5/9/13. Part 10. Graphs. Outline. Circuits. Introduction Terminology Implementing Graphs

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1 Prt 10. Grphs CS 200 Algorithms n Dt Struturs 1 Introution Trminology Implmnting Grphs Outlin Grph Trvrsls Topologil Sorting Shortst Pths Spnning Trs Minimum Spnning Trs Ciruits 2 Ciruits Cyl A spil yl tht psss through vry vrtx (or g) in grph xtly on n rturns k to th pl it strt. 1

2 5/9/13 Eulr s rig prolm (Brigs of Konigsrg Prolm) Rivr Btwn Eg Bnk D A An Eg Btwn Eg Btwn A n CA n C Eg Btwn Isln C C n D Isln D Eg Btwn Rivr Eg Btwn B n Eg C Btwn B n D Bnk B n C B Eulr Is it possil to trvl ross vry rig without rossing ny rig mor thn on? h?p://yskrthi.worprss.om/2006/07/31/ulr- n- th- rigs- of- konigsrg 4 Eulr s rig prolm (Brigs of Konigsrg Prolm) Eulr Is it possil to trvl ross vry rig without rossing ny rig mor thn on? h?p://yskrthi.worprss.om/2006/07/31/ulr- n- th- rigs- of- konigsrg 5 Eulr pths/iruits Eulr pth: A pth tht visits h g only on in th grph Eulr iruit: A yl tht visits h g only on in th grph 2

3 Exmpl: Dos ny grph hv n Eulr iruit? 7 Exmpl: Dos ny grph hv n Eulr pth? 8 Exmpl: Dos ny grph hv n Eulr iruit? f g 9 3

4 Eulr pths/iruits Is thr simpl ritrion tht llows us to trmin whthr grph hs n Eulr iruit or pth? Eulr Pths Thorm: A onnt multigrph hs n Eulr pth.iff. it hs xtly two vrtis of o gr Eulr Ciruits Thorm: A onnt multigrph with t lst two vrtis hs n Eulr iruit.iff. h vrtx hs n vn gr. 4

5 Mohmm s Simitrs f Cn Mohmm s simitrs rwn without liqing pnil n th rwing gins n ns t th sm point? - - f- i g- h- j- i- k- g- - - i g h j k 13 Hmiltonin Pths/Ciruits A Hmiltonin pth/iruit: pth/iruit tht visits vry vrtx xtly on. Dfin for irt n unirt grphs. Is thr n ffiint wy to trmin whthr grph hs Hmiltonin iruit? Dos ny grph hv Hmiltonin iruit or Hmiltonin pth? 15 5

6 DIRAC s Thorm If G is simpl grph with n vrtis with n 3 suh tht th gr of vry vrtx in G is t lst n/2, thn G hs Hmiltonin iruit. 16 Or s Thorm If G is simpl grph with n vrtis with n 3 suh tht g(u)+g(v) n for vry pir of nonjnt vrtis u n v in G, thn G hs Hmiltonin iruit. 17 Hmiltonin Pths/Ciruits Dir n Or s thorms provi suffiint onition for onnt simpl grph to hv Hmiltonin iruit. Thy o NOT provi nssry onition for th xistn of Hmiltonin iruit This prolm longs to lss of prolms for whih it is liv thr is no ffiint (polynomil running tim) lgorithm. 6

7 5/9/13 Th Trvling Slsmn Prolm (TSP) Cn w us th Shortst pth lgorithm? Cn w us th Eulr Ciruit? TSP: Givn list of iss n thir pirwis Cn tw uts th Hmilton Ciruit? istns, fin shortst possil our ht visits h ity xtly on. 13,509 iss n towns in th US tht hv mor thn 500 rsints An opsml TSP tour through Grmny s 15 lrgst iss (on out of 14!/2) h?p:// Using Hmiltonin Ciruits Exmin ll possil Hmiltonin iruits n slt on of minimum totl lngth With n itis.. (n-1)! Diffrnt Hmiltonin iruits Ignor th rvrs orr iruits (n-1)!/2 With 50 itis 12,413,915,592,536,072,670,862,289,047,373, 375,038,521,486,354,677,760,000,000,000 routs 20 Th thr utilitis prolm Hous A Hous B? Hous C 21 7

8 Dsigning Mirohip You r signing mirohip onntions twn ny two units nnot ross h?p:// Plnr Grphs You r signing mirohip onntions twn ny two units nnot ross Th grph sriing th hip must plnr plnr non- plnr h?p://n.wikipi.org/wiki/plnr_grph Is this grph plnr? 24 8

9 Chip Dsign You wnt mor thn plnrity: th lngths of th onntions n to s short s possil (fstr, n lss ht is gnrt) h?p:// Eulr s Formul Lt G onnt plnr simpl grph with gs n v vrtis. Lt r th numr of rgions in plnr rprsnttion of G. Thn r=-v+2 26 Exmpl Suppos tht onnt plnr simpl grph hs 20 vrtis, h of gr 3. Into how mny rgions os rprsnttion of this plnr grph split th pln? 27 9

10 Grph Coloring A oloring of simpl grph is th ssignmnt of olor to h vrtx of th grph so tht no two jnt vrtis r ssign th sm olor 28 Mp n grph B A C D B C D F G A E F E G 29 Chromti numr Th lst numr of olors n for oloring of this grph. Th hromti numr of grph G is not y χ(g) 30 10

11 Th four olor thorm Th hromti numr of plnr grph is no grtr thn four 31 Exmpl 32 Exmpl Wht is th hromti numr of th grph Cn, whr n>=3? (Cn is th yl with n vrtis) 33 11