A Contraction Pro cedure for Planar Directed Graphs y Stephen Guattery Gary L Miller Scho ol of Computer Science Carnegie Mellon University Pittsburgh PA lem Many graph algorithms either implicitly or explic Abstract itly solve this problem For sequential algorithm design We show that testing reachability in a planar DAG the two classic paradigms for solving this problem are can b e p erformed in parallel in O log n log n time BFS and DFS They only require time at most prop or O log n time using randomization using O n pro tional to the size of the graph Parallel p olylogarithmic cessors In general we give a paradigm for contracting time algorithms for the problem now use approximately a planar DAG to a p oint and then expanding it back O M n pro cessors where M n is the numb er of pro This paradigm is develop ed from a prop erty of planar di cessors needed to multiply two n n matrices together rected graphs we refer to as the Poincare index formula in parallel Ullman and Yannakakis give a probabilis p Using this new paradigm we then overlay our appli tic algorithm which that works in O n time using n cation in a fashion similar to parallel tree contraction pro cessors for sparse graphs UY This blowup in MR MR We also discuss some of the changes the amount of work for parallel algorithms makes work needed to extend the reduction pro cedure to work for with general directed graphs on negrain parallel ma general planar digraphs Using the stronglyconnected chines virtually impossible One p ossible way around components algorithm of Kao Kao we can compute this dilemma is to nd useful classes of graphs for which multiplesource reachability for general planar digraphs the problem can b e solved eciently In pioneering 3 in O log n time using O n pro cessors This improves pap ers Kao and Shannon KS and Kao and Klein the results of Kao and Klein KK who showed that KK showed that the reachability problem and many 5 this problem could b e p erformed in O log n time using related problem could b e solved in p olylogarithmic time O n pro cessors This work represen ts initial results of using only a linear numb er of pro cessors Their meth an eort to develop ecient algorithms for certain prob o ds require one to solve each of many related problems lems encountered in parallel compilation by reducing one problem to another Each reduction in tro duces more logarithmic factors to the running time 5 In the end they used O log time to solve the planar Intro duction reachability problem for multiple start vertices In this pap er we give a general paradigm for contract Testing if there exists a path from a vertex x to a vertex ing planar directed acyclic graphs DAGs to a p oint y in a directed graph is known as the reachability prob We will show that after O log n rounds of contraction an nno de directed planar DAG will b e reduced to a This work supp orted in part by the Avionics Lab oratory Wright Research and Development Center Aeronautical Sys p oint There have b een several contraction rules pro tems Division AFSC US Air Force WrightPatterson AFB p osed for undirected planar graphs Phi Gaz but Ohio under Contract FC ARPA Or this is the rst set for a class of directed planar graphs der No and by National Science Foundation Award CCR After we presen t the rules for contraction it will b e a y This work supp orted in part by National Science Foundation relatively simple matter to overlay rules necessary to grant DCR compute multiplesource reachability These results are part of a larger eort to develop a set of reduction rules for arbitrary planar directed graphs ie those with cycles as well as DAGs The algorithm for the general case is more complicated and is not pre sented here though we discuss changes involved in ex tending the reduction pro cedure to the general case We is the head of the arc We say that an arc is out of feel that the class of directed planar graphs are impor its tail and into its head An arc a is incident to a tant for at least two reasons First the class includes vertex v if v is the head or the tail of a The degree several important classes including tree and series paral of a vertex v is the numb er of arcs incident to it we lel graphs Second the ow graph for many structured represen t this numb er as deg r eev The indegree of programming languages without function calls is planar a vertex v is the numb er of arcs that have v as their Our goal is to develop the basic algorithmic foundation head the outdegree of v is the numb er of arcs with v for a class of planar graphs so that a theory of planar as their tail ow graphs could b e based on it For any directed graph G we can dene an undirected 0 In the interest of simplicity we only present the de graph G on the same set of vertices in the following way 0 tails of the DAG case here On the other hand we feel for each arc u v in G we include an edge u v in G 0 that our algorithm for planar DAGs is interesting in its We refer to G as the underlying graph of G In this own right First ignoring our algorithm for the general pap er we will distinguish b etween edges and arcs edges case we can improve the computation of manysource are undirected and lie in the underlying graph while 2 reachability by a factor of log n time by simply using arcs are directed When we refer to arcs in G as edges 0 the strong connectivity of Kao Kao Our algorithm we are actually referring to the asso ciated edges in G for general planar digraphs removes one further log n A directed path is a sequence of vertices factor Second it uses new top ological techniques in v v v such that the v s are distinct with the 0 1 k i particular the Poincare index formula This should b e exception that we might have v v and for all 0 k of interest in parallel algorithm design for digraphs i k we have the arc v v in A A directed i1 i Throughout the pap er we will assume that the graph cycle is a directed path such that v v A digraph 0 k G V A is a directed emb edded planar graph If that contains no directed cycles is called a directed an an emb edding is not given we can construct one acyclic graph DAG in O log n time using n pro cessors using the work of A planar directed graph is a directed graph that Gazit Gaz and Ramachandran and Reif RR We can b e drawn in the plane in such a way that its arcs assume that the emb edding is given in some nice com intersect only at vertices A sp ecication of some par binatorial way such as the cyclic ordering of the arcs ticular way in which such a graph can b e drawn in the radiating out of each vertex plane is called a planar emb edding of the digraph In This pap er is divided into seven sections The sec an emb edded planar digraph we dene parallel arcs as ond gives the main denitions necessary to dene and two arcs u v and u v such that either u u 1 1 2 2 1 2 analyze the directed graph contraction algorithm The and v v or u v and v u and the arcs are 1 2 1 2 1 2 third gives the contraction algorithm for sp ecial case of consecutive in the cyclic order at b oth u and v Par 1 1 of a planar DAG The theorems in Sections and show allel edges in the underlying graph are edges asso ciated that the reduction algorithm for planar DAGs works with parallel arcs in the graph in a logarithmic numb er of reduction steps The sixth If the p oints corresp onding to the arcs in an emb ed section explains how the reduction pro cedure can b e ded planar digraph are deleted the plane is divided into applied to the manysources reachability problem and a numb er of connected regions These regions are called calculates the running time Finally in Section we faces The b oundary of a face is the set of arcs that are discuss work in progress including some of the steps adjacent to that face We denote the set of faces by F necessary to extend this result to the case of general Eulers formula which holds for emb edded connected planar digraphs planar graphs relates the numb ers of arcs vertices and faces Preliminaries jV j jE j jF j If the graph also has or more vertices no selfloops Planar Directed Graphs and no parallel edges then each face will have at least We will assume that the reader is familiar with basic three edges in its b oundary and it is easy to prove the denitions and results from graph theory that apply to following inequality undirected graphs see for example textb ooks such as the one by Bondy and Murty BM jE j jV j A directed graph digraph GV A is a set of vertices V and a set of arcs A Each arc a A is The formula corresp onding to with jAj substituted an ordered pair drawn from V V We say that arc for jE j holds for emb edded planar digraphs that have a u v is directed from u to v u is the tail and v a connected underlying graph since the orientations of the arcs do not aect the quantities involved The in not distinguish b etween the notation used in these two equality corresp onding to with jAj substituted for cases jE j holds for an emb edded planar digraph G if Gs un Our approach dep ends on combinatorial arguments 0 derlying graph G is connected and has no selflo ops or based on the following simple but fundamental theorem parallel edges which we refer to as the Poincare index formula Theorem For every embedded connected planar di The Poincare Index Formula graph the fol lowing formula holds Let GV A b e a connected emb edded planar digraph X X with faces F We say that a vertex of G is a indexv indexf sourcesink if its indegreeoutdegree is zero The v 2V f 2F alternation numb er of a vertex is the