O n processors in CRCW PRAM

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1 PARALLEL COMPLEXITY OF SINGLE SOURCE SHORTEST PATH ALGORITHMS Mshra, P. K. Department o Appled Mathematcs Brla Insttute o Technology, Mesra Ranch-8355 (Inda) & Dept. o Electroncs & Electrcal Communcaton Engneerng Computer Vson Lab Indan Insttute o Technology, Kharagpur West Bengal-730 Inda Emal: pkmshra@eee.org, pkmsra@gmal.com Abstract In ths paper we have presented ecent parallel algorthms or all par shortest paths problem and dscussed the complexty o algorthms. The parallel complexty o O n O n processors n CRCW PRAM algorthm acheves ( log ) tme when usng ( ) model. We acheve overall complexty o SSSP s O( n n) log. Keywords: All par shortest paths, Complexty analyss, Parallel algorthms.. Introducton Dkstra s lnear breadth rst search algorthm s the best known sequental algorthm or ndng the shortest path between any gven par o vertces. However, Dkstra s algorthm cannot be parallelzed easly to run n less tme than lnear tme and has a lmted speed. ASSSP problem s that o bndng the mnmum weght path to every vertex o a graph startng rom a sngle source vertex. It s undamental problem n graph theory wth numerous applcatons. For example the vertces o a graph represent ctes and the edge weghts represent the cost o transportaton between the ctes they connect the SSSP rom a source to vertex A gve the mnmum transportaton costs rom cty A to every other cty. Graphs are abstract data structure consstng o vertces (nodes) and edges that connect pars o them. In a drected graph the connecton s n one drecton only. Adacency matrces [,4] and adacency lsts are two popular computer representaton o graphs. For a graph o n vertces the adacency matrx [, 8] s an n n array o bts wth entry (, ) set to vertces and are connected by an edge. Another n n array stores the weghts assocated wth the edges. The adacency lst or the same graph conssts o n lsts [3,5,6,7]. Lstng the vertces each vertex s connected to n some say ascendng order o ther labels. The lengths o lsts sum to e, were e s no o vertex. The storage requrement o adacency lsts typcally s less than that o adacency matrces. From adacency lst, connectvty can be determned only n O(n) tme n the worst case [,5] where as the degree o a vertex can be compute n O() tme, so adacency lst s preerred over adacency matrx. We represent graphs on processor array usng adacency lsts. The lst o neghbors o each vertex s stored n PE (Processng Elements). To access a partcular vertex, the lst must be accessed undependably n each PE or nstance searchng a neghbor vertex and ndng mnmum cost edge searchng and sortng should be done ndependently on the lsts present on PE s. So addressng memory helps n graph operaton. In parallel algorthm two varable are used, watng s used an array that keeps tracks o processors watng or work halt s set to true when all the processors watng and queue s

2 empty. The queue must be locked beore enqueung. Beore a process compares a new dstance to dstance(v). Varable dstance(v) should be locked. Fnally a process nds a queue empty, t sets watng true, and checks other processors nds watng sets halt true. The queue should be locked whle process checks other processes [8,].. Sequental Sngle Source Shortest Path : Agan we present a sequental soluton based on dvde and conquer whch we shall parallelze. The sequental algorthm we shall be denotng by SSSP(G). Beore descrbng the algorthm we ntroduce some notatons. The source vertex s denoted by s. We shall be reerrng to the same dvde and conquer tree [6] or ths problem. The tree s dened n the same ashon as t was dened earler. We shall be denotng ts level by l, startng rom zero at the topmost level. Inormally the nvarant o our dvde and conquer s the mantenance o the sngle source shortest path normaton n each sub problem rom those vertces actng as the source, whch had been a separator vertex on the path startng orm the parent o that sub problem to the root o the dvde and conquer tree or t s s. More precsely or we mantan the shortest path normaton rom each source (dened below) to every vertex where v { S } {}} s V ( l,, () m l ) ( l), ( ) where the uncton s dened as ollows, ( ) ( ) ( ) ( ) / ; ( l+ ) ( ) / or l >. () v n V, For a more pctoral descrpton o what the algorthm s dong please reer to the correctness o the algorthm. In our algorthm we shall assume that every vertex s assocated wth a number, whch we shall be denotng ths as ts mask. Maskng operaton on a vertex wll mply settng ths number to one whch denotes the lowest level n the dvde and conquer tree at whch that vertex rst became a separator vertex. It s ntroduced to help n marnatng the nvarant descrbed above and ts use wll be clear as we go nto the detals o the algorthm. We may assume that unmaskng operaton s equvalent to settng the number o the vertex to a negatve n tag. Intally all vertces are unmasked, besdes s whose mask s set to zero. Now we gve the detals o the algorthm. Algorthm SSSP ( G, l ) Comments : l stands or the depth o recurson or the level n our dvde and conquer tree. k s the maxmum clque sze. l wll be ntalzed to one whle we nvoke ths procedure. Step I. I c log n ext. (c s any constant ) then apply AP to get the transtve closure and Step. Ths step s exactly smlar to step I o AP.e., denty the good separator S. Mask all unmasks vertces o S wth l. I any vertex o S s already masked they are let unchanged. Then buld up two nduced sub graphs G V S, ) and o ( E

3 G where G ( V S, E ) o G where V and V are two sets o vertces o the two components ater the decomposton o G wth S. Step 3. Recursvely solve SSSP (G, l+). Step 4. Recursvely solve SSSP (G, l+). Step 5. Ths step s the mergng step to mantan the nvarant. (a) Buld up a dstance matrx S where each entry { sd ( v ), sd ( v )} d ( v ) mn G G (3) or all pars v,v S. To obtan where sd G (v ) perorm a transtve closure on that matrx. (b) For every vertex v V n G nd the shortest path to every separator vertex by the ollowng operaton { sd ( v, t) sd ( t, )} d( v, u) mn t S G + u G. (4) In the above operaton sd G (t, u) s already dented n step 5(a) and sd G (v, t) was mantaned as our nvarant. The same operaton s done on each vertex o V n G also. (c) Unmask all vertces o S whch has number l. For each o the masked vertces taken as the source compute the shortest dstance to every other vertex. I a masked vertex v S then the shortest dstance rom t to every other vertex s already computed n step5(b). I v V s a masked vertex then the shortest dstance o any vertex v V can be computed as { sd ( v, u) sd ( u, v )} sd G ( v ) mn u S G + G. (5) Here sd G (v, u) and sd G (u ) were already dented n step 5( b). I v V s a masked vertex and s any other vertex then the shortest dstance can be computed as { sd ( v, u) sd ( u, v )}, sd ( v )} sd G ( v ) mn{mn u S G + G G. (6) The same operaton wll be done on masked vertces o V. Note: From now on n subsequent, dscussons the word node wll be or the dvde and conquer tree and the word vertex wll be used or the graph G. Lemma The algorthm SSSP correctly determnes the sngle source shortest dstance rom s to every other vertex. Proo : Here we shall elaborate what the algorthm SSSP s dong. We shall argue on the act that ths dvde and conquer mantans the nvarant whch s the mantenance o the shortest path normaton rom those source vertces at a node o the dvde and conquer tree whch are present n that node and had been separator vertces at levels less than or equal to the current level. Let us consder a part o the dvde and conquer tree startng rom the root. The source vertex s s or whch we want to buld up the shortest path normaton. Now B s one o the chld o A obtaned through splttng steps as descrbed n step and. At the node B only s s masked wth 0. All other vertces are unmasked. Then n C 3

4 vertces s and s 3 are masked wth and s has retaned ts mask 0. Smlarly at D, s s masked wth. At E, s 4 and s 6 s masked wth and s 3 retans ts mask. At F, s s masked wth 0, s and s 3 are masked wth and s 5 s masked wth. Now we assume that the nvarant s preserved at E and F. Ths means at E we know the shortest dstances rom s 3, s 4, s 5, s 6 and s 7 to every other vertex n E through paths whch uses vertces o E n G. Smlarly at F we know the shortest dstances o s, s 3, s 5, s 7, and s to every other vertex n F. To prove the correctness we have to show that ths nvarant wll be preserved at C also. Now snce every separator vertex ( here s 3, s 7 ) at E or F the shortest dstance to every other vertex s known mples dstance between them (.e. here dstance between s 3 and s 7 ) wll be known. Thus we can clearly see that step5(a) correctly computes the all par shortest dstance among separator vertces at the node C. In our nvarant we know that the shortest dstance o every vertex n E and F rom s 3 and s 7 s known through paths whch are solely contaned n sub graphs nduced by vertces o E and F respectvely n F. So ater the computaton o the all par shortest dstances n step5(a) among separator vertces t s qute obvous to see that step5(b) computes the shortest dstance o every vertex to each o separator vertces whch are here s 3 and s 7. Lastly n step 5(c) only s 7 s unmasked but s 3 retans ts mask snce t s masked wth. Then the shortest dstance o s, s 4, s 5, s 6 to every other vertces n C are determned snce they are only masked vertces besdes s 3 or whch the shortest dstance to every other vertces n C were already determned n step5(b). Thus we see that the nvarant s preserved at the node C o the dvde and conquer tree. 3. Parallel Complexty o Sngle Source Shortest Path : Now we shall be showng how to parallelze SSSP. We shall be denotng t by PSSSP. For PSSSP we shall be usng only O(n) processors. Here ollows the detals o the algorthm. Descrpton o PSSSP ( G) : Here we shall be takng the same approach whch we took whle we parallelzed AP. The buldng up o the elmnaton tree wll be done only once. Ths would take O (n) processors and O(log n) tme. The algorthm conssts o two phases. The rst phase s the dvde phase n whch the dvde and conquer tree dsplayed wll be bult up n the same ashon as descrbed n PAP and wll take O(n) processors and O(log n) tme. In the second phase at the maxmum level o the dvde and conquer tree we perorm step. Here we have at mosto sub problems each o sze log n O(log n). We solve each o ths sub problem n parallel allocatng O(log n) processors to each. So accordng to Theorem, ths level can be completed n O (log n) tme and wth O(n) processors. Now n general or level where each sub problem correspondng to s o sze O and we have O processors avalable or t. Clearly step5(a) wll take O() tme snce k S, 4

5 where k s a constant. Step 5 (b) wll take O() tme by assgnng one processor to each vertex and parallelly executng ths step or each o them. In G, number o masked vertces can be at most k + ( ncludng s ). So t wll take O() tme to execute step5(c) or G,. Ths ollows rom the observaton that or each o the masked vertces o G, to buld up sngle source shortest path normaton to all other vertces o G, as descrbed n step5(c) amounts to k O n work. Hence applyng Theorem knowng the act that total number o processors avalable or G, s O we establsh our clam. Snce all sub problems at level are executed n parallel, the tme requred to execute level s also O () wth O(n) processors. The overall tme complexty o the entre problem s log n O( ) O(log n). (7) Thus we have establshed the ollowng Theorem. Theorem Sngle source shortest path problem on chordal graphs wth maxmum clque sze bounded by constant can be solved n parallel wth O (n) processors n O(log n) tme on CRCW PRAM model o computaton. 7. Concluson In ths paper we have presented ecent parallel algorthms and complexty or shortest path problems n graphs. The algorthm or the sngle source problem, acheves near optmal processor tme product n contrast to general graphs where we ace the transtve closure bottleneck. Reerences:. Akl. S. G., The Desgn and Analyss o Parallel Algorthms, Prentce Hall, New Jersey, Berkman, O. and Vshkn, U., Fndng level-ancestors n trees, Tech. Report. UMIACS -TR 9-9, Unversty o Maryland, Bertoss, A. A and Bonuccell, M.A., Some parallel algorthms on nterval graph, Dscr. Appl. Math. 6 ( 987 ) Booth, S. K. and Colbourn, C. J., Problems polynomally equvalent to graph somorphsm, Techncal Report CS-77-04, Computer Scence Dept. Unv. o Waterloo, Canada, Chen, C.Y. and Das, S. K., Parallel algorthms or level order traversals o general trees, Journal o Combnatorcs, Vol.4. No , pp Chen, L., Parallel complexty o dscrete problems, Ph D thess, Oho State Unv Homann, C. M., Group-theoretc algorthms and graph somorphsm, Vol. 36. Lecture Notes n Computer Scence, Sprnger Verlag. 8. Horowtz, E. and Sahn, S., Fundamentals o Computer Algorthms, Computer Scence Press,

Odd Graceful Labeling of Some New Type of Graphs

Odd Graceful Labeling of Some New Type of Graphs Internatonal Journal o Mathematcs Trends and Technology IJMTT) Volume 41 Number 1- January 2017 Odd Graceul Labelng o Some New Type o Graphs Sushant Kumar Rout 1 Debdas Mshra 2 and Purna chandra Nayak