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[(mea)-9.4(sure)-338.1(\(Letouze)]TJ
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()Tj
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T*
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T*
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/F5 1 Tf
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[(phasi)-7.8(ng)]TJ
/F1 1 Tf
3.1753 0 TD
[(\).)-293.3(Th)-7(eir)-299(algorithm)]TJ
/F5 1 Tf
7.5627 0 TD
[(MED)-8.8(ICC)]TJ
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-16.0075 -1.2007 TD
[(\()-88.1(Mohri,)-301.1(2003\))-296(to)-294.5(mode)-8.4(l)-293.7(the)-297.1(proles)-302.5(and)-294.3(efcient)-12.5(ly)-294.5(com)-7.6(pute)-297.8(the)-297.1(distance.)-303.1(Howeve)-10.3(r,)-293.3(the)-297.1(com)-7.6(plexity)-299.9(of)-293.7(this)]TJ
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0 -1.2007 TD
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(B)Tj
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[(time)-8.2(s,)-326.7(resulting)-333.1(in)-323(a)-326.9(transduc)-9.9(er)-324(with)-331.2(3)]TJ
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[(states)-329(\()-82.4(Mohr)-7(i,)-328.3(2004;)-325.2(Schw)-8(arz)-326.5(et)-330.4(al.,)]TJ
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[(20)-7.2(14\).)-272.1(Here,)]TJ
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/F5 1 Tf
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[(1.1.)-502(Copy)-328.8(number)-333.3(transformation)]TJ
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-1.0015 -1.2007 TD
[(CNP)-10.1(s,)]TJ
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[(.)-233.7(W)0(e)-239.2(call)-239.8(this)-237.4(probl)-8.6(em)-234.5(the)-240.2(Copy)-241.1(Number)-243.4(Transfor)-9.1(mation)-237.3(Probl)-7.8(em)-240.2(\(CNTP\).)-244.2(A)-232.3(CNP)-243.5(is)-233.1(repr)-8.8(esented)-242.3(by)]TJ
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[(a)-355.4(vect)-9.4(or)-356.3(of)-350.6(no)-7.2(n-negative)-361.9(integer)-10.7(s)-354.8(\(the)-356.7(numb)-6.6(er)-352.4(of)-356.3(copi)-7.6(es)-351.6(of)-356.3(each)-361.2(segment)-10.7(\).)-355.9(A)-351.8(segm)-9.6(ental)-360(deletion)-359.4(\(ampl)-8.2(i-)]TJ
T*
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T*
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/F5 1 Tf
10.959 0 0 10.959 74.0976 255.2315 Tm
[(1.2.)-502(Our)-331.8(contribution)]TJ
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[(We)-341.6(show)-338.2(that)-340.4(the)-342.6(CNTP)-340.8(is)-335.5(solv)-7(able)-339.4(in)-334.3(linear)-345.7(time)-338.2(and)-339.8(constant)-346.4(space.)-340.8(The)-339.6(algo)-7.6(rithm)-339.2(relies)-341.2(on)-337.3(sever)-10.6(al)]TJ
-1.0015 -1.195 TD
[(prope)-7.6(rties)-350(of)-344.9(the)-348.3(problem)-351.1(that)-346.1(we)-348.7(establish)-351.2(in)-345.7(Sect)-8.6(ion)-340.8(3)-6.5(.1,)-343(which)-348(may)-349(also)-350.2(be)-344.7(relevan)-9.7(t)-345(t)0(o)-345.7(the)-342.6(anal)-9.4(ysis)-343.9(of)]TJ
0 -1.2007 TD
[(othe)-7.6(r)-361.2(p)-6.5(roblems)-369.4(involvi)-7.9(ng)-365.7(CNPs.)-371(By)-364.7(exploitin)-8.3(g)-365(these)-369.8(prope)-7.6(rties,)-367.5(we)-365.7(obta)-7.6(in)-362.8(a)-366.7(pseudo)-7.6(polyn)-6.6(omial)-367.5(dynam)-9.1(ic)]TJ
T*
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T*
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T*
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1.0015 -1.2007 TD
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[(2.)-501.9(PRELIMINAR)6.4(IES)]TJ
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[(\),)-373(where)]TJ
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[(correspo)-9.1(nds)-373.4(to)-368.5(a)-366.7(segment)-374.9(in)-374.2(the)]TJ
-29.9266 -1.2007 TD
[(no)-7.2(rmal)-257.8(genome,)-262.4(whe)-8(re)-255.7(the)-257.2(segment)-10.7(s)-258(are)-258.2(ordered)-263.2(as)-254.9(in)-260.4(the)-257.2(normal)-259.3(genom)-9.1(e.)-259(For)-253.9(simpl)-8.4(icity)-258.6(we)-257.6(call)-262.5(a)-252.9(p)0(o)-7.2(sition)]TJ
T*
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[(.)-267.8(A)-272.2(CN)-266.3(opera)-9.4(tion)-270.3(\(CNO\))-276.4(is)-267.2(a)-270(triple)]TJ
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[(ZEIR)-11.9(A)-334.8(E)0(T)-335.3(AL.)]TJ
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[(\)\))-511.9(as)-516.7(follows.)-519.1(For)-515.6(each)]TJ
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6.9738 0 0 6.9738 259.5401 541.2471 Tm
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6.9738 0 0 6.9738 320.485 541.2471 Tm
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[(,)-415.8(othe)-7.6(rwise)-419.5(\(i.e.,)-419.7(if)]TJ
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(i)Tj
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[(.)-376(A)-380.3(triple)]TJ
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[(\))-378.3(with)]TJ
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[(has)-380.8(no)-377.1(eff)-8(ect)-378.5(on)-382.8(the)-382.4(CNP,)-380.4(that)-385.9(is,)]TJ
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(V)Tj
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(\))Tj
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(V)Tj
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[(.)-376(Given)-387.9(two)-378.9(CNPs,)]TJ
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[(\))-429.5(\(sour)-9(ce\))-434.6(and)]TJ
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(t)Tj
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(1)Tj
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(2)Tj
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(n)Tj
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[(\))-429.5(\(targe)-9.2(t\),)-433.4(a)-429.3(C)0(N)-431.3(transf)-8.6(ormat)-8.2(ion)-431.8(\(CNT\))-437.3(is)-426.6(a)-435(vector)]TJ
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(m)Tj
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[(\),)-259.2(where)]TJ
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(N)Tj
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[(and)-260.2(each)]TJ
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(i)Tj
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[(\))-253.1(i)0(s)-255.9(a)-258.6(CNO,)-259.9(such)-262.1(that)]TJ
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()Tj
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(c)Tj
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(1)Tj
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(\()Tj
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(\)\)\)\))Tj
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[(,)-415.8(denoted)]TJ
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(j)Tj
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(C)Tj
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(j)Tj
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[(,i)415.8(s)]TJ
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[(.)-415.8(A)0(n)-420.9(exampl)-8.7(e)-412.3(i)0(s)-420.9(given)-418.1(in)-414(Figur)-7.8(e)-412.3(1)0(.)-416.6(Finall)-8.2(y,)-416.6(we)-417(denot)-8.4(e)-412.3(the)-422.3(number)-421.6(of)]TJ
-13.7825 -1.2007 TD
[(operations)-241.8(of)-236.8(weight)]TJ
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(2f)Tj
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[(1)-168.5(1)]TJ
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(g)Tj
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7.4718 0 0 6.641 288.5101 481.3227 Tm
(i)Tj
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9.9626 0 0 9.9626 292.9322 482.9668 Tm
(by)Tj
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()Tj
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(\))Tj
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()Tj
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(h)Tj
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(w)Tj
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(:)Tj
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(i)Tj
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()Tj
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(h)Tj
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(gj)Tj
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[(.)-233.7(For)-236.8(examp)-10.9(le,)-234.1(in)]TJ
-36.4138 -1.2007 TD
[(Figure)-340.4(1,)]TJ
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(op)Tj
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(\()Tj
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(C)Tj
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()Tj
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(-)Tj
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[(1)-168.5(2\))]TJ
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(=)Tj
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(1.)Tj
-9.1789 -1.2007 TD
[(The)-294.1(CN)-289(distanc)-9.9(e)-292.8(from)]TJ
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(S)Tj
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(T)Tj
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[(,)-290.6(dist\()]TJ
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(S)Tj
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()Tj
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[(\),)-293.3(is)-290(the)-297.1(smallest)-297(size)-295.1(of)-293.7(a)-287.1(CNT)]TJ
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(C)Tj
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[(that)-294.9(satises)]TJ
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5.2638 0 TD
(C)Tj
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(\()Tj
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(S)Tj
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(\))Tj
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(=)Tj
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(T)Tj
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[(,)-290.6(whe)-8(re)-289.8(if)]TJ
-39.2989 -1.195 TD
[(no)-337.3(such)-341.8(CNT)-346.5(exists,)-339(d)-6.5(ist\()]TJ
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()Tj
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(T)Tj
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(\))Tj
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(=)Tj
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(1)Tj
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[(.)-341.8(Note)-341.6(that)]TJ
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(dist)Tj
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[(is)-341.2(not)-335.1(sym)-7(metric.)-343.9(For)-339.2(exam)-10.2(ple,)-337.3(for)]TJ
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(=)Tj
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[(\(1\))-336.3(and)]TJ
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(T)Tj
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(\(0\),)Tj
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(\))Tj
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[(1)-421.9(but)-426.2(dist\()]TJ
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(T)Tj
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()Tj
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(S)Tj
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(\))Tj
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(=)Tj
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(1)Tj
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[(.)-421.5(G)0(i)-8.2(ven)-425.2(two)-430.1(CNPs,)]TJ
/F5 1 Tf
8.4789 0 TD
(S)Tj
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(=)Tj
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(\()Tj
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(s)Tj
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6.9738 0 0 6.9738 341.7448 433.644 Tm
(1)Tj
9.9626 0 0 9.9626 345.77 435.1747 Tm
()Tj
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(s)Tj
/F1 1 Tf
6.9738 0 0 6.9738 354.6141 433.644 Tm
(2)Tj
9.9626 0 0 9.9626 358.5826 435.1747 Tm
()Tj
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.1682 Tc
(...)Tj
/F1 1 Tf
1.3145 0 TD
0 Tc
()Tj
/F5 1 Tf
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(s)Tj
6.9738 0 0 6.9738 387.1558 433.644 Tm
(n)Tj
/F1 1 Tf
9.9626 0 0 9.9626 391.1243 435.1747 Tm
[(\))-429.5(and)]TJ
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2.6347 0 TD
(T)Tj
/F9 1 Tf
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(=)Tj
/F1 1 Tf
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(\()Tj
/F5 1 Tf
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(t)Tj
/F1 1 Tf
6.9738 0 0 6.9738 438.6896 433.644 Tm
(1)Tj
9.9626 0 0 9.9626 442.6581 435.1747 Tm
()Tj
/F5 1 Tf
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(t)Tj
/F1 1 Tf
6.9738 0 0 6.9738 450.4818 433.644 Tm
(2)Tj
9.9626 0 0 9.9626 454.4503 435.1747 Tm
()Tj
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(...)Tj
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1.3145 0 TD
0 Tc
()Tj
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(t)Tj
6.9738 0 0 6.9738 482.003 433.644 Tm
(n)Tj
/F1 1 Tf
9.9626 0 0 9.9626 485.9715 435.1747 Tm
[(\),)-424.2(the)-427.9(CNTP)]TJ
-37.9787 -1.2007 TD
[(seeks)-561.8(dist\()]TJ
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4.5069 0 TD
(S)Tj
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()Tj
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(T)Tj
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[(\))-554.7(\(if)-555.3(one)-556.1(exists\).)-558(We)-557.8(say)-557.3(that)-556.7(a)-554.5(CNT)]TJ
/F5 1 Tf
17.2822 0 TD
(C)Tj
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1.2178 0 TD
(is)Tj
/F5 1 Tf
1.2235 0 TD
(optimal)Tj
/F1 1 Tf
3.6192 0 TD
[(if)-552.6(it)-559(realizes)-562.5(dist\()]TJ
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7.6993 0 TD
(S)Tj
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()Tj
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(T)Tj
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[(\))-722.5(that)-556.7(is,)]TJ
/F7 1 Tf
-38.8266 -1.2007 TD
(j)Tj
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(C)Tj
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(j)Tj
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(=)Tj
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1.7811 0 TD
(S)Tj
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()Tj
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(T)Tj
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[(\))-321.4(\(there)-327.8(may)-326.3(exist)-325.3(several)-329.6(optimal)-326.2(CNTs\))-9.7(.)-319.1(W)0(e)-324.5(let)]TJ
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20.5087 0 TD
(B)Tj
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(=)Tj
/F1 1 Tf
.882 0 TD
(max)Tj
/F7 1 Tf
1.7356 0 TD
(f)Tj
/F1 1 Tf
.5008 0 TD
(max)Tj
/F5 1 Tf
6.9738 0 0 6.9738 423.3259 414.8786 Tm
(n)Tj
0 -.9186 TD
(i)Tj
/F9 1 Tf
.4552 0 TD
(=)Tj
/F1 1 Tf
.7154 0 TD
(1)Tj
/F7 1 Tf
9.9626 0 0 9.9626 435.5148 411.2503 Tm
(f)Tj
/F5 1 Tf
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(s)Tj
6.9738 0 0 6.9738 444.3022 409.7762 Tm
(i)Tj
/F7 1 Tf
9.9626 0 0 9.9626 446.7967 411.2503 Tm
(g)Tj
/F1 1 Tf
.5008 0 TD
[()-332.8(max)]TJ
/F5 1 Tf
6.9738 0 0 6.9738 475.7101 414.8786 Tm
(n)Tj
0 -.9186 TD
(i)Tj
/F9 1 Tf
.4471 0 TD
(=)Tj
/F1 1 Tf
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(1)Tj
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(f)Tj
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(t)Tj
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(i)Tj
/F7 1 Tf
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(gg)Tj
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1.3202 0 TD
[(denot)-8.4(e)]TJ
-40.5167 -1.2007 TD
[(the)-336.9(maximum)-339.1(CN)-340.2(in)-334.3(the)-336.9(input.)-339.8(Finally,)-339.4(for)-336.3(all)-334(1)]TJ
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[(,)-336.1(w)0(e)-337.3(dene)]TJ
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(i)Tj
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[(3.)-496.7(AN)-333.2(ALGORITHM)-328.8(FOR)-333.9(CNTP)]TJ
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9.9626 0 0 9.9626 117.5811 339.5338 Tm
[(We)-330.2(rst)-326.9(present)-333.5(an)]TJ
/F5 1 Tf
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(O)Tj
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(nB)Tj
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6.9738 0 0 6.9738 219.9117 343.1621 Tm
(2)Tj
9.9626 0 0 9.9626 223.937 339.5338 Tm
[(\)-time)-332.3(and)]TJ
/F5 1 Tf
4.5638 0 TD
(O)Tj
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-17.9082 -1.2007 TD
[(\(Secti)-9.2(ons)-362(3.1)-366.1(and)-362.6(3.2\).)-363.5(Re)-8(call)-365(that)]TJ
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14.5337 0 TD
(B)Tj
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[(is)-364(the)-365.3(maximal)-368.6(integer)-369.2(in)-362.8(the)-365.3(input,)-368.2(so)-361.2(that)-368.9(algorithm)-369.6(is)-364(pseudo)-7.5(po-)]TJ
-15.5124 -1.2007 TD
[(lynomi)-7.9(al.)-342.2(Then,)-346.5(we)-343(modi)-9.3(fy)-339.3(this)-345.5(algori)-8.2(thm)-344.3(to)-340(run)-345.7(in)-340(linear)-351.4(time)-343.9(\(Secti)-9.2(on)-343(3.3\).)-346.5(On)-341.2(a)-344(high)-341.6(leve)-9.4(l,)-339.7(the)-348.3(modi-)]TJ
T*
[(cation)-303.9(is)-290(b)-6.5(ased)-298(on)-291.7(the)-297.1(obser)-8.8(vation)-296.4(that)-300.6(the)-297.1(table)-297.4(used)-301.9(by)-291.7(the)-297.1(algo)-7.6(rithm)-299.4(to)-294.5(store)-296.1(valu)-7.6(es)-294.7(of)-293.7(p)-6.5(artial)-297.2(solu)-7(tions)]TJ
T*
[(can)-296.1(be)-293.5(describ)-9.1(ed)-293.5(by)]TJ
/F5 1 Tf
8.3651 0 TD
(O)Tj
/F1 1 Tf
.7227 0 TD
(\()Tj
/F5 1 Tf
.3357 0 TD
(n)Tj
/F1 1 Tf
.5008 0 TD
[(\))-293(piecewis)-10.7(e)-292.8(linear)-300.2(functions,)-296.6(whe)-8(re)-295.5(each)-298.6(function)-299.9(encapsul)-9.7(ates)]TJ
/F5 1 Tf
25.6473 0 TD
(O)Tj
/F1 1 Tf
.7227 0 TD
(\()Tj
/F5 1 Tf
.3301 0 TD
(B)Tj
/F1 1 Tf
.6146 0 TD
[(\))-293(entr)-9.6(ies)-292.6(of)-293.7(the)]TJ
-37.239 -1.2007 TD
[(table.)-297.8(We)-296.1(show)-298.4(that)-294.9(each)-298.6(functi)-8.2(on)-291.7(has)-295.5(only)-296(three)-296.6(linear)-300.2(segment)-10.7(s,)-292.6(and)-294.3(so,)-293.3(the)-297.1(computa)-9.5(tion)-293.1(of)-293.7(an)-293.5(entry)-300.6(can)]TJ
T*
[(be)-407.3(perf)-8.8(ormed)-409.4(in)-408.3(time)]TJ
/F5 1 Tf
9.3382 0 TD
(O)Tj
/F1 1 Tf
.7227 0 TD
[(\(1\))-410.3(rather)-413.2(than)]TJ
/F5 1 Tf
6.4702 0 TD
(O)Tj
/F1 1 Tf
.7227 0 TD
(\()Tj
/F5 1 Tf
.33 0 TD
(B)Tj
/F1 1 Tf
.6203 0 TD
[(\).)-407.2(Furthermor)-9.5(e,)-412.7(since)-409.7(each)-412.4(funct)-10.3(ion)-409.1(can)-409.9(be)-413(represente)-11.3(d)-404.8(i)0(n)-408.3(a)]TJ
-18.2041 -1.195 TD
[(compact)-301.5(man)-7.6(ner,)-296.6(the)-297.1(size)-300.8(of)-293.7(table)-303.1(shrinks)-302.7(from)]TJ
/F5 1 Tf
19.4674 0 TD
(O)Tj
/F1 1 Tf
.7227 0 TD
(\()Tj
/F5 1 Tf
.3301 0 TD
(nB)Tj
/F1 1 Tf
1.121 0 TD
.293 Tc
[(\)t)293(o)]TJ
/F5 1 Tf
1.7015 0 TD
0 Tc
(O)Tj
/F1 1 Tf
.7227 0 TD
(\()Tj
/F5 1 Tf
.3357 0 TD
(n)Tj
/F1 1 Tf
.5008 0 TD
[(\).)-293.3(The)-299.8(precise)-301.2(deni)-7.6(tions)-295.1(of)-299.4(the)-297.1(table)-297.4(and)-300(the)]TJ
-24.9018 -1.2007 TD
[(functi)-8.2(ons)-401.8(are)-411.9(given)-406.7(in)-402.6(Sect)-8.6(ions)-405.3(3.2)-405.9(and)-408.1(3.3.)-406.3(Our)-406.5(proof)-406.1(of)-407.5(the)-405.2(correctnes)-11.3(s)-406(o)0(f)-401.8(the)-410.9(use)-403.6(of)-407.5(these)-409.7(functions)]TJ
T*
[(require)-9.9(s)-332(a)-332.6(som)-7(ewhat)-338.4(extensi)-9.9(ve)-333.4(case)-339.7(anal)-9.4(ysis)-332.6(that)-340.4(is)-335.5(present)-9.1(ed)-333.4(separat)-10.9(ely)-336.9(in)-334.3(Section)-338(3.4.)]TJ
/F5 1 Tf
10.959 0 0 10.959 107.6031 208.0062 Tm
[(3.1.)-496.9(Key)-333.4(propositions)]TJ
/F1 1 Tf
9.9626 0 0 9.9626 117.5811 190.0913 Tm
[(We)-318.8(start)-321.6(by)-320.2(devel)-10.1(oping)]TJ
/F12 1 Tf
7.9702 0 0 7.9702 215.8865 190.0913 Tm
[(DpCntpA)14.7(lg)]TJ
/F1 1 Tf
9.9626 0 0 9.9626 269.518 190.0913 Tm
.3191 Tc
[(,a)319.1(n)]TJ
/F5 1 Tf
1.838 0 TD
0 Tc
(O)Tj
/F1 1 Tf
.717 0 TD
(\()Tj
/F5 1 Tf
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(nB)Tj
/F1 1 Tf
6.9738 0 0 6.9738 309.4298 193.7196 Tm
(2)Tj
9.9626 0 0 9.9626 313.455 190.0913 Tm
[(\)-time)-321(dynam)-9.1(ic)-319.1(progr)-7.8(amming)-323.5(algori)-8.2(thm)-315.9(for)-324.9(CNTP.)-324.1(Let)]TJ
-20.6624 -1.2007 TD
(\()Tj
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/F9 1 Tf
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(=)Tj
/F1 1 Tf
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(s)Tj
/F1 1 Tf
6.9738 0 0 6.9738 131.9244 176.6551 Tm
(1)Tj
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()Tj
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(s)Tj
/F1 1 Tf
6.9738 0 0 6.9738 144.737 176.6551 Tm
(2)Tj
9.9626 0 0 9.9626 148.7622 178.1291 Tm
()Tj
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1.3145 0 TD
0 Tc
()Tj
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(s)Tj
6.9738 0 0 6.9738 177.3354 176.6551 Tm
(n)Tj
/F1 1 Tf
9.9626 0 0 9.9626 181.3039 178.1291 Tm
(\))Tj
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(T)Tj
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/F1 1 Tf
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(t)Tj
/F1 1 Tf
6.9738 0 0 6.9738 213.6755 176.6551 Tm
(1)Tj
9.9626 0 0 9.9626 217.7007 178.1291 Tm
()Tj
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(t)Tj
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6.9738 0 0 6.9738 225.4677 176.6551 Tm
(2)Tj
9.9626 0 0 9.9626 229.4929 178.1291 Tm
()Tj
/F10 1 Tf
.6658 0 TD
.1682 Tc
(...)Tj
/F1 1 Tf
1.3145 0 TD
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()Tj
/F5 1 Tf
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(t)Tj
6.9738 0 0 6.9738 256.9889 176.6551 Tm
(n)Tj
/F1 1 Tf
9.9626 0 0 9.9626 261.0141 178.1291 Tm
[(\)\))-278.6(be)-282.2(the)-280(input.)-282.9(Obs)-7.4(erve)-281.7(that)-283.5(there)-285.3(exis)-8.8(ts)-278.6(a)-281.4(CNT)]TJ
/F5 1 Tf
19.5584 0 TD
(C)Tj
/F1 1 Tf
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[(such)-284.9(that)]TJ
/F5 1 Tf
3.9094 0 TD
(C)Tj
/F1 1 Tf
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(\()Tj
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(S)Tj
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(\))Tj
/F9 1 Tf
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(=)Tj
/F5 1 Tf
.7227 0 TD
(T)Tj
/F1 1 Tf
-42.5653 -1.2007 TD
[(if)-262.3(and)-271.5(only)-261.9(if)-268(there)-268.2(does)-267.8(not)-266.8(exist)-268.4(an)-265.1(inde)-7.6(x)-262.5(1)]TJ
/F7 1 Tf
18.0959 0 TD
()Tj
/F5 1 Tf
1.0471 0 TD
(i)Tj
/F7 1 Tf
.5634 0 TD
()Tj
/F5 1 Tf
1.0471 0 TD
(n)Tj
/F1 1 Tf
.7625 0 TD
[(such)-267.8(that)]TJ
/F5 1 Tf
3.881 0 TD
(s)Tj
6.9738 0 0 6.9738 364.422 164.6929 Tm
(i)Tj
/F9 1 Tf
9.9626 0 0 9.9626 368.5605 166.1669 Tm
(=)Tj
/F1 1 Tf
.7227 0 TD
[(0)-262.5(and)]TJ
/F5 1 Tf
2.4754 0 TD
(t)Tj
6.9738 0 0 6.9738 403.2566 164.6929 Tm
(i)Tj
/F11 1 Tf
9.9626 0 0 9.9626 408.4723 166.1669 Tm
(>)Tj
/F1 1 Tf
1.0471 0 TD
[(0.)-268.6(Since)-265.4(the)-268.6(existe)-9.2(nce)-267.6(of)-265.3(such)]TJ
-31.2468 -1.2007 TD
[(an)-276.5(index)-281.5(can)-279(be)-276.5(determin)-9.3(ed)-276.5(in)-277.4(linear)-283.1(time)-275.7(\(w)-7.4(here,)-276.4(if)-279.4(such)-279.2(an)-276.5(inde)-7.6(x)-273.9(i)0(s)-272.9(found,)-285(we)-274.7(return)]TJ
/F7 1 Tf
35.0993 0 TD
(1)Tj
/F1 1 Tf
1.0015 0 TD
[(\),)-276.3(we)-280.4(will)-277(assum)-11.5(e)]TJ
-36.1008 -1.2007 TD
[(that)-255.1(dist\()]TJ
/F5 1 Tf
3.5452 0 TD
(S)Tj
/F1 1 Tf
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()Tj
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(T)Tj
/F1 1 Tf
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(\))Tj
/F11 1 Tf
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(<)Tj
/F7 1 Tf
1.0528 0 TD
(1)Tj
/F1 1 Tf
1.0015 0 TD
[(.)-250.8(T)0(o)-251.7(simpl)-8.4(ify)-251.7(the)-251.5(present)-9.1(ation,)-256.2(we)-251.9(further)-254.8(assum)-11.5(e)-247.2(w.l.o.g.)-256(that)]TJ
/F5 1 Tf
24.9872 0 TD
(t)Tj
/F1 1 Tf
6.9738 0 0 6.9738 437.5558 140.7684 Tm
(1)Tj
9.9626 0 0 9.9626 441.5243 142.2425 Tm
()Tj
/F5 1 Tf
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(t)Tj
6.9738 0 0 6.9738 449.3479 140.7684 Tm
(n)Tj
/F7 1 Tf
9.9626 0 0 9.9626 456.0944 142.2425 Tm
(6)Tj
T*
()Tj
/F1 1 Tf
1.0471 0 TD
[(0.)-251.5(Indee)-9.4(d,)-251.5(if)]TJ
/F5 1 Tf
5.1044 0 TD
(t)Tj
/F1 1 Tf
6.9738 0 0 6.9738 520.1573 140.7684 Tm
(1)Tj
/F9 1 Tf
9.9626 0 0 9.9626 525.8267 142.2425 Tm
(=)Tj
/F1 1 Tf
.717 0 TD
(0)Tj
-42.6962 -1.2007 TD
(or)Tj
/F5 1 Tf
1.1836 0 TD
(t)Tj
6.9738 0 0 6.9738 122.2299 128.8062 Tm
(n)Tj
/F9 1 Tf
9.9626 0 0 9.9626 127.8992 130.2803 Tm
(=)Tj
/F1 1 Tf
.717 0 TD
[(0,)-354(we)-354.4(can)-358.7(solve)-356.7(the)-354(input)-356.5(\()]TJ
/F5 1 Tf
11.1762 0 TD
(S)Tj
/F9 1 Tf
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(=)Tj
/F1 1 Tf
1.1381 0 TD
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1.3373 0 TD
(s)Tj
/F1 1 Tf
6.9738 0 0 6.9738 279.8928 128.8062 Tm
(1)Tj
9.9626 0 0 9.9626 283.8614 130.2803 Tm
()Tj
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(s)Tj
/F1 1 Tf
6.9738 0 0 6.9738 292.7054 128.8062 Tm
(2)Tj
9.9626 0 0 9.9626 296.6739 130.2803 Tm
()Tj
/F10 1 Tf
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(...)Tj
/F1 1 Tf
1.3145 0 TD
0 Tc
()Tj
/F5 1 Tf
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(s)Tj
6.9738 0 0 6.9738 325.2471 128.8062 Tm
(n)Tj
/F1 1 Tf
9.9626 0 0 9.9626 329.2723 130.2803 Tm
[()-167.8(1\))]TJ
/F5 1 Tf
1.838 0 TD
(T)Tj
/F9 1 Tf
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(=)Tj
/F1 1 Tf
1.1381 0 TD
0 Tc
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1.3373 0 TD
(t)Tj
/F1 1 Tf
6.9738 0 0 6.9738 381.3164 128.8062 Tm
(1)Tj
9.9626 0 0 9.9626 385.3416 130.2803 Tm
()Tj
/F5 1 Tf
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(t)Tj
/F1 1 Tf
6.9738 0 0 6.9738 393.1086 128.8062 Tm
(2)Tj
9.9626 0 0 9.9626 397.1337 130.2803 Tm
()Tj
/F10 1 Tf
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(...)Tj
/F1 1 Tf
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()Tj
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(t)Tj
6.9738 0 0 6.9738 424.6298 128.8062 Tm
(n)Tj
/F1 1 Tf
9.9626 0 0 9.9626 428.655 130.2803 Tm
[()-167.8(1\)\))-353.4(instead,)-357.4(since)-358.5(it)-354.2(holds)]TJ
-32.2256 -1.195 TD
[(that)-414.4(dist\()]TJ
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3.7045 0 TD
(S)Tj
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()Tj
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(T)Tj
/F1 1 Tf
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(\))Tj
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1.7811 0 TD
(S)Tj
/F9 1 Tf
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/F1 1 Tf
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()Tj
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(T)Tj
/F9 1 Tf
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()Tj
/F1 1 Tf
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[(\).)-412.8(Finally,)-413.3(we)-417(assume)-418.1(w.l.o.g.)-415.3(for)-410.3(all)-413.6(1)]TJ
/F7 1 Tf
16.6392 0 TD
()Tj
/F5 1 Tf
1.0527 0 TD
(i)Tj
/F7 1 Tf
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()Tj
/F5 1 Tf
1.0471 0 TD
(n)Tj
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(,)Tj
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(s)Tj
6.9738 0 0 6.9738 419.7542 116.8441 Tm
(i)Tj
/F11 1 Tf
9.9626 0 0 9.9626 425.0266 118.3748 Tm
(>)Tj
/F1 1 Tf
1.0471 0 TD
[(0.)-410.9(Indee)-9.4(d,)-410.9(if)-410.3(there)-416.2(exists)]TJ
-32.9084 -1.2007 TD
(1)Tj
/F7 1 Tf
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()Tj
/F5 1 Tf
1.0471 0 TD
(i)Tj
/F7 1 Tf
.5634 0 TD
()Tj
/F5 1 Tf
1.0471 0 TD
(n)Tj
/F1 1 Tf
1.0471 0 TD
[(such)-552.3(that)]TJ
/F5 1 Tf
4.4386 0 TD
(s)Tj
6.9738 0 0 6.9738 200.3527 104.8819 Tm
(i)Tj
/F9 1 Tf
9.9626 0 0 9.9626 204.4913 106.4126 Tm
(=)Tj
/F1 1 Tf
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[(0,)-547.4(then)-553.9(also)]TJ
/F5 1 Tf
5.7418 0 TD
(t)Tj
6.9738 0 0 6.9738 271.6157 104.8819 Tm
(i)Tj
/F9 1 Tf
9.9626 0 0 9.9626 275.7543 106.4126 Tm
(=)Tj
/F1 1 Tf
.717 0 TD
[(0,)-547.4(and)-550.4(we)-547.8(can)-552.2(solve)-550.2(the)-547.4(input)-549.9(\()]TJ
/F5 1 Tf
14.3402 0 TD
(S)Tj
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(=)Tj
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1.1381 0 TD
0 Tc
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(s)Tj
/F1 1 Tf
6.9738 0 0 6.9738 449.2345 104.8819 Tm
(1)Tj
9.9626 0 0 9.9626 453.203 106.4126 Tm
()Tj
/F10 1 Tf
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(...)Tj
/F1 1 Tf
1.3145 0 TD
0 Tc
()Tj
/F5 1 Tf
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(s)Tj
6.9738 0 0 6.9738 481.7762 104.8819 Tm
(i)Tj
/F9 1 Tf
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(-)Tj
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(1)Tj
9.9626 0 0 9.9626 493.9652 106.4126 Tm
()Tj
/F5 1 Tf
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(s)Tj
6.9738 0 0 6.9738 502.7526 104.8819 Tm
(i)Tj
/F9 1 Tf
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(+)Tj
/F1 1 Tf
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(1)Tj
9.9626 0 0 9.9626 514.9416 106.4126 Tm
()Tj
/F10 1 Tf
.6658 0 TD
.1682 Tc
(...)Tj
/F1 1 Tf
1.3145 0 TD
0 Tc
()Tj
/F5 1 Tf
-42.8669 -1.2007 TD
(s)Tj
6.9738 0 0 6.9738 111.4582 92.9197 Tm
(n)Tj
/F1 1 Tf
9.9626 0 0 9.9626 115.4267 94.4504 Tm
(\))Tj
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[(2.)-501.9(\()]TJ
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[(are)-326.5(disjoint.)-326(Thus,)-328.8(we)]TJ
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[(1)-168.5(1\).)-817.6(Fo)-6.4(r)-816.5(any)-817.8(CNP)]TJ
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[(,)-336.1(fol-)]TJ
-38.7413 -1.2007 TD
[(lowed)-296.8(by)-297.4(an)-293.5(applica)-10.5(tion)-293.1(of)]TJ
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[(,)-290.6(does)-296.2(not)-295.3(change)-300.1(any)-294.3(entr)-9.6(y)]TJ
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[(such)-296.2(that)]TJ
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[(.)-290.6(W)0(e)-296.1(have)-296.8(that)]TJ
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[(:)-253.9(Dene)]TJ
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[(1\))-253.9(and)]TJ
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9.9626 0 0 9.9626 397.5873 380.7495 Tm
[()-167.8(1\).)-254.3(As)-251.3(in)-254.7(the)-257.2(second)-259.7(case,)]TJ
-29.7104 -1.2007 TD
[(we)-337.3(obtain)-336.2(an)-339.1(optimal)-337.6(CNT)-340.8(that)-340.4(satises)-342.3(Conditio)-7.7(n)-330.8(1)0(.)]TJ
-1.7527 -1.2007 TD
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