Shortest Paths. CSE 421 Algorithms. Bottleneck Shortest Path. Negative Cost Edge Preview. Compute the bottleneck shortest paths

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Aldous Goodman
5 years ago
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1 Shor Ph CSE Alorihm Rihr Anron Lr 0- Minimm Spnnin Tr Ni Co E Dijkr lorihm m poii o For om ppliion, ni o mk n Shor ph no wll in i rph h ni o yl Ni Co E Priw Topoloil Sor n or olin h hor ph prolm in ir yli rph Bllmn-For lorihm in hor ph in rph wih ni o (or rpor h xin o ni o yl). Bolnk Shor Ph Din h olnk in or ph o h mximm o lon h ph x Comp h olnk hor ph Dijkr Alorihm or Bolnk Shor Ph S = {}; [] = ni ininiy; [] = ininiy or!= - - Whil S!= V Choo in V-S wih minimm [] A o S For h w in h nihorhoo o [w] = min([w], mx([], (, w))) y x z

2 Minimm Spnnin Tr Minimm Spnnin Tr Inro Prolm Dmonr hr irn ry lorihm Proi proo h h lorihm work Gry Alorihm or Minimm Spnnin Tr Exn r y inlin h hp o oin A h hp h join ijoin omponn Dl h mo xpni h o no ionn h rph 0 8 Gry Alorihm Prim Alorihm Exn r y inlin h hp o oin Conr h MST wih Prim lorihm rin rom rx Ll h in orr o inrion Gry Alorihm Krkl Alorihm A h hp h join ijoin omponn Gry Alorihm Rr-Dl Alorihm Dl h mo xpni h o no ionn h rph Conr h MST wih Krkl lorihm Ll h in orr o inrion Conr h MST wih h rrl lorihm Ll h in orr o rmol

3 Why o h ry lorihm work? For impliiy, m ll o r iin L S o V, n ppo = (, ) i h minimm o o E, wih in S n in V-S i in ry minimm pnnin r Proo Sppo T i pnnin r h o no onin A o T, hi r yl Th yl m h om = (, ) wih in S n in V-S T = T { } + {} i pnnin r wih lowr o Hn, T i no minimm pnnin r Opimliy Proo Prim Alorihm omp MST Rr-Dl Alorihm Lmm: Th mo xpni on yl i nr in minimm pnnin r Krkl Alorihm omp MST Dlin wih h mpion o no ql wih For h wih o iin A mll qnii o h wih Gi i rkin rl or ql wih Dijkr Alorihm or Minimm Spnnin Tr S = {}; [] = 0; [] = ininiy or!= Whil S!= V Choo in V-S wih minimm [] A o S For h w in h nihorhoo o [w] = min([w], (, w)) y x z

4 Minimm Spnnin Tr Unir Grph G=(V,E) wih wih Gry Alorihm or Minimm Spnnin Tr [Prim] Exn r y inlin h hp o oin [Krkl] A h hp h join ijoin omponn [RrDl] Dl h mo xpni h o no ionn h rph 0 8 Why o h ry lorihm work? For impliiy, m ll o r iin E inlion lmm L S o V, n ppo = (, ) i h minimm o o E, wih in S n in V-S i in ry minimm pnnin r o G Or qilnly, i i no in T, hn T i no minimm pnnin r S V - S i h minimm o wn S n V-S Proo Opimliy Proo Sppo T i pnnin r h o no onin A o T, hi r yl Th yl m h om = (, ) wih in S n in V-S S V - S T = T { } + {} i pnnin r wih lowr o Hn, T i no minimm pnnin r Prim Alorihm omp MST Krkl Alorihm omp MST Show h whn n i o h MST y Prim or Krkl, h i h minimm o wn S n V-S or om S.

5 S = { }; T = { }; whil S!= V Prim Alorihm Pro Prim lorihm omp n MST Show n i in h MST whn i i o T hoo h minimm o = (,), wih in S, n in V-S o T o S Krkl Alorihm L C = {{ }, { },..., { n }}; T = { } whil C > Pro Krkl lorihm omp n MST Show n i in h MST whn i i o T L = (, ) wih in C i n in C j h minimm o joinin iin in C Rpl C i n C j y C i U C j A o T Rr-Dl Alorihm Lmm: Th mo xpni on yl i nr in minimm pnnin r Dlin wih h mpion o no ql wih For h wih o iin A mll qnii o h wih Gi i rkin rl or ql wih

6 Appliion: Clrin Gin ollion o poin in n r- imnionl p, n n inr K, ii h poin ino K h r lo ohr Din lrin Dii h ino K o mximiz h in wn ny pir o i (S, S ) = min {i(x, y) x in S, yins} Dii ino lr Dii ino lr Dii ino lr Din Clrin Alorihm L C = {{ }, { },..., { n }}; T = { } whil C > K L = (, ) wih in C i n in C j h minimm o joinin iin in C Rpl C i n C j y C i U C j

7 K-lrin

CSE 421 Algorithms. Warmup. Dijkstra s Algorithm. Single Source Shortest Path Problem. Construct Shortest Path Tree from s

CSE 421 Algorithms. Warmup. Dijkstra s Algorithm. Single Source Shortest Path Problem. Construct Shortest Path Tree from s CSE Alorihm Rihr Anron Dijkr lorihm Sinl Sor Shor Ph Prolm Gin rph n r r Drmin in o ry r rom Iniy hor ph o h r Epr onily hor ph r Eh r h poinr o pror on hor ph Conr Shor Ph Tr rom Wrmp - - I P i hor ph