Эффективные алгоритмы, осень 2007: Приближённые алгоритмы

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    .

    Computer Science http://logic.pdmi.ras.ru/infclub/

    . (CS ) 2. 1 / 15

    http://logic.pdmi.ras.ru/~infclub/

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    . (CS ) 2. 2 / 15

  • A (relative performance guarantee) , I

    I A(I )/OPT(I ) ;I A(I )/OPT(I ) .

    -.

    1, 1. , 1.

    . (CS ) 2. 3 / 15

  • A (relative performance guarantee) , I

    I A(I )/OPT(I ) ;I A(I )/OPT(I ) .

    -.

    1, 1. , 1.

    . (CS ) 2. 3 / 15

  • A (relative performance guarantee) , I

    I A(I )/OPT(I ) ;

    I A(I )/OPT(I ) . -.

    1, 1. , 1.

    . (CS ) 2. 3 / 15

  • A (relative performance guarantee) , I

    I A(I )/OPT(I ) ;I A(I )/OPT(I ) .

    -.

    1, 1. , 1.

    . (CS ) 2. 3 / 15

  • A (relative performance guarantee) , I

    I A(I )/OPT(I ) ;I A(I )/OPT(I ) .

    -.

    1, 1. , 1.

    . (CS ) 2. 3 / 15

  • A (relative performance guarantee) , I

    I A(I )/OPT(I ) ;I A(I )/OPT(I ) .

    -.

    1, 1. , 1.

    . (CS ) 2. 3 / 15

  • 1

    . (CS ) 2. 4 / 15

  • {s1, . . . , sn}. (shortestcommon superstring) u, si .

    NP-.

    . (CS ) 2. 5 / 15

  • {s1, . . . , sn}.

    (shortestcommon superstring) u, si .

    NP-.

    . (CS ) 2. 5 / 15

  • {s1, . . . , sn}. (shortestcommon superstring) u, si .

    NP-.

    . (CS ) 2. 5 / 15

  • {s1, . . . , sn}. (shortestcommon superstring) u, si .

    NP-.

    . (CS ) 2. 5 / 15

  • U = {u1, . . . , un} = {S1, . . . , Sk}, Si U. U: U =

    S S .

    Si pi 0. (set cover problem) , U .

    Hn- .

    . (CS ) 2. 6 / 15

  • U = {u1, . . . , un} = {S1, . . . , Sk}, Si U.

    U: U =

    S S . Si pi 0. (set cover problem) , U .

    Hn- .

    . (CS ) 2. 6 / 15

  • U = {u1, . . . , un} = {S1, . . . , Sk}, Si U. U: U =

    S S .

    Si pi 0. (set cover problem) , U .

    Hn- .

    . (CS ) 2. 6 / 15

  • U = {u1, . . . , un} = {S1, . . . , Sk}, Si U. U: U =

    S S .

    Si pi 0.

    (set cover problem) , U .

    Hn- .

    . (CS ) 2. 6 / 15

  • U = {u1, . . . , un} = {S1, . . . , Sk}, Si U. U: U =

    S S .

    Si pi 0. (set cover problem) , U .

    Hn- .

    . (CS ) 2. 6 / 15

  • U = {u1, . . . , un} = {S1, . . . , Sk}, Si U. U: U =

    S S .

    Si pi 0. (set cover problem) , U .

    Hn- .

    . (CS ) 2. 6 / 15

  • , i , j , k , si k sj k , wijk = si sj [k + 1..|sj |]:

    s6 = abcab X

    bbca Y

    , s2 = bbca Y

    aaccbcac Z

    w6,2,4 = abcab X

    bbca Y

    aaccbcac Z

    s set(s) = {si |si s} set(s) s : U = {s1, . . . , sn}, = {set(si )} {set(wijk)}

    . (CS ) 2. 7 / 15

  • ,

    i , j , k , si k sj k , wijk = si sj [k + 1..|sj |]:

    s6 = abcab X

    bbca Y

    , s2 = bbca Y

    aaccbcac Z

    w6,2,4 = abcab X

    bbca Y

    aaccbcac Z

    s set(s) = {si |si s} set(s) s : U = {s1, . . . , sn}, = {set(si )} {set(wijk)}

    . (CS ) 2. 7 / 15

  • , i , j , k , si k sj k , wijk = si sj [k + 1..|sj |]:

    s6 = abcab X

    bbca Y

    , s2 = bbca Y

    aaccbcac Z

    w6,2,4 = abcab X

    bbca Y

    aaccbcac Z

    s set(s) = {si |si s} set(s) s : U = {s1, . . . , sn}, = {set(si )} {set(wijk)}

    . (CS ) 2. 7 / 15

  • , i , j , k , si k sj k , wijk = si sj [k + 1..|sj |]:

    s6 = abcab X

    bbca Y

    , s2 = bbca Y

    aaccbcac Z

    w6,2,4 = abcab X

    bbca Y

    aaccbcac Z

    s set(s) = {si |si s} set(s) s : U = {s1, . . . , sn}, = {set(si )} {set(wijk)}

    . (CS ) 2. 7 / 15

  • , i , j , k , si k sj k , wijk = si sj [k + 1..|sj |]:

    s6 = abcab X

    bbca Y

    , s2 = bbca Y

    aaccbcac Z

    w6,2,4 = abcab X

    bbca Y

    aaccbcac Z

    s set(s) = {si |si s} set(s) s : U = {s1, . . . , sn}, = {set(si )} {set(wijk)}

    . (CS ) 2. 7 / 15

  • , i , j , k , si k sj k , wijk = si sj [k + 1..|sj |]:

    s6 = abcab X

    bbca Y

    , s2 = bbca Y

    aaccbcac Z

    w6,2,4 = abcab X

    bbca Y

    aaccbcac Z

    s set(s) = {si |si s}

    set(s) s : U = {s1, . . . , sn}, = {set(si )} {set(wijk)}

    . (CS ) 2. 7 / 15

  • , i , j , k , si k sj k , wijk = si sj [k + 1..|sj |]:

    s6 = abcab X

    bbca Y

    , s2 = bbca Y

    aaccbcac Z

    w6,2,4 = abcab X

    bbca Y

    aaccbcac Z

    s set(s) = {si |si s} set(s) s

    : U = {s1, . . . , sn}, = {set(si )} {set(wijk)}

    . (CS ) 2. 7 / 15

  • , i , j , k , si k sj k , wijk = si sj [k + 1..|sj |]:

    s6 = abcab X

    bbca Y

    , s2 = bbca Y

    aaccbcac Z

    w6,2,4 = abcab X

    bbca Y

    aaccbcac Z

    s set(s) = {si |si s} set(s) s : U = {s1, . . . , sn}, = {set(si )} {set(wijk)}

    . (CS ) 2. 7 / 15

  • ()

    , , si , 2Hn-

    .

    . (CS ) 2. 8 / 15

  • ()

    ,

    , si , 2Hn-

    .

    . (CS ) 2. 8 / 15

  • ()

    , , si

    , 2Hn-

    .

    . (CS ) 2. 8 / 15

  • ()

    , , si , 2Hn-

    .

    . (CS ) 2. 8 / 15

  • ()

    , , si , 2Hn-

    .

    . (CS ) 2. 8 / 15

  • ()

    , , si , 2Hn-

    .

    . (CS ) 2. 8 / 15

  • si (-) s, : s1 si , s1 w1,j ,k , , , .. ,

    . (CS ) 2. 9 / 15

  • si (-) s

    , : s1 si , s1 w1,j ,k , , , .. ,

    . (CS ) 2. 9 / 15

  • si (-) s,

    : s1 si , s1 w1,j ,k , , , .. ,

    . (CS ) 2. 9 / 15

  • si (-) s, : s1 si , s1

    w1,j ,k , , , .. ,

    . (CS ) 2. 9 / 15

  • si (-) s, : s1 si , s1 w1,j ,k

    , , , .. ,

    . (CS ) 2. 9 / 15

  • si (-) s, : s1 si , s1 w1,j ,k , , , ..

    ,

    . (CS ) 2. 9 / 15

  • si (-) s, : s1 si , s1 w1,j ,k , , , .. ,

    . (CS ) 2. 9 / 15

  • ()

    , s , s

    .

    . (CS ) 2. 10 / 15

  • ()

    , s

    , s

    .

    . (CS ) 2. 10 / 15

  • ()

    , s , s

    .

    . (CS ) 2. 10 / 15

  • ()

    , s , s

    .

    . (CS ) 2. 10 / 15

  • 1

    . (CS ) 2. 11 / 15

  • , :s1 = abc , s2 = bac , s3 = bcd , s4 = cde wijk :w132 = abcd , w141 = abcde, w241 = bacde, w342 = bcde set(si ) set(wijk), :S1 = set(s1) = {s1}, p1 = 3,S2 = set(s2) = {s2}, p2 = 3,S3 = set(s3) = {s3}, p3 = 3,S4 = set(s4) = {s4}, p4 = 3,S5 = set(w132) = {s1, s3}, p5 = 4S6 = set(w141) = {s1, s3, s4}, p6 = 5,S7 = set(w241) = {s2, s4}, p7 = 5,S8 = set(w342) = {s3, s4}, p8 = 4

    . (CS ) 2. 12 / 15

  • , :s1 = abc , s2 = bac , s3 = bcd , s4 = cde

    wijk :w132 = abcd , w141 = abcde, w241 = bacde, w342 = bcde set(si ) set(wijk), :S1 = set(s1) = {s1}, p1 = 3,S2 = set(s2) = {s2}, p2 = 3,S3 = set(s3) = {s3}, p3 = 3,S4 = set(s4) = {s4}, p4 = 3,S5 = set(w132) = {s1, s3}, p5 = 4S6 = set(w141) = {s1, s3, s4}, p6 = 5,S7 = set(w241) = {s2, s4}, p7 = 5,S8 = set(w342) = {s3, s4}, p8 = 4

    . (CS ) 2. 12 / 15

  • , :s1 = abc , s2 = bac , s3 = bcd , s4 = cde wijk :w132 = abcd , w141 = abcde, w241 = bacde, w342 = bcde

    set(si ) set(wijk), :S1 = set(s1) = {s1}, p1 = 3,S2 = set(s2) = {s2}, p2 = 3,S3 = set(s3) = {s3}, p3 = 3,S4 = set(s4) = {s4}, p4 = 3,S5 = set(w132) = {s1, s3}, p5 = 4S6 = set(w141) = {s1, s3, s4}, p6 = 5,S7 = set(w241) = {s2, s4}, p7 = 5,S8 = set(w342) = {s3, s4}, p8 = 4

    . (CS ) 2. 12 / 15

  • , :s1 = abc , s2 = bac , s3 = bcd , s4 = cde wijk :w132 = abcd , w141 = abcde, w241 = bacde, w342 = bcde set(si ) set(wijk), :S1 = set(s1) = {s1}, p1 = 3,S2 = set(s2) = {s2}, p2 = 3,S3 = set(s3) = {s3}, p3 = 3,S4 = set(s4) = {s4}, p4 = 3,S5 = set(w132) = {s1, s3}, p5 = 4S6 = set(w141) = {s1, s3, s4}, p6 = 5,S7 = set(w241) = {s2, s4}, p7 = 5,S8 = set(w342) = {s3, s4}, p8 = 4

    . (CS ) 2. 12 / 15

  • , U = {si}i[1..4], = {Si}i[1..8], Si pi

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