On Algebraic Expressions of Generalized Fibonacci Graphs

Transcription

1 On Algebraic Expressions of Generalized Fibonacci Graphs MARK KORENBLIT and VADIM E LEVIT Department of Computer Science Holon Academic Institute of Technology 5 Golomb Str, PO Box 305, Holon 580 ISRAEL {korenblit, levitv}@haitacil Abstract: - The paper investigates relationship between algebraic expressions and generalized Fibonacci graphs We propose an algorithm for constructing expressions of generalized Fibonacci graphs On every step it divides the graph into two parts of the same size We show that this algorithm applied to an n-vertex log ( k + ) generalized Fibonacci graph of degree k generates an expression of O ( n ) k length When k equals, O n length the algorithm generates expressions of Key-Words: - complexity of an algebraic expression, generalized Fibonacci graph, optimal representation of the algebraic expression, st-dag, st-dag expression, subexpression, subgraph Introduction A graph G=(V,E) consists of a vertex set V and an edge set E, where each edge corresponds to a pair (v,w) of vertices If the edges are ordered pairs of vertices (ie, the pair (v,w) is different from the pair (v,w), then we call the graph directed or digraph; otherwise, we call it undirected A path from vertex v 0 to vertex v j in a graph G=(V,E) is a sequence of its vertices v 0, v, v,, v j, v j, such that (v i,v i ) E for i j A graph G =(V,E ) is a subgraph of G=(V,E) if V V and E E A graph G is homeomorphic to a graph G (homeomorph of G ) if G can be obtained by subdividing edges of G via adding new vertices A two-terminal directed acyclic graph (st-dag) has only one source s and only one sink t In an st-dag, every vertex lies on some path from the source to the sink An st-dag that has only one path between the source and the sink is known as a path graph A complete graph is an undirected graph that has an edge between any two vertices An algebraic expression is called an st-dag expression if it is algebraically equivalent to the sum of products corresponding to all possible paths between the source and the sink of the st-dag [] This expression consists of terms (edge labels), and the operators + (disjoint union) and (concatenation, also denoted by juxtaposition when no ambiguity arises) An st-dag expression of an st-dag G will be further denoted by Ex(G) We define the complexity of an algebraic expression in two ways The complexity of an algebraic expression is (i) a total number of terms in the expression including all their appearances (the first complexity characteristic) or (ii) a number of plus operators in the expression (the second complexity characteristic) The first and the second complexity characteristics of an st-dag expression are denoted by T(n) and P(n) respectively, where n is the number of vertices of the graph (the size of the graph) An equivalent expression with the minimum complexity is called an optimal representation of the algebraic expression A series-parallel (SP) graph is defined recursively as follows: (i) A single edge (u,v) is a series-parallel graph with source u and sink v (ii) If G and G are series-parallel graphs, so is the graph obtained by either of the following operations: (a) Parallel composition: identify the source of G with the source of G and the sink of G with the sink of G (b) Series composition: identify the sink of G with the source of G As shown in [] and [7], a SP graph expression has a representation in which each term appears only once We proved in [7] that this representation is an optimal representation of the SP graph expression from the first complexity characteristic point of view For example, the expression of the SP graph presented in Fig is abd+abe+acd+ace+fe+fd a f b c Fig A series-parallel graph d e

2 b b b i b i+ b n a a 3 a a i a i+ a i+ a n 3 i i+ i+ n n Fig An n-vertex Fibonacci graph This expression can be reduced to (a(b+c)+f)(d+e), where each term appears once The notion of a Fibonacci graph (FG) was introduced in [6] A Fibonacci graph has vertices {,, 3,, n} and edges {(v,v+) v=,,,n } U {(v,v+) v=,,,n } That is, in an n-vertex FG (n ), two edges leave each of its n vertices except the two final vertices (n and n) Two edges leaving the v vertex ( v n ) enter the v+ and the v+ vertices The single edge leaving the n vertex enters the n vertex No edge leaves the n vertex This graph is illustrated in Fig A generalized Fibonacci graph [6] of degree k (FG k ) has vertices {,, 3,, n} and edges FG a {(v,w) v < w n and w v k} FG a a a 3 a 4 a b b b 3 b 4 a a 3 a FG 3 b b b 3 b 4 a c c a a 3 a c 3 Fig3 The hierarchy of generalized 6-vertex Fibonacci graphs a 5 a That is, in an n-vertex FG k (n k), k edges leave each of its n vertices except the k final vertices (n k+, n k+,, n, n); k edges leaving the v vertex ( v n k) enter the v+, the v+,, the v+k vertices respectively; k edges leaving the n k+ vertex enter the n k+, the n k+3,, the n vertices respectively; k edges leaving the n k+ vertex enter the n k+3, the n k+4,, the n vertices respectively, etc; the single edge leaving the n vertex enters the n vertex; no edge leaves the n vertex Fig 3 presents the hierarchy of 6-vertex generalized Fibonacci graphs FG is a path graph (it is a SP graph) The Fibonacci graph defined above is FG The next graph is FG 3 FG n is the last nontrivial graph in the hierarchy of n-vertex generalized Fibonacci graphs (its undirected version is a complete graph) Hence, for n = 6 it is FG 5 (this graph is not shown in Fig 3) As shown in [6], an n-vertex FG k has k( k +) m = kn edges It is proved in [3] that an st-dag is SP if and only if it does not contain a subgraph homeomorphic to the forbidden subgraph enclosed between vertices and 4 of an FG from Fig Thus, generalized Fibonacci graphs (k ) are of interest as ''through'' non-sp st-dags Mutual relations between graphs and algebraic expressions are discussed in [], [7], [8], [9], [0], [], [], [3], and other works Specifically, [0], [], and [3] consider the correspondence between SP graphs and read-once functions A Boolean function is defined as read-once if it may be computed by some formula in which no variable occurs more than once (read-once formula) On the other hand, a SP graph expression can be reduced to the representation in which each term appears only once Hence, such a representation of a SP graph expression can be interpreted as a read-once formula (Boolean operations are replaced by arithmetic ones) An expression of a homeomorph of the forbidden subgraph belonging to any non-sp st-dag has no representation in which each term appears once For example, consider the subgraph enclosed between vertices and 4 of an FG from Fig Possible

3 optimal representations of its expression are a (a a 3 +b )+b a 3 or (a a +b )a 3 +a b For this reason, an expression of a non-sp st-dag can not be represented as a read-once formula However, for arbitrary functions, which are not read-once, generating the optimum factored form is NPcomplete [4] Some heuristic algorithms developed in order to obtain good factored forms are described in [4], [5], and other works Therefore, generating an optimal representation (from both complexity characteristics point of view) for a non-sp st-dag expression is a highly complex problem In [9] we presented a heuristic algorithm based on the decomposition method built to simplify the expressions of FG While [7] is a substantially improved version of [9], in [9], [7] we show that complexity characteristics of an n-vertex FG can be estimated as O ( n ) Our intent in this paper is to simplify the expressions of FG k for an arbitrary k The Decomposition Method for Constructing Expressions of Generalized Fibonacci Graphs of Degree In this section we sketch some of the results received in [9], [7] Consider the n-vertex FG presented in Fig Denote by E(p,q) a subexpression related to its subgraph (which is an FG too) having a source p ( p n) and a sink q ( q n, q p) If q p, then we choose any decomposition vertex i (p + i q ) in a subgraph, and, in effect, split it at this vertex (Fig 4) Otherwise, we assign final values to E(p,q) As follows from the FG structure any path from vertex p to vertex q passes through vertex i or avoids it via edge b i- Therefore, E(p,q) is generated by the following recursive procedure (decomposition procedure): case q = p: E(p,q) case q = p + : E(p,q) a p 3 case q p + : choice(p,q,i) 4 E(p,q) E(p,i)E(i,q)+ E(p,i )b i- E(i+,q) Lines and contain conditions of exit from the recursion The special case when a subgraph consists of a single vertex is considered in line It is clear that such a subgraph can be connected to other subgraphs only serially For this reason, it is accepted that its subexpression is, so that when it is multiplied by another subexpression, the final result is not influenced Line describes a subgraph consisting of a single edge The corresponding subexpression consists of a single term equal to the edge label The common case is processed in lines 3 and 4 The procedure choice(p,q,i) in line 3 chooses an arbitrary decomposition vertex i on the interval (p,q) so that p < i < q A current subgraph is decomposed into four new subgraphs in line 4 Subgraphs described by subexpressions E(p,i) and E(i,q) include all paths from vertex p to vertex q passing through vertex i Subgraphs described by subexpressions E(p,i ) and E(i+,q) include all paths from vertex p to vertex q passing via edge b i- E(,n) is the expression of the initial n-vertex FG (Ex(FG )) Hence, the decomposition procedure is initially invoked by substituting parameters and n instead of p and q, respectively The following theorem that determines an optimal location of the decomposition vertex i in an arbitrary interval (p,q) of a Fibonacci graph is proved in [7] Theorem The representation with a minimum total number of terms among all possible representations of Ex(FG ) derived by the decomposition method is achieved if and only if on q p + each recursive step i is equal to for odd q p + q p + 3 q p+ and to or for even q p+, ie, when i is a middle vertex of the interval (p,q) This decomposition method is called optimal It can be shown that the optimal decomposition method also brings about the minimum to the number of plus operators of Ex(FG ) Thus, the total number of terms in the n-vertex FG expression derived by the optimal decomposition method is defined recursively as follows: b p b i b q a p a i p p+ i i i+ q q Fig4 Decomposition of a Fibonacci subgraph at vertex i 3 a i a q

4 T () = 0, T () = n = T T n n T + +, 3 T n The number of plus operators in this expression is defined similarly: and P() = 0, P() = 0 n P( n) = P P For large n n n P + +, 3 P n 4T + P ( n) 4P + By the master theorem [] the recurrence as n = αt + f ( n), β n where α and β > are constants, and is β n interpreted as either or β asymptotically as follows: n, can be bounded β = Θ n log α ε if β ( n) = O n α log β f for some constant ε > 0 Θ For the 8-vertex FG, the possible expression derived by the optimal decomposition method is Therefore, T(n) and P(n) are ( n ) (a (a a 3 +b )+b a 3 )((a 4 a 5 +b 4 )(a 6 a 7 +b 6 )+a 4 b 5 a 7 )+ (a a +b )b 3 (a 5 (a 6 a 7 +b 6 )+b 5 a 7 ) It contains 5 terms and 9 plus operators We conjecture that the optimal decomposition method provides an optimal representation (for both our complexity characteristics) of an algebraic expression of a Fibonacci graph 3 The Decomposition Method for Constructing Expressions of Generalized Fibonacci Graphs of Arbitrary Degrees We intend to generate an algebraic expression for FG k and apply the decomposition method with that end in view The method is the decomposition method described in the previous section extended for a generalized Fibonacci graph We consider the n-vertex FG k (k ) and, as in section, denote by E(p,q) a subexpression related to its subgraph (which is FG k too) having a source p ( p n) and a sink q ( q n, q p) If q p (k ), then we choose any decomposition vertex i (p + k i q k + ) in a subgraph and split it at this vertex Otherwise, we assign final values to E(p,q) Any path from vertex p to vertex q passes through vertex i or avoids it via connecting edges (labeled b, c, d, with indexes) For example, as follows from the FG 3 structure (Fig 5) any path from vertex p to vertex q passes through decomposition vertex i or avoids it via connecting edges: b i-, c i- or c i- Therefore, E(p,q) can be generated for FG 3 by the following c p c i c i c q 3 b p b i b q a p a i p p+ i i i+ q q Fig5 Decomposition of a generalized Fibonacci subgraph of degree 3 at vertex i a i a q 4

5 decomposition procedure: case q = p: E(p,q) case q = p + : E(p,q) a p 3 case q = p + : E(p,q) a p a p+ +b p 4 case q = p + 3: E(p,q) (a p a p+ +b p )a p+ + a p b p+ +c p 5 case q p + 4: choice(p,q,i) 6 E(p,q) E(p,i)E(i,q)+ E(p,i )b i- E(i+,q)+ E(p,i )c i- E(i+,q)+ E(p,i )c i- E(i+,q) For an arbitrary k (k ), the number of connecting edges increases as an arithmetical progression from level to level (b-edges are located on the second level, c-edges are located on the third one, etc) Hence, the number of subgraphs revealed after the decomposition is equal to k + y= y = k k + The number of connecting edges and the number of additional plus operators between parts of the derived expression are equal to k y= k y = ( k ) We will choose the decomposition vertex i in the middle of a decomposed subgraph on each recursive step (we proceed from the supposition that it is the best way not only for k = but for any k and, also, call this decomposition method optimal) In such a case, for an n-vertex FG k, the total number of terms T(n) in the expression Ex(FG k ) derived by the optimal decomposition method can be estimated as follows: T n k k + T + k k Analogously, for an n-vertex FG k, the number of plus operators P(n) in the expression Ex(FG k ) derived by the optimal decomposition method can be estimated as follows: P k k n k k + P + Hence, by the master theorem and log ( k k + n = O n ) T () log ( k k + n = O n ) P () Specifically, for k =, when our graph is a path graph, () and () give O(n) For k = we have a O n complexity Fibonacci graph, and receive 3 Substituting 3 for k in () and () gives ( n ) O Further increase of k leads to slower increase of the order of complexity characteristics For instance, for log 4 O n, ie, less than k = 4, T(n) and P(n) are 4 O ( n ) 4 Conclusion and Future Work The paper investigates relationship between algebraic expressions and generalized Fibonacci graphs The optimal decomposition method applied to an n-vertex generalized Fibonacci graph of degree log ( k k + ) k (FG k ) generates an expression of O ( n ) complexity This result is a generalization of a previous result received for an n-vertex Fibonacci graph (FG ), where complexity characteristics are estimated as O ( n ) We conjecture that splitting of FG k in the middle of a decomposed subgraph on each recursive step yields the shortest expression possible (as in Theorem, where k equals two) Our experimental calculations show that the order of complexity characteristics of an n-vertex complete st-dag (FG n ) does not exceed n log n Based on the fact that any n-vertex st-dag (a two-terminal directed acyclic graph) is a subgraph of an n-vertex complete st-dag we are going to estimate the complexity of the shortest expression for any st-dag References: [] W W Bein, J Kamburowski, and M F M Stallmann, Optimal Reduction of Two-Terminal Directed Acyclic Graphs, SIAM Journal of Computing, Vol, No 6, 99, pp -9 [] T H Cormen, C E Leiseron, and R L Rivest, Introduction to Algorithms, The MIT Press, Cambridge, Massachusetts, 994 5

6 [3] R J Duffin, Topology of Series-Parallel Networks, Journal of Mathematical Analysis and Applications 0, 965, pp [4] M C Golumbic and A Mintz, Factoring Logic Functions Using Graph Partitioning, Proc IEEE/ACM Int Conf Computer Aided Design, November 999, San Jose, California, USA, pp [5] M C Golumbic, A Mintz, and U Rotics, Factoring and Recognition of Read-Once Functions using Cographs and Normality, Proc 38th Design Automation Conference, Las Vegas, Nevada, USA, June 00, pp 09-4 [6] M C Golumbic and Y Perl, Generalized Fibonacci Maximum Path Graphs, Discrete Mathematics 8, 979, pp [7] M Korenblit and V E Levit, On Algebraic Expressions of Series-Parallel and Fibonacci Graphs, in: Discrete Mathematics and Theoretical Computer Science, Proc 4th Int Conf, DMTCS 003, LNCS 73, 003, pp 5-4 [8] V E Levit and M Korenblit, Symbolic PERT and its Generalization, in: Intelligent Scheduling of Robots and Flexible Manufacturing Systems, Proc Int Workshop held at the Center for Technological Education Holon, Holon, Israel, July 995, pp [9] V E Levit and M Korenblit, A Symbolic Approach to Scheduling of Robotics Lines, in: Intelligent Scheduling of Robots and Flexible Manufacturing Systems, The Center for Technological Education Holon, Holon, Israel, 996, pp 3-5 [0] D Mundici, Functions Computed by Monotone Boolean Formulas with no Repeated Variables, Theoretical Computer Science 66, 989, pp 3-4 [] D Mundici, Solution of Rota s Problem on the Order of Series-Parallel Networks, Advances in Applied Mathematics, 99, pp [] V Naumann, Measuring the Distance to Series-Parallelity by Path Expressions, in: Graph-Theoretic Concepts in Computer Science, Proc 0th Int Workshop, WG 94, LNCS 903, 994, pp 69-8 [3] P Savicky and A R Woods, The Number of Boolean Functions Computed by Formulas of a Given Size, Random Structures and Algorithms 3, 998, pp [4] A R R Wang, Algorithms for Multilevel Logic Optimization, PhD Thesis, University of California, Berkeley, 989 6