PARALLEL NESTED DISSECTION i 3 R PATH ALGEBRA COMPUTATIONS

Th/s paper extends the authors" parallel nested dissection algorithm of [13] originally devised for solving sparse lhteaf systems. We present a class of new applications of the nested dissection method, this time to path algebra compulal~,.hs (in both car~ of single source and all pair paths), where the, ~ ~ problem is defined by a symmetric malrix .4 who~ asem¢is4~i graph G with n verticex is planar. We substamlially improce the known algorithms for path algebra problems of that l~mnml class; this has further applications to maxinmm flow and minimum cut woblem~ in an undirected planar network and to the feasibility testing of a multicommodity flow hi a planar networlL graph computations * path algebras * parallel algorithms • network flow I. Intro&actkm In this ~*,per we substantially improve the known par~.el algorithms for several problems of practical ~aterest which can be reduced to path algebra computations. Gondran and Min~ux [4, pp. 41-42, 75-81] list the applications of path algebras to the problems of: veh/cle routing,, investment and stock control, dynamic progrmn-ming with discrete states and discrete time, network optiniization, artificial intelligence and pattern refognition, labyrinths and mathematical <~games, enc~di~ and decoding of information: ~ompare also Lawler [8], Tarjan [17,18]. [4, pp. 84-102] thoroughly investigates general algorithms for such problems based on matrix oper-atio~ in d/o/ds, see next sections. We propose a substantial improvement of Ihese. gco~ algorithms in the important case where-the input matrix A is associated with an uadirected planar graph or, more generally, wi~ a graph from the class of graphs having small separator families; see Definition 2 of Section 4 and compare Lipton, Rose and Tarjan [10], Pan and Reif [13,15]. Our improventent relies on our extension of the nested dissection parallel algorithm of ll31 to path alSe-bra problems (originally the al~ithm was applied in [13] to linear systems of ~tious to extend the sequential algorithm of [10] for the same problem; then, in [15], the algorithm was extended to the kast-apmres and ~ear pr~ cmnpma-tions). Our new extens/oa is somewhat smpfisiag because the divisions and subtractimm of the mil~ hal algorithm of 1131 are am genera~ aUowed in the dioid: furtlg.mm~ for that reason we can extend (to dioids) neither the special" rco.u'sive facto~/zation of the haput matrix A from |131 ~./'s factodzation of A from [10L but we do extend the special teem'sly© factodzation of the inverse matrix A-1 of [13] to the ~hnilar …