AbstractAn interesting graph theoretic problem, called the orderability problem for programmable logic array (PLA) folding, was formulated and shown to be NP-complete. We show that a closely related problem, called the orderability problem for bipartite folding, is solvable in linear time

Ravi, S.S.

Elsevier - Publisher Connector 

Discrete Applied Mathematics 25 (1989) 317-320 
North-Holland 
317 
NOTE 
ON THE ORDERABILITY PROBLEM FOR PLA FOLDING* 
S.S. RAVI 
Department of Computer Science, SUNY at Albany, 1400 Washington Avenue, Albany, 
NY 12222, USA 
Received 29 June 1987 
Revised 26 July 1988 
An interesting raph theoretic problem, called the orderability problem for programmable logic 
array (PLA) folding, was formulated and shown to be NP-complete. We show that a closely 
related problem, called the orderability problem for bipartite folding, is solvable in linear time. 
1. Definitions and problem formulation 
We consider an interesting raph theoretic problem that arises in conjunction with 
PLA folding [2,5,7]. This problem involves mixed graphs, that is, graphs that con- 
tain both directed and undirected edges. Given a directed edge (x,~), we refer to the 
nodes x and y as tail and head nodes respectively. Using this terminology, the prob- 
lem can be formulated as follows. 
Orderability for bipartite folding (OBF) 
Instance: An undirected graph G(V,E) and a set F= { (*Di, qi): 1 ~ilr} of un- 
directed edges none of which is in E and which form a matching.’ 
Question: Can the edges of F be oriented in such a way that for every tail node 
t and every head node h, the edge {t, h) is not in E? 
For a given OBF instance, any orientation of F that satisfies the above condition 
is called a valid orientation; if there is a valid orientation, the set F is said to be 
orientable. 
In order to place the above problem in proper perspective, we use some defini- 
tions from [5]. An alternating path in a mixed graph is a simple path whose first 
and last edges are directed edges2 and in which directed and undirected edges alter- 
nate. An alternating cycle in a mixed graph is an alternating path such that there 
* Research supported in part by the National Science Foundation under grants MCS-8103713 and 
DCI-8603318. 
t A set of edges forms a matching if no two edges in the set are incident on the same node. 
2 Each directed edge must be traversed along its direction. 
0166-218X/89/.$3.50 0 1989, Elsevier Science Publishers B.V. (North-Holland) 
318 S.S. Ravi 
is an undirected edge between the initial and the terminal nodes of the path. The 
reader can easily verify that the OBF problem defined above is equivalent o the 
problem of determining whether the edges of Fcan be oriented in such a way that 
the resulting mixed graph does not have an alternatingpath. We will show in Section 
2 that this problem can be solved in linear time. In contrast, the orderabifityprob- 
[em for PLA folding (OF) [5], in which we are required to determine whether the 
edges in F can be oriented in such a way that the resulting mixed graph does not 
have an alternating cycle, is NP-complete3 [5]. Thus although the specifications of 
OBF and OF problems are very similar, their complexities are very different if 
P#NP. 
2. A linear time algorithm for the OBF problem 
Throughout this section, we use standard graph theoretic terminology [a]. We 
begin our discussion with the observation that the nodes of G which do not appear 
in any edge of F have no role in determining the orientability of F and so may be 
deleted from G. We therefore assume that each node of G appears in some edge of 
F. With this assumption, the following (easily verified) lemma points out a relation- 
ship between connected components of G and the orientability of F. This lemma 
plays a key role in our algorithm. 
Lemma 1. Let c n undirected graph G( V, E) and set F constitute an instance of the 
OBFproblem with each node of G appearing in some edge of F. Then, in any valid 
orientation of F, each connected component of G must be composed entirely of 
head nodes or of tail nodes. 
The above lemma indicates that for purposes of determining orientability, we can 
treat each connected component of G as a single unit. To understand the role played 
by this lemma, let us consider an edge (pi, qi) in F. In any valid orientation of F, 
one of Pi and qi must be a head node and the other a tail node. Thus Lemma 1 im- 
plies that the nodes pi and qt must be in different connected components of G (i.e., 
if for some edge (pj, qj) in F, pi and qj are both in the same connected component 
of G, then F is not orientable). Suppose there is a valid orientation of F with Pi as 
a tail node and qi as a head node. If pi is in the connected component Cr and qi is 
in C,, then by Lemma 1, all of the nodes in Cr must be tail nodes and all of the 
nodes in C, must be head nodes. Therefore, every edge in F induces constraints on 
a pair of connected components of G. 
1’0 handle these constraints, we construct an auxiliary undirected graph, 
GC(V,,EC).4 Each node in V, corresponds to a connected component of G. There is 
3 For NP-completeness and related topics, see [4]. 
4 We will permit “self loops” in CC 
On the orderability problem for PLA folding 319 
an edge between odes x and y of V, if and only if for some edge (pi, qi) in F, pi 
is in the connected component corresponding to x and qi is in the connected com- 
ponent corresponding to y. The following theorem shows how the graph G, can be 
used to determine the orientability of F. 
Theorem 2. Let the undirected graph G(V, E) and set F constitute an instance of 
the OBFproblem such that each node of G appears in some edge of F. Then, F is 
orientabIe iJ’ and only if the auxiliary graph G&.,E,) obtained from G and F is 
bipartite. 
Proof. (c’) Suppose G, is bipartite. Let V,, and V, represent a bipartition of V,. 
We obtain an orientation of F by assigning all of the nodes of G which are in the 
connected components corresponding to the nodes in V,, as head nodes and all of 
the nodes which are in the connected components corresponding to the nodes in 
V,, as tail nodes. It is easy to verify that this is a valid orientation of F. 
(a) Suppose F is orientable. Consider any valid orientation of F. In that orienta- 
tion, each connected component of G must be composed entirely of tail nodes or 
of head nodes (Lemma 1). Therefore, we can assign (unambiguously) a label T or 
H to each node x of the auxiliary graph G, depending upon whether the connected 
component of G corresponding to x is composed of tail nodes or of head nodes. It 
is easy to verify that this labeling provides a 2-coloring of G,. Thus G, is bipar- 
tite. Cl 
Theorem 2 leads directly to an algorithm for the OBF problem. 
Algorithm Orient 
Input. Undirected graph G(V,E) and set F= {(pi,qi): 1 sizer}. 
Output. A valid orientation of the edges of F, if one exists. 
begin 
Obtain the connected components of G; 
Construct the auxiliary graph GJC’,, EC); 
if G, is bipartite 
then Orient the edges in F according to the bipartition 
else print “there is no solution”; 
end 
We conclude by pointing out that the above algorithm can be implemented to run 
in O(l V I+ IE I) time. It is well known that the connected components of a graph 
G(V, E) can be obtained in O(l VI + I E 1) time using depth-first search [l]. The aux- 
iliary graph G, contains at most 1 VI nodes and at most IF I = 0( I V 1) edges (because 
each pair in F adds one edge to EC). Testing whether G, is bipartite can be done us- 
320 S.S. Ravi 
ing breadth-first search in O(l V, I+ j&I) = O(l VI) time [3]. Thus the overall 
running time of the algorithm is O(/ VI + [El). 
Acknowledgment 
I would like to express my sincere thanks to the referees for their valuable sugges- 
tions. I am indebted to my thesis advisor Prof. Errol L. Lloyd (University of Pitts- 
burgh) for several helpful discussions and for his comments on the earlier drafts of 
this paper. 
References 
[l] A.V. Aho, J.E. Hopcroft and J.D. Ullman. The Design and Analysis of Algorithms (Addison- 
Wesley, Reading, MA, 1974). 
[2] J.R. Egan and C.L. Liu, Optimal bipartite folding and partitioning of a PLA, IEEE Trans. CAD 
Integrated Circuits and Syst. 3(3) (1984) 191-199. 
[3] S. Even, Graph Algorithms (Computer Science Press, Rockville, MD, 1979). 
[4] M.R. Garey and D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP- 
Completeness (Freeman, San Francisco, CA, 1979). 
[5] G.D. Hachtel, A.R. Newton and A.L. Sangiovanni-Vincentelli, An algorithm for optimal PLA 
folding, IEEE Trans. CAD Integrated Circuits and Syst. l(2) (1982) 63-77. 
[6] F. Harary, Graph Theory (Addison-Wesley, Reading, MA, 1969). 
[7] S.S. Ravi and E.L. Lloyd, On the complexity of near-optimal PLA folding, Tech. Rept. 84-18. 
Department of Computer Science, SUNY at Albany, NY (1984); also: SIAM J. Comput. 17 (1988) 
696-710. 


On the orderability problem for PLA folding 

file:///data/remote/core/dit/data/elsevier/pdf/054/aHR0cDovL2FwaS5lbHNldmllci5jb20vY29udGVudC9hcnRpY2xlL3BpaS8wMTY2MjE4eDg5OTAwMTAz.pdf

On the orderability problem for PLA folding

Abstract

Similar works

Full text

Available Versions

Elsevier - Publisher Connector

Elsevier - Publisher Connector