Graphs. Minimum Spanning Trees. Slides by Rose Hoberman (CMU)

Transcription

1 Graphs Miimum Spaig Trees Slides by Rose Hoberma (CMU)

2 Problem: Layig Telephoe Wire Cetral office 2

3 Wirig: Naïve Approach Cetral office Expesive! 3

4 Wirig: Better Approach Cetral office Miimize the total legth of wire coectig the customers

5 Miimum Spaig Tree (MST) (see Weiss, Sectio 2.2.2) A miimum spaig tree is a subgraph of a udirected weighted graph G, such that it is a tree (i.e., it is acyclic) it covers all the vertices V cotais V - 1 edges the total cost associated with tree edges is the miimum amog all possible spaig trees ot ecessarily uique

6 How Ca We Geerate a MST? a 2 d 9 6 b a 2 d 9 6 b c e c e 7

7 Prim s Algorithm Iitializatio a. Pick a vertex r to be the root b. Set D(r) = 0, paret(r) = ull c. For all vertices v Î V, v ¹ r, set D(v) = d. Isert all vertices ito priority queue P, usig distaces as the keys a 2 d 9 6 b e a b c d 0 Vertex Paret e - c e 8

8 Prim s Algorithm While P is ot empty: 1. Select the ext vertex u to add to the tree u = P.deleteMi() 2. Update the weight of each vertex w adjacet to u which is ot i the tree (i.e., w Î P) If weight(u,w) < D(w), a. paret(w) = u b. D(w) = weight(u,w) c. Update the priority queue to reflect ew distace for w 9

9 Prim s algorithm a c 2 d 9 6 e b e 0 d b c a d b c a Vertex Paret e - b - c - d - Vertex Paret e - b e c e d e The MST iitially cosists of the vertex e, ad we update the distaces ad paret for its adjacet vertices 10

10 Prim s algorithm a c 2 d 9 6 e b d b c a a c b 2 Vertex Paret e - b e c e d e Vertex Paret e - b e c d d e a d 11

11 Prim s algorithm a 2 d 9 6 b a c b 2 Vertex Paret e - b e c d d e a d c e c b Vertex Paret e - b e c d d e a d 12

12 Prim s algorithm a 2 d 9 6 b c b Vertex Paret e - b e c d d e a d c e b Vertex Paret e - b e c d d e a d 13

13 Prim s algorithm a c 2 d 9 6 e b b The fial miimum spaig tree Vertex Paret e - b e c d d e a d Vertex Paret e - b e c d d e a d 1

14 Ruig time of Prim s algorithm (without heaps) Iitializatio of priority queue (array): O( V ) Update loop: V calls Choosig vertex with miimum cost edge: O( V ) Updatig distace values of ucoected vertices: each edge is cosidered oly oce durig etire executio, for a total of O( E ) updates Overall cost without heaps: O( E + V 2 ) Whe heaps are used, apply same aalysis as for Dijkstra s algorithm (p.69) (good exercise) 1

15 Prim s Algorithm Ivariat At each step, we add the edge (u,v) s.t. the weight of (u,v) is miimum amog all edges where u is i the tree ad v is ot i the tree Each step maitais a miimum spaig tree of the vertices that have bee icluded thus far Whe all vertices have bee icluded, we have a MST for the graph! 16

16 Correctess of Prim s This algorithm adds -1 edges without creatig a cycle, so clearly it creates a spaig tree of ay coected graph (you should be able to prove this). But is this a miimum spaig tree? Suppose it was't. There must be poit at which it fails, ad i particular there must a sigle edge whose isertio first preveted the spaig tree from beig a miimum spaig tree. 17

17 Correctess of Prim s Let G be a coected, udirected graph Let S be the set of edges chose by Prim s algorithm before choosig a errorful edge (x,y) x y Let V' be the vertices icidet with edges i S Let T be a MST of G cotaiig all edges i S, but ot (x,y). 18

18 Correctess of Prim s Edge (x,y) is ot i T, so there must be a path i T from x to y sice T is coected. Isertig edge (x,y) ito T will create a cycle There is exactly oe edge o this cycle with exactly oe vertex i V, call this edge (v,w) x y v w 19

19 Correctess of Prim s Sice Prim s chose (x,y) over (v,w), w(v,w) >= w(x,y). We could form a ew spaig tree T by swappig (x,y) for (v,w) i T (prove this is a spaig tree). w(t ) is clearly o greater tha w(t) But that meas T is a MST Ad yet it cotais all the edges i S, ad also (x,y)...cotradictio 20

20 Aother Approach Create a forest of trees from the vertices Repeatedly merge trees by addig safe edges util oly oe tree remais A safe edge is a edge of miimum weight which does ot create a cycle a 2 d 9 6 b forest: {a}, {b}, {c}, {d}, {e} c e 21

21 Kruskal s algorithm Iitializatio a. Create a set for each vertex v Î V b. Iitialize the set of safe edges A comprisig the MST to the empty set c. Sort edges by icreasig weight a c 2 d 9 6 e b F = {a}, {b}, {c}, {d}, {e} A = Æ E = {(a,d), (c,d), (d,e), (a,c), (b,e), (c,e), (b,d), (a,b)} 22

22 Kruskal s algorithm For each edge (u,v) Î E i icreasig order while more tha oe set remais: If u ad v, belog to differet sets U ad V a. add edge (u,v) to the safe edge set A = A È {(u,v)} b. merge the sets U ad V F = F - U - V + (U È V) Retur A Ruig time bouded by sortig (or fidmi) O( E log E ), or equivaletly, O( E log V ) (why???) 23

23 Kruskal s algorithm a 2 d 9 6 b E = {(a,d), (c,d), (d,e), (a,c), (b,e), (c,e), (b,d), (a,b)} c e Forest {a}, {b}, {c}, {d}, {e} {a,d}, {b}, {c}, {e} {a,d,c}, {b}, {e} {a,d,c,e}, {b} {a,d,c,e,b} A Æ {(a,d)} {(a,d), (c,d)} {(a,d), (c,d), (d,e)} {(a,d), (c,d), (d,e), (b,e)} 2

24 Kruskal s Algorithm Ivariat After each iteratio, every tree i the forest is a MST of the vertices it coects Algorithm termiates whe all vertices are coected ito oe tree 2

25 Correctess of Kruskal s This algorithm adds -1 edges without creatig a cycle, so clearly it creates a spaig tree of ay coected graph (you should be able to prove this). But is this a miimum spaig tree? Suppose it was't. There must be poit at which it fails, ad i particular there must a sigle edge whose isertio first preveted the spaig tree from beig a miimum spaig tree. 26

26 Correctess of Kruskal s K T S Let e be this first errorful edge. Let K be the Kruskal spaig tree Let S be the set of edges chose by Kruskal s algorithm before choosig e Let T be a MST cotaiig all edges i S, but ot e. e 27

27 Correctess of Kruskal s Proof (by cotradictio): Lemma: w(e ) >= w(e) for all edges e i T - S Assume there exists some edge e i T - S, w(e ) < w(e) Kruskal s must have cosidered e before e However, sice e is ot i K (why??), it must have bee discarded because it caused a cycle with some of the other edges i S. But e + S is a subgraph of T, which meas it caot form a cycle...cotradictio K e S T 28

28 Correctess of Kruskal s Isertig edge e ito T will create a cycle There must be a edge o this cycle which is ot i K (why??). Call this edge e e must be i T - S, so (by our lemma) w(e ) >= w(e) We could form a ew spaig tree T by swappig e for e i T (prove this is a spaig tree). w(t ) is clearly o greater tha w(t) But that meas T is a MST Ad yet it cotais all the edges i S, ad also e...cotradictio 29

29 Greedy Approach Like Dijkstra s algorithm, both Prim s ad Kruskal s algorithms are greedy algorithms The greedy approach works for the MST problem; however, it does ot work for may other problems! 30