J.P. Schmidt, A. Siegel and A. Srinivasan. Chernoff-Hoeffding bounds for applications with limited independence. SIAM Journal on Discrete Mathematics 8 (1995), no. 2, 223--250.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....real valued random variables with distribution PX , expected value E[X] and range in the interval [a; b] Then for any t 2 R hfi fi X i Gamma E[X] fi fi t 2e 2t 2 (b Gammaa) 2 Deltan : 2. 8) Similar bounds hold also for sums of random variables with limited independence [SSS95] Let X 1 ; Xn be k wise independent random variables with distribution PX , alphabet in the interval [0; 1] and expectation E[X] Then fi fi X i Gamma n fi kn Gammabk=2c : 2.9) 2.3 Entropy and Information Theory The (Shannon) entropy [Sha48] of a random variable ....

Jeanette P. Schmidt, Alan Siegel, and Aravind Srinivasan, Chernoff-Hoeffding bounds for applications with limited independence, SIAM Journal on Discrete Mathematics 8 (1995), no. 2, 223--250.

....in the rest of this section. Note that the monotonicity is immediate. When a new bucket is added, the only items that move are those that are now closest to that bucket s associated points. No items move between old buckets The proof of theorem 4.4. 1 requires the following technical lemma from [12] that gives upper bounds on a sum of Bernoulli variables when these variables are only k way independent. Lemma 4.4.2 If X is the sum of k wise independent binary random variables, with = E[X] then (I) for ffi 1 and k bffi Gammabk=2c (II) for ffi 1 and k = bffie Gammabffi=3c We ....

Jeanette Schmidt, Alan Siegel and Aravind Srinivasan. Chernoff-Hoeffding Bounds for Applications with Limited Independence. In Proc. 4th ACS-SIAM Symposium on Discrete Algorithms, 1993.

....do not hold for dependent random variables. The well known Chernoff Hoeffding bounds from the theory of large deviations are an excellent example in this respect. Much effort has been made to salvage these sharp bounds in the more general situation under consideration; see, for example, 11, 13, 3, 8] It turns out that one can apply the Chernoff Hoeffding bounds to sums of strongly negatively dependent random variables just as one would apply them to independent random variables; see Section 6. Hence, it is useful to establish negative dependence among random variables. However, ....

....following correlation inequality on the random variables B i , i 2 [n] Theorem 12 Let I; J [n] be index sets such that either I J = or 11 I [ J = n] and let t I ; t J be arbitrary non negative integers. Then Pr i2I B i t I ; Pr ( i2I B i t I ) Delta Pr (5. 2) Remark 13 (5.2) is referred to as the negative quadrant dependence condition for X : i2I B i and Y : j2J B j . It is known to be equivalent to the negative association condition ( GammaA) for X;Y , 7] This can also be easily seen by replacing f; g in the proof of Theorem 12 by arbitrary ....

Schmidt, J. P., Siegel, A. and Srinivasan, A. (1995) Chernoff--Hoeffding bounds for applications with limited independence. SIAM J. Discrete Math. 8 223--250 14

....3.1. In general, the flow shop problem, where the objective is to minimize the makespan, is NP complete even on three machines ( 15] The best result known is O(log 2 (m) log log(m) approximation algorithm, where is the maximum number of operations per job, and m is the number of machines ([39, 42]) When m is fixed (but arbitrary) Hall [23] gave a PTAS for this problem. Online scheduling of intervals. Lipton and Tomkins [32] considered the problem of online scheduling of intervals on a single machine (resource) where the objective is to maximize the resource utilization. They gave an ....

J.P. Schmidt, A. Siegel, and A. Srinivasan. "ChernoffHoeffding Bounds for Applications with Limited Independence ". In SIAM J. Discrete Math., vol. 6, 1995, pp. 223--250.

....of roots of a random polynomial as a sum of random variables. We will use the fact that there is some independence between the random variables to prove that the probability that a polynomial has k roots vanishes exponentially in k. Our technical tool will be the following lemma: 20 Lemma 3.3. 1 ([SSS93], Theorem 1) Let X 1 ; X q 2 f0; 1g be random variables, X = P q i=1 X i and = E[X] For every k 1, if the X i s are d wise independent for d h = h(q; k) def = k Gamma 1) 1 Gamma =q then, P r(X k) Gamma q h Delta ( q) h Gamma k h Delta Intuitively, this ....

....the variables in the polynomial. Since the number of roots k, can be much more than d (it is bounded by dq n Gamma1 , where n is the number of variables) we cannot use lemma 3.3.1. Instead, we use a different lemma, which requires less independence between the random variables: Lemma 3.3. 3 ([SSS93], Lemma 3) Let X 1 ; X q 2 f0; 1g be random variables , X = P q i=1 X i and = E[X] For every k 1, if the X i s are d wise independent for d h(q; k) then, P r(X k) Gamma q d Delta ( q) d Gamma k d Delta This yields the following result: The fraction of ....

Jeanette P. Schmidt, Alan Siegel and Aravind Srinivasan. Chernoff-Hoeffding bounds for applications with limited independence. Proceedings of the 4th Annual ACM Symposium on Discrete Algorithms, 1993, pp. 331-340.

....fX 1 ; X l g. The property we will require is: Lemma 5 Let T f1; 2; ng, jT j=n ffi. Suppose k ffil=6. If S is chosen at random as described above, then P r[jS T j ffil=2] 1 Gamma e Gammabk=2c : We use the following lemma, which is a special case of Theorem 2. 5 from [SSS]: Lemma 6 Let Y 1 ; Y l be k wise independent 0 1 random variables, Y = P l i=1 Y i , and = EY . Let ff = q ke 1=3 = and suppose ff 1. Then P r[jY Gamma j ff] e Gammabk=2c : 7 Proof of Lemma 5: Define the random variables Y i to be 1 iff X i 2 T , and 0 otherwise. Let ....

J.P. Schmidt, A. Siegel, A. Srinivasan, Chernoff-Hoeffding Bounds for Applications with Limited Independence, Proceedings of the 4th Annual ACM-SIAM Symposium on Discrete Algorithms, 1993, pp. 331-340.

....to the one used in [13] see also [17] to calculate the tail bounds of a well known occupancy problem. We exploit the properties of special sequences of random variables called martingales, using Azuma s inequality [2] for their analysis. Similar results in a more general context are presented in [21]. Consider the following process. We have a collection of n balls, of which ffn are red and (1 Gamma ff)n are black (0 ff 1) We select without replacement uniformly at random fin balls (0 fi 1) Let Omega 1 be the random variable representing the number of red balls that are selected; it ....

J. P. Schmidt, A. Siegel, A. Srinivasan. Chernoff--Hoeffding Bounds for Applications with Limited Independence. In Proc. of the 3rd Annual ACM--SIAM Symposium on Discrete Algorithms (SODA '93), 1993, pp. 331--340.

....laws that do not hold for dependent random variables. The well known Chernoff Hoeffding bounds from the theory of large deviations are an excellent example in this respect. Much effort has been made to salvage these sharp bounds in the more general situation under consideration; see, for example, [11, 13, 3, 8]. It turns out that one can apply the Chernoff Hoeffding bounds to sums of strongly negatively dependent random variables just as one would apply them to independent random variables; see Section 6. Hence, it is useful to establish negative dependence among random variables. However, this can ....

Schmidt, J. P., Siegel, A. and Srinivasan, A. (1995) Chernoff--Hoeffding bounds for applications with limited independence. SIAM J. Discrete Math. 8 223--250

....number of roots of a random polynomial as a sum of random variables. We will use the fact that there is some independence between the random variables to prove that the probability that a polynomial has k roots vanishes exponentially in k. Our technical tool will be the following lemma: Lemma 7 ([11], Theorem 1) Let X 1 ; X q 2 f0;1g be random variables, X = q i=1 X i and = E[X] For every k 1, if the X i s are d wise independent for d h = h(q; k) def = k Gamma 1) 1 Gamma =q then, Pr(X k) Gamma q h Delta ( q) h Gamma k h Delta Intuitively, this lemma ....

....of all the variables in the polynomial. Since the number of roots k, can be much more than d (it is bounded by dq n Gamma1 , where n is the number of variables) we cannot use lemma 8. Instead, we use a different lemma, which requires less independence between the random variables: Lemma 9 ([11], Lemma 3) Let X 1 ; X q 2 f0;1g be random variables , X = q i=1 X i and = E[X] For every k 1, if the X i s are d wise independent for d h(q; k) then, Pr(X k) Gamma q d Delta ( q) d Gamma k d Delta This yields the following result: The fraction of multivariate ....

Jeanette P. Schmidt, Alan Siegel and Aravind Srinivasan. ChernoffHoeffding bounds for applications with limited independence. Proceedings of the 4th Annual ACM Symposium on Discrete Algorithms, 1993, pp. 331340.

....It is worth noting that Chebychev inequality tends to provide very large bounds, particularly for large values of k. Other inequalities exist which provide tighter bounds. Examples of these are the one sided Chebychev inequality, and the Chernoff Hoeffding bounds (Chernoff 1952, Hoeffding 1963, Schmidt et al. 1992) which provide bounds for the probability tails of sums of binary random variables (many thanks to Gunter Rudolf for pointing this out to us) These inequalities can all lead to interesting new schema theorems. Unfortunately, the left hand sides of these inequalities (i.e. the bound for the ....

Schmidt, J. P., A. Siegel and A. Srinivasan (1992). Chernoff-hoeffding bounds for applications with limited independence. Technical Report 92-1305. department of computer science, Cornell University.

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Jeanette P. Schmidt, Alan Siegel, and Aravind Srinivasan. Chernoff-Hoeffding bounds for applications with limited independence, SIAM J. Discrete Math., 8 (1995), pp. 223--250.

....] e Gammata . Let X 1 ; X 2 ; X n be independent r.v.s that take on values in [0; 1] with E[X i ] p i , 1 i n. Let X : P n i=1 X i , and : E[X] P n i=1 p i . We want good upper bounds on Pr[X (1 ffi) for ffi 0; we recall some such bounds now, as presented in [61]. Chernoff [18] showed that for identically distributed f0; 1g r.v.s X 1 ; X 2 ; X n and for a , min t E[e tX ] e at (n; a) a ) a ( n Gamma n Gamma a ) n Gammaa : Hoeffding [33] extended this by showing that (n; a) is an upper bound for the above minimum ....

....that: G( ffi) i) decays exponentially in ffi 2 for small ffi (ffi 1) and (ii) decays exponentially in (1 ffi) ln(1 ffi) for larger ffi (ffi 1) Also, some of the constants such as 3 and 4 in the exponents above, can be improved slightly. 8 We next present a useful result from [61], which offers a new look at the ChernoffHoeffding (CH) bounds. For real x and any positive integer r, let i x r j : x(x Gamma1) Delta Delta Delta(x Gammar 1) r as usual, with i x 0 j : 1. Define, for z = z 1 ; z 2 ; z n ) 2 n , a family of symmetric polynomials ....

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J. P. Schmidt, A. Siegel, and A. Srinivasan. Chernoff-Hoeffding bounds for applications with limited independence. SIAM J. Discrete Math., 8:223--250, 1995.

.... considering E[ Z Gamma E[Z] k ] or similar expectations, which look at the Z i k or fewer at a time (via linearity of expectation) The main application of this has been that the Z i s can then be sampled using few random bits, yielding a de randomization pseudo randomness result (e.g. 1 [3, 22, 8, 23, 24, 29]) Our results show that such ideas can in fact be used to show that some structures exist This is one of our main contributions. However, while many applications of Lemma 1.1 have been constructivized (Beck [5] Alon [1] our MIP result is only existential. For PIPs and CIPs, we present a ....

....i;j that satisfy the conditions of the theorem, then there is the possibility of effectively reducing the dependency by a lot (d 0 can be replaced by the value d) Concrete instances of this will be studied in later sections. The tools behind our MIP application are our new LLL, and a result of [29]. Define, for z = z 1 ; z n ) 2 n , a family of polynomials S j (z) j = 0; 1; n, where S 0 (z) j 1, and for j 2 [n] S j (z) X 1i1 i2 Delta Delta Delta i j n z i 1 z i 2 Delta Delta Delta z i j : The relevant theorem of [29] is Theorem 3.2. 29] Given r.v.s ....

[Article contains additional citation context not shown here]

J. P. Schmidt, A. Siegel, and A. Srinivasan, ChernoffHoeffding bounds for applications with limited independence, SIAM Journal on Discrete Mathematics, 8 (1995), pp. 223--250.

....] p i . Let X : P i2[n] X i and : E[X ] P i2[k] p i . Let S b be the bth elementary symmetric polynomial; i.e. S b (p 1 ; pn ) X I [k] jIj=b Y i2I p i : We summarize the large deviation bounds of Chernoff [5] Hoeffding [10] and Schmidt, Siegel, and Srinivasan [22] in the following: Lemma 2.1. For 0 ffi 1, Pr[jX Gamma j ffi] 2e Gammaffi 2 =3 . Lemma 2.2. For ffi 0, Pr[X (1 ffi) S dffie (p 1 ; pn ) Gamma (1 ffi) dffie Delta H( ffi) where H( ffi) e ffi = 1 ffi) 1 ffi) 4 For p 2 (0; 1) let L( p) be ....

J. P. Schmidt, A. Siegel, and A. Srinivasan, Chernoff-Hoeffding bounds for applications with limited independence, SIAM J. Discrete Mathematics, 8 (1995), pp. 223--250.

....jjX jj CH fflE[X] e Gammaffl 2 E[X] 3 , for 0 ffl 1. Comparable datings can be attained for the approximation jjX jj CH fflE[X] e GammafflE[X ] log(1 ffl) 2 , for ffl 1. Recently, similar bounds have been established for limited independence; the strongest to date appear in [SSS 93], where the first bound is shown to hold when every subset of ffl 2 E[X] Bernoulli trials is guaranteed to be mutually independent, for ffl 1, and the CH estimate bound jjX jj CH fflE[X] as defined for full independence is shown to hold provided the mutual independence occurs for any set ....

J. P. Schmidt, A. Siegel and A. Srinivasan. Chernoff-Hoeffding Bounds for Applications with Limited Independence. Proc. 4th Ann. ACM-SIAM Symp. on Discrete Algorithms, 1993, 331--340.

....enough that the ChernoffHoeffding bound for fully random Bernoulli Trials holds with limited independence. Bound (19) was attained by modeling independent probes as independent Bernoulli trials. From Theorem 5 in 43 Double hashing is computable and randomizable with universal hash functions [15], it can be seen that (much more than) sufficient independence is achieved for an independence Gamma 3jI j Gamma hjW j = e L, for the maximum e L value used in (23) Alternatively, Lemma A2 can be used, with an increase in by factor of five, and a nominal change in the size of the bound. ....

....that exhibit weak correlations and that might be supported only by a source of limited randomness. 5. Appendix This section contains two technical Lemmas, which can simplify large deviation calculations in cases of full and limited independence. Lemma A. 2 is a special case of Theorem 6 in [15]. Lemma A1. Let X = P n i=1 X i be the sum of n mutually independent Bernoulli trials X 1 ; Xn , where P robfX i = 1g = p i . Then for any C 0; and 0 ffl 1, P robfX (1 ffl)E[X] Cg e Gamma1:25fflC : Proof: Let p = E[X] n . According to Hoeffding, Ho 63] P robfX (1 ....

[Article contains additional citation context not shown here]

J.P. Schmidt, A. Siegel, and A. Srinivasan. Chernoff-Hoeffding Bounds for Applications with Limited Independence. Proc. 4th Ann. ACM-SIAM Symp. on Discrete Algorithms, 1993, 331--340. To appear SIAM J. Discrete Math.

....least I items in D have their first probe in [1; I]g, since the factor of n accounts for all shifts of [1; I] But this probability describes a large deviation for a sum of ffn wise independent Bernoulli trials, each having probability I n of success. From a weak application Theorem 5 of [17], we attain that if I(1 Gamma ff) then Probfat least I items in D have their first probe in [1; I]g e Gamma(1 Gammaff) 2 I 3 : It follows that X(I) is nominal in LP and LP when = c(log n) for suitably large fixed c. 35 Closed hashing is computable and optimally randomizable with ....

J.P. Schmidt, A. Siegel, and A. Srinivasan. Chernoff-Hoeffding Bounds for Applications with Limited Independence. Proc. 4th Ann. ACM-SIAM Symp. on Discrete Algorithms, 1993, 331--340.

No context found.

J.P. Schmidt, A. Siegel and A. Srinivasan. Chernoff-Hoeffding bounds for applications with limited independence. SIAM Journal on Discrete Mathematics 8 (1995), no. 2, 223--250.

No context found.

J. Schmidt, A. Siegel, A. Srinivasan. Chernoff-Hoeffding bounds for Applications with Limited Independence. Proceedings of the 4th Annual Symposium on Discrete Algorithms, ACM-SIAM, 1993.

No context found.

Schmidt, J.P., A. Siegel, A. Srinivasan, `Chernoff-Hoeffding Bounds for Applications with Limited Independence,' Proc. 4th Symp. on Discrete Algorithms, pp. 331--340, ACM-SIAM, 1993.

No context found.

J. P. Schmidt, A. Siegel, and A. Srinivasan. Chernoff-Hoeffding bounds for applications with limited independence, SIAM J Discrete Math 8 (1995), 223--250.

No context found.

J.P. Schmidt, A. Siegel and A. Srinivasan. Chernoff-Hoeffding bounds for applications with limited independence. In Proceedings of the Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 331--340, 1993.

No context found.

J.P. Schmidt, A. Siegel, and A. Srinivasan. Chernoff-Hoeffding bounds for applications with limited independence. SIAM Journal on Discrete Mathematics, 8:223--250, 1995.

No context found.

J. Schmidt, A. Siegel, and A. Srinivasan. Chernoff-Hoeffding bounds for applications with limited independence. SIAM Journal on Discrete Mathematics, 8(2):223--250, 1995.

No context found.

J.P. Schmidt, A. Siegel, A. Srinivasan, Chernoff-Hoeffding bounds for applications with limited independence, SIAM Journal Discrete Mathematics 6 (1995), 223250.

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