# SRC Technical Note June 17, Tight Thresholds for The Pure Literal Rule. Michael Mitzenmacher. d i g i t a l

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1 SRC Techical Note Jue 17, 1997 Tight Thresholds for The Pure Literal Rule Michael Mitzemacher d i g i t a l Systems Research Ceter 130 Lytto Aveue Palo Alto, Califoria Copyright cdigital Equipmet Corporatio All rights reserved

4 It is clear that dl =, sice at each step, two literals are removed. Hece, as L 0 =, we have L = t. Whe a radom pure literal is chose, the expected umber of times it appears i the formula is simply the average umber of clauses a radom variable appears i (up to a O( 1 ) additive error). Oe ca see this by otig that ay give pure literal is equally likely to be ay of the remaiig variables; the fact that its egatio appears 0 times does ot affect the coditioal distributio of its umber of appearaces, give the curret state (X 0 (t), X 1 (t),...). (The O( 1 ) discrepacy is caused by the fact that a pure literal is slightly less likely that a radom literal to appear 0 times, as we kow that oe literal, its egatio, appears 0 times; this, however, oly chages thigs by a O( 1 ) term, which ca be safely dismissed i the limit as. From ow o, we igore this discrepacy i establishig the differetial equatios.) Hece dc i 0 = A = ix i. L Makig use of the idetity = i 0 ix i, which expresses the total umber of remaiig variables i the formula i two differet ways, ad our kowledge of the form of L, we may rewrite this as dc = t, from which it is easily derived that C = C 0 (1 t) k/. The equatios describig the behavior of the X i are slightly more complex. First, ote that the pure literal deleted durig a time step appears i times with probability Xi. Now, suppose the pure literal occurs j times. The we L lose a literal that appears i times wheever oe of j clauses cotaiig that variable cotais a literal that appears exactly i times. Note that there are j (k 1) variables deleted, as there are k 1 variables per clause (1 variable for each clause is take by the pure literal!). The probability that each such variable is oe that appears i times is ixi. (Agai, ote that we have here igored additive O( 1 ) terms, such as whe a two appearaces of a literal are deleted.) Hece the expected loss of literals of size i is Xi Ai Xi L. Oe ca similarly determie the expected gai i X i durig a time step from all literals that appear i + 1 times ad have 1 appearace deleted. The result yields: dx i = A(k 1)iX i + A(k 1)(i + 1)X i+1 X i L for i 1. Note the case of X 0 is special, sice we always remove the egatio of a pure literal, which by defiitio appears 0 times, at each step: dx 0 = A(k 1)X 1 X 0 L 1. 3

5 3 The Solutio Recall that, oce X 0 = 0, the process stops. Hece our goal is to determie a explicit equatio for X 0, ad use it to determie what values of m guaratee that X 0 > 0 for t [0, 1). Oce we have solved this determiistic case give by the differetial equatios, we ca use this iformatio to make statemets regardig the limitig case of the radom process as. (Note that, for techical reasos, we also require k 3; see Lemma 4.4 of [1].) For the equatios below, we use c = m which is a fixed costat. Oe may check that the solutios for the X i, i 1, are give by the followig formulas: where X i (t) = ( ) C j (k 1)/k λ i, j (1 t) 1/, c j=i λ i, j = ( ck ) j ( ( 1) i+ j j) i. j! X 0 ca be solved for explicitly, or by otig that X 0 = L i 1 X i, yieldig X 0 (t) = t i=1 ( ) C j (k 1)/k λ i, j (1 t) 1/. c j=i We ow fid a coveiet form for X 0 (t): ( ) C j (k 1)/k X 0 (t) = t λ i, j (1 t) 1/ c i=1 j=i ( ck ) j = (1 t) [(1 ( 1/ t) 1/ ( 1) i+ j j) ] i (1 t) (k 1) j/ j! i=1 j=i ( = (1 t) 1/ j ( ) ) ( ) j ck(1 t) (k 1)/ j (1 t) 1/ ( 1) i+ j i j! j=1 = (1 t) 1/ (1 t) 1/ + j=1 i=1 ( ) ck(1 t) (k 1)/ j ( (1 t) = (1 t) [(1 1/ t) 1/ (k 1)/ ) ] ck 1. Hece, to determie whe X 0 (t) >0, it suffices to examie the expressio ( (1 t) (1 t) 1/ (k 1)/ ) ck 1, 4 j!

7 k c k Table 1: The thresholds for the pure literal rule for k-sat. These values match simulatios quite well eve for a very small umber of clauses (i the tes of thousads). [4] R. M. Karp ad M. Sipser. Maximum matchigs i sparse radom graphs. I Proceedigs of the d IEEE Symposium o Foudatios of Computer Sciece (1981) pp [5] T. G. Kurtz. Approximatio of Populatio Processes. SIAM (1981) [6] M. Mitzemacher. Load balacig ad desity depedet jump Markov processes. I Proc. of the 37 th IEEE Symp. o Foudatios of Computer Sciece (1996) pp. 13. [7] M. Mitzemacher. The Power of Two Choices i Radomized Load Balacig Ph.D. thesis, Uiversity of Califoria, Berkeley. (September 1996) [8] N.C. Wormald. Differetial equatios for radom processes ad radom graphs. The Aals of Applied Probability 5 (1995) pp

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