chapter4 do phuc tap cua cac giai thuat do thi

8/14/2019 Chapter4 Do phuc tap cua cac giai thuat do thi

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Chng 4

phc tp ca cc gii thut th


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Ni dung

1. Cc gii thut th cn bn2. th c trng s

3. th c hng


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1.Cc gii thut th cn bn

C nhiu bi ton c nh ngha theo i tng v cc

kt ni gia cc i tng y.

Mt th l mt i tng ton hc m m t nhng biton nh vy.Cc ng dng trong cc lnh vc:

Giao thngVin thngin lcMng my tnh

C s d liuTrnh bin dchCc h iu hnhL thuyt th


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Mt th d

A

B GC

F

D E

L M

KJ

IH

Hnh 4.1a Mt th th d


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Thut ng

Mt th l mt tp cc nh v cc cnh. Cc nh lnhng i tng n m c th c tn v c mt s tnhcht khc v cnh l ng kt ni gia hai nh.

Mt li it x n y trong mt th l mt danh sch

nhng nh m nhng nh k tip nhau c kt ni nhvo nhng cnh trn th.

Mt th l lin thngnu c mt li i t mi nt nmt nt khc trong th. Mt th m khng lin thng

th bao gm nhiu thnh phn lin thng.

Mt li i n l mt li i m trn khng c nh nolp li.


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Thut ng (tt.)

Mt chu trnh (cycle )l mt li i n ngoi tr nh u

tin v nh cui cng trng nhau (mt li i t mt nhquay v chnh n).

Mt th khng c chu trnh c gi l cy (tree). Mtnhm cc cy khng lin thng c gi l rng(forest).

Cy bao trm ca mt th l mt th con m cha tt ccc nh trong cy nhng mt s cnh chm n mi nh.

Gi s nh trong mt th l V, s cnh l E, s cnh ca

th c th c t 0 n V (V-1)/2. (Chng minh truy chng).

th c tt c mi cnh hin din gia mi cp nh cgi l th y (complete graphs).


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Thut ng (tt.)

Cc th vi s cnh tng i t c gi l th tha;cc th vi ch mt s t cnh mt i c gi l th dy.

Cc th m t cho n gi l nhng th v hng(undirected graphs). Trong cc th c trng s (weighted

graphs), nhng gi tr s (trng s) c gn vo mi cnh din t th d khong cch hay chi ph.

Trong th c hng(directed graphs) cc cnh l mt

chiu: mt cnh i txsangy ch khng phi i ty sangx.Cc th c hng, c trng s cn c gi l ccmng(networks).


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Cch biu din th

Ta phi nh x cc tn nh thnh nhng s nguyn trongtm tr gia 1 v V.

Gi s c tn ti hai hm:

- hm index: chuyn i t tn nh thnh s nguyn- hm name: chuyn i s nguyn thnh tn nh.

C hai cch biu din th:- dng ma trn k cn

- dng tp danh sch k cn


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Cch biu din ma trn k cn

A B C D E F G H I J K L MA 1 1 1 0 0 1 1 0 0 0 0 0 0B 1 1 0 0 0 0 0 0 0 0 0 0 0C 1 0 1 0 0 0 0 0 0 0 0 0 0D 0 0 0 1 1 1 0 0 0 0 0 0 0

E 0 0 0 1 1 1 1 0 0 0 0 0 0F 1 0 0 1 1 1 0 0 0 0 0 0 0G 1 0 0 0 1 0 1 0 0 0 0 0 0H 0 0 0 0 0 0 0 1 1 0 0 0 0I 0 0 0 0 0 0 0 1 1 0 0 0 0J 0 0 0 0 0 0 0 0 0 1 1 1 1

K 0 0 0 0 0 0 0 0 0 1 1 0 0L 0 0 0 0 0 0 0 0 0 1 0 1 1M 0 0 0 0 0 0 0 0 0 1 0 1 1

Mt ma trn Vhng V ct chacc gi tr Booleanm a[x, y]l true if

nu tn ti mtcnh t nhxnnhy vfalse nungc li.

Hnh 4.1b: Ma trn kcn ca th hnh4.1a


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Ghi ch v cch biu din ma trn k cn

Mi cnh tng ng vi 2 bit trong ma trn: micnh ni giaxvy c biu din bng gi tr trueti c a[x, y]v a[y, x].

tin li gi nh rng c tn ti mt cnh ni minh v chnh n.

Li biu din ny ch thch hp khi cc th l dy.


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Gii thutprogram adjmatrix (input, output);const maxV = 50;

var j, x, y, V, E: integer;a: array[1..maxV, 1..maxV] ofboolean;

beginreadln (V, E);

for x: = 1 to V do /*initialize the matrix */ for y: = 1 to V do a[x, y]: = false; for x: = 1 to V do a[x, y]: = true; for j: = 1 to E do begin

readln (v1, v2);x := index(v1); y := index(v2);

a[x, y] := true; a[y, x] := true

end;end.


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Cch biu din bng tp danh sch k cn

Trong cch biu din ny, mi nh m ni ti mtnh c kt thnh mt danh sch k cn(adjacency-list) cho nh .

program adjlist (input, output);const maxV = 100;type link = node

node = record v: integer; next: linkend;var j, x, y, V, E: integer;

t, x: link;

adj: array[1..maxV] oflink;


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beginreadln(V, E);

new(z); z.next: = z;

for j: = 1 to V do adj[j]: = z; for j: 1 to E do begin

readln(v1, v2);

x: = index(v1); y: = index(v2);

new(t); t.v: = x; t.next: = adj[y];

adj[y]: = t; /* insert x to the first element of

ys adjacency list */

new(t); t.v = y; t.next: = adj[x];

adj[x]:= t; /* insert y to the first element ofxs adjacency list */

end;end.


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a b c d e f g h i j k l m

f

c

b

g

a a f g a e i h k j j j

e f e a l

m

m l

d d

Hnh 4.1c: Biu din bng tp danhsch k cn ca th hnh 4.1


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Cc phng php duyt th

Duyt hay tm kim trn th: ving mi nh/nttrong th mt cch c h thng.

C hai cch chnh duyt th:

- duyt theo chiu su trc (depth-first-search )- duyt theo chiu rng trc (breadth-first-search).


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Duyt theo chiu su trc gii thut quy

procedure dfs; procedure visit(n:vertex);

beginadd n to the ready stack;

while the ready stack is not empty doget a vertex from the stack, process it,and add any neighbor vertex that has not been processed

to the stack.if a vertex has already appeared in the stack, there is no

need to pushit to the stack, but it is moved to the top of the stack.

end;beginInitialize status;

for each vertex, say n, in the graph do ifthe status of n is not yet visited then visit(n)

end;


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Tm kim theo chiu su trc biu dindanh sch k cn

procedure list-dfs;var id, k: integer;

val: array[1..maxV] ofinteger;procedure visit (k: integer);var t: link;beginid: = id + 1; val[k]: = id; /* change the status of k

to visited */t: = adj[k]; / * find the neighbors of the vertex k */

while t z do begin ifval[t .v] = 0 then visit(t.v);

t: = t.next end

end;


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beginid: = 0;

for k: = 1 to V do val[k]: = 0; /initializethe status of all vetices */

for k: = 1 to V do ifval[k] = 0 then visit(k)end;

Ghi ch: Mng val[1..V]cha trng thi ca ccnh.val[k] = 0 nu nh kcha h c ving (not yet visited),val[k] 0 nu nh k c ving.val[k]: = j ngha l nhjth m c ving trong qu trnhduyt l nh k.


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Nh vy kt qu ca gii thut duyt DFS trn th cho hnh 4.1a vi tp danh sch k cn cho hnh 4.1c l

A F E G D C G

Lu : th t ca cc nh trong cc danh sch k cn c nh

hng n th t duyt ca cc nh khi p dng DFS.


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phc tp ca duyt theo chiu su trc

Tnh cht 5.1.1Duyt theo chiu su trc mt th biu din bng cc danh sch k cn i hi thi

gian t l V+ E.

Chng minh: Chng ta phi gn tr cho mi phnt ca mng val(do t l vi O(V)), v xt mint trong cc danh sch kt cn biu din th (do

t l vi O(E)).


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Duyt theo chiu su trc biu din bng matrn k cn

Cng mt phng php c th c p dng cho th c biu din bng ma trn k cn bngcch dng th tc visitsau y:

procedure visit(k: integer);var t: integer;begin

id: = id + 1; val[k]: = id;

for t: = 1 to V do ifa[k, t] then

ifval[t] = 0 then visit(t)end;


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phc tp ca duyt theo chiu su trc biu din ma trn

Tnh cht 5.1.2Duyt theo chiu su trc mt th biu din bng ma trn k cn t l vi V2.

Chng minh: Bi v mi bit trong ma trn k cnca th u phi kim tra.


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Duyt theo chiu su trc khng quy

procedure list-dfs;varid, k: integer; val: array[1..max V] of integer;

procedure visit(k: integer);vart: link;begin

push(k); repeat

k: = pop; id:= id + 1; val[k]: = id; /* change the status of k to visited */

t =: adj[k]; /* find the neighbors of the vertex k */. while t z do begin

ifval[t.v] = 0 then begin

push(t.v); val[t.v]: = -1 /* change the status of t.v to ready */ end

else ifval[t.v] = -1 then shift t.v to the top of the stack;t: = t.next

enduntil stackempty

end;


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beginid: = 0; stackinit;

fork: = 1 to V do val[k]: = 0; /* initialize the

status of all vertices */ fork: = 1 to V do

ifval[k] = 0 then visit(k)end;

Vi gii thut khng quy, ta cn dng mt stack c gil ready stack.Ghi ch:

val[k] = 0 nu nh kl cha c ving thm,

val[k] = -1 nu nh kang trong ready stackval[k] l mt tr dng nu nh k c ving thm.


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A

B C G

F

A

E

F

B C

A

D

G

F

B C

A

E

Hnh 4.3a DFS da vo stack

G


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G E DB B B B M

C C C C C L L

A F F F F F F H I J K K K

Hnh 4.3b Ni dung ca stack khi thc hin duyt theo chiu su trc


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Duyt theo chiu rng trc

Khi duyt th nu ta dng mt queue thay v mt stack, ta s

i n mt gii thut duyt theo chiu rng trc(breadth-first-search).

procedure bfs; procedure visit(n: vertex); begin

add n to the ready queue;while the ready queue is not empty doget a vertex from the queue, process it, and add any neighborvertex that has not been processed to the queue andchange their status to ready.

end;beginInitialize status;

foreach vertex, say n, in the graphifthe status ofn is not yet visited then visit(n)

end;


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procedure list-bfs;var id, k: integer; val: array[1..max V] ofinteger;

procedure visit(k: integer);var t: link;

beginput(k); /* put a vertex to the queue */

repeatk: = get; /* get a vertex from the queue */

id: = id + 1; val[k]: = id; /* change the status of k to visited */

t: = adj[k]; /* find the neighbors of the vertex k */ while t z do begin ifval[t .v] = 0 then begin

put(t.v); val [t.v]: = -1 /* change the vertex t.v to ready */ end;t: = t.next

enduntil queueempty

end;


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beginid: = 0; queue-initialze;

for k: = 1 to V do val[k]: = 0; /initialize thestatus of all vertices */ for k: = 1 to V do

ifval[k] = 0 then visit(k)end;

Duyt theo chiu su trc v duyt theo chiu rng trcch khc nhau ch gii thut u dng stack v gii thutsau dng hng i. Do , phc tp tnh ton ca DFS v

BFS l nh nhau.


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A

Hnh 4.4 Ni dung ca hng i khi thc hin BFS

C

F

C

BB B

GG G

K

G

D

L

D D DD

E EE

M

L

M

J

H

I

E

M


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2. th c trng s

Chng ta mun m hnh ha nhng bi ton thc t s dng

nhng trng s(weights ) hay chi ph (costs) c gn vomi cnh.

C hai bi ton c gii quyt trong phn ny:

- tm mt cch t tn ch ph nht ni tt c cc nhtrong th.- Tm mt li i vi tng chi ph t nht gia hai nh

cho.

Bi ton th nht l bi ton cy bao trm ti thiu; bi tonth hai l bi ton tm ng i ngn nht.


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Cy bao trm ti thiu

Mt cy bao trm ti thiu (minimum spanning tree ) ca mt

th c trng s l mt tp cc cnh chm tt c cc nhsao cho tng trng s ca cc cnh l nh nht.

Cy bao trm ti thiu khng nht thit l duy nht trongmt th.

cb d

ai e

fgh

8 7

9

14

10

4

6

1 2

7

2

4

8

11

Hnh 4.5 Cy baotrm ti thiu


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Gii thut Prim

Gii thut Prim l gii thut gii bi ton cy bao trm

ti thiu. Gii thut ny cng trnh by mt chin lc gii mt bi ton ti u ha: gii thut tham lam(greedy):

Ti mi bc ca gii thut, ta phi chn mt trong mt

s kh nng la chn. Chin lc tham lam xut vicla chn kh nng m xem ra tt nht ti lc .

Mt chin lc nh vy thng khng m bo em li ligii ti u ton cc cho cc bi ton.

Tuy nhin i vi bi ton cy bao trm ti thiu, ta c thchng minh rng chin lc tham lam c th em licy bao trm vi tng trng s ti thiu.


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Xy dng cy bao trm ti thiu

Gi s chng ta c mt th v hng, lin thng

G = (V, E) vi hm gn trng s w: E R v mun tmmt cy bao trm ti thiu cho G.

C mtgii thut tng qutm to dng dn cy bao trm

ti thiu mt lc thm mt cnh.

Gii thut ny duy tr mt tp A m lun lun l mt tpcon ca mt cy bao trm ti thiu.

Ti mi bc, mt cnh (u, v) c chn thm vo tp Am khng vi phm h thc bt bin A {(u, v)} lun lun lmt tp con ca mt cy bao trm ti thiu.


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Ta gi mt cnh nh th l cnh an ton cho tp A v n cth c thm vo A mt cch an ton m khng vi phm h

thc bt bin nu trn.procedure GENERIC_MST(G, w);/* G is a weighted graph with the weight function w */

begin

A :=; while A does not form a spanning tree do begin

find an edge (u,v) that is safe for A;

add (u,v) to A.

end/* the set A at this point is the result */

end;


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Gii thut Prim

Trong gii thut Prim, tp A hnh thnh mt cu trc cy. Mt

cnh an ton c a vo A thng l cnh c trng s nhnht ni cy A vi mt nh khng thuc v cy.

Cy bao trm khi i t mt nt r bt k rv pht trin chon khi cy ph tt c mi nh trong V. Ti mi bc, mt

cnh nh (lightedge) ni mt nh trong A vi mt cnh trongV - A.Trong qu trnh thc hin gii thut ny, tt c nhng nhm khng thuc v cy A c t trong mt hng i c

u tin Q da trn mt trng l key. Vi mi nh v, key[v]lcnh c trng s nh nht ca nhng cnh ni v vi mt nhtrong cy. Theo qui c key[v] = nu khng tn ti mtcnh nh vy.


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Trng p[v] lu tn ca nh cha ca nh v trong cy.

procedure MST-PRIM (G, w, r);/* G is weighted graph with the weight function w, and r is an arbitrary

root vertex */begin

Q: = V[G]; foreach u Q do key[u]: = ;

key[r]: = 0; p[r]: = NIL;

while Q is not empty dobeginu: = EXTRACT-MIN(Q);

foreach v Q and w(u, v) < key[v] then/ * update the key field of vertice v */

beginp[v] := u; key[v]: = w(u, v) end

endend;


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cb d

a i e

h g f

8 7

8

10

14

4

21

2

67

11

8

4

cb d

a i e

h g f

8 7

8

10

14

4

21

2

67

11

8

4

Hnh 4.6. Mt thd v gii thutPrim.


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cb d

a i e

h g f

8 7

8

10

14

4

21

2

67

11

8

4

cb d

a i e

h g f

8 7

8

10

14

4

21

2

67

11

8

4


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cb d

a i e

h g f

8 7

8

10

14

4

21

2

67

11

8

4

cb d

a i e

h g f

8 7

8

10

14

4

21

2

67

11

8

4


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cb d

a i e

h g f

8 7

8

10

14

4

21

2

67

11

8

4

phc tp ca gii thut Prim ty thuc vo cch

m chng ta thi cng hng i c u tin.


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3. th c hng

Cc th c hng l cc th trong cc cnh ni vi

cc nt c hng.

A

B GC

F

D E

L M

KJ

IH

Hnh 4.7. Mt th d v th c hng


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Thng th hng ca cc cnh biu th mi lin h trc

sau (precedence relationship) trong ng dng c m hnhha.

Th d, th c hng c th c dng m hnh ha

mt ng dy sn xut (assembly line).Trong phn ny, chng ta xem xt cc gii thut sau:

- tnh bao ng truyn (transitive closure)- sp th t topo (topological sorting)


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Tnh bao ng truyn

Trong th c hng, chng ta quan tm n tp nhm n c t mt nh no bng cch duyt cccnh trong th theo mt hng c n nh.

Mt tc v m ta mun thc hin l thm mt cnh tx

ny nu tn ti mt cch no i txny th to ra bng cch thm tt c cc cnh c tnh

cht trn c gi l bao ng truyn ca th.

V th bao ng truyn th thng l th dy, do chng ta dng cch biu din ma trn k cn.


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Gii thut Warshall

C mt gii thut n gin tnh bao ng truyn ca

mt th c biu din bng ma trn k cn.for y : = 1 to V do

for x : = 1 to V doifa[x, y] then

for j: = 1 to V do ifa[y, j] then a[x, j]: = true;

S. Warshall ra gii thut ny nm 1962, da trn mtquan st n gin: Nu tn ti mt cch i t nt x nnt y v cch i t nt y n nt j, th s c cch i tnt x n nt j.


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Mt th d tnh bao ng truyn ( cho th hnh 4.7)

A B C D E F G H I J K L MA 1 1 0 0 0 1 1 0 0 0 0 0 0B 0 1 0 0 0 0 0 0 0 0 0 0 0C 1 0 1 0 0 0 0 0 0 0 0 0 0D 0 0 0 1 0 1 0 0 0 0 0 0 0E 0 0 0 1 1 0 0 0 0 0 0 0 0

F 0 0 0 0 1 1 0 0 0 0 0 0 0G 0 0 1 0 1 0 1 0 0 1 0 0 0H 0 0 0 0 0 0 1 1 1 0 0 0 0I 0 0 0 0 0 0 0 1 1 0 0 0 0

J 0 0 0 0 0 0 0 0 0 1 1 1 1K 0 0 0 0 0 0 0 0 0 0 1 0 0L 0 0 0 0 0 0 0 0 0 0 0 1 1M 0 0 0 0 0 0 0 0 0 0 0 1 1

Ma trn k cn ngvi bc khi uca gii thut

Warshall


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Bi ton cc li i ngn nht

i vi th c trng s (c hng hoc khng) ta c th

mui xy dng mt ma trn cho php ngi ta tm c li ingn nht txny i vi mi cp nh. y l bi tonnhng li i ngn nht cho mi cp nh (all-pairs shortest

path problem).

A

B GC

F

D E

L M

KJ

IH

2

3

1

11

4

2

5

3

2

1

2

2

1

1

11

Gii h Fl d


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Gii thut Floyd

C th dng mt phng php tng t nh phng phpWarshall, m c a ra bi R. W. Floyd:

for y : = 1 to V dofor x : = 1 to V do

ifa [x, y] > 0 thenfor j: = 1 to V do

ifa [y, j] > 0 thenif(a[x, j] = 0) or (a[x, y] + a[y, j] < a [x, j])

thena[x, j] = a[x, y] + a[y, j];


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Mt th d dng gii thut Floyd (cho thi hnh 4.7)

A B C D E F G H I J K L M

A 0 1 0 0 0 2 4 0 0 0 0 0 0B 0 0 0 0 0 0 0 0 0 0 0 0 0C 1 0 0 0 0 0 0 0 0 0 0 0 0D 0 0 0 0 0 1 0 0 0 0 0 0 0E 0 0 0 2 0 0 0 0 0 0 0 0 0

F 0 0 0 0 2 0 0 0 0 0 0 0 0G 0 0 1 0 1 0 0 0 0 1 0 0 0H 0 0 0 0 0 0 3 0 1 0 0 0 0I 0 0 0 0 0 0 0 1 0 0 0 0 0

J 0 0 0 0 0 0 0 0 0 0 1 3 2K 0 0 0 0 0 0 0 0 0 0 0 0 0L 0 0 0 0 0 5 5 0 0 0 0 0 1M 0 0 0 0 0 0 0 0 0 0 0 1 0

Ma trn k cnng vi bckhi u cagii thut Floyd


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A B C D E F G H I J K L MA 6 1 5 6 4 2 4 0 0 5 6 8 7

B 0 0 0 0 0 0 0 0 0 0 0 0 0C 1 2 6 7 5 3 5 0 0 6 7 9 8D 0 0 0 5 3 1 0 0 0 0 0 0 0E 0 0 0 2 5 3 0 0 0 0 0 0 0F 0 0 0 4 2 5 0 0 0 0 0 0 0

G 2 3 1 3 1 4 6 0 0 1 2 4 3H 5 6 4 6 4 7 3 2 1 4 5 7 6I 6 7 5 7 5 8 4 1 2 5 6 8 7J 10 11 9 11 9 12 8 0 0 9 1 3 2

K 0 0 0 0 0 0 0 0 0 0 0 0 0L 7 8 6 8 6 9 5 0 0 6 7 2 1M 8 9 7 9 7 10 6 0 0 7 8 1 2

Ma trn k cn ngvi bc cui ca

gii thut Floyd

Tnh cht 5.3.2 GiithutFloyd gii biton nhng li i

ngn nht gia nhngcp c phc tptnh ton O(V3).

Gii th h ii th t Fl d


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Gii thch gii thut Floyd

Gii thut Floyd lp Vbc trn ma trn k cn a.

Sau bc lp thy, a[x, j] s cha chiu di nh nht cabt k li i no t nhxn nhjm khng i quanhng nh mang ch s ln hny. Ngha l,xvjc thbt k nh no nhng nhng nh trung gian trn li i

phi nh hn hay bngy.

Ti bc lp thy, ta tnh cc phn t ca ma trn a bngcng thc sau:

ay[x,j] = min( ay-1[x,j], ay-1[x, y] + ay-1[y, j])

Ch sy ch tr ca mt phn t trong ma trn a sau bclp thy.


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y

xj

ay-1[x,y] ay-1[y,j ]

ay-1[x,j ]

Cng thc ny c minh ha bng hnh v sau y.

X th t t


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Xp th t tp

th c hng khng chu trnh (Directed Acyclic Graph)

th c hng m khng c chu trnh c gi l cc th c hng khng chu trnh (dags).

Tp th t ring phn v xp th t tp

Cho G l mt th c hng khng chu trnh. Xt quanh th t< c nh ngha nh sau:u < v nu c mt li i tu n v trong G.

Quan h ny c 3 tnh cht:(1) Vi mi nh trong V[G], not (u < u). (khng phn x)(2) nu u < v, th not( v < u) . (khng i xng)(3) nu u < v v v < w, th u < w. (Truyn)

T th t i h


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Tp th t ring phn

Mt quan h < trn tp S m tha hai tnh cht bt i xngv truyn l mt quan h th t ring phn (partial ordering)ca S, v mt tp c trn n mt quan h th t ring phnth c gi l tp th t ring phn (partially order set).

Nh vy, bt k th c hng khng chu trnh c th ccoi nh l mt tp th t ring phn.

T th t i h (tt )


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Tp th t ring phn (tt.)

Mt khc, gi s S l mt tp th t ring phn vi quan h

th t ring phn


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Xp th t tp

Cho G l mt th c hng khng chu trnh. Mtth t tp (topological sort)T ca G l mt th ttuyn tnh m bo ton th t ring phn ban utrong tp nh V[G].

Ngha l: nuu < v trong V(tc l nu c mt li it u n v trong G), th u xut hin trc v trong th

t tuyn tnh T.


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A

B GC

F

D E

L M

KJ

IH

Cc nt trong th hnh trn c th c sp th t tp

theo th t sau:J K L M A G H I F E D B C

Sp th t tp1 (Phng php 1)


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Sp th t tp1 (Phng php 1)

C vi phng php sp th t tp.Phng php 1

tng cn bn l tm mt nt khng c nt i sau (nosuccessor) loi b n ra khi th v a n vo mt danhsch.Lp li qu trnh ny cho n khi th rng th s sinh ra mt

danh sch. o ngc danh sch ny ta s c th t tp.Algorithm:while the graph has a node with no successors do

remove one such node from the graph and

add it to the end of a list.ifthe loop terminates with the graph emptythen the list shows the reserse of a topological order.else the graph contains a cycle.

Hnh 4 8 Sp th t tp bng phng php 1


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A

B GC

F

D E

L M

KJ

IH

C B D E F I H G A M L K JJ K L M A G H I E F D B C

Hnh 4.8 Sp th t tp bng phng php 1

Phng php 2 (sp xp tp)


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Phng php 2 (sp xp tp)

Phng php th hai l thc hin theo kiu tm kim theo

chiu su trc v thm mt nt vo danh sch mi khi cnthit ly mt nt ra khi stack tip tc. Khi gp mt ntkhng c nt i sau th ta s ly ra (pop) mt phn t tnh stack.

Algorithm:Start with nodes with no predecessor, put them in the stack.while the stack is not empty do ifthe node at top of the stack has some successors

thenpush all its successors onto the stack

elsepop it from the stack and add it to the list.


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Bi ton nhng li i ngn nht t mt nh ngun


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Bi ton nhng li i ngn nht t mt nh ngun

Th xt bi ton nhng li i ngn nht t mt nh ngun

(single-source shortest-paths problem):Cho mt th G = (V, E), chng ta mun tm mt li i ngnnht t mt nh ngun no s V n mi nt v V.

Biu din nhng li i ngn nhtChng ta mun tnh khng ch trng s ca nhng li i ngnnht m cn xc nh nhng nh trn nhng li i ngn nht.Cho mt th G = (V, E), ta duy tr cho mi nh v V mt

nh i trcp[v]m l mt nh khc hay l NIL. Gii thutgn tr cho thuc tnhp sao cho dy cc nh i trc xutpht t nh v s cho ra li i ngn nht ts n v.

Nhng li i ngn nht v s ni lng


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Gii thut tm nhng li i ngn nht thng dng mt k thutc gi l s ni lng(relaxation).

S ni lng l mt phng php m lin tip gim cn trn catrng s ca li i ngn nht thc s ca mi nh cho n khicn trn bng vi trng s li i ngn nht.

Nh vy, vi mi nh v V, ta duy tr thuc tnh d[v], m l cntrn ca trng s ca mt li i ngn nht t nh ngun s n v.Ta gi d[v]l c lng li i ngn nht(shortest path estimate).

Qu trnh ni lng mt cnh (u, v) bao gm vic th xem ta c

th ci thin li i ngn nht n v m ang tm thy bng cch iqua u v nu nh vy, ta cp nht d[v]vp[v]. Mt bc ni lngs lm gim c lng li i ngn nht d[v]v cp nht thuctnhp[v].

Nhng li i ngn nht v s ni lng


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5 9

7

6

65

55

2

2

Relax(u,v)

u v

u v

(a)

2

2u

u

v

v

Relax(u,v)

(b)

Hnh 4.10: S ni lng ca mt cnh

Gii thut Dijkstra


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Gii thut ny gii bi ton nhng li i ngn nht t nhngun cho mt th c hng, c trng s G = (V, E) trong

trng hp cc trng s ca cc cnh l tr khng m.

Gii thut ny duy tr mt tp S ca cc nh m trng s cacc li i ngn nht t nh ngun c xc nh. Nghal, vi mi nh v S, ta c

d[v] = min(c lng li i ngn nht ts n v)

Gii thut lin tin chn nh u V S vi thuc tnh dnhnht, a u vo S, v ni lng mi cnh i ra t u.

Trong gii thut sau y, ta dng hng i c u tin Qcha tt c cc nh trong V-S, lp kha theo thuc tnh d. Vgi nh th G c din t bng cc danh sch k cn.

Gii thut Dijkstra

procedure dijkstra(G w s);


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procedure dijkstra(G, w, s);/* G is a graph, w is a weight function and s is the source node */

begin

for each vertex v V[G] do /* initialization */ begin d[v]: = ; p[s]: = NIL end;d[s]: = 0; S: = ; Q: = V[G]

while Q is not empty do begin

u: = EXTRACT-MIN (Q); S: = S {u};

for each vertex v Adj [u] do /* relaxation */ ifd[v] > d [u] + w(u, v) then begin d[v]: = d[u] + w(u, v); p[v]: = u end

endend

Gii thut Dijkstra


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V gii thut Dijkstra lun lun chn nh gn nht trong

V-S a vo tp S, n thc s s dng chin lc thamlam.

Gii thut tham lam thng khng m bo em li li giiti u trong trng hp tng qut, nhng i vi bi ton

ny, gii thut Dijkstra thc s em li nhng li i ngnnht.

Gii thut Dijkstra tng t nh gii thut Prim dng

tnh cy bao trm ti thiu. N cng dng mt hng i c u tin tm mt nh nh nht bn ngoi mt tp, anh vo tp, v iu chnh li trng s ca nhng nhcn li bn ngoi tp.

Gii thut Dijkstra

Mt th d


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Mt th d

5

10

00

10

5s

u v

x y

10

5

s

u v

x y


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8

7

14

5

0

8

5 7

13

0

93

2

7

2

1

10

5

4

6

u v

x y

ss

10

1

2

7

2

39

4

6

u v

x y


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8

7

9

5

0 0

8 9

75

x y

u v u v

x y

1

2

s

10

5

2

9

7 4

6

1

2

4

69

10

5

s 72

3

(s, u): (s, v):

(s, x): (s, y):

phc tp ca gii thut Dijkstra


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Nu hng i c u tin Q = V S c thc hin bi

mt mng tuyn tnh, mi thao tc EXTRACT-MIN tnO(V), v c tt c |V| thao tc nh vy, do ta c mt chiph tnh ton cho thao tc ny l O(V2).

Mi nh v V c a vo tp S ng mt ln, do mi

nh trong cc danh sch k cnAdj[v]c xt ng mtln trong sut tin trnh ca gii thut.

V tng s nh trong tt cc cc danh sch k cn l |E|, ctt c |E| bc lp cho vng lpfor, vi mi ln lp tn O(1).

Thi gian tnh ton ca gii thut lO(V2 + E) = O(V2).




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Nu hng i c u tin Q c hin thc bi mt cu

trc heap, mi thao tc EXTRACT-MIN tn chi ph O(lgV),v c tt c |V| thao tc ny, do ta c tng chi ph chothao tc EXTRACT-MIN l O(VlgV).

Php gn d[v]: = d[u] + w(u, v) i hi mt thao tc cpnht kha ca nh v trong heap v n tn O(lgV). C tt c|E| thao tc nh vy. Do tng thi gian tnh ton ca gii

thut l

O(V lgV + E lgV).


chapter4 do phuc tap cua cac giai thuat do thi

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