15-853: Algorithms in the R eal W orld Lecture 6 , September 30, 2002 Graph S eparators Part II L ect ur er : Prof. Guy Blelloch S cr ibe: Flavio L er da 1. Separator theorems At the end of last class we intr oduced the concept of a s eparator theorem: a s epar at or theorem proves that it is possible to obtain a good s epar ator for a cer t ai n class of graphs . A good s epar ator was defined as having a cut of size s maller than a fi x ed cons tant times a funct i on of the s iz e of the gr aph, and s uch that the s iz e of the bigger of the two s ub-graphs generated by the cut is smaller than a fi x ed fraction of the s iz e of the or iginal graph. I t is important to notice that it does not make s ens e to have a s epar at or theorem for a s i ngl e gr aph because it is always possible to find a bi g enough bound s uch that any s eparator is good enough. What we need i s a cl ass of graphs where the s iz e of the gr aphs can vary with a parameter n. We defined a cl ass 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 appl i cati on this is not always the cas e: however, in many cas es , 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 ub- graphs that are highly connected and they do not have a Page 1 of 14 Algorithms in the R eal World – Graph S eparators II good s epar ator . The algor ithms based on graph separators s till 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 . Def init 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 t ex -s eparator theorem as follows: Def init ion 1.2: A class of graphs S s atis fies a f (n) - vertex-s eparator theorem if there ar e cons tants α <1 and β > 0 s uch that for every graph G = (V , E) in the cl ass S there ex is ts a cut set C ⊆ V which partitions the gr aph G in two s ub-graphs A and B s uch that C ≤ β f ( G ) , A ≤ α G and B ≤ α G . And anal ogous l y an edge-s eparator theorem: Def init ion 1.3: A class of graphs S s atis fies a f (n) -edge- s eparator theorem if there ar e cons tants α <1 and β > 0 s uch that for every graph G = (V , E) in the clas s S there exis ts a cut set C ⊆ E which partitions the gr aph G in two s ub-graphs A and B s uch that C ≤ β f ( G ) , A ≤ α G and B ≤ α G . As we s howed i n the pr evious class, every good edge- s eparator can be tur ned i nto a good vertex-s eparator. T herefore if a cl ass of graphs has an f (n) -edge-s eparator theorem, it also has an f (n) -vertex-s eparator theorem. However the other way around i s not always true. For ins tance, planar graphs (as we will see below) have a f (n) -vertex-s eparator theorem: however, if we cons ider Page 2 of 14 Algorithms in the R eal World – Graph S eparators II the planar graph with n vertices obtained connecting one vertex to all the other s (to for m a s t ar ) , it has a good vertex-s eparator (the cut which contains only the center of the s tar is a ver t ex -s eparator), 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 s eparator. We will show that planar graphs satisfy a n -vertex- s eparator theorem. Also a par t i cul ar class of d - dimens ional mes hes satisfies a n(d −1) d -vertex-s eparator of which planar graphs represent the cas e d = 2 . T heor em 1.4: Any graph from a cl ass with a n1−ε - s eparator theorem with ε > 0 has O(n) edges . T his means that if a gr aph is from a cl ass that has a l es 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 s eparators until you are left with single ver tices . The r oot Page 3 of 14 Algorithms in the R eal World ± Graph S eparators II of the tr ee contains the or iginal graph; each node of the tree is either a l eaf 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 at or (either edge- or vertex-s eparator). G C A 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 i f the two children are equal sized ( wi thi n one ver tex difference). T heor em 2.1: For a cl as s of graphs S s atis fying an (α, β ) n1−ε -edge-s eparator theorem, we can generate a per fect l y balanced s epar ator tree with separator size C ≤ kβ f ( G ) . Figure 2 . 1 ± Unbalanced and bal anced s epar ator tree. Page 4 of 14 Algorithms in the R eal World ± Graph S eparators II P r oof: T he s epar ator tree obtained by the (α, β ) n1−ε -edge- s eparator has a linear order (n ) of leafs, which correspond to the ver tices of the gr aph. First find a pat h in the s eparator tree fr om the r oot to the middl e leaf ( n 2 ). Cons ider 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: βn1−ε + β (αn)1−ε + β (α 2n)1−ε +... = βn1−ε (1+α +α 2 +...)1−ε T he ter m (1+α +α 2 + ...)1−ε is constant, assuming that α and ε are cons tants and s i nce ther e ar e at most n levels in the tree. So we can define: 1−ε 2 1−ε 1 k = (1+α +α + ...) ≤ 1+α and we have: C ≤ kβn1−ε and s i nce G = n and f (n) = n1−ε : C ≤ kβ f ( G ) W What this theorem means is that if the s iz e of the cut is s ub-linear to the s iz e of the s iz e of the gr aph, given a (α, β ) f (n) -s eparator theorem we can convert it into a (1 2,kβ ) f (n) -s eparator theorem with k > 1 (and ther efor e allowing a bi gger cut size). 3. Planar Separator Theorem We ar e going to for mulate now a s epar at or theorem for planar graphs. First we need to define the clas s of planar graphs formally. Def init ion 3.1: T he s et of planar graphs is the s et of graphs that can be embedded i n a pl ane or in a s pher e s o that no two edges cross. Page 5 of 14 Algorithms in the R eal World ± Graph S eparators II I t is easy to s ee that this is a cl ass of graphs since if a graph can be embedded i n a pl ane ( or a s pher e) without any edges crossing, any sub-graph of this graph can be embedded i n 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 cl ass of graphs .