On the Minimum Spanning Tree for Directed Graphs with Potential Weights

On the Minim um Spanning T ree for Directed Graphs with P oten tial W eigh ts V. A. Buslo v ∗ , V. A. Kh udobakhsho v † Dep artment of Computational Physics St.-Petersbur g State University Septem b er 7 , 2021 Abstract In general the problem of find ing a mi minum sp anning tree fo r a w eighted directed graph is difficult b ut solv able. There are a lot of differ- ences b etw een problems for directed and undirected graphs, therefore the algorithms for undirected graphs cannot usually b e applied to th e directed case. In this pap er we examine th e k in d of w eigh ts such th at th e problems are equiv alen t and a minimum spanning tree of a directed graph may b e found by a simple algorithm for an un d irected graph. 1 In tro duction The pr o blem of finding a minimum spanning tree is well known to graph theo - rists as well as to programmer s who deal with graph theory applications. F or an undirected case there are a lot of simple algor ithms such a s Pr im’s algorithm [5]. F or directed gr aphs, a genera l solution also exists, see pap er s [1, 2, 3] for more information. F urthermor e, there are s ome optimizations for directed and undirected cases based o n Fibo nacci heaps [4]. In c ertain pr oblems in physics we deal with directed graphs whose weigh ts by definition sa tisfy the given r elation. Sometimes the pro pe r ties of weigh ts provide a p ossibility of simplifying the pr oblem to an undir ected ca se. F or example, cons ide r a dir e cted graph whose weight s satisfy the equation Q ij = ϕ ij − ϕ ii . (1) The main result of this pa p e r is that for the gra ph whose a weigh t matrix is Q with p ositive entries a minimum spanning tree may be found by a simple alg o- rithm for a corr esp onding undirec ted graph whose a weight matrix is symmetric matrix ϕ . ∗ ab v ab v@bk.ru † vitaly .khudoba khshov@gmail.com 1 One can think ab out the weight ϕ ij as a height o f a p otential ba rrier which has to b e surmount ed in o rder to ge t to p oint i from point j . F rom this p oint of view ϕ ii is a lo ca l minimum of a po tential well. So weight s whic h s atisfy Equation (1) will b e called p otential . 2 Definitions Depending on the problem, directed tr ees may be defined in tw o ways. Usua lly it is supp ose d that in-degree id ( v ) ≤ 1 for all vertices v . Howev er we define tree such that o ut-degree od ( v ) ≤ 1 a nd od ( v ) = 1 iff v is the ro ot. W e will deno te G ( V , E , ω ) for an undirected gr aph wher e V is a vertex set, E is an edge set and ω is a weight matrix cor resp onding to the edge set. Clearly ω is always symmetric. In the c a se of a directed gra ph we will use the same notation except a prime to denote a directed gr aph a nd we will us e A instead of E in order to note that in the directed case we will hav e a n arc set. Also in the undirected case the weight matr ix is not neccessar y symmetric. 3 Minim um Spanning T rees Suppo se we have an undirected g raph G ( V , E , ϕ ), where ϕ is a symmetric matrix whose diago nal en tries are no t neccess ary eq ual to ze r o, but inequa tions ϕ ii < ϕ ik and ϕ ii < ϕ ki hold for all k 6 = i . F or every G we can define a direc ted graph G ′ ( V , A, Q ) wher e Q is a w eight matrix which satisfies Equation (1). W e must find a tree T whic h minimizes the following expres sion w ( T ) = X ( i,j ) ∈ T Q ij . (2) This tree is called a minim um s panning tree for the weigh ted directed graph G ′ . The minimum s pa nning tre e for undirected graph G is defined in a similar wa y . F or this class of weighted dir ected graphs the follo wing prop osition can b e po sed. Prop ositi on 1. A minimu m sp anning tr e e of a dir e cte d gr aph G ′ c oinci des with a minimum sp anning tr e e for an un dir e cte d gr aph G with a r o ot in vertex k for which ϕ kk is minimum. Pr o of. The minum um fo r (2) can be written, acco rding for Q weigh ts prop er ties in following form: min T ⊂ G ′ w ( T ) = min T ⊂ G ′ X ( i,j ) ∈ T ( ϕ ij − ϕ ii ) . Allowing fo r the facts that the num ber of arcs in spanning tr ee T equa ls | V | − 1 and the off-diago nal en tries do not dep end on diago nal ones, the minimum of 2 the prev ious expressio n equals min T ⊂ G ′ X ( i,j ) ∈ T ϕ ij − max k X i 6 = k ϕ ii . It is clear that if ϕ ii = 0 for all i then G ′ changes 1 to G . Therefore the fir s t summand in the previous expression equals to the weigh t of the co rresp onding a minum um spanning tree of the undire cted gr aph G . min T ⊂ G X { i,j }∈ T ϕ ij − max k X i 6 = k ϕ ii It follows min T ⊂ G X { i,j }∈ T ϕ ij − | V | X i =1 ϕ ii + min k ϕ kk As a result, the minimum spanning tree of G ′ can b e g iven from the minimum spanning tree o f G b y fixing a ro ot in vertex k with a minimum v alue o f ϕ kk . 4 Conclusion and F uture W ork In the previo us section it is shown tha t a lgorithms for undirected gra phs can be applied to a direc ted case if weights of a directed g raph hav e certain sp ecial prop erties. In the future we plan to g et a solution for a more difficult problem, finding the minimum s panning fore s t for g r aphs whose weights a re p otential. References [1] F. Bo c k. An algor ithm to constr uct a minimum directed spanning tr ee in a directed net work. In Developments in Op er ations Rese ar ch , pag es 29–44 , New Y or k, 197 1. Go rdon and Breach. [2] Y. J. Chu a nd T. H. Liu. On the shortest arb or escence of a dir ected gra ph. Sci. Sin., S er. A , 14 :1396 –1400 , 1965. [3] J. Edmonds . Optimum br a nchings. J. R es. Nat. Bur. St andar ds , B7 1:233 – 240, 196 7. [4] Harold N. Gab ow, Zvi Ga lil, Thomas H. Sp encer , and Rob ert E. T ar jan. Efficient algor ithms for finding minim um spanning tr ees in undir ected and directed gr a phs. Combinatoric a , 6 :109– 122, 1 986. [5] R. C. Prim. Shortest co nnec tion netw orks and some generaliz a tions. Bel l Systems T e chnolo gy Journal , 3 6:138 9–140 1, 1957. 1 In this case w e can cha nge the ordered pair ( i, j ) to a 2-elemen ts subset { i, j } . 3