August (Correct)

No context found.

D. Angluin and L. Valiant. Fast probabilistic algorithms for hamiltonian circuits and matchings. J. Comput. Syst. Sci., 18:155--193, 1979.

Optimal Parallel Construction of Hamiltonian Cycles and.. - MacKenzie, Stout (1993) (1 citation) (Correct)

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D. Angluin and L. G. Valiant. Fast probabilistic algorithms for hamiltonian circuits and matchings. J. Comput. System Sci., 18:155--193, 1979.

Concentration Bounds for Unigrams Language Model - Drukh, Mansour (2004) (Correct)

No context found.

D. Angluin and L. G. Valiant. Fast Probabilistic Algorithms for Hamiltonian Circuits and matchings, In Journal of Computer and System Sciences, 18:155-193, 1979.

Learning in the Presence of Malicious Errors - Michael Kearns Att (1993) (81 citations) (Correct)

No context found.

D. Angluin, L.G. Valiant. Fast probabilistic algorithms for Hamiltonian circuits and matchings. Journal of Computer and Systems Sciences, 18, 1979, pp. 155-193. 32

Efficient Distribution-free Learning of Probabilistic Concepts - Kearns, Schapire (1993) (108 citations) (Correct)

No context found.

Dana Angluin and Leslie G. Valiant. Fast probabilistic algorithms for Hamiltonian circuits and matchings. Journal of Computer and System Sciences, 18(2):155--193, April 1979.

The Efficiency of Resolution and Davis-Putnam Procedures - Beame, Karp, Pitassi, Saks (1999) (8 citations) (Correct)

No context found.

D. Angluin and L. Valiant. Fast probabilistic algorithms for Hamiltonian circuits and matchings. Journal of Computer and System Sciences, 18:155-193, 1979.

Deterministic Design for Neural Network Learning: - An Approach Based (Correct)

No context found.

D. Angluin and L. Valiant, "Fast probabilistic algorithms for hamiltonian circuits and matchings," Journal of Computer and System Science, vol. 18, pp. 155--193, 1979.

A Distributed Algorithm To Find Hamiltonian Cycles In.. - Levy, Louchard, Petit (2000) (Correct)

No context found.

D. Angluin and L. G. Valiant. Fast probabilistic algorithms for hamiltonian circuits and matchings. Journal of Computer and System Sciences, 18:155-193, 1979.

Unknown - (Correct)

No context found.

D. Angluin and L. G. Valiant. Fast probabilistic algorithms for hamiltonian circuits and matchings. J. Computer and System Sciences, 18:155--193, 1979.

The Resolution Complexity of Random Graph k-Colorability - Beame, Culberson, al. (2003) (Correct)

No context found.

D. Angluin and L. Valiant. Fast probabilistic algorithms for Hamiltonian circuits and matchings. Journal of Computer and System Sciences, 18:155--193, 1979.

Almost All - Graphs Are Easy (Correct)

No context found.

Angluin, D., L. G. Valiant. "Fast Probabilistic Algorithms for Hamiltonian Circuits and Matchings." In Journal of Computer and System Sciences 18 , 155--193, 1979.

A probabilistic analysis of the Floyd-Rivest expected.. - Gerbessiotis, Siniolakis (2002) (Correct)

No context found.

D. Angluin and L. G. Valiant. Fast probabilistic algorithms for Hamiltonian circuits and matchings. Journal of Computer and System Sciences, 18:155-193, 1979.

A Sharp Threshold in Proof Complexity Yields Lower.. - Achlioptas, Beame.. (Correct)

No context found.

D. Angluin and L. Valiant. Fast probabilistic algorithms for Hamiltonian circuits and matchings. Journal of Computer and System Sciences, 18:155--193, 1979.

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