Te???a%20G??f??%20Te

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Te???a G??f??Teµe???se??-???????µ??-?fa?µ?????e
f??a?? 4 S??desµ???t?ta
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??sa????

• ?e??t? ßa?µ?? s??desµ???t?ta? e??? ???f??
• ?fa?µ???? se ?p??asd?p?te µ??f?? d??t?a
  (t??ep????????a??, s?????????a?? ?.a.)

???? e??a? t? ?a??te?? d??t?? ap? ?p??? a?t????
se ????
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S??desµ???t?ta ????f??

• S????? ap???pt??s?? ????f?? V? (vertex cut set,
  vertex separating set) e??? s??dedeµ???? ???f?? G
  e??a? t? s????? t?? ????f?? ?ste ? ???f?? GV? ?a
  µ?? e??a? s??dedeµ???? ?a? ?a µ?? ?p???e? ???s??
  ?p?s????? t?? V? µe t?? ?d?a ?d??t?ta
• S??desµ???t?ta ????f?? (vertex connectivity),
  VC(G), e??a? t? e????st? kV?, ?ste ? ???f??
  G ?a e??a? s??dedeµ???? a? d?a??af??? ????te?e?
  ap? k ????f??.
• ? G ???eta? k-s??dedeµ???? k-connected a? VC(G)k

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S??desµ???t?ta ????f?? ?s??s?
???? e??a? ? t?µ? VC t?? ???f?? N5, K5, S6, W6,
C4, K4,3, ???ß??(5,3) ?
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?e???? Te???µata

• Te???µa ??a ????f? v e??? d??d??? e??a?
  ap???pt??sa a? ?a? µ???? a? d(v)gt1
• ????sµa ???e µ? as?µa?t?? ap??? s??dedeµ????
  ???f?? ??e? t??????st?? 2 ????f?? p?? de? e??a?
  ap???pt??se?
• Te???µa ??a ????f? v e??a? ap???pt??sa a? ?a?
  µ???? a? ?p?????? 2 ????f?? u ?a? w (u,w?v), ?ste
  ? v ?a ß??s?eta? se ???e µ???p?t? ap? t?? u p???
  t?? w

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S??desµ???t?ta ??µ??

• S????? ap???pt??s?? a?µ?? ?? edge cut set,
  edge separating set e??a? t? s????? t?? a?µ??
  ?ste ? ???f?? G-E? ?a µ?? e??a? s??dedeµ????,
  ????? ?a ?p???e? ???s?? ?p?s????? t?? ?? µe t??
  ?d?a ?d??t?ta
• S??desµ???t?ta a?µ?? EC(G) edge connectivity
  e??? ???f?? G e??a? t? e????st? kE?, ?ste ? G
  ?a pa?aµ??e? s??dedeµ???? ?pe?ta ap? d?a??af? k-1
  a?µ??
• ??a? ???f?? G ???eta? k-s??dedeµ???? ?? p??? t??
  a?µ?? edge k-connected a? EC(G)k

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S??desµ???t?ta ??µ?? ?s??s?
???? e??a? ? t?µ? ?C t?? ???f?? N5, K5, S6, W6,
C4, K4,3, ???ß??(5,3) ?
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?e???? Te???µata

• Te???µa ??a a?µ? e e??a? ap???pt??sa a? ?a?
  µ???? a? ?p?????? 2 ????f?? u ?a? w, t?t??e? ?ste
  ? e ?a ß??s?eta? se ???e µ???p?t? ap? t?? u p???
  t?? w.
• Te???µa ??a a?µ? e??a? ap???pt??sa a? ?a? µ????
  a? de? pe????eta? se ?????
• Te???µa Whitney VC(G) EC(G) d(G)
• ????sµa EC(G) floor(2m/n)
• Te???µa ?st? 1?n-1. ?? d(G)?(n?-2)/2, t?te ?
  G e??a? ?-s??dedeµ????

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?a??de??µa
VC(G)? ?C(G)?
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?eµ???a G??f??

• ??a? d?s??dedeµ????-biconnected ???f?? de? ??e?
  ap???pt??se? ????f??.
• ??a? t?t???? ???f?? ap?te?e? ??a teµ????-block ?
  µ?a d?s???st?sa-bicomponent (biconnected
  component)
• ?eµ???? e??? ???f?? ???eta? ??a? ?p????f?? p??
  e??a? d?s??dedeµ???? ?a? ??e? t? µ???st? a???µ?
  ????f??.
• ??sa teµ???a ??e? ? d?p?a??? ???f???

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?eµ???a G??f?? s????e?a

• ??? teµ???a e??? ???f?? ????? t? p??? µ?a ?????
  ????f?.
• ???e ???f?? ta?t??eta? µe t?? ???s? t?? teµa????
  t??.
• ?a teµ???a e??? ???f?? µp????? ?a ß?e???? µe DFS
• ?e? µp??e? µ?a ????f? ?a e??a? ????? se d??
  teµ???a e??? ???f??.
• ?a teµ???a e??? ???f?? ???????? t?? a?µ?? se
  a?e???t?ta s????a.

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?s?te???? ???a µ???p?t?a

• ?s?te???? ???a µ???p?t?a (internally disjoint
  paths) e??a? d?? µ???p?t?a µe ?????? te?µat????
  ????f??, ????? ???e? ?????? ????f??.

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0
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Te???µa Whitney

• Te???µa ??a? ???f?? G µe n3 e??a?
  d?-s??dedeµ???? a? ?a? µ???? a? d?? ?p??esd?p?te
  ????f?? t?? e??a? s??dedeµ??e? µe t??????st?? d??
  es?te???? ???a µ???p?t?a.
• ????sµa ?? ??a? ???f?? G e??a? d?s??dedeµ????,
  t?te d?? ?p??esd?p?te ????f?? t?? a?????? se ??a?
  ?????.
• ????sµa ?? ??a? ???f?? ap?te?e?ta? ap? ??a
  teµ???? µe n3, t?te d?? ?p??esd?p?te a?µ?? t??
  a?????? se ??a? ?????.

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Te???µa Menger

• Te???µa ? µ???st?? a???µ?? es?te???? ?????
  µ???pat??? ap? µ?a ????f? u se µ?a ????f? v e???
  s??dedeµ???? ???f?? ?s??ta? µe t?? e????st?
  a???µ? ????f??, p?? ???????? t?? ????f?? u ?a? v.
• 11 d?at?p?se?? t?? Te???µat?? Menger
• Te???µa Whitney ??a? ???f?? e??a? k-s??dedeµ????
  a? ?a? µ???? a? ??a ta ?e??? ????f?? e?????ta? µe
  t??????st?? k es?te???? ???a µ???p?t?a.

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????sµata (?? p??? a?µ??)

• ????sµa ? µ???st?? a???µ?? es?te???? ?????
  µ???pat??? ap? µ?a ????f? u se µ?a ????f? v e???
  s??dedeµ???? ???f?? ?s??ta? µe t?? e????st?
  a???µ? a?µ??, p?? ???????? t?? ????f?? u ?a? v.
• ????sµa ??a? ???f?? G e??a? k-s??dedeµ???? ??
  p??? t?? a?µ??, a? ?a? µ???? a? ??a ta ?e???
  ????f?? e?????ta? µe t??????st?? k es?te???? ???a
  µ???p?t?a.

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Ge???e?µ??? p??ß??µa s??d?sµ??

• ?? ?e??µe?? p??ß??µa t?? s??d?sµ?? a?af??eta? st?
  p??ß??µa t?? e??es?? e????st?? ?e??????t??
  d??d???.
• ??a e????st? ?e?????? d??d?? µe n3 ??e? ECVC1.
  
• ?? ?e???e?µ??? p??ß??µa t?? s??d?sµ?? e??a? ?a
  ß?e?e? ?p????f?? d????t?? ???f?? µe e????st?
  ß????, ?ste ? s??desµ???t?ta ?a ?s??ta? µe l
• ?? l1, t?te ta p??ß??µata ta?t????ta?.

???? e??a? p?? a???p?st? ?n8
VCEC4
VC1 EC3
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Ge???e?µ??? p??ß??µa s??d?sµ??

• Ta ?e????e? ? pe??pt?s? µ? ????sµ???? ???f??.
• S??p?? e??a? ? e??es? e??? ???f?? µe n ????f??
  ?a? ep???af?? 0..n-1), l-s??ededeµ???? ?a? µe
  t?? e????st? a???µ? a?µ??.
• ? ???f?? a?t?? s?µß????eta? µe Hl,n
• ??a??????ta? t?e?? pe??pt?se??
• l ??t?? (l2r).
• l pe??tt? (l2r1), n ??t??.
• l pe??tt? (l2r1), n pe??tt?.

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Ge???e?µ??? p??ß??µa s??d?sµ??

• l ??t?? (l2r). ??? ??µß?? i ?a? j e??a?
  ?e?t??????, a? ir j ir
• l pe??tt? (l2r1), n ??t??. ?atas?e???eta? ?
  ???f?? H2r,n (?p?? ???). ?p?s?? d?? ??µß?? i ?a?
  in/2 e?????ta? ??a 1i n/2.
• l pe??tt? (l2r1), n pe??tt?. ?atas?e???eta? ?
  ???f?? H2r,n. ?p?s?? e???eta? ? ??µß?? 0 µe t???
  (n1)/2 ?a? (n1)/2 ?a? ? ??µß?? i µe t?? ??µß?
  i(n1)/2 ??a 1i(n1)/2.

H4,8 H5,8 H5,9
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Ge???e?µ??? p??ß??µa s??d?sµ??

• Te???µa ? ???f?? ?l,n e??a? l-s??dedeµ????.
• Te???µa ? e????st?? a???µ?? a?µ?? t?? ???f??
  ?l,n e??a? ceil(ln/2).

H4,8
H5,8
H5,9
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?s?µ??f?sµ??

• ???sµ?? ??? ???f?? G1(V1,E1) ?a? G2(V2,E2)
  ?????ta? ?s?µ??f????-isomorphic a? ?p???e? µ?a
  aµf?µ???s?µa?t? a?t?st????a f ap? t? t? s????? V1
  st? s????? V2 µe t?? ?d??t?ta ?t? ?? ????f?? a, b
  e??a? ?e?t?????? st? G1 a? ?a? µ??? a? ?? ????f??
  f(a), f(b) e??a? ?e?t?????? st? G2, ??a ???e
  ?e???? a,b t?? V1.
• ? s????t?s? f ???µ??eta? ?s?µ??f???- isomorphism.

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??ap?st?s? ?s?µ??f?sµ?? 1? ??s?

• ???????µ?? e?????? ?s?µ??f?sµ??
• ??s?d?? ???f?? G ?a? ???f?? ?
• ???d?? ??? ? ???
• ?? V(G)? V(H), t?te return OXI
• Te????µe µ?a d??ta?? t?? ????f?? t?? G
• ?ata???f??µe t?? p??a?a ?e?t???a? ?G t?? G
• G?a ???e d??ta?? t?? ????f?? t?? H
• ?ata???f??µe t?? p??a?a ?e?t???a? ?H
• ?? ?G?H t?te return NAI
• return OXI

????p????t?ta ?
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??ap?st?s? ?s?µ??f?sµ?? 2? ??s?

• ?st? ?t? ?? ???f?? a?apa??st??ta? µe t? µ???d?
  t?? p?????? p??spt?s?? (incidence matrix).
• ?p??e? ? ??a? p??a?a? p??spt?s?? ?a
  µetas??µat?s?e? st?? ???? µ?s? a?t?µeta??se??
  ??aµµ?? ?/?a? st????. ?p?s?? µ? ap?te?esµat???
  ??s?.
• ?p?????? ap?te?esµat???? a??????µ?? µ??? ??a
  e?d???? pe??pt?se?? ???f??.
• ?p?s??, µp??e? ?a e??a? e????? ? ap?de??? ?t? d??
  ???f?? de? e??a? ?s?µ??f????

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?µet?ß??te? invariants

• S?????e? ??a t?? e????? d?ap?st?s? a? d?? ???f??
  de? e??a? ?s?µ??f????
• ?d?a t????
• ?d?? µ??e????
• ?d?a a???????a ßa?µ???
• ?d??? a???µ?? s???st?s???
• G?a ???e s???st?sa t?? (4) apa?t??ta? ?et??? ??
  p??te? t?e?? e??t?se??
• ????? ?? d?? ???f?? t? ?d?? ???µat??? p??????µ?
• G?a nlt8, a? ??e? ?? e??t?se?? apa?t????? ?et???,
  t?te ?? ???f?? e??a? ?s?µ??f???? (??e?
  ap?de???e?).

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?a?ade??µata - ?s??s?
f(a) e f(b) a f(c) b f(d) c f(e) d
f(a) 6 f(b) 1 f(c) 3 f(d) 5 f(e) 2 f(f)
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?a?ade??µata - ?s??s?
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Ge???e?s? ?s?µ??f?sµ??

• ??? ???f?? e??a? 1-?s?µ??f???? a? ?a??sta?ta?
  ?s?µ??f???? µet? t?? epa?e???µµ??? d??spas? t??
  ap???pt??s?? ????f??.
• ?a teµ???a e??? ???f?? e??a? ?s?µ??f??? p??? t??
  s???st?se? e??? ????? ???f??.

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?a??de??µa
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?a??de??µa ?s??s?
0
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9
0
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9
1
7
7
1
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5
1
7
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6
2
3
3
5
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????? e??a? ? 1-?s?µ??f???? µe d??spas?
ap???pt??s?? ????f?? ?
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Ge???e?s? ?s?µ??f?sµ??

• ?? ? ???f?? ap?te?e?ta? ap? ??a teµ????, t?te ?
  ?????a t?? 1-?s?µ??f???t?ta? ta?t??eta? µe t??
  ?????a t?? ?s?µ??f???t?ta?.
• Te???µa ?? d?? ???f?? e??a? 1-?s?µ??f????, t?te
  ? se??? ?a? ? µ?de????t?t? t??? e??a? ?se?.

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??t?st????a ??????

• ??? ???f?? ????? a?t?st????a ?????? circuit
  correspondence, a? ?p???e? aµf?µ???s?µa?t?
  a?t?st????a a?µ?? ?a? ??????, ?ts? ?ste ??a?
  ?????? t?? p??t?? ???f?? ?a ??e? a?t?st???? ?????
  st? de?te?? ???f?, p?? ap?te?e?ta? ap? t??
  a?t?st???e? a?µ??.
• Te???µa ??? 1-?s?µ??f???? ???f?? ?????
  a?t?st????a ??????

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2-?s?µ??f?sµ??

• ??? ???f?? e??a? 2-?s?µ??f???? a? ?a??sta?ta?
  ?s?µ??f???? µet? t?? epa?e???µµ??? efa?µ??? µ?a?
  ? d?? ap? t?? e??? p???e??
• d?ad????? d??spas? ap???pt??s?? ????f??
• d?a????sµ?? t?? e??? ???f?? se d?? ??????
  ?p????f???, p?? ????? 2 ?e??? ?????? ????f?? ?a?
  epa?as??des? t?? ?p????f?? ta?t?p????ta? t??
  ????f?? µe d?af??et??? t??p?

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2-?s?µ??f?sµ?? ?a??de??µa
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2-?s?µ??f?sµ??

• ??? ?s?µ??f???? ???f?? e??a? 1-?s?µ??f????, d??
  1-?s?µ??f???? ???f?? e??a? 2-?s?µ??f????. ??
  a?t??et? de? ?s??e?.
• Te???µa (Whitney) ??? ???f?? e??a? 2-?s?µ??f????
  a? ?a? µ???? a? ????? a?t?st????a ??????

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?a??de??µa
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???????µ??

• ???????µ?? d??s??s?? ???f??
• BFS, a?a??t?s? ?at? p??t??
• DFS, a?a??t?s? ?at? ß????
• ?? a??????µ?? a?t?? ???s?µ?p?????ta?
• ??a t?? e??es? ap?st?se?? ap? ??p??a ????f?
• ??a t? d?ap?st?s? a? ???f?? e??a? s??dedeµ????
• ??a t?? e??es? (?s???? s??dedeµ????) s???st?s??
• ??a t?? t?p??????? ta????µ?s?
• ??a t?? e??es? t?? s?µe??? ?????s??
• ??a t?? e??es? ?ef????
• ??a t? d?ap?st?s? ??????
• ??a t? d?ap?st?s? ep?ped???t?ta?

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BFS vs DFS

• BFS
• 1959 Moore
• ?p?s?ept?µaste t?? p?? ???? ??µß?
• ???p???s? µe ????.

• DFS
• 1973 Hopcroft-Tarjan
• ?p?s?ept?µaste t?? p?? ßa?? ??µß?
• ???p???s? µe st??ßa.

? d?af??? ???e?ta? st? d?µ? ded?µ????!
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BFS se d??d??
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DFS se d??d??
?a?t??eta? µe t?? p??d?ateta?µ??? (preorder)
d??s??s?
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BFS vs DFS
dfs(G) list L empty tree T
empty choose a starting vertex x
visit(x) while(L nonempty)
remove edge (v,w) from end of
L if w not visited
add (v,w) to T visit(w)
bfs(G) list L empty tree T
empty choose a starting vertex x
visit(x) while(L nonempty)
remove edge (v,w) from
beginning of L if w not visited
add (v,w) to T
visit(w)
Visit ( vertex v) mark v as visited
for each edge (v,w) add edge
(v,w) to end of L
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BFS ???es? ap?st?se??

• ??s?d?? ???f?? G µe ep???af?? ?a? ????f? x?V
• ???d?? ?? ap?st?se?? ap? t?? ????f? x p??? ??e?
  t?? ????f?? p?? e??a? p??spe??s?µe? ap? a?t??
• T?t??µe i?0. St?? ????f? x ??t??µe t?? ep???af?
  i.
• ???s???µe ??e? t?? ????f?? ????? ep???af?? p??
  e??a? ?e?t?????? p??? t??????st?? µ?a ????f? µe
  ep???af? i
• ?? de? ?p???e? ??p??a t?t??a ????f?, t?te ?
  ???f?? e?a?t?????e.
• T?t??µe t?? ep???af? i1 se ??e? t?? ????f?? p??
  ß?????a? st? ??µa 2
• T?t??µe i?i1. ???a????µe st? ??µa 2

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BFS ???es? ap?st?se??, ? ?d?a

• Se ???e ??????? st??µ? ?p???e? ??a µ?t?p?
  ????f?? p?? t?? ????µe a?a?a???e?, a??? de? t??
  ????µe a??µ? epe?e??as?e?
• ?aµß????µe d?ad????? t?? ????f?? t?? µet?p??
  ?a? a?a?a??pt??µe t??? ?e?t??e? d?µ???????ta? ??a
  ??? µ?t?p?

s

• ?sp?e? ????f?? µ? µa??a??sµ??e? ?a? e?t?? ?????
• G??? ????f?? µa??a??sµ??e? ?a? e?t?? ?????
• ?a??e? ????f?? µa??a??sµ??e? ?a? e?t?? ?????

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??a??t??? BFS a?????p???s?

• procedure BFS(Ggraph snodevar colorcarray
  distiarray parentparray)
• for each vertex u do
• coloruwhite
• distu8
• parentunil
• colorsgray dists0
• init(Q) enqueue(Q, s)

?(n)
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??a??t??? BFS main

• while not (empty(Q)) do
• uhead(Q)
• for each v in adju do
• if colorvwhite then
• colorvgray
• distvdistu1
• parentvu
• enqueue(Q,v)
• dequeue(Q) colorublack
• end BFS

O(m)
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?a??de??µa BFS
r s t u
r s t u


0
1
0



s
w r




1



v w x y
v w x y r s
t u r s
t u
1
2


0
1
0
2
r t x
t x v
2


2

1
1
2
v w x y v
w x y
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?a??de??µa BFS s????e?a
r s t u
r s t u
3
1
0.
0
3
2
2
1
x v u
v u y
2
2

1
1
2
2
3
v w x y
v w x y r s
t u r s
t u
3
1
0
2
1
0
2
3
u y
y
3
2
3
2
2
1
1
2
v w x y v
w x y
?????, ? ????f? y ß?a??e? ap? t?? ???? ?a?
µa????eta?
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????p????t?ta t?? BFS

• ??????p???s? T(n).
• ???e ??µß?? µpa??e? st?? ???? µ?a f??? (ap?
  ?sp??? ???eta? ????) ?a? ? ??sta ?e?t??as??
  d?as???eta? µ?a f???.
• ?? p???, ??e? ?? ??ste? d?as?????ta?.
• ???e a?µ? ?aµß??eta? d?? f????, ?p?te ? ß?????
  epa?a?aµß??eta? t? p??? 2E f????.
• ?e???te?? pe??pt?s? O(nm)
• ?? ? ???f?? e??a? ???p???µ???? µe p??a?a
  ?e?t??as??, t?te ? p???p????t?ta e??a? ?(n2).

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BFS tree
?se? a?µ?? t?? ???f?? G pa???s?????ta? st? BFS
tree ???µ????ta? de?d????? tree edges, ??
?p????pe? ???µ????ta? d?asta????µe?e? cross
edges.
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DFS ??a??t?s? ?at? ß????

• ???????µ?? DFS t?? Hopcroft-Tarjan (1973)
• ??s?d?? ???f?? G µe ep???af?? ?a? ????f? x?V
• ???d?? s????? ? de?d????? ????f?? ?a? a???µ?s?
  dfi(v)
• T?t??µe T?Ø, i?1.
• G?a ???e v?V, ??t??µe dfi(v)?0
• G?a ???e u µe dfi(u) e?te?e?ta? DFS(u).
• St?? ???d? d??eta? t? s????? ?.
• ??ad??as?a DFS(v)
• T?t??µe dfi(v)?i, i?i1
• G?a ???e u ?N(v) e?te????ta? ?? e?t????
• ?? dfi(u)0, t?te ??t??µe T?TUe, ?p??
  e(u,v) µ?a µ? ???s?µ?p???µ??? p??sp?pt??sa
  a?µ?, ?a? l(e)?used
• ?a???µe t?? DFS(u)

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DFS tree
?se? a?µ?? t?? ???f?? G pa???s?????ta? st? DFS
tree ???µ????ta? de?d????? tree edges, ??
?p????pe? ???µ????ta? ?p?s??e? back edges.
49
50
?p?s??e? a?µ??
0
1
4
2
9
5
11
7
6
8
10
DFS Tree

• Te???µa ???e ?p?s??a a?µ? (u,v) p?? p????pte?
  ?at? t?? a?a??t?s? ?at? ß???? (DFS) e??? µ?
  ?ate?????µe??? ???f?? e???e? ????f?? p??
  ß??s???ta? se s??s? ap??????/p???????.

50
51
??a??t??? DFS

• procedure DFS(Ggraph var color carray
  d,fiarray parentparray)
• for each vertex u do
• coloruwhite parentunil
• time0
• for each vertex u do
• if coloruwhite then DFS-Visit(u)
• end DFS

?(n)
52
??a??t??? DFS-visit(u)

• colorugray timetime1 dutime
• for each v in adju do
• if colorvwhite then
• parentvu DFS-Visit(v)
• colorublack timetime1
• futime
• end DFS-Visit

53
?a??de??µa DFS
u v w
u v w
1/
1/
2/
x y z
x y z
u v w
u v w
1/
2/
1/
2/
B
3/
3/
4/
x y z
x y z
54
?a??de??µa DFS
u v w
u v w
1/
2/
1/
2/
B
B
3/
3/6
4/5
4/5
x y z
x y z
u v w
1/
2/7
B
3/6
4/5
x y z
55
?a??de??µa DFS
u v w
1/8
2/7
9
C
B
F
3/6
4/5
10
x y z
x y z
u v w
u v w
1/8
2/7
9
9/12
1/8
2/7
B
F
C
B
F
C
3/6
4/5
10/11
3/6
4/5
10/11
x y z
x y z
56
????p????t?ta t?? DFS

• ??????p???s? T(n).
• ? DFS-visit ?a?e?ta? µ?a f??? ??a ???e ??µß? v.
• ? ß????? for e?te?e?ta? dv f????.
• S??????? ? DFS-visit ?a?e?ta? T(m) f????.
• ?e???te?? pe??pt?s? T(nm)

57
?fa?µ???? t?? DFS

• O ???f?? G e??a? s??dedeµ????? ??te???µe
  DFS-Visit(v). ?? p??se??????µe ??e? t?? ????f??,
  t?te NAI, a????? OXI. O(nm)
• ? ???f?? G e??a? d??d??? ??te???µe
  DFS-Visit(v). ?? p??se??????µe ??e? t?? ????f??
  ?a? de? ?p?????? ?p?s??e? a?µ??, t?te ???, a?????
  OXI. O(n)
• ???es? teµa????. ??te???µe DFS. ??a??t??µe st??
  ????f?? e??? teµa???? ??a id. T(nm)
• ???es? ??????. ??te???µe DFS. ?? ?p??????
  ?p?s??e? a?µ??, t?te NAI, a????? ???. O(n) d??t?
  p??se??????ta? t? p??? n a?µ??. ???s??? ?? a?µ??
  (u,v) ?a? (v, u) de? ap?te???? ?????.

58
???????µ?? e??es?? teµa???? µe DFS

• ???e? ?a e?t?p?st??? ?? ap???pt??se? ????f?? ??
  e???
• ?? µ?a ap???pt??sa ????f? v e??a? ? ???a t??
  d??d??? DFS, t?te ? v p??pe? ?a ??e? pe??ss?te??
  ap? ??a ???.
• ?? µ?a ap???pt??sa ????f? v de? e??a? ???a, t?te
  p??pe? ? v ?a ??e? ??a ??? s, t?? ?p???? ??p????
  ap?????? (s?µpe???aµßa??µ???? t?? s) ?a s??d?eta?
  µe ??a? p?????? t?? v µ?s? 1 ?p?s??a? a?µ?? t?
  p???.
• G?a ???e ????f? v ????eta? e?t?? ap? t? d(v) ?a?
  µ?a ep?p???? µetaß??t?, ? l(v), p?? d????e? t?
  µ????te?? ap? t?? ep???af?? d(v) ?a? d(s), ?p?? s
  e??a? e?te ap?????? t?? v, e?te p??????? µ?s?
  µ?a? t? p??? ?p?s??a? a?µ??, p?? e???e? t??
  p?????? a?t? µe ??a? ap????? t?? v.
• ??a ? µ???st? t?µ? p?? µp??e? ?a p??e? ? l(v)
  e??a? d(v).
• ??a ??a ?a ?s??e? t? 2, p??pe? l(s)?d(v)

58
59
?p?????sµ?? pa?aµ?t??? l(v)

• ?????? l(v)d(v).
• ? ?p?????sµ?? t?? l(v) ???eta? ??t??ta? t?? t?µ?
  t?? ?s? µe t? e????st? st???e?? t?? s??????
• d(v)?l(s) s ???? t?? v?d(w) (s,w) µ?a
  ?p?s??a a?µ?
• ? l(v) e??µe???eta? ?p?te p??spe???eta? ??a? ????
  s, t?t???? ?ste l(v)?l(s) ? ?p?te ß??s?eta? µ?a
  ?p?s??a a?µ?. ??t? µp??e? ?a ep?te???e? µe t?
  ß???e?a µ?a? st??ßa?.
• ?p???pt??se? ????f?? e??a? ?se? l(u)?d(v)

59
60
???????µ??

• ??s?d?? ??a? ???f?? G µe ep???af?? ?a? µ?a
  ????f?
• ???d?? ?? ????f?? se ???e teµ???? t?? G
• 1. T?t??µe i?1 ?a? ade?????µe t? st??ßa.
• 2. G?a ???e v?V ??t??µe d(v)?0
• 3. ??? d(v)0 ??a ??p??? v ?a???µe t??
  findblocks(v,0).
• ??ad??as?a findblocks(v,w)
• 1. T?t??µe, d(vi)?1, l(vi) ?d(vi), i?i1
• 2.G?a ???e u?N(v)
• ?? df i(v)0, t?te e?te????ta? ta e??? ß?µata
• ? a?µ? (u,v) µpa??e? st? st??ßa a? de? e??a?
  a??µ? e?e?
• ?a?e?ta? ? findblocks(u,0)
• T?t??µe l(v)?min(l(v),l(u))

60
61
???????µ?? (s????e?a)

• ?? l(u)dfi(v), t?te ap?????ta? ap? t? st??ßa
  ?a? d????ta? st?? ???d? ??e? ?? a?µ?? ap? t??
  ????f? t?? st??ßa? µ???? ?a? t?? a?µ? (u,v) (st?
  st?d?? a?t? ? v e??a? e?te ???a e?te ap???pt??sa
  ????f?)
• ??????, a? df i(u)ltdf i(v) ?a? u ? w t?te
  e?te????ta?
• O???µe t?? a?µ? (v,w) st? st??ßa
• T?t??µe l(v)?min(l(v),l(w))

61
62
?a??de??µa
?e?????ta? ap? t?? ????f? 1, ?a d????? ta dfi
?a? l ??a ??e? t?? ????f??
62
63
?a??de??µa
St? ???f? (a) ?e?????µe ap? t?? ????f? ?. ?a
e???????? d?ad????? ?? t?µ?? t?? dfi ?a? l.
63
64
?a??de??µa
?a e???????? ?? t?µ?? dfi ?a? l.
64