, ??? a ? ran f t??r? ???sts b ? ran g s??? t??t ?? ???? ?? t?? ???q????t? width F ? |D| ????s? ??t??? t??t t?? ?rst ??? t?? ??st r?s??ts ?r?s??t?? ????? ??r? ??? r???? ????? s???? t??? ?r? ????r?? r?s???t????? ?? t?? ????t???s t???r?
, ran g) ? f (D)? ??r????r? f (ran g) ?s ran(f ? g) ??? f (D) ?s ran f ? ???r???r? t?? ?????s??? ran(f ? g) ? ran f ????s? ??t x, y ? ker g? ?? ???? g(x) = g(y)? t??r???r?, ? g)(x) = (f ? g)(y)? ???? x, y ? ker
, ?s ?? F s??? t??t ker f ?g ? ker f ? ????? ??r ??? x, y ? D s??? t??t f ? g(x) = f ? g(y)? ?? ???? t?? ?q??t??? f (x) = f (y)? ??rst? ??t??? t??t f (x) = f (y) ? f ? g(x) = f ? g(y) ? g(x) = g(y) ???? ??r ??? x, y s??, g)(x 1 ) = (f ? g)(y 1 ) ??? t?? ?q??t??? f (x 1 ) = f
, = f (y 2 )? ??? ?? ?????? ?r?? ker f ? g ? ker f t??t t?? ???q????t? f ? g(x 2 ) = f ? g(y 2 ) ????s? ?? ??? t??? ?????? x 3 ??? y 3 s??? t??t g(x 2 ) = x 3 ??? g(y 2 ) = y 3 ? ??? s? ??? ??? ??t??? t??t ???r? x i , y i ?r? ??t????? ?r?? t?? ????t??? g? ????ss?r???? t??r? ???st n 0 ? N ??? k ? N s??? t??t? ??r ???r? n ? n 0 ?? ???? g k (x n ) = x n ??? g k (y n ) = y n ? ??t? x n ??? y n ??t????? ?s ?r?????s??? ?? ???? t??? g k?1 (x n ) = x n+k?1 ??? g k?1, ) = g k ? (x n 0 ) ??? g j (y 1 ) = g k ?
, ? g j (x 1 ) = f ? g k ? (x n 0 ) ??? f ? g j (y 1 ) = f ? g k ? (y n 0 )? ????? ?? ???? t?? ?????s??? ker f ? g ? ker f ? ?? ?????? t?? ?q??t???s f ? g j?1
, ?s f (x 1 ) = f ?g k ? ?j (x n 0 ) ??? f (y 1 ) = f ?g k ? ?j (y n 0 ) ????? ??t x ? , y ? ? D ?? t?? ?????s s??? t??t x ? = g k ? ?j
, k]) ??? y ? = y n 0 +((k ? ?j)[k]) ? ?? ????t??s?s? ?? ???? t??t f (x 1 ) = f (y 1 )? t??? ?? ??? ?????? t?? ?q??t??? f (x ? ) = f (y ? ) ????? ?s ? ???tr????t??? ??t? (i)? s???? x ? ??? y ? ?r? ????t t?r???? t?? ????t??? g st?rt??
, x k ? ker f ?? x 1, t? ?? x 1 , . . . , x k ? ker f ?? t??r? ?s ?t ???st t?? ??????ts x i ??? x j ????? x 1
, ??? x 3 ?r? ???r??s? ????r??t? ?? ???? t??? t?? s?????s?s? y 1 , y 2 ??? y 3 ?r? ???r??s? ????r??t ?r ??t? ?ss??? t??t t??r? ???st i ??? j? i = j s??? t??t y i = y j ? ????? ?? ???? x i , x j ? ker h? ??t? h t?? ????t??? s??? t??ty i , y j , y k )? ?ss??? ?????? t??t y 1 , y 2 ??? y 3 ?r? ???r??s? ????r??t? ?? ????? ?????? ?? ???? t??t t??r? ???sts f ? [F ] s??? t??t x i , x j ? ker f ?? ??? ???? ?? t??r? ???sts g ? [F ] s??? t??t y i , y j ? ker g? ??r ???r? i, j ? {1, M (y i , y j , y k )) = (x j, p.3
, ?s s(x) = y? s(y) = x ??? s(z) = z? ??r ??st???t x, y, z ? D? ??? s?t {x, y} ?s ?????? swap s?
, ?? ran g ker f ??? |ker f | = 2 t??? ran(f ? g) = ran f ? ???? ?? ran g ker f ???
, ?? ??t ran g ker f ??? |ker f | = 2? ??rst? r???r? t??t |ker
, ? g)? ??t y ? ran f ? ?????s? t??t ??r ??? x ? ran g? t?? ???q????t? f (x) = y ????s? ??t z ? ran f ? ??t? z = y? ?? ??r ??? x ? ran g? ?? ???? f (x) = z s???? |ran f | = 2? ????? ??????s ran g ? ker f ? ???tr????t??? ??t? t?? ?ss???t???? ??t ran g ker f ? ?? ???? t? s??? t??t ker(f ? g) ? ker g? ?????s? t??t t??r? ???st x, y ? ker(f ? g) s??? t??t g(x) = g(y)? ????? |ran g| = 2? ?? ???? ??t ?s r????? s?
, t? ??t? x? y ??? z ???r??s? ????r??t ??? t s??? t??t t = ker f 1 ? ker f 2 ????s ??t? f 1 ] s??? t??t f ? = {(a 1 , a 2 ); (b 1 , b 2 ); (c 1 , c 2 )}? ?ss??? ??t???t ??ss ?? ????r???t? t??t ker f 1 = {0, 1} ??? ker f 2 = {1)) = 1 ??r ???r? ??r??t?t??? ?? ???s? ?? ???? t?? ?q??t??? M (a 1 , b 1 , c 1 ) = 1? ?? s? ) = 1? ???s ?s t?? ??s? ?? ??? ???? ?? f ?s t?? ????t?t? ???r?t??? ?r ? s??? ??r??t?t??? s??? t??t swap f = {0
, ? s s??? t??t ker f ? ker g ? swap s? ????? ??r ???r? f, g ? F ? ?? ???? ??t??r ker f ? ker g ?r ker g ? ker f
, ?t???s ? s??? ??r??t?t??? s s??? t??t ker F ? ker g / ? swap s? ??r ??? ker f ??? ker g ???????r???? ??r???s? ??? width F = 2 ??? t??r? ???sts f ? F s??? t??t | ker f | = 2? ??? F ???t???s ????t?? ??? s??? ??r??t?t??? s s??? t??t swap s = ker f ????s? ??t ???s ??t ???t??? ??? ??r????r ??r??t?t???s? ??? width F = 1 ??? t??r? ?? ????t???s f ? F s??? t??t | ker f | = 2? ??? width F = 1 ??? t??r? ???sts f ? F s??? t??t | ker f, ????t??? s 1 ? s 2 ????s t
, ?r?? ?t ?s ??s? t? s?? t??t [F ] = F ?????s? ??r ???r? f i , f j ? F ??t? f i / ? {f 0 , f 1 }? t?? ?q??t??? f i ? f j = f j ????s ?t??s ??? ??? s?? t??t ?? n ?s t?? ?????stst??t ???r?t???s s???? F ???t???s ???st??t ????t???s f 0 ??? f 1 ? ??? ?r??????? r?s??t ?? t??s s??t??? ?s t??, |D|? t??? ?s?(F ? ) ?s NP???????t?? ??t??? t??t t??s r?s??t ???s ??t ?r??? ?s ? ??? ????????? ?? t?? ????????t? ??
, ?s s??? t??t ??r ???r? f ? F ? ?q??t???s f (0) = 0 ??? f (1) = 1 ????? ?????t ??r t?? t?? ???st??t ????t???s f 0
, ? r ??t? t?? ??r??st ?????r ?? 1?s? ????????? t?? 1????t?r? ??t k t?? ?????r ?? 1?s ?? r? ?? s??? t??t k s????? ?? ?q??? t? n ? 1? ?ss??? t??t k ? n ? 2? t??? t??r? ???st ?t ???st t?? 0 ?? r? ??t c 1 ??? c 2 t?? ??????s ??r? t??r? ?s ? 0 ?? r? ?? ????
, ??? ?t ???st n r??s t? ????t s??? ? ??tr??? ??t ??t???t t?? 0????t?r r
, ?s s??? t??t ??r ???r? f ? F ? ?q??t???s f (0) = 0 ??? f (1) = 1 ????? ?????t ??r t?? t?? ???st??t ????t???s f 0
, ?s?? ??t? t?? 0????t?r? ?t ?s ?q???????t t? s?? t??t t?? ??tr?? ?s ???s??s?? ??t? t?? 0????t?r? ?t ?s ?q???????t t? s?? t??t t?? ??tr?? ?s
, ?s s??? t??t ??r ???r? f ? F ? ?q??t???s f (0) = 0 ??? f (1) = 1 ????? ?????t ??r t?? t?? ???st??t ????t???s f 0
, r) ????s ?s t?? 0????t?r? ???s???r t?? s?t S = (r 1
, ? r = ? 1 r 1 ? ? 2 r 2 ? . . . ? ? k r k ??t? ? t?? ??r ???r?t???? ????? ?t ?s ????r t??t ?? ???? ????t?? 2 k ??????ts ?? G s???? ???? r??s ?r? ???r??s? ????r??t? ????r? t??t ?? t??r?
, ?? ???r??s? ????r??t? ?t ?s ??s? ????r t??t ?? ???? ?t ???st log(n) r??s t? ???? t?? ????r???? ??t???? n ??????s? ????? t?? ??s?s S ???t???s ?t ???st log(n) r??s? ??? G? t??t ?s t? s?? t?? ??tr?? F H rhs ?ts???? ??st ?t ???st ???t???s 2 log(n) r??s? ????
, ?s s??? t??t ??r ???r? f ? F ? ?q??t???s f (0) = 0 ??? f (1) = 1 ????? ?????t ??r t?? t?? ???st??t ????t???s f 0
, ?s f 0 , f 1 ? F ??t??tt??? r?s???t????? 0 ??? 1? ???s r?s??t ????s ??r ???r? ???t? ?????? D ?? t?? ??r??? ???t? ?s ?????r t??? ?r ?q???s t? t?? ??r??????t? ?? D? ?t ????s ?s t? t?? ??????t?r? t??t? ?s t??s ?r?t?r??? ?s ??r????? t?? ?s?(F ? )