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Minimum-Cost Spanning Tree As a Path-Finding Problem 1 Introduction MinimumCost Spanning Tree as a PathFinding Problem Bruce M Maggs Serge A Plotkin Laboratory for Computer Science MIT Cambridge MA July Abstract In this pap er we show that minimumcost spanning tree is a sp ecial case of the closed semiring pathnding problem This observation gives us a nonrecursive algorithm for nding minimum cost spanning trees on meshconnected computers that has the same asymptotic running time but is much simpler than the previous recursive algorithms Intro duction In this pap er we show that minimumcost spanning tree is a sp ecial case of the closed semiring pathnding problem sections For a graph of n vertices the pathnding problem can 3 b e solved sequentially in O n steps by a dynamic programming algorithm of which the algorithms of Floyd and Warshall are sp ecial cases This dynamic programming algorithm has a well known O n step implemen tation on an n n meshconnected computer Previously known minimumcost spanning tree algorithms for the mesh are based on the recursive algorithm of Boruvka also attributed to Sollin pp which is complicated to implemen t For example the algorithm of achieves O n steps by reducing the fraction of the mesh in use by a constant factor at each recursive call The dynamic programming algorithm has the same asymptotic running time but is much simpler 0 This research was supp orted in part by the Defense Advanced Research Pro jects Agency under Contract N C and by the Oce of Naval Research under Contract NK Bruce Maggs is supp orted in part by an NSF graduate fellowship The rest of this pap er consists of two short sections In section we show how to cast minimum cost spanning tree as a pathnding problem In section we briey describ e an O n step mesh algorithm to solve the problem Minimumcost spanning tree In this section we dene the minimumcost spanning tree problem and a related pathnding prob lem We give a recurrence for solving the pathnding problem via dynamic programming We then prove that the solution to the pathnding problem contains the solution to the minimumcost spanning tree problem 1 Given an nno de connected undirected graph G V E where V is the set f ng and 0 0 the minimumcost spanning tree problem is to nd a C where each edge fi j g in E has cost C j i ij subgraph that connects the vertices in V such that the sum of the costs of the edges in the subgraph is minimum We assume that the edge costs are unique If not lexicographical information can b e added to make them unique For convenience we also assume that if fi j g is not in E then it 0 0 C has cost C j i ij k for each i j k n of the shortest The pathnding problem is to compute the cost C ij lowestcost path from i to j that passes through vertices only in the set f k g where the cost of a path is dened to be the highest cost of any edge on the path For any i and j the shortest path from i to j with no intermediate vertex higher than k either passes through k or do es not In the rst case the cost of the shortest path from i to j is either the cost of the shortest path from i to k or the cost of the shortest path from k to j whichever is higher In the second case we have k 1 k k C C Thus C can b e computed by the recurrence ij ij ij k 1 k k 1 k 1 gg C min fC C maxfC ij ij k j ik The following theorem shows that the unique minimumcost spanning tree can b e recovered from the costs of the shortest paths 0 n Theorem An edge fi j g is in the unique minimumcost spanning tree if and only if C C ij ij 1 For simplicit y we assume that the graph is connected The same technique will nd a minimumcost spanning forest of a disconnected graph 0 n Pro of The pro of has two parts We rst show that if fi j g is a tree edge then C C We ij ij 0 n then show that if C C then the edge fi j g is in the tree First assume that fi j g is a tree ij ij 0 n edge but that C C Consider the cut of the graph that fi j g crosses but no other tree edge ij ij 0 n crosses Since C C there must b e some path from i to j whose highestcost edge has cost ij ij n 0 0 C C Hence every edge on this path has cost less than C This path must cross the cut at ij ij ij least once Replacing the edge fi j g by any edge on the path that crosses the cut reduces the cost 0 n of the tree a contradiction Conversely assume that C C but that fi j g is not a tree edge ij ij 0 Adding the edge fi j g to the tree forms a cycle whose highestcost edge costs more than than C ij Replacing this edge by fi j g yields a tree with smaller cost a contradiction Implementation on a meshconnected computer In this section we give a short description of an O n step algorithm for solving the minimumcost spanning tree problem on an n n meshconnected computer We assume that the diagonal element in each mesh row can broadcast a value to the other elements of the row in a single step This type of broadcast can b e simulated by a mesh without this capability by slowing the algorithm down by a constant factor The algorithm pro ceeds as follows We assume that the input graph 0 is given in the form of a matrix of edge costs C which enters rowbyrow through the top of the mesh Matrix row i is mo died as it passes over rows through i and is stored when it reaches k 1 is broadcast right and left mesh row i When matrix row i passes over mesh row k the value C ik k 1 from the diagonal cell k k Each cell k j j n knows the value of C and computes k j k 1 k 1 k 1 k maxfC C gg C min fC ij ij ik k j which is passed down to the next mesh row After reaching mesh row i matrix row i stays there until each matrix row l i l n ab ove it has passed over it and then continues to propagate n down passing over the rest of the matrix rows The output matrix C exits rowbyrow from the b ottom of the mesh By theorem the adjacency matrix of the minimumcost spanning tree can b e constructed by comparing the input and output matrices Acknowledgments We would like to thank Tom Leighton and Charles Leiserson for their helpful suggestions References A V Aho J E Hop croft and J D Ullman The Design and Analysis of Computer Algorithms AddisonWesley Reading MA M J Atallah and S R Kosara ju Graph problems on a meshconnected pro cessor array Journal of ACM July T W Christopher An implementation of Warshal ls algorithm for transitive closure on a cellular computer Technical Rep ort Inst for Comp Research Univ of Chicago Chicago IL E Dekel D Nassimi and S Sahni Parallel matrix and graph algorithms SIAM Journal on Computing November R W Floyd Algorithm shortest path Communications of the ACM L J Guibas H T Kung and C D Thompson Direct VLSI implemen tation for combinatorial algorithms In Proc of the Caltech Conference on Very Large Scale Integration pages January S C Kleene Representation of events in nerve nets and nite automata In C E Shannon and J McCarthy editors Automata Studies pages Princeton University Press F T Leighton Introduction to the Theory of Networks Parallel Computation and VLSI Design Unpublished manuscript C E Leiserson F M Rose and J B Saxe Optimization of synchronous circuitry by retiming In Third Caltech Conference on VLSI pages March C E Leiserson and J B Saxe Optimizing synchronous systems Journal of VLSI and Computer Systems Spring K N Levitt and W H Kautz Cellular arrays for the solution of graph problems Communi cations of the ACM September R McNaughton and H Yamada Regular expressions and state graphs for automata IRE Trans on Electronic Computers F L Van Scoy The parallel recognition of classes of graphs IEEE Transactions on Computers C July RE Tarjan Data Structures and Network Algorithms SIAM Philadelphia PA S Warshall A theorem on b o olean matrices Journal of ACM .
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