MA015: Graph Algorithms

Transcription

1 MA Minimum cuts 1 MA015: Graph Algorithms 3. Minimum cuts (in undirected graphs) Jan Obdržálek obdrzalek@fi.muni.cz Faculty of Informatics, Masaryk University, Brno

2 All pairs flows/cuts MA Minimum cuts 2

3 MA Minimum cuts 3 Computing the cut/flow for all pairs In this section, let G be an undirected network: i.e. G = (V, E) a weighted (capacitated) graph with a capacity function c : E R + Question How much time we need to find the minimum s t cut (maximum flow) for all pairs of s, t V (G)? for a single pair we can do it, e.g., in time τ = O(n 2 m) = n 2 pairs: O(n 2 τ) can we do better? YES: n 1 maximal-flow computations are enough!

4 MA Minimum cuts 4 Gomory-Hu trees Definition For a network G = (V, E) with capacities c, the Gomory-Hu tree is a tree T = (V, F ) with capacities w such that: 1 s, t V (G).f (s, t) = f T (s, t) (flow property) 2 A minimum s t cut in T corresponds to a minimum s t cut in G (cut property) Notation for U V we use δ(u) = δ(v \ U) to denote the set of edges between U and V \ U (the cut given by U) f (s, t) the minimum capacity of an s t cut in G f T (s, t) the corresponding value in T consequence of the cut property for each edge e = {s, t} F, δ(u) is a minimum capacity s t cut of G, where U is any of the two components of T \ e

5 MA Minimum cuts 5 Gomory-Hu trees example network G 10 a 8 b 3 f c 4 e d Tree T d 14 a 18 b 17 f 13 e 15 c

6 MA Minimum cuts 6 Use of Gomory-Hu trees Definition For a network G = (V, E) with capacities c, the Gomory-Hu tree is a tree T = (V, F ) with capacities w such that: 1 s, t V (G).f (s, t) = f T (s, t) (flow property) 2 a minimum s t cut in T corresponds to a minimum s t cut in G (cut property) What this means? assume we have a Gomory-Hu tree T to find the value of f (s, t), it is enough to find the minimum value of w(e) for edges e on the unique s t path in T Questions 1 does a Gomory-Hu tree always exist? 2 can we efficiently compute it?

7 MA Minimum cuts 7 Gomory-Hu tree construction: outline the algorithm maintains: a partition of V (sets S 1, S 2,..., S k ) a spanning tree T on the vertex set of {S 1, S 2,..., S k } initial partition: S 1 = V n 1 split operations (rounds): choose S i s.t. S i 2 split S i into two non-empty parts S a i, S b i remove S i from T and replace it by S a i, S b i, connected by an edge connect S a i and S b i to other S j s (according to the connectivity of vertices in S a i and S b i ) result: the tree T

8 MA Minimum cuts 8 Splitting S i compute the connected components of the forest obtained from T by deleting S i contract the vertices of G in each of these components to a single vertex, obtaining the graph H choose two vertices a, b of S i compute the minimum a b cut in H let A and B be the two sides of this cut let S a i = S i A and S b i = S i B connect S a i and S b i with an edge of the same capacity as the minimum cut replace an edge {S i, S j } by {S a i, S j } if S j A and by {S b i, S j } if S j B (capacities are kept the same)

9 MA Minimum cuts 9 Construction example: iteration 1 a 10 8 b 3 f c 4 e d initial partition: {a, b, c, d, e, f } select b and f (arbitrary choice) minimum b f cut: 17 partition: {a, b}, {c, d, e, f } a,b 17 c,d,e,f

10 MA Minimum cuts 10 Construction example: iteration 2 a 10 8 b 9 cdef initial partition: {a, b}, {c, d, e, f } select a and b in {a, b} minimum a b cut: 18 partition: {a}, {b}, {c, d, e, f } a b c,d,e,f 18 17

11 MA Minimum cuts 11 Construction example: iteration 3 a,b 11 8 f c 4 e d initial partition: {a}, {b}, {c, d, e, f } select c and f in {c, d, e, f } minimum c f cut: 13 partition: {a}, {b}, {c, d, e}, {f } a b f c,d,e

12 MA Minimum cuts 12 Construction example: iteration 4 c a,b,f e 5 7 d initial partition: {a}, {b}, {c, d, e}, {f } select d and e in {c, d, e} minimum d e cut: 14 partition: {a}, {b}, {c, e}, {d}, {f } d 14 a 18 b 17 f 13 c,e

13 MA Minimum cuts 13 Construction example: iteration 5 c a,b,f e 5 7 d initial partition: {a}, {b}, {c, e}, {d}, {f } select c and e in {c, e} minimum c e cut: 15 partition: {a}, {b}, {c}, {d}, {e}, {f } d 14 a 18 b 17 f 13 e 15 c

14 MA Minimum cuts 14 Basic lemmas Lemma A For u, v, w V the following inequality holds: f (u, w) min(f (u, v), f (v, w)) Lemma B For u, v 1, v 2,..., v r, w V the following inequality holds: f (u, w) min(f (u, v 1 ), f (v 1, v 2 ),..., f (v k,, w))

15 MA Minimum cuts 15 Main lemma G-H Lemma Let δ(s) be some minimum r s cut for some vertices r, s V (s S), and let v, w S. Then there is a minimum v w cut δ(t ) with T S. Meaning if we have a graph G = (V, E) and we contract a subset X V that corresponds to some mincut, then the value of f (s, t) does not change for two nodes s, t X. we will show (later) that the connected components that we contract during a split-operation each correspond to some mincut and, hence, f H (s, t) = f (s, t) (where f H (s, t) is the value of a minimum s t mincut in graph H)

16 MA Minimum cuts 16 Main lemma proof G-H Lemma Let δ(s) be some minimum r s cut for some vertices r, s V (s S), and let v, w S. Then there is a minimum v w cut δ(t ) with T S. Let δ(x) be a minimum v w cut, with X S and X (V \ S). both δ(s \ X) and δ(s X) are v w cuts inside S. w.l.o.g s X. case r X: c(δ(x \ S)) + c(δ(s \ X)) c(δ(s)) + c(δ(x)) c(δ(x \ S)) c(δ(s)) (because X \ S is a r s cut) together c(δ(s \ X)) c(δ(x)) case r X: c(δ(x S)) + c(δ(s X)) c(δ(s)) + c(δ(x)) c(δ(x S)) c(δ(s)) (because X S is a r s cut) together c(δ(s X)) c(δ(x))

17 MA Minimum cuts 17 Invariant Invariant (existence of representatives) For any edge {S i, S j } in T, there are vertices a S i and b S j such that w(s i, S j ) = f (a, b) and the cut defined by the edge {S i, S j } is a minimum a b cut in G. Invariant implies that, at the end of the algorithm, T is a Gomory-Hu tree. not difficult to show

18 MA Minimum cuts 18 Invariant proof obviously true for the initial two-node tree we need to show it is maintained by the split operation split operation node S i, vertices a, b in S i, will split into Si a and Si b invariant says all edges leaving S i correspond to some mincuts by G-H Lemma, the capacity of the mincut between a and b is the same as in G choosing representatives for edges of T a and b for the {S a i, S b i } edge keep the old representatives for edges not incident to S i otherwise {X, S i } E(T ), w.l.o.g. to be replaced by {X, S a i } assume old representatives x X, s S i if s S a i, keep x and s as representatives if s S b i, choose x and a as representatives

19 MA Minimum cuts 19 Invariant proof For the new pair of representatives x and a, we need to show f (x, a) = f (x, s): by invariant, {X, S i } induces a cut of capacity f (x, s) x and a on the opposite sides of this cut = f (x, a) f (x, s) B forms a mincut separating a and b by contracting B into v b we obtain G by G-H Lemma, f (x, a) = f (x, a) as x, a B by Lemma A, f (x, a) min(f (x, v b ), f (v b, a)) since s B, we have f (v b, x) f (s, x) the a b cut splitting S i into S a i and S b i also separates s and x = f (a, v B ) f (a, b) f (x, s).

20 Global Minimum Cuts MA Minimum cuts 20

21 MA Minimum cuts 21 Global Minimum Cut so far, we were interested in so-called minimum s t cuts between a given pair of vertices s, t (source/sink) corresponds to the maximum flow between those two vertices (Ford-Fulkerson theorem) now we want to find a minimal cut for the whole graph G MINIMUM CUT Given a connected, undirected graph G = (V, E) and a capacity function c : E R +, find a set A = δ(s) s.t. S V and c(a) is minimum. c(a) = a A c(a) λ(g) capacity of a minimum cut of G λ(g; v, w) capacity of a minimum v w cut of G

22 MA Minimum cuts 22 Global Minimum Cut motivation A 75-year-old Georgian woman cut-off the internet in an entire country for more than 12 hours with only her garden spade.

23 Global Minimum Cut motivation The Global Internet Is Being Attacked by Sharks, Google Confirms MA Minimum cuts 23

24 MA Minimum cuts 24 Global Minimum Cut motivation Russia May Be Planning To Cut Internet Cables Under Sea, US Intelligence Shows Concern

25 MA Minimum cuts 25 Global Minimum Cut solving n 1 flow problems (minimum s t cuts) is enough δ(s) is a v w cut if exactly one of v, w is in S finding a minimum v w cut requires solving one max-flow problem (replace each edge by two oppositely directed arcs) if we fix one vertex r of G, then every cut is a r s cut for some s in G = solve n 1 max-flow problems one for each choice of s and take the minimum. Using O(n 3 ) max-flow algorithm, we can find a minimum cut in time O(n 4 ).

26 MA Minimum cuts 26 Edge contraction let e = uv be an edge in G the graph G uv = G e obtained from G by contracting e is defined as: V (G uv) = (V (G) \ {u, v}) {x} (where x V (G)) the edges: E(G uv) = {ab E(G) {a, b} {u, v} = } (1) {xw uw E(G) vw E(G)} (2) comments the capacities of edges are same as in the original graph the resulting graph can have multiple edges (but no new loops) for purposes of flow algorithms we can replace parallel edges with a single edge, of capacity equal to the sum of their capacities

27 MA Minimum cuts 27 Edge contraction Theorem Let vw E(G). Then 1 Every cut of G vw is a cut of G. 2 Every cut of G which does not separate v from w is a cut of G vw. Corollary: λ(g) = min(λ(g vw ), λ(g; v, w)) simple algorithm 1 choose an edge vw 2 compute a minimum v w cut of G 3 replace G by G vw 4 repeat steps 1-3 (n 2)-times (the last cut is trivial) 5 the best of these cuts is a minimum cut of G

28 Node Identification Algorithm MA Minimum cuts 28

29 MA Minimum cuts 29 Node Identification Algorithm Nagamochi-Ibaraki 1992, Stoer-Wagner 1994 does not use maximum flow computation first discovered by Nagamochi and Ibaraki Stoer and Wagner gave a simpler description and proof Main idea using the theorem on the previous slide choose v and w in such a way that the minimum v w cut is easy to find

30 MA Minimum cuts 30 Legal ordering Definition A legal ordering of G is an ordering v 1, v 2,..., v n of the vertices of G such that c(δ(v i 1 ) δ(v i )) c(δ(v i 1 ) δ(v j )) i, j.2 i < j n where V i = {v 1,..., v i } The usefulness of the legal ordering is demonstrated by the following theorem: Theorem (N-I) If v 1,... v n is a legal ordering of G, then δ(v n ) is a minimum v n 1 v n cut of G.

31 MA Minimum cuts 31 The min-cut algorithm MINIMUMCUT(G,c) 1 M := 2 while V > 1 do 3 v 1,..., v n := FINDLEGALORDERING(G,c) 4 if c(δ(v n )) < M then 5 M := c(δ(v n )) 6 A := δ(v n ) 7 G := G vn 1 v n 8 return A

32 MA Minimum cuts 32 Finding a legal ordering start with any vertex a V keep adding the most tightly connected vertex: z A such that c(a, z) = max{c(a, y) y A} c(a, y) the sum of the weights of edges between A and y FINDLEGALORDERING(G,c) 1 a := some vertex of G 2 A := {a} 3 while A V do 4 z := vertex most tightly connected to A 5 A := A {z} 6 return vertices in order they were added to A

33 MA Minimum cuts 33 Example DEMONSTRATION

34 MA Minimum cuts 34 Proof of Theorem N-I Theorem If v 1,... v n is a legal ordering of G, then δ(v n ) is a minimum v n 1 v n cut of G. we know that δ(v n ) is a v n 1 v n cut it remains to show that c(δ(v n )) λ(g; v n 1, v n ) by induction on E and V base case: trivial for V = 2 or E = 0 inductive step case 1: e = v n 1 v n E(G) let G = G \ e, and δ be δ on G v 1,..., v n is still a legal ordering of G we have c(δ(v n )) = c(δ (v n ))+c(e) IH = λ(g ; v n 1, v n )+c(e) = λ(g; v n 1, v n )

35 Q.E.D. MA Minimum cuts 35 case 2: v n 1 v n E(G) Proof of Theorem N-I Lemma A For all u, v, w V we have λ(g; u, w) min(λ(g; u, v), λ(g; v, w)) We therefore need to prove 1 c(δ(v n )) λ(g; v n 2, v n ) we apply induction to G = G \ v n 1 : v 1,..., v n 2, v n is a legal ordering of G we have: c(δ(v n)) = c(δ (v n)) IH = λ(g ; v n 2, v n) = λ(g; v n 2, v n) 2 c(δ(v n )) λ(g; v n 2, v n 1 ) we apply induction to G = G \ v n: v 1,..., v n 2, v n 1 is a legal ordering of G we have: c(δ(v n)) c(δ(v n 1 )) = c(δ (v n 1 )) = λ(g ; v n 2, v n 1 ) = λ(g; v n 2, v n 1 )

36 MA Minimum cuts 36 Complexity analysis the main loop of MINIMUMCUT is executed n 1 times the cost of FINDLEGALORDERING bottleneck: choosing the most tightly connected vertex z vertices not in A reside in a priority queue key: sum of the weights connecting it to A each time a vertex is added we need to update the queue V EXTRACT-MAX and E INCREASE-KEY operations using Fibonacci heaps we get time O( E + V log V ) we can do contractions in time O(n) TOTAL TIME: O(nm + n 2 log n)

37 Random Contractions MA Minimum cuts 37

38 MA Minimum cuts 38 Karger s algorithm Karger 1993 CONTRACT (G) 1 while V > 2 do 2 e := edge chosen from E uniformly at random 3 G := G e 4 return the unique cut of G notes correct result with small probability can be iterated works for unweighted multigraphs for weighted graphs choose edge e with probability c(e) c(e) complexity n 2 iterations of the main loop each contraction can be done in time O(n) TOTAL TIME: O(n 2 )

39 MA Minimum cuts 39 Example DEMONSTRATION

40 MA Minimum cuts 40 Probability of an error for a fixed cut C note: for unweighted multigraphs suppose the minimum cut C has k edges a minimum degree of any vertex in G must be at least k the probability an edge of C was chosen for the first contraction is k E k nk 2 = 2 n similarly for the (i + 1)-th contraction the probability is at most 2 n i the probability none of the edges of C was chosen (the correct answer) is therefore at least n 3 ( 1 2 ) n 3 ( ) 1 n i 2 n = = n i n i 2 i=0 i=0

41 MA Minimum cuts 41 Iterating the CONTRACT algorithm the probability of finding a minimal cut in one step is at least 2/(n(n 1)) but we can run the algorithm multiple times and select the minimum cut Theorem Let C be a minimum cut of G and k N. The probability that the algorithm CONTRACT does not return C in one of kn 2 runs is at most e 2k. Proof ( ) kn (1 2n ) kn 2 ) n(n 1) 2 (e 2 kn 2 n 2 = e 2k using 1 x e x

42 MA Minimum cuts 42 Iterating the CONTRACT algorithm Theorem Let C be a minimum cut of G and k N. The probability that the algorithm CONTRACT does not return C in one of kn 2 runs is at most e 2k. Practical consequences: 10n 2 -times: correct with probability more than n 2 times: correct with probability more than 1 1/e n 2 log n-times: correct with probability more than 1 1/n 2 Theorem Running Karger s algorithm O(n 2 log n)-times gives a minimum cut with high probability and takes time O(n 4 log n).

43 MA Minimum cuts 43 Improving Karger s algorithm Karger-Stein 1996 Clifford Stein the S in [CLRS] probability of contracting a minimum cut edge: 2 n i this probability increases with each step: first step: 2/n last step: 2/3 solution: algorithm CONTRACT(G,t) stop when t edges are left solve the remaining graph exactly probability of avoiding a cut C is at least: n t 1 i=0 ( 1 2 ) = n i ( ) t / 2 ( ) n 2 becomes less than 1/2 at around t = 1 + n/ 2

44 MA Minimum cuts 44 FASTMINCUT(G) Improving Karger s algorithm 2 Karger-Stein if V 6 then 2 return MINIMUMCUT(G) 3 else 4 t := 1 + n/ 2 5 C 1 := FASTMINCUT(CONTRACT(G,t)) 6 C 2 := FASTMINCUT(CONTRACT(G,t)) 7 return min(c 1, C 2 ) time complexity in each iteration the size decreases by a factor of 2 = O(log n) iterations in i-th level: 2 i subproblems, of size n/2 i/2 calling CONTRACT twice, each call O(n/2 i/2 ) = O(n 2 /2 i ) = O(n 2 ) over each level TOTAL TIME: O(n 2 log n)

45 MA Minimum cuts 45 Improving Karger s algorithm 2 Karger-Stein 1996 bounding the error probability We obtain the following recurrence for the lower bound on the probability of success on a graph of size n: Solving this recurrence we obtain: P(n) 1 (1 1 2 P( n/ )) 2 P(n) = Ω(1/log n) finding all minimum cuts Theorem The algorithm FASTMINCUT finds all minimum cuts with a high probability in time O(n 2 log 3 n).