Question

Modify the statement of the cut property so that it is true, even for the case of edge weights that are not necessarily distinct. Prove that your modified statement is true when edge weights may be repeated.

Answer 1

ns Using a bucket im plementaltion (also knoun as Dol's Implemen tatton) Djstna algoi lhm Can be Made to aun i,n o(VIE). idea is to find an upp bound on shostest distance lobclu . Since # c aximum edge weight a veatez a bepotd Mot v- times, we a bound of O(Wvi) on Soatest (poth di pmopoty op. Dijkst기a algo». th m that the distance 広at cae desineded penmanent core non - decaeasing we can pie sent the followin mes, we g&a stance Using the xe non-decatasing we can presunt procedure efficient., Dijkshal (G.1, s Inpul Graph G;(M).etge uet]ht te Mat edge weight W, vole SEV ouspu Shontest path distonce labely to叽all v㎝tices ve V: dist (v):o bucret (v)- nill BIil keep vente of distome labd i 8Io]: S, dist (5)- o, bucket Cs)-o CS Scanned with CamScanner der - o

Incsement indey Uindes fon all edges Cu.v)EE: Hemp -dist Cv) if dist Cv)> dist(u) + dit (v): dist (u)+ I(Uv) Remove Vf:nom B [temp] Douobly tihnked list shoud be used fosn csnay B ohich clows us to do the Pollouoing oponations in oC)time checking whethes a buckef ts émpty or nonempty () eletinq n element from a bucket (3) Addinq an Element to The bucke Scanned with CS Camscanner