Ivan Hal Sudborough

On the Tape Complexity of Deterministic Context-Free Languages

Let DSPACE(L(n)) denote the family of languages recognized by deterministic L(n)-tape bounded Turmg machines The pnnopal result described m this paper is the equivalence of the following statements (l) The determtmsttc context-free language L~ 2) (described m the paper) is m DSPACE(Iog(n)) (2) The simple LL(I) languages are m DSPACE(tog(n)) (3) The simple precedence languages are in DSPACE(Iog(n)). (4) DSPACE(Iog(n)) is identical to the famdy of languages recogmzed by deterministic two-way multlhead pushdown automata m polynomml tmae These results are obtained by constructing a determlmstlc context-free language L~ 2~ which is log(n)-complete for the family of determlmstlc context-free languages In other words, a tape hardest deterministic context-free language is described The best upper bound known on the tape complexity of (deterministic) context-free languages is (log(n)) 2

The vertex separation and search number of a graph

Abstract We relate two concepts in graph theory and algorithmic complexity, namely the search number and the vertex separation of a graph. Let s ( G ) denote the search number and vs ( G ) denote the vertex separation of a connected, undirected graph G . We show that vs ( G ) ≤ s ( G ) ≤ vs ( G ) + 2 and we give a simple transformation from G to G′ such that vs ( G′ ) = s ( G ). We characterize those trees having a given vertex separation and describe the smallest such trees. We also note that there exist trees for which the difference between search number and vertex separation is indeed 2. We give algorithms that, for any tree T , compute vs ( T ) in linear time and compute an optimal layout with respect to vertex separation in time O ( n log n ). Vertex separation has previously been related to progressive black / white pebble demand and has been shown to be identical to a variant of search number, node search number , and to path width , which has been related to gate matrix layout cost . All these properties are known to be computationally intractable. For fixed k , an O ( n log 2 n ) algorithm is known which decides whether a graph has path width at most k .

Improved dynamic programming algorithms for bandwidth minimization and the MinCut Linear Arrangement problem

The dynamic programming algorithm of J. B. Saxe (SIAM J. Algebraic Discrete Methods (1980)) for the bandwidth minimization problem is improved. It is shown that, for all k > 1, BANDWIDTH(k) can be solved in O(nk) steps and simultaneous O(nk) space, where n is the number of vertices in the graph, and that each such problem is in NSPACE(log n). The same improved dynamic programming algorithm approach works to show that the MINCUT LINEAR ARRANGEMENT problem restricted to the fixed value k, denoted by MINCUT(k), is solvable in O(nk) steps and simultaneous O(nk) space and is in the class NSPACE(log n).

A Note on Tape-Bounded Complexity Classes and Linear Context-Free languages

Let LINEAR denote the family of hnear context-free languages and DSPACE(L(n)) [NSPACE(L(n))] denote the family of languages recognized by determmmtm [nondetermimstie] off-line L(n)-tape bounded Turmg machines. The equivalence of the following statements, for 0 g ~ < 1, m shown by describing a log(n)-complete hnear language. (1) LINEAR C DSPACE(logl+(n)). (2) The hnear context-free language L(B1) ~s in DSPACE(logl+(n)). (3) NSPACE(L(n)) C DSPACE([L(n)]~+), for all L(n) > log(n). All the above statements are known to be true when e = 1

Minimizing Width in Linear Layouts

A (linear) layout of an undirected graph G is a one-to-one function mapping the vertices of G to integers. The cutwidth of G under a linear layout L, denoted by cw(G,L), is the maximum, taken over all possible i, of the number of edges connecting vertices assigned to integers less than i to vertices assigned to integers at least as large as i. The cutwidth of a graph G, denoted by cw(G), is the minimum of cw(G,L), taken over all possible linear layouts L. The problem of determining the cutwidth of a graph, called the Min Cut Linear Arrangement problem, has applications in VLSI, for example in the minimization of interconnection channels in Weinberger arrays [16].

Complexity and decidability for chain code picture languages

Abstract A picture description is a word over the alphabet &{u, d, r, l} , where u means “go one unit line up from the current point”, and d , r , and l are interpreted analogously with down, right, and left instead of up. By this, a picture description describes a walk in the plane—its trace is the picture it describes. A set of picture descriptions describes a (chain code) picture language. This paper investigates complexity and decidability questions for these picture languages. Thus it is shown that the membership problem is NP-complete for regular picture languages (i.e., picture languages described by regular languages of picture descriptions), and that it is undecidable whether two regular picture description languages describe a picture in common. After this we investigate so-called stripe picture languages (all pictures are within a stripe defined by two parallel lines), providing ‘better’ complexity and decidability results: Membership is decidable in linear time for regular stripe picture languages. Emptiness of intersection and equivalence is decidable for regular stripe picture languages.

Time and tape bounded auxiliary pushdown automata

We consider language families defined by nondeterministic and deterministic log(n)-tape bounded auxiliary pushdown automata within polynomial time. It is known that these families are precisely the set of languages which are (many-one) log tape reducible to context-free languages and deterministic context-free languages, respectively. The results described here relate questions concerning these classes to other complexity classes and to questions concerning the tape complexity of context-free languages, resolution based proof procedures, solvable path systems, and deterministic context-free languages.

Polynomial time algorithms for the min cut problem on degree restricted trees

Polynomial algorithms are described that solve the MIN CUT LINEAR ARRANGEMENT problem on degree restricted trees. For example, the cutwidth or folding number of an arbitrary degree d tree can be found in O(n(logn)d-2) steps. This also yields an algorithm for determining the black/white pebble demand of degree three trees. A forbidden subgraph characterization is given for degree three trees having cutwidth k. This yields an interesting corollary: for degree three trees, cutwidth is identical to search number.

Efficient algorithms for path system problems and applications to alternating and time-space complexity classes

Let SPS(f(n)) denote the solvable path system problem for path systems of bandwidth f(n) and SPS (f(n)) the corresponding problem for monotone systems. Let DTISP (poly, f(n)) denote the polynomial time and simultaneous f(n) space class and SC = UkDTISP (poly, logkn). Let ASPACE (f(n)) denote the sets accepted by f(n) space bounded alternating TMs and ASPACE (f(n)) the corresponding one-way TM family. Then, for well-behaved functions fεO(n)-o(log n), (1) SPS (f(n)) is ≤log-complete for DTISP (poly, f(n)), (2) {SPS(f(n)k)}k≥1 is ≤log-complete for ASPACE (logf(n)), (3) {SPS (f(n)k)}k≥1 is ≤log-complete for ASPACE (log f(n)), (4) SPS(f(n)) ε DSPACE(f(n) × log n), (5) ASPACE(log f(n)) ⊆ UkDSPACE(f(n)k), and (6) SC = CLOSURE ≤log(ASPACE(log log n)).

Bandwidth and pebbling

The main results of this paper establish relationships between the bandwidth of a graphG — which is the minimum over all layouts ofG in a line of the maximum distance between images of adjacent vertices ofG — and the ease of playing various pebble games onG. Three pebble games on graphs are considered: the well-known computational pebble game, the “progressive” (i.e., no recomputation allowed) version of the computational pebble game, both of which are played on directed acyclic graphs, and the quite different “breadth-first” pebble game, that is played on undirected graphs. We consider two costs of a play of a pebble game: the minimum number of pebbles needed to play the game on the graphG, and the maximumlifetime of any pebble in the game, i.e., the maximum number of moves that any pebble spends on the graph. The first set of results of the paper prove that the minimum lifetime cost of a play of either of the second two pebble games on a graphG is precisely the bandwidth ofG. The second set of results establish bounds on the pebble demand of all three pebble games in terms of the bandwidth of the graph being pebbled; for instance, the number of pebbles needed to pebble a graphG of bandwidthk is at most min (2k2+k+1, 2k log2|G|); and, in addition, there are bandwidth-k graphs that require 3k−1 pebbles. The third set of results relate the difficulty of deciding the cost of playing a pebble game on a given input graphG to the bandwidth ofG; for instance, the Pebble Demand problem forn-vertex graphs of bandwidthf(n) is in the class NSPACE (f(n) log2n); and the Optimal Lifetime Problem for either of the second two pebble games is NP-complete.ZusammenfassungDie Hauptergebnisse dieser Arbeit ergeben Beziehungen zwischen der „Bandwidth” eines GraphenG — die das Minimum ist, über alle Projektionen vonG auf eine Linie, von dem maximalen Abstand zwischen Bildern benachbarter Knoten vonG — und der Leichtigkeit, verschiedene „Pebble Games” aufG zu spielen. Es werden drei Pebble Games auf Graphen betrachtet: das wohlbekannte „computational” Pebble Game, die „progressive” (d. h. keine Wiederberechnung erlaubt) Version des computational Pebble Game, von denen beide auf directed acyclic Graphen gespielt werden, und das ziemlich verschiedene „breadth-first” Pebble Game, das auf undirected Graphen gespielt wird. Wir betrachten zwei verschiedene Kosten für das Pebble Game: die minimale Anzahl von Pebbles, die man braucht, um das Pebble Game auf einem GraphenG zu spielen, und die maximaleLebensdauer eines Pebble in einem Spiel, d. h. die maximale Anzahl von Zügen während denen ein Pebble auf dem Graphen verweilt. Die erste Gruppe von Hauptergebnissen in dieser Arbeit zeigt, daß die minimalen Lebensdauer-Kosten eines Spielverlaufs in einem der beiden letzten Pebble Games auf einem Graphen genau die Bandwidth vonG ist. Die zweite Gruppe von Ergebnissen stellt obere Schranken auf für die Anzahl von benötigten Pebbles in Abhängigkeit von der Bandwidth des betrachteten Graphen, z. B. um einen GraphenG mit Bandwidthk zu pebblen, braucht man höchstens min (2k2+k+1, 2klog2|G|) Pebbles; ferner gibt es GraphenG von Bandwidthk für die man 3k−1 Pebbles braucht. Die dritte Gruppe von Ergebnissen setzt die Schwierigkeit, die Kosten eines Pebble Game auf einem gegebenen input-GraphenG festzustellen, in Beziehung zur Bandwidth vonG, z.B. das „Pebble Demand Problem” für Graphen mitn vertices von Bandwidthf(n) ist in der Klasse NSPACE (f(n)log2n); und das „Optimal Lifetime Problem” ist für jedes der beiden letzten Pebble Games NP-vollständig.