Graph S eparators Part II

Transcription

1 15-853: Algorithms in the R eal World Lecture 6, September 30, 2002 Graph S eparators Part II L ect ur er : Prof. Guy Blelloch S cr ibe: Flavio Ler da 1. Separator theorems At the end of last class we intr oduced the concept of a separator theorem: a s epar ator theorem proves that it is possible to obtain a good s epar ator for a cer tain class of graphs. A good s epar ator was defined as having a cut of size smaller than a fix ed cons tant times a function of the s ize of the gr aph, and s uch that the s ize of the bigger of the two sub-graphs generated by the cut is smaller than a fix ed fraction of the s ize of the or iginal graph. I t is important to notice that it does not make s ens e to have a s epar ator theorem for a s ingle gr aph because it is always possible to find a big enough bound s uch that any separator is good enough. What we need is a class of graphs where the s ize of the gr aphs can vary with a parameter n. We defined a class of graphs so that each sub-graph of a graph that belongs to the class also belongs to that class. T his is useful for developing the theor y, but for some application this is not always the cas e: however, in many cases, the r es ults are s till applicable in practice. For instance, if we cons ider the r outing gr aph of the I nternet, it has a good s epar ator, but there ar e s ome s ubgraphs that are highly connected and they do not have a Page 1 of 14

2 good s epar ator. The algor ithms based on graph separators still work well in practice in cases like this, as long as the highly connected s ub-graphs are s mall enough. We will start refreshing s ome of the definitions. Definit ion 1.1: A class of graphs is a s et S of graphs that is closed under the s ub-graph relation. We can define a ver tex -separator theorem as follows: Definit ion 1.2: A class of graphs S satisfies a f ( n) - vertex-separator theorem if there ar e cons tants α < 1 and 0 G = V, E in the class S β > such that for every graph ( ) there exis ts a cut set C V which partitions the gr aph G in two s ub-graphs A and B such that C β f ( G ), A α G and B α G. And analogous ly an edge-separator theorem: Definit ion 1.3: A class of graphs S satis fies a f ( n) -edgeseparator theorem if there ar e cons tants α < 1 and β > 0 such that for every graph G ( V, E) = in the clas s S there exists a cut set C E which partitions the gr aph G in two sub-graphs A and B such that C β f ( G ), A α G and B α G. As we s howed in the pr evious class, every good edgeseparator can be tur ned into a good vertex-separator. T herefore if a class of graphs has an f ( n) -edge-separator theorem, it also has an f ( n) -vertex-separator theorem. However the other way around is not always true. For instance, planar graphs (as we will see below) have a f ( n) -vertex-separator theorem: however, if we cons ider Page 2 of 14

3 the planar graph with n vertices obtained connecting one vertex to all the other s (to for m a s tar ), it has a good vertex-separator (the cut which contains only the center of the s tar is a ver tex -separator), but it does not have any good edge s epar ator : in fact, in order to divide the ver tices in half it is necessary to have a cut that contains n 2 edges. Figure 1.1 Planar graph with a good vertex separator but not a good edge separator. We will show that planar graphs satisfy a n -vertexseparator theorem. Also a par ticular class of d - dimensional meshes satisfies a n ( d 1) d -vertex-separator of which planar graphs represent the cas e d = 2. T heor em 1.4: Any graph from a class with a separator theorem with ε > 0 has O( n ) edges. 1 n ε - T his means that if a gr aph is from a class that has a les s than linear separator theorem the aver age degr ee of the graph is constant. T he pr oof is left as an exercise. 2. Separator Trees A s epar ator tree is the tr ee induced by recursively finding separators until you are left with single ver tices. The r oot Page 3 of 14

4 of the tr ee contains the or iginal graph; each node of the tree is either a leaf if the node contains only a ver tex, or it has two childr en: the children contain the par titions of the graph in the par ent node defined by a s epar ator (either edge- or vertex-separator). G A C B Figure 2.1 A s epar ator tree. S ometimes the ver tices in the cut are carried to both children, sometimes only to one of them, sometimes to neither. A s epar ator tree is fully balanced if the two children are equal sized (within one ver tex difference). T heor em 2.1: For a clas s of graphs S s atisfying an ( α, β ) 1 n ε -edge-separator theorem, we can generate a per fectly balanced s epar ator tree with separator size C kβ f ( G ). Figure 2.1 Unbalanced and balanced s epar ator tree. Page 4 of 14

5 P r oof: T he s epar ator tree obtained by the ( α, β ) n 1 ε -edgeseparator has a linear order (n ) of leafs, which correspond to the ver tices of the gr aph. First find a path in the separator tree fr om the r oot to the middle leaf ( n 2 ). Consider cutting the tr ee s o that all the nodes on the left of the middle node ar e on one s ide and the r es t are on the other side. It is possible to s epar ate thes e two halves by cutting all edges in the nodes on the s elected path: the maximum number of edges cut in the nodes, starting at the top, is: n ε ( n) ε ( n) ε β β α β α... βn ε = (1 + α + α +...) ε 2 1 T he ter m (1 + α + α +...) ε is constant, assuming that α and ε are cons tants and s ince ther e ar e at most n levels in the tree. So we can define: and we have: and s ince G W = n and k = (1 + α + α +...) 1 α + f ( n) 2 1 ε 1 1 C kβn ε 1 = n ε : C kβ f ( G ) What this theorem means is that if the s ize of the cut is sub-linear to the s ize of the s ize of the gr aph, given a ( α, β ) f ( n) -separator theorem we can convert it into a (1 2, kβ ) f ( n) -separator theorem with k > 1 (and ther efor e allowing a bigger cut size). 3. Planar Separator Theorem We ar e going to for mulate now a s epar ator theorem for planar graphs. First we need to define the clas s of planar graphs formally. Definit ion 3.1: T he s et of planar graphs is the s et of graphs that can be embedded in a plane or in a s pher e s o that no two edges cross. 1 ε Page 5 of 14

6 I t is easy to s ee that this is a class of graphs since if a graph can be embedded in a plane (or a s pher e) without any edges crossing, any sub-graph of this graph can be embedded in the s ame plane (or sphere) without any edges crossing as well. Therefore the s et of planar graphs is closed under the s ub-graph relation and for ms a class of graphs. Also note that if a gr aph is planar, an embedding in a plane (or in a s pher e) can be found in linear time. T heor em 3.2 (Planar Separators L ipt on-t ar j an, 1977): T he class of planar graphs obeys a (2 3,4) n - vertex-separator theorem. P r oof: T he pr oof of this theorem is going to be cons tr uctive, in which we s how that there exis ts such a s epar ator by giving an algorithm constructing it. T he algor ithm, which is linear-time, is described in the following and a pseudo-code for it is given in Figure 3.1. First of all we need to find an embedding of the planar graph in the plane: then we per for m a br eadth first search starting fr om a r andom vertex. The s tar ting ver tex can be any vertex, even if in practice it is better to choos e a peripheral vertex, since this improves the per for mance of the algor ithm. I f the number of levels that are necessary to vis it the whole gr aph is less than n (Figure 3.1), then we call a sub-procedure called CUT S HALLOW, which gives a separator which satisfies the cons tr aints we will describe this procedure later on. T his represents the cas e wher e the gr aph is not very deep. Page 6 of 14

7 I f there ar e mor e than n levels, we will look for the level j in the B FS which contains the n 2 th vertex. If this level contains less than n vertices (Figure 3.2), then the vertices in the level j are a s epar ator for the gr aph because they split the s pace in two halves, one befor e level j and one after : since level j contains the n 2 vertex, there ar e at most n 2 1 vertices in the levels before (and after ) level j. These two s ets will be s ets A and B in which the gr aph is partitioned whos e car dinality is at most n 2 1. This would be enough to s atis fy the bounds imposed on the s ize of the s ub-graphs. Figure 3.1 A S hallow graph. T his represents the cas e wher e the gr aph is not very thick, where at each level of the gr aph there ar e not many vertices. Figure 3.2 A thin graph. What we s till have to take car e of are thos e gr aphs that are too deep to fall in the firs t case but have s ome levels Page 7 of 14

8 around the middle of the gr aph that contain too many vertices to be us ed to par tition the gr aph (Figure 3.3). I f there ar e mor e than n vertices in level j we will look for levels i and k, such that i < j < k, k i < n, and ther e are less than n vertices in levels i and k. i Figure 3.3 Levels i, j, and k. By a s imple counting ar gument it is possible to s how that such level must exist. If there wer e no s uch i and k, then for every value of i and k we would have that there ar e more than n vertices in both levels, which means that for each level l such that i < l < k there ar e at least n vertices on each level. Since i and k can be at most j levels away, there mus t be n levels with more than n vertices in each of them, which means that there ar e mor e than n n = n vertices in the gr aph. But the gr aph has exactly n vertices so that s a contradiction and levels i and k which satisfy the given properties must exist. I f the number of vertices before level i is less than n 3, then we can use level i as a s epar ator (the s ame applies if the number of vertices after level k is less than n 3 ). Otherwise we can extract the gr aph between levels i and k and s ubs titute level i with a s ingle r oot vertex: calling CUT S HALLOW on this graph will return a cut of this subgraph. We can piece together the two par ts of the gr aph obtained with CUT S HALLOW with the two par ts obtained k n Page 8 of 14

9 removing levels i through j to for m a partition of the gr aph, adding the bigger of the left and r ight components to the bigger of the two par titions. Let R be the number of vertices in the levels before i ; let S be the number of vertices in the levels after k. Then the number of nodes between level i and k is n R S : the procedure CUT S HALLOW computes a par titioning with is in the wor s t-case If we assume, without loss of generality, that R < S we have that: 2 A = R + ( n R S ) 3 1 B = S + ( n R S ) 3 s ince we as sumed R < S and fr om before R < n 3 and S < n 3, we have: 2 2n 2R 2S 2n R 2S A = R + ( n R S ) = R + = and: R 2S < 0 because R < S, so: 2n R 2S 2n A < + < similarly for B : 1 n R S n 2S R B = S + ( n R S) = S + = but: n 2 2S R 2S 3 2n n < < = < because R < S and S < n 3, so: n n 2n B < + = W SEPARATOR(G) Find an embedding of G in the plane. Perform BFS. Page 9 of 14

10 If number of levels < n Then C =CUTSHALLOW(G). Done. Fi. Find level j containing n 2 vertex. If L j C = L j. Done. Fi. < n Then Find i and j such that: i < j < k ; k i < n ; Li Lk < n ; < n. If the number of vertices before level i is greater than n 3 Then C = L i. Done. Fi. If the number of vertices after level k is greater than n 3 Then C = L k. Done. Fi. Extract the part of the graph between levels i and k. Replace level i with a single root vertex. Call CUTSHALLOW. Add the bigger of the left and right components to the smaller of the two partitions and vice versa. Done. Figure 3.4 Lipton-T arj an algorithm. T he s ubroutine CUT S HALLOW assumes that the DFS of the graph has d levels and r etur ns a vertex-separator of size at most 2d + 1. Page 10 of 14

11 Here s the intuition behind the CUT S HALLOW procedure: let s consider the B FS tree: this tree is embedded in the planar graph, and s ince the gr aph is planar, it is embedded in a plane (or sphere). We can therefore find a path from the r oot to one of the vertices including at most one ver tex from each level: therefore this path will have at most d + 1 vertices, including the r oot. Taking another such path we s plit the tree (and the gr aph) in two and we have a cut of size at most 2d + 1, because they share the r oot. I t is interesting to notice that a ver s ion of this theorem for slightly loos er bounds, a n log n -vertex-separator theorem, dates back to Ungar in Kernighan-Lin Heuristic Even if the planar graph separator theorem by Lipton and T arjan proposes an algorithm to find s uch a s epar ator, this is not the currently most used algor ithm for this purpose. Kernighan and Lin proposed a heur is tic to impr ove a given edge s epar ator based on hill climbing which is most used in practice. The initial partitioning can be obtained us ing a different algorithm (for example the CUT S HALLOW presented befor e), or by randomly partitioning the gr aph. T his algorithm was developed in the context of circuit design for solving the pr oblem of routing connection between boards: a s ystem would contain different boards that need to be inter connected. If we can find an edge separator (with a s mall amount of crossing edges ) we can reduce the number of interconnections that are necessary among the boar ds. T his algorithm allows for weighted edges, which means that each edge is assigned a weight and the pr oblem we want to s olve is to find an edge s epar ator with a s mall Page 11 of 14

12 weight, where the weight of the s epar ator is the s um of the weight of the edges that belong to the cut. T he algorithm assumes we have an initial cut, with two equal sized par titions A and B. We want then to s wap a subset X of A with a s ubs et Y of B, where X and Y have the s ame s ize: the hope is that this switch will reduce the cut size. Finding the optimal subset would s olve the optimal separator program itself, therefore it must be NP-hard. I nstead, the Ker nighan and Lin propose a heur is tic which swaps a s ingle pair of vertices at a time, swapping that pair that at the time mos t decreases the cut size, or less increases it. T he idea is to impr ove the cut size when possible with a greedy scheme. When this is not impossible, instead of stopping, the algor ithm allows a s wap to incr eas e the cut size: this is useful when we r eached a local minimum, since it might take us out of it being able to r each a better result later on. Here ar e s ome definitions that we will use dur ing the explanation of the algor ithm: w( u, v ) the weight of the edge between u and v. C( A, B ) the weighted cut between A and B. I ( v ) E( v ) G( v) = E( v) I( v) the s um of the weights of the edges incident in v which stay within the s ame partition. the s um of the weights of the edges incident in v that go to the other partition. Page 12 of 14

13 the s ingle ver tex gain, the r eduction of the cut size achieved by putting v on the other side. G( u, v) = G( u) + G( v) 2 w( u, v) the pair wise gain, the r eduction of the cut size achieve by swapping u and v. A B A B u u Figure 4.1 I (u)= 1, E(u)=3, G(u)=2 on a gr aph with unit weights. Given a gr aph G = ( V, E) and a par titioning ( A, B ) of its vertices in two equal size s ubs ets, the Ker nighan-lin algorithm puts every pair ( u, v ) where u A and v B in a priority queue Q based on the pr ior ity G( u, v ), the pair wise gain. T hen it repeats V 2 times the following s teps : it extract from Q the pair ( u, v ) with the highes t priority, removes from Q every pair involving either u or v ; updates the priorities of the neighbor s of u and v, updating their position in the queue if necessary: if w is a neighbor of u, its priority is increased by 2 w( u, w ) if w is on the s ame s ide as u before changing the par titions, decreases it by the same quantity otherwise; swaps the ver tices u and v. At the end, the par tition obtained corresponds to the original one wher e the s ubs ets A and B have been swapped. We s elect the bes t intermediate cut C. This requires to be able to r econs tr uct the bes t cut, either by undoing s ome of the s waps, redo the s equence of swaps k Page 13 of 14

14 that led to s uch a cut, or saving the bes t cut during processing. 2 We can estimate the r unning time as O( n ) for each iteration of the loop, so the total running time would be 3 O( n ). Fiduccia-Mattheyses is a var iation of Kernighan-Lin where we cons ider individual vertices instead of pairs. First we build two queues Q a and Q b which contain the vertices in A and B respectively, based on the pr ior ity G( u ). T hen at each step, that we r epeat again V 2 times, we select the two ver tices at the top of the two queues, and we s wap them. We don t need to r emove any other vertices, but we s till need to update the pr ior ities for the neighbors of the two chos en vertices. Again, at the end, we have to s elect the bes t cut among the inter mediate ones. T his version of the algor ithm has a r unning time that is O( V + E ) using appr opr iate data s tructur es : remember that each queue now has a s iz e that is linear in the number of vertices, while befor e it was quadratic. Page 14 of 14