C.S. 252 Pro. Rorto Tmssi Computtionl Gomtry Sm. II, 1992 1993 Dt: My 3, 1993 Sri: Shmsi Moussvi Plnr Upwr Drwings 1 Thorm: G is yli i n only i it hs upwr rwing. Proo: 1. An upwr rwing is yli. Follow th hin rom th sour, sin it is upwr thn th Y oorint will only inrs; thror it n nvr rt yl. Not tht th rwing n not plnr. Figur 1: An upwr rwing. 2. An yli grph G hs n upwr rwing. On wy to show th xistn o n upwr rwing or G, woul to rw ll th vrtis o G long vrtil lin, with sour vrtis hving smllr Y vlu, whih woul mk th grph upwr. Th lgorithm woul s ollows: Fin th sours (vrtis with no inomming g), st it s Y vlu. 1
Do DFS srh to in th nxt vrtis (tr rmoving th sours) n ssign th nxt Y vlu to it. Although this rwing is not vry ni rwing o G, ut in it is upwr n th proo is omplt. 2 g 0 h 5 1 4 3 6 ) gs: g h 6 5 4 3 2 1 Y = 0 Figur 2: ) n yli rwing, ) it s upwr rwing. ) 2
2 Upwr Plnrity Tsting Not vry plnr DAG mits plnr upwr rwing. Thr r thr irnt lgorithms or tsting th xistn o n upwr plnr rwing o igrph, howvr thir omplxity n improv. 1. Di Bttist n Tmssi s lgorithm whih provs th igrgh is upwr plnr i n only i it is sugrph o plnr st-igrph. St-igrphs hv th ollowing hrtristis: yli plnr on sour (s) on sink (t) with g (s,t) 2. Hutton n Luiw s O(n 2 ) tim omplxity upwr plnrity tsting or singl-sour igrphs. A rnt rsrh hs improv th tim omplxity to O(n). 3. Brtolzzi, Di Bttist, Liott n Mnnino hv sign n O(n 2 ) tim lgorithm whih tsts upwr plnrity o igrphs with ix ming. this lgorithm woul xplor ltr. 2.1 Dinitions In ny upwr rwing, th inommimg n outgoing gs o h vrtx v ppr onsutivly in th str o v. S() = (numr o sours o ) = (numr o sinks o ). Smll ngls S r onir thos lss thn 180 gr. Lrg ngls L r thos grtr thn 180 gr. A pol is th sour or sink o th. In plnr stright lin upwr rwing : * Eh pol hs xtly on lrg ngl inint on it. * Eh intrnl hs S() - 1 lrg intrnl ngls. * Th xtrnl h hs S(h) + 1 lrg intrnl ngls. 2.2 Assignmnt Mol Now tht w r l to in th numr o lrg ngls or h, n ssignmnt o pols to s my xist suh tht: 3
Eh pol v is ssign to xtly on inint on v. Eh intrnl hs S() - 1 pols ssign to it. Th xtrnl h hs S(h) + 1 pols ssign to it. A onsistnt ssignmnt o pols to s n oun y using low lgorithm sussivly whih tks O(n 2 ) tim. Th prour woul to in th mximum low rom vrtis(sours n sinks) to s. Thn sssign pol to suh tht thr xists th mximum low rom th pol to. In this omputtion th supply o h pol is 1, n mn o o h is S() or S(h). Extrnl Intrnl Tru sour Tru sink # pols = s() - 1 = 2-1 = 1 # pols = s() - 1 = 1-1 = 0 # pols = s() - 1 = 3-1 = 2 # pols = s() + 1 = 2 + 1 = 3 Figur 3: Assignmnt o pols to s using low lgorithm. 2.3 Thorm An m igrph mits n upwr rwing i n only i it mits onsistnt ssignmnt o pols to s. 4
Proo: I onsistnt ssignmnt o pols to s xists (sri in Assignmnt mol stion), thn th pols ssign to h r rwn with lrg ngl in tht, n th igrph woul om upwr. Extrnl Intrnl Tru sour Tru sink ) ) Figur 4: ) Th signmnt o pols to s, ) Th rsult upwr rwing. 2.4 Anothr xmpl: Thn gin NOT vry plnr DAG mits n upwr plnr rwing. In th ollowing xmpl onsistnt ssignmnt o pols n not oun, sin th xtrnl ns two pols, ut th grph only hs on pol to or. Thror this DAG n not rwn upwr. 5
# pols = 2 # pols = 0 tru sink # pols = 0 # pols = 0 tru sour Figur 5: No upwr rwing or this grph. 6