????t??? ??t s 1 ??? s 2 ?? t?? t?? s??r??s ??t? s 1 < s 2 ? ??rst? ?? ???? t?? ???? ?? t?? ????r?t??? t??? t?? r???rs??? ??r????? ?s?? t? ?????t? OP T (s 1 , s 2 ; [s 1 ,
???s ??t?r ?(i) ?s r??t?? ??? t?? t????? (i, ?(i))? ?? st?rt??? ?t s 1 ? ?(s 1 ) ??ts ?s ? s??r?? ??r t?? ????s ?? [?(s 1 ), n]? ??r?? ??s?s ??? ??????? ?? ?? ?(s 1 ) < s 2 ? t??? ?? ???s???r t?? s??? ?r????? ??t? s??r??s ?(s 1 ) ???rr???? ,
s 1 ) = s 2 ? t??? ?? ???s???r t?? ?r????? ??t? ???? ??? s??r?? s 2 ??rr???? t?? tr??? ?r?? ??t? s??r??s s 1 ??? s 2 ? ,
> s 2 ? t??? ?? ???s???r t?? ?r?????s ??t? t?? s??r??s s 1 ??? s 2 ?? ????t? [s 1 , ?(s 1 ) ? 1] ??? t?? t?? s??r??s s 2 ??? ?(s 1 ) ?? ????t? [?(s 1 ), n ,
) ??t? ??t? tr???? ????r? t??t ?(s 2 ) = ?(s 1 )??r?s ?(s 1 )? s 2 ??s?rts ??r??t?? t?? tr??? ?? (s 1 , ?(s 1 ))s 2 ) > ?(s 1 )? ??? tr??? ?r?? s 2 t? ????s ??t?r ?(s 2 ) ?s r??t?? ??? t?? t????? (s 2 , ?(s 2 ))? ??r t?? ??t?r???, ?? ???? t? ???? ??t? ? ?r????? ??t? t?? s??r??s ?(s 1 ) ??? ?(s 2 )? ?? r????t t?? ?r?????t ??t? ? 2 (s 1 ) ??r t?? tr??? t???r?s ????s ??t?r ? ,
s 1 ) < ?(s 2 ) ?? ?????t t?? tr??? ?? ?(s 1 ) ?? t?? t????? (s 2 , ?(s 2 )) ,
??t? t?? s??r??s ?(s 2 ) ??? ? 2 (s 1 )? ??? s? ??? ,
? s 1 ?s r??t?? t? ????s ???r ????? ??? t?????s (s 1 , ?(s 1 ))? (?(s 1 ), ? 2 (s 1 ))? (? 2 (s 1 ), ? 3 (s 1 ))? . . . ??? s????t?????s?? ??r s 2 ??? t?????s ,
?? ? h (s 1 ) < ? h?1 (s 2 ) ??r ? h ? (s 2 ) < ? h ? (s 1 )) ??? t?? ?r????? ??????s ? ?r????? ??t? ??? s??r?? ?? ? h?1 (s 2 ) ?r?s?? ? h ? (s 1 )?? ,
2 ; [i, u])? t?? ??st ?? ?? ??t???? s???t??? ?? s??????t? [i, u] ??t? s??r??s i ??? s 2 ???r? s 1 ? i ? s 2 ? i < u ? n ??? i ??rr??s t?? tr??? ?? s 1 ? t??t ?s? m i,x = m s1,x ??r i < x ? u? ?? OP T ? ? (i, j; [j, u])? ??t? ? = 1 ?r 2 t?? ??st ?? ?? ??t???? s???t??? ?? s??????t? [j, u] ??t? s??r??s i ??? j? ???r? s 2 ? i < j < u ? n ??? ???r? t??r? ???sts ? t????? (?(j), j) ??t? ?(j) < i? ??r OP T ? 1 (i, j, [j, u])? i ??rr??s t?? tr??? ?? s 2 ,
t? ??s?s ?? i = s 2 ?r i = j t?? ?r????? ?s r?????? t? t?? ??s? ?? ??? s??r?? i ??rr???? t?? tr??? ?? ??t? s??r??s s 1 ??? s 2 ? t??t ?s? m i,x = m s1,x + m s2 ,
? ??t? t??t i ?s ?? ??r? ? h (s ? ) ??t? ? = 1 ?r (s ) ? ?s t?? ????? ?s ??t tr??? t??s s???t??? ?s?s t? ?r??? t?? tr??? ?? i t? s??? ???? x? ??t? ?(i) < x < u? ? t????? (k, l) ??t? k < ?(i) < l < u? ?? t?? ????????t? ?? i? k ? i? ?t??r??s?? k ???? ??rr? t?? tr??? ?? s??? s??r?? ??? k = ? h ? (s ? )? ??t t??? ?(k) ? j < l ?r???s ? ???tr????t??? ?s t?? t????? (k, l) ?s ?????r t??? (k, ?(k))? ??rt??r??r? k = i? ?t??r??s? (i, ?(i)) ????? ??t ?? t?? ?????st t????? ?r?? i? ???r???r?? k > i ??? t?? s???t??? ?s?s t ? 1 t?????s t? r??t? t?? tr??? ?r?? i t? k? ??r??t? t?? tr??? ?r?? i t? x ?? ?s??? ?rst t?? t????? (i, ?(i)) ??? t??? ?????t??? t?? tr??? ?rr???? ?? ?(i) ??t? t?? t????? (k, l)i)) ,
k, l) ?r?ss??? ?(i) ??t? k < ?(i) < l ?????t ??r???s ??r k = j? ?? s??? ??s?s t?? ??st ?? ?s??? ??r??t?? ? t????? (j, l)? t??t ?s m j,l + ?(j, l) ? 1? ????t ?? ??ss t??? t??t ?? ?s??? t?? t?(i), l)? ?? ????? 2m j,l + ?(?(i), l) ? 1? ??t ?s ,
i) ? 1]? ????? ?s ???? t??t ?? i ?t??t ?s? ?? s 1 ? ?s t?? s??r?? s 2 ?s ??ts??? t?? ??t?r???? ???s t?? tr??? ??st???? ??r t?? ????s ?? [?(i, ????? ?s?s ?s ?rst t????? (i, ?(i))? ?? ???? t? ??? t?? ??st ?? t?? t????? (i, ?(i))? t??t ?s x?[?(i),u] m s1,x + ?(i, ?(i)) ? 1? ,
?(i), u] ?s?s ?rst t?? t????? (i, ?(i)) ,
?? s???t??? ??r t?? tr??? t? [?(i), u] ??t? t?? t?? s??r??s s 2 ??? ?(i)? ???r? ?(i) ??rr??s t?? tr??? ?r?? i? t??t ?s ?? s 1 ? t??t ?s OP T ? OPT(1;[1,1]) and OPT ,
?r??s s 1 ??? s 2 ? ???r? ?(i) ??? ??rr??s t?? tr??? ?? i t? x s??? t??t ?(i) < x ? u? ???s ??? t?? s??r??s s 1 = 1 ???, 2005. ,
i h , s h+1 , . . . , s k ; [i h , u])? ???r? i 1 < i 2 < . . . < i h < s h+1 < . . . < s k ? s 2 < i 1 ? s 3 < i 2 ? . . .? s h < i h?1 ? ???r? t??r? ???sts ? t????? (?(i h ), i h ) ??t? ?(i h ) ? i 1 ? ??? ???? i j ??ts ?s ? s??r?? ??? ??rr??s t?? tr??? ?? t?? s??r?? s ?(j) ??t? ?? ? ??r??t?t??? ?? {1, 2 . . . , k}? ?? ???t? ??r j ? h + 1? i j = s j ? s? t?? s??r?? i j ??rr??s t?? tr??? ?? s j ??? t??r???r? ??r j ? h + 1? ?(j) = j? ?s? ,
s k ; [i h , ?(i 1 ,
s h+m , ?(i 1 ), s h+m+1 ,
?? s h+m ???? ??rr? t?? tr??? ?? ??t? s??r??s s h+m ??? s ?(1) ? ??? ?? ????? t?? ????r?t?? ??t? k ? 1 s??r??s? ?? t??r? ?s ?? t????? ?r?? i 1 ?????? ?? [i h , u]? ???? t?? tr??? ?? i 1 ??? ?? ??r??t?? s??t t? i h ??? t?? t????? (?(i h ), i h ), ?? ?(i 1 ) = s ,
i * h = i 1 , s h+1 , . . . , s k ; [i h ,
1 ??rr??s t?? tr??? ?? ??t? s??r??s s ?(1) ??? s ?(h) ? t?? ?t??r ????s ??rr???? t???r ?r?????s tr???? ??r? ,
route des Lucioles -BP 93 -06902 Sophia Antipolis Cedex (France) Unité de recherche INRIA Futurs : Parc Club Orsay Université -ZAC des Vignes 4, rue Jacques Monod -91893 ORSAY Cedex (France) Unité de recherche INRIA Lorraine : LORIA, Technopôle de Nancy-Brabois -Campus scientifique 615, rue du Jardin Botanique -BP 101 -54602 Villers-lès-Nancy Cedex (France) Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu -35042 Rennes Cedex (France) Unité de recherche INRIA Rhône-Alpes : 655, avenue de l'Europe -38334 Montbonnot Saint, Ismier (France) Unité de recherche INRIA Rocquencourt : Domaine de Voluceau -Rocquencourt -BP 105 -78153 Le Chesnay Cedex, 2004. ,
BP 105 -78153 Le Chesnay Cedex (France) ?tt??????????r????r ,