Uniform 2D-Monotone Minimum Spanning Graphs

CCCG 2018, Winnipg, Cn, August 8 10, 2018 Uniorm 2D-Monoton Minimum Spnning Grphs Konstntinos Mstks Astrt A gomtri grph G is xy monoton i h pir o vrtis o G is onnt y xy monoton pth. W stuy th prolm o prouing th xy monoton spnning gomtri grph o point st P tht (i) hs th minimum ost, whr th ost o gomtri grph is th sum o th Eulin lngths o its gs, n (ii) hs th lst numr o gs, in th ss tht th Crtsin Systm xy is spii or rly slt. Builing upon prvious rsults, w sily otin tht th two solutions oini whn th Crtsin Systm is spii n r oth qul to th rtngl o inlun grph o P. Th rtngl o inlun grph o P is th gomtri grph with vrtx st P suh tht two points p, q P r jnt i n only i th rtngl with ornrs p n q os not inlu ny othr point o P. Whn th Crtsin Systm n rly hosn, w not tht th two solutions o not nssrily oini, howvr w show tht thy n oth otin in O( P 3 ) tim. W lso giv simpl 2 pproximtion lgorithm or th prolm o omputing th spnning gomtri grph o k root point st P, in whih h root is onnt to ll th othr points (inluing th othr roots) o P y y monoton pths, tht hs th minimum ost. 1 Introution A squn o points in th Eulin pln q 0, q 1,..., q t is ll y monoton i th squn o thir y oorints, i.. y(q 0 ), y(q 1 ),..., y(q t ), is ithr rsing or inrsing, with y(p) noting th y oorint o th point p. A gomtri pth Q = (q 0, q 1,..., q t ) is ll y monoton i th squn o its vrtis, i.. th squn q 0, q 1,..., q t, is y monoton. I Q is y monoton or som xis y thn Q is ll monoton. Lt G = (P, E) gomtri grph. I h p, q P r onnt y y monoton pth thn G is ll y monoton. I G is y monoton or som xis y thn G is ll uniorm monoton (ollowing th trminology o [22]). Uniorm monoton grphs wr ll This rsrh ws innilly support y th Spil Aount or Rsrh Grnts o th Ntionl Thnil Univrsity o Athns. Shool o Appli Mthmtil n Physil Sins, Ntionl Thnil Univrsity o Athns, Athns, Gr, kmst@mth.ntu.gr 1 monoton grphs y Anglini [3]. I h p, q P r onnt y monoton pth, whr th irtion o monotoniity might ir or irnt pirs o vrtis, thn G is ll monoton. Monoton grphs wr introu y Anglini t l. [4]. Drwing n (strt) grph s monoton (gomtri) grph hs n topi o rsrh [3, 4, 5, 13, 24]. Th Monoton Minimum Spnning Grph prolm, i.. th prolm o onstruting th monoton spnning gomtri grph o givn point st tht hs th minimum ost, whr th ost o gomtri grph is th sum o th Eulin lngths o its gs, ws rntly introu (ut not solv) in [22] n it rmins n opn prolm whthr it is NP-hr. Sin th mor gnrl (without th rquirmnt o monotoniity) Eulin Minimum Spnning Tr prolm n solv in Θ( P log P ) tim [27], this onstituts grt irntition tht is inu y th ition o th proprty o monotoniity. A point st P is k root i thr xist k points r 1, r 2,..., r k P istinguish rom th othr points o P whih r ll th roots o P. A gomtri grph G = (P, E) is ll k root i P is k root n its roots r th roots o P. A k root gomtri grph G is k root y monoton i h root r P n h point p P \ {r} r onnt y y monoton pths. Similrly, G is k root uniorm monoton (ollowing th trminology o [22]) i it is k root y monoton or som xis y. For simpliity, w my lso not point sts or gomtri grphs tht r 1 root simply s root. A polygon tht is 2 root y monoton, in whih its roots r its lowst n highst vrtis, n tringult in linr tim [11]. L n Prprt [16] prpross suivision S o th pln suh tht th rgion in whih qury point longs n oun quikly, y (i) xtning th gomtri grph ouning S to 2 root y monoton plnr gomtri grph in whih th roots r th highst n lowst vrtis o S, n (ii) onstruting st o pproprit y monoton pths rom th lowst to th highst vrtx o S. Aitionlly, L n Prprt [16] not tht 2 root plnr gomtri grph, whr ll vrtis hv irnt y oorints, in whih th roots r th highst n lowst vrtis o th grph is 2 root y monoton i n only i h non-root vrtx hs oth nighor ov it n nighor low it. Furthrmor, root gomtri grph G = (P, E), whr ll vrtis hv irnt y oorints, with

30 th Cnin Conrn on Computtionl Gomtry, 2018 (singl) root r tht is not th highst or lowst point o P is root y monoton i n only i h non-root vrtx p hs nighor q suh tht y(q) is twn y(r) (inlusiv) n y(p) [22]. Aitionlly, root uniorm monoton grphs n iintly rogniz [22]. Th k root y monoton (uniorm monoton) minimum spnning grph (ollowing th trminology o [22]) o k root point st P is th k root y monoton (uniorm monoton) spnning grph o P tht hs th minimum ost. Th root y monoton (uniorm monoton) minimum spnning grph 1 o root point st P n prou in O( P log 2 P ) (rsp., O( P 2 log P )) tim [22]. Th prolm o rwing root tr s root y monoton minimum spnning grph is stui in [20]. Th ( P root) y monoton minimum spnning grph o point st P is th gomtri pth tht trvrss ll th points o P y moving north, rom th lowst point to th highst point o P [22]. Rgring th prolm o prouing th k root y monoton minimum spnning grph o k root point st P, with 1 < k < P, it is n opn prolm, pos in [22], whthr it is NP-hr. Th rstrit thrs tr prolm ws introu in [12] n is rlt to th root y-monoton minimum spnning grph prolm onstrin to root point sts P in whih th y oorint o th root is zro n th y oorints o th othr points o P r ll ngtiv (or ll positiv). Th input o th rstrit thrs tr prolm is omplt grph with root whr h g hs ost n h vrtx hs vlu n th gol is to output th spnning tr in whih th pth rom th root to h vrtx rss in vlu tht hs th minimum ost. Th rstrit thrs tr prolm is grily solvl [12, Corollry 2.6]. A gomtri pth Q = (q 0, q 1,..., q t ) is xy monoton i th squn o its vrtis is oth x monoton, i.. th squn x(q 0 ), x(q 1 ),..., x(q t ), is monoton, n y monoton. Q is 2D-monoton (ollowing th trminology o [22]) i it is x y monoton or som orthogonl xs x, y. A gomtri grph G = (P, E) is 2D-monoton (ollowing th trminology o [22]) i h pir o points o P is onnt y 2D-monoton pth. 2D-monoton pths/grphs wr ll ngl-monoton pths/grphs y Bonihon t l. [8]. Bonihon t l. [8] show tht iing i gomtri grph G = (P, E) is 2D-monoton n on in O( P E 2 ) tim. Tringultions with no otus intrnl ngls r 2D-monoton grphs [10, 19]. Thr xist point sts or whih ny 2D-monoton spnning grph is not plnr [8]. Th prolm o onstruting 2D-monoton grphs with symptotilly lss thn qurti gs ws stui y Luiw n Monl [18]. It is n opn prolm, pos in [22], whthr th 2Dmonoton spnning grph o point st P tht hs th 1 In [22] it is shown tht it is tully tr. minimum ost n iintly omput. Th (root) xy monoton n (root) uniorm 2Dmonoton (using th trminology o [22]) grphs r in similr to th (root) y monoton n (root) uniorm monoton grphs. Diing i root gomtri grph G = (P, E) is root xy monoton (uniorm 2D-monoton) n on in O( E ) (rsp., O( E log P )) tim [22]. Aitionlly, th root xy monoton (uniorm 2D-monoton) spnning grph o root point st P tht hs th minimum ost 2 n omput in O( P log 3 P ) (rsp., O( P 2 log P )) tim [22]. W ous on th proution o th xymonoton minimum spnning grph (xy MMSG) o point st P, i.. th xy-monoton spnning grph o P tht hs th minimum ost, n th proution o th uniorm 2D monoton minimum spnning grph (2D UMMSG) o point st P, i.. th uniorm 2D monoton spnning grph o P tht hs th minimum ost. W lso stuy th orrsponing prolms rgring th proution o th spnning grphs with th lst numr o gs, i.. th proution o th xy-monoton spnning grph with th lst numr o gs n th proution o th uniorm 2D monoton spnning grph with th lst numr o gs. A urv C is inrsing-hor [15, 26] i or h p 1, p 2, p 3, p 4 trvrs in this orr long it, th lngth o th lin sgmnt p 1 p 4 is grtr thn or qul to th lngth o p 2 p 3. Almri t l. [1] introu inrsinghor grphs whih r th gomtri grphs or whih h two vrtis r onnt y n inrsing-hor pth. Inrsing-hor grphs r wily stui [1, 6, 10, 21, 23]. Th prolm o prouing inrsing-hor spnning grphs (whr Stinr points my ) o point st P ws stui in [1, 10, 21]. Th pproh mploy in [1, 10, 21], ws to onnt th points o P y 2D-monoton pths sin s not y Almri t l. [1] 2D-monoton pths r lso inrsing-hor pths. Lt P point st n lt p, q P thn p n q r rtngulrly visil i th rtngl with ornrs p n q os not inlu ny othr point o P. Furthrmor, th rtngl o inlun grph o P is th gomtri grph spnning P suh tht pq is n g o th grph i n only i p n q r rtngulrly visil. Alon t l. [2] not rtngulrly visil points s sprt points n th rtngl o inlun grph s th sprtion grph. Computing th rtngl o inlun grph G = (P, E) o P n on in O( P log P + E ) tim [25]. Thr xist point sts P or whih th numr o gs o thir rtngl o inlun grph is Ω( P 2 ) [2]. Th rtngl o inlun grph os not rmin th sm i th Crtsin Systm is rott [14, Proposition 3]. 2 In [22] it is shown tht it is tully tr, not s th root xy monoton (uniorm 2D-monoton) minimum spnning tr in [22] n rvit s th root xy MMST (rsp., root 2D UMMST ) in [22].

CCCG 2018, Winnipg, Cn, August 8 10, 2018 Drwing n strt grph s rtngl o inlun grph hs n stui [17]. Our Contriution. Builing upon prvious rsults, w sily otin tht givn point st P th xy MMSG o P is qul to th xy monoton spnning grph o P tht hs th lst numr o gs n r oth qul to th rtngl o inlun grph o P. W not tht givn point st P th 2D UMMSG o P os not nssrily oini with th uniorm 2D-monoton spnning grph o P tht hs th lst numr o gs. W lso show tht oth th 2D UMMSG o P n th uniorm 2D-monoton spnning grph o P tht hs th lst numr o gs n prou in O( P 3 ) tim. Aitionlly, w giv simpl 2 pproximtion lgorithm or th prolm o prouing th k root y monoton minimum spnning grph o k root point st. 2 Prliminris 2.1 xy Monoton Minimum Spnning Grphs Anglini [3] not th ollowing Ft rgring y monoton grphs. Ft 1 (Anglini [3]) Lt G = (P, E) y monoton grph whr no two points o P hv th sm y oorint n lt p, q P suh tht or h r P \ {p, q} th squn p, r, q is not y monoton. Thn, p n q r jnt in G. Ft 1 is sily xtn in th ontxt o xy monoton grphs. Mor spiilly, lt G = (P, E) xy-monoton grph n p, q P suh tht or h r P \ {p, q} th squn o points p, r, q is not xy monoton, thn p n q r jnt in G. Alon t l. [2] not tht th points p, q o point st P r rtngulrly visil i n only i or h r P \{p, q} th squn o points p, r, q is not xy monoton. Hn, th rtngl o inlun grph o P is sugrph o G. Liott t l. [17, Lmm 2.1] show tht th rtngl o inlun grph o point st is xy monoton grph 3. From th prvious two sntns, rgring th rtngl o inlun grph, w otin th ollowing Corollry. Corollry 1 Lt P point st. Th xy MMSG o P n th xy monoton spnning grph o P tht hs th lst numr o gs oini n thy r oth qul to th rtngl o inlun grph o P. 3 Thnilly spking, Liott t l. [17] show tht th rtngl o inlun grph o point st is grph suh tht h two vrtis r onnt y pth lying insi th rtngl in y ths vrtis ut upon rul ring th pth tht is otin in thir proo is xy monoton. W rll tht th rtngl o inlun grph G = (P, E) o P n prou in O( P log P + E ) tim [25] whih is optiml [25] n tht thr xist point sts P or whih th rtngl o inlun grph hs siz Ω( P 2 ) [2] s wll s point sts or whih it hs linr siz [2]. 2.2 Root Uniorm 2D-Monoton Grphs Mstks n Symvonis [22] stui th prolm o rognizing root uniorm 2D-monoton grphs. Thy initilly not th ollowing Ft. Ft 2 (Osrvtion 8 in [22]) Lt G gomtri grph G = (P, E) with root r. I on rotts Crtsin Systm x y, thn G my om root x y monoton whil prviously it ws not, or vi vrs, only whn th y xis oms (or lvs th position whr it prviously ws) prlll or orthogonl to 1. lin pssing through r n point p P \ {r}. 2. n g pq E, whr p, q r. Bs on Ft 2, Mstks n Symvonis [22] gv rottionl swp lgorithm not s th root uniorm 2D-monoton rognition lgorithm in [22]. Ft 3 ([22]) Th root uniorm 2D-monoton rognition lgorithm i) omputs, in O( E log P ) tim, st o suiint Crtsin Systms, o siz O( E ), whih r ssoit with (1) lins pssing through r n point p P \ {r} n (2) gs pq E, whr p, q r. ii) tsts, in O( E ) totl tim 4, i G is root x y monoton or som Crtsin Systm x y in th prviously omput st o suiint Crtsin Systms. Ft 4 (Thorm 1 in [22]) Lt P root point st thn th root y monoton minimum spnning grph o P n otin in O( P log 2 P ) tim. 3 Th 2D-UMMSG Prolm W now l with th onstrution o th 2D UMMSG n th uniorm 2D monoton spnning grph with 4 Thnilly spking in [22] it is shown tht th rmining stps, i.. th stps tr th omputtion o th suiint Crtsin Systms, o th root uniorm 2D-monoton rognition lgorithm tk O( E log P ) totl tim. Intrnlly in th root uniorm 2D-monoton rognition lgorithm givn in [22], or h p P \ {r} it is stor th st o jnt points to p tht r in th rtngl w.r.t. th Crtsin Systm x y with ornrs p n r, whih is not s A(p, x, y ) in [22]. Furthrmor, it is stor th st o points p P \ {r} or whih A(p, x, y ) > 0 whih is not s B(x, y ) in [22]. Howvr, only th rinlitis o ths sts r nssry [22, Lmm 9], hn i inst o th sts A(p, x, y ), p P n B(x, y ) thir rinlitis r stor, th rmining stps o th root uniorm 2D-monoton rognition lgorithm tk O( E ) totl tim.

30 th Cnin Conrn on Computtionl Gomtry, 2018 th lst numr o gs. W initilly show tht th 2D UMMSG o point st P n otin in O( P 3 ) tim. For this, w mploy rottionl swp thniqu. Our pproh rgring th onstrution o th 2D UMMSG is similr to th pproh mploy or th lultion o th root uniorm 2D monoton spnning grph tht hs th minimum ost in [22]. W ssum tht no thr points o P r ollinr n no two lin sgmnts pq n p q, p, p, q, q P, r prlll or orthogonl. Lt P point st n p point o P. Lt RV (p, x, y ) not th sust o points o P tht r rtngulrly visil rom p w.r.t. th Crtsin Systm x y. S or xmpl, Figur 1(). Proposition 2 I w rott Crtsin Systm x y ountrlokwis, thn th x y MMSG o P hngs only whn y rhs or movs wy rom lin prpniulr or prlll to lin pssing through two points o P. Proo. I w rott th Crtsin Systm x y ountrlokwis thn th RV (p, x, y ) or point p P hngs only whn y rhs or movs wy rom lin prpniulr or prlll to lin pssing through two points o P ;.g. s Figur 1. From th prvious n sin th RV (p, x, y ), p P, quls to th st o jnt vrtis o p in th x y MMSG o P (Corollry 1), w otin th Proposition. y p () g h i y j x p () g h i j x y p () Figur 1: In () RV (p, x, y ) = {,,,, g, h, i}. In () th y oms prlll to th n now is not rtngulrly visil rom p. Finlly, in () th y hs lt th position whr it prviously ws orthogonl to th n now oms rtngulrly visil rom p. Lt S = {s [0, π 2 ) : lin o slop s is prpniulr or prlll to lin pssing through two points o P }. g h i j x Lt S = {s 1, s 2,..., s l } with l = ( ) P 2 suh tht 0 s1 < s 2 <... < s l < π 2. W now in th st S suiint s to qul to {s 1, 1+s 2 s 2, s 2, 2+s 3 s 2,..., s l, l + π 2 2 }. Lt x 1 y 1, x 2 y 2,..., x 2l y 2l th Crtsin Systms in whih th vrtil xis hs slop in S suiint, orr w.r.t. th slop o thir vrtil xis. Thorm 3 Th uniorm 2D monoton minimum spnning grph o point st P n omput in O( P 3 ) tim. Proo. From Proposition 2 n th prvious initions w otin th ollowing Proposition. Proposition 4 Th uniorm 2D monoton minimum spnning grph o P is on o th x y MMSG o P ovr ll Crtsin Systms x y with y o slop in S suiint. W now giv O( P 3 ) tim rottionl swp lgorithm. Th lgorithm initilly omputs th x 1 y 1 MMSG o P n thn it otins h x i+1 y i+1 MMSG o P rom th x i y i MMSG o P. Throughout th prour th Crtsin Systm x opt y opt in whih th lgorithm nountr th minimum ost solution so r is stor. In its lst stp, th lgorithm romputs th x opt y opt MMSG o P, whih sin it is qul to th rtngl o inlun grph G = (P, E) w.r.t. th Crtsin Systm x opt y opt (Corollry 1) it n omput in O( P log P + E ) tim [25]. Th ruil proposition (whih w show ltr) tht mks th tim omplxity o th lgorithm qul to O( P 3 ) is tht h trnsition rom th x i y i MMSG o P to th x i+1 y i+1 MMSG o P tks O( P ) tim. For h two points p, q o P lt I(p, q, x i, y i ) th numr o points o P \ {p, q} tht r inlu in th rtngl w.r.t. th Crtsin Systm x i y i with opposit vrtis p n q. Thn, RV (q, x i, y i ) n quivlntly in using th quntitis I(p, q, x i, y i ), p P \ {q}, s ollows: p RV (q, x i, y i ) i I(p, q, x i, y i ) = 0. W stor th RV (q, x i, y i ), q P, i = 1,2,..., 2l in th t strutur rv(q) whih is implmnt s n rry o P oolns. W lso stor th I(p, q, x i, y i ), p, q P, i = 1,2,..., 2l in th vril i(p, q). Computing th Crtsin Systms x i y i, i = 1, 2,..., 2l n on in O( P 2 log P ) tim. Aompni with h Crtsin Systm x i y i is th pir o points (p i, q i ) suh tht p i q i is ithr prlll or prpniulr to th y i xis or th y i 1 xis. Ihino n Sklnsky [14] not tht mploying rng tr [7, 9] tht ontins th points o P on n lult i) th rtngl o inlun grph o P, n ii) th I(p, q, x, y), p, q, P, or Crtsin Systm xy. Applying th prviously mntion pproh, not y Ihino n Sklnsky [14], r otin i) th rtngl o inlun grph o P w.r.t. th Crtsin Systm x 1 y 1 (whih y Corollry 1 quls to th x 1 y 1 MMSG o P ), n ii) th I(p, q, x 1, y 1 ), p, q, P.

CCCG 2018, Winnipg, Cn, August 8 10, 2018 W now show tht w n upt ll th rv(p), p P, suh tht rom qul to RV (p, x i 1, y i 1 ), p P, thy om qul to RV (p, x i, y i ), p P, in O( P ) totl tim. For h p P \ {p i, q i } th upt o rv(p) tks O(1) tim. This is tru, sin only th points p i n q i hv to tst or inlusion to or rmovl rom rv(p). Mor spiilly, w hv to tst i or on o thm, sy p i, th rtngl with ornrs p n p i ontins (or it os not ontin) q i w.r.t. th Crtsin Systm x i y i whil it i not ontin (or it ontin) it w.r.t. x i 1 y i 1. I this is tru, thn th i(p i, p) hngs n p i hs to tst or mmrship in rv(p) n inlu to or rmov rom rv(p). Rgring rv(p i ), th upt tks O( P ) tim, sin or h othr point q P \ {p i, q i } w hv to tst i th rtngl with ornrs q n p i ontins (or it os not ontin) q i w.r.t. th Crtsin Systm x i y i whil it i not ontin it (or it ontin it) w.r.t. th x i 1 y i 1 n i so upt oth th i(q, p i ) n th xistn o q in rv(p i ) i nssry. Similrly, rv(q i ) n upt in O( P ) tim. W not tht th prour o otining th 2D UMMSG n trivilly moii suh tht th uniorm 2D-monoton spnning grph o point st P with th lst numr o gs n otin in O( P 3 ) tim. Sin or n ritrry Crtsin Systm x y th x y MMSG o P is qul to th x y monoton spnning grph o P with th lst numr o gs (Corollry 1), th only moiition whih is nssry is tht in th trnsition rom th Crtsin Systm x i y i to th Crtsin Systm x i+1 y i+1 w hk i th x i+1 y i+1 monoton spnning grph o P with th lst numr o gs hs th lst numr o gs mong ll th prou solutions so r. In Figur 2 is givn point st P or whih th 2D UMMSG o P is irnt rom th uniorm 2D monoton spnning grph o P with th lst numr o gs. In Figur 3 w giv point st P or whih th (nonuniorm) 2D monoton spnning grph o P with th lst numr o gs os not oini to th (nonuniorm) 2D monoton spnning grph o P tht hs th minimum ost. Rgring rognizing uniorm 2D-monoton grphs, w not tht th O( E log P ) tim rottionl swp lgorithm givn in [22], whih is i gomtri grph G = (P, E) with spii vrtx r s root is root uniorm 2D monoton, n sily xtn into O( P 2 log P + P E ) tim rottionl swp lgorithm tht is i G is uniorm 2D monoton. Mor spiilly, in orr to i i G is uniorm 2D monoton, th P root gomtri grphs (p 1, G), (p 2, G),..., (p P, G) whr (p i, G) is th gomtri grph G with root p i n {p 1, p 2,..., p P } is th vrtx st o G, r onsir. A Crtsin Systm x y is rott ountrlokwis. From Ft 2, it ol () () Figur 2: Th points, n orm right ngl. Aitionlly, th points, n orm right ngl. Th slop o is smllr thn th slop o. Th uniorm 2D monoton spnning grph with th lst numr o gs is otin whn th y xis oms prpniulr to th n is shown in (). On th othr hn th 2D UMMSG is otin whn th y xis oms prpniulr to th n is shown in (). () () Figur 3: Th slop o is π 4 whil th slop o is 3π 4. In () is pit th 2D monoton spnning grph o P with th lst numr o gs. In () is illustrt th 2D monoton spnning grph o P tht hs th minimum ost. lows tht on o ths P root gomtri grphs oms root x y monoton whil prviously it ws not, or vi vrs, only whn th y xis oms (or lvs th position whr it ws prviously) prlll or orthogonl to lin pssing through two points o P. Hn, O( P 2 ) Crtsin Systms n to onsir, whih n omput in O( P 2 log P ) tim. Whn th y oms (or lvs th position tht it prviously ws) prlll or prpniulr to lin pssing through th points p, q P thn y Ft 2 th sttus, i.. ing root x y monoton, o th root gomtri grphs (p, G) n (q, G) my hng. Hn, th stps o th root uniorm 2D-monoton rognition lgorithm givn in [22] or hnling th vnt ssoit with th urrnt Crtsin Systm x y rgring th root gomtri grphs (p, G) n (q, G), r ppli. Furthrmor, i pq E thn y Ft 2 it ollows tht th sttus, i.. ing root x y monoton, o h (r, G), r P \ {p, q}, my lso hng. Hn, or h (r, G), r P \ {p, q}, th stps o th root uniorm 2D-monoton rognition lgorithm givn in [22] or hnling th vnt ssoit with th urrnt Crtsin Systm x y r ppli. Sin, th rmining stps, i.. tr th lultion o th suiint xs, o th root uniorm 2D-monoton rognition lgorithm, givn in [22], rgring ny o ths P root

30 th Cnin Conrn on Computtionl Gomtry, 2018 gomtri grphs tk O( E ) tim (Ft 3), pplying th rmining stps rgring ll ths P root gomtri grphs, tks O( P E ) totl tim. 4 A 2 Approximtion Algorithm or th k Root y Monoton Minimum Spnning Grph Prolm W now stuy th prolm o prouing th k root y monoton minimum spnning grph o k root point st P, whr 1 < k < P. W ssum tht no two points hv th sm y oorint. Lt P point st n, R thn P y> is th sust o points o P whos y oorint is grtr thn. Similrly r in P y, P y< n P y. P <y< is th sust o points o P whos y oorint is twn n. Similrly r in P <y, P y< n P y. In [22, Lmm 1] it is not tht th root y monoton minimum spnning grph o root point st P with root r is th union o th root y monoton minimum spnning grphs o (i) P y y(r) n (ii) P y y(r). Th prvious Ft is xtn to th ollowing Lmm. Lmm 5 Lt P k root point st, with 1 < k < P, whr r 1, r 2,..., r k r th roots o P suh tht y(r 1 ) < y(r 2 ) <... < y(r k ). Th k root y monoton minimum spnning grph o P is th union o 1. th root y monoton minimum spnning grph o P y y(r1). 2. th root y monoton minimum spnning grph o P y y(rk ). 3. th 2 root y monoton minimum spnning grph o P y(ri) y y(r i+1), 1 i k 1. Thorm 6 Givn k root point st P, with 1 < k < P, w n otin in O( P log 2 P ) tim k root y monoton spnning grph o P with ost t most twi th ost o th k root y monoton minimum spnning grph o P. Proo. For 2 root point st P with roots r 1 n r 2 tht r th lowst n highst points o th point st, rsptivly, w prov th ollowing Lmm. Lmm 7 Givn 2 root point st P with roots r 1 n r 2 tht r th lowst n highst points o th point st, rsptivly, w n otin in O( P log 2 P ) tim 2 root y monoton spnning grph o P with ost t most twi th ost o th 2 root y monoton minimum spnning grph o P. Proo. Initilly, w mploy Ft 4 to P onsiring it to hv only th root r 1 n otin th gomtri grph G 1. Thn, w mploy Ft 4 to P onsiring it to hv only th root r 2, otining G 2. In th inl stp w rturn th union o G 1 n G 2. G 1 G 2 is 2 root y monoton sin G 1 (G 2 ) is root y monoton with root r 1 (rsp., r 2 ). W now show tht G 1 G 2 hs ost t most twi th ost o th 2 root y monoton minimum spnning grph G opt o P. Sin, in G opt ll th points p r onnt with r 1 (r 2 ) y y monoton pths it ollows tht its ost is grtr thn or qul to th ost o G 1 (rsp., G 2 ). Hn, th ost o G 1 G 2 whih is lss thn or qul to th sum o th osts o G 1 n G 2 is t most twi th ost o G opt. From Lmm 5, Ft 4 n Lmm 7 w otin th Thorm. A 2 root plnr gomtri grph G = (P, E) with roots r 1, r 2 s.t. y(r 1 ) < y(p) < y(r 2 ), p P \ {r 1, r 2 }, is 2 root y monoton i n only i or h p P \ {r 1, r 2 } thr xist q 1, q 2 Aj(p) with y(q 1 ) < y(p) < y(q 2 ) [16]. Furthrmor, root gomtri grph G = (P, E) with (singl) root r tht is not th highst or lowst point o P is root y monoton i n only i or h p P \ {r} thr xists q Aj(p) suh tht y(q) is twn y(r) (inlusiv) n y(p) [22]. W xtn th prvious two Propositions to th ollowing quivlnt hrtriztion o k root y monoton grphs whr th lttr implis n iint rognition lgorithm or k root y monoton grphs. Proposition 8 Lt G = (P, E) k root gomtri grph, whr 1 < k < P, with roots r 1, r 2,..., r k suh tht y(r 1 ) < y(r 2 ) <... < y(r k ). G is k root y monoton i n only i 1. or h p P y<y(r1) thr xists q Aj(p) s.t. y(q) (y(p), y(r 1 )]. 2. or h p P y>y(rk ) thr xists q Aj(p) s.t. y(q) [y(r k ), y(p)). 3. or h p P y(ri)<y<y(r i+1) thr xist q 1, q 2 Aj(p) s.t. y(q 1 ) [y(r i ), y(p)) n y(q 2 ) (y(p), y(r i+1 )], i = 1, 2,..., k 1. 4. thr xists q Aj(r 1 ) s.t. y(q) (y(r 1 ), y(r 2 )]. 5. thr xists q Aj(r k ) s.t. y(q) [y(r k 1 ), y(r k )). 6. thr xist q 1, q 2 Aj(r i ) s.t. y(q 1 ) [y(r i 1 ), y(r i )) n y(q 2 ) (y(r i ), y(r i+1 )], 2 i k 1. 5 Furthr Rsrh Dirtions Givn point st P n th 2D monoton spnning grph o P tht hs th lst numr o gs prou in polynomil tim? Dos thr xist t pproximtion lgorithm, t < 2, or th k root y monoton minimum spnning grph prolm? Aknowlgmnt: I woul lik to thnk Prossor Antonios Symvonis or his vlul ontriution in vloping th rsults prsnt in this ppr.

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