S. Cook and C. Racko. Space lower bounds for maze threadability on restricted machines. SIAM Journal on Computing, 9(3):636-652, 1980.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....that is, one whose basic move is based on the adjacencies of the graph, as opposed to one whose basic move is based on the bits in the graph s encoding. An appropriate structured model for proving such a tradeoff is some variant of the JAG ( jumping automaton for graphs ) of Cook and Rackoff [20]. Such an automaton has a set of states, and a limited supply of pebbles that it can move from vertex to adjacent vertex ( walk ) or directly to a vertex containing another pebble ( jump ) The purpose of its pebbles is to mark certain vertices temporarily, so that they are recognizable when some ....

....when some other pebble reaches them. The pebbles represent vertex names that a structured algorithm might record in its workspace. Walking represents replacing a vertex name by some adjacent vertex found in the input. Jumping represents copying a previously recorded vertex name. Rabin (see [20]) Savitch [31] Blum and Sakoda [13] Blum and Kozen [12] Hemmerling [24] and others have considered similar models; see Hemmerling s monograph for an extensive bibliography (going back over a century) emphasizing results for labyrinths graphs embedded in two or three dimensional ....

[Article contains additional citation context not shown here]

S. A. Cook and C. W. Rackoff. Space lower bounds for maze threadability on restricted machines. SIAM Journal on Computing, 9(3):636--652, Aug. 1980.

....A one way local ordering (1LO) on a graph defines, for each vertex v, a total ordering on the edges leaving v. A two way local ordering (2LO) provides, in addition, an ordering on incoming edges. We showed that the language (FO DTC 1LO) extends the Jumping Automata on Graphs (JAG) model [CR80] to a more robust complexity class that still permits interesting lower bounds on graph reachability. On the other hand, we showed that the language (FO TC 1LO) is strong enough to express a total ordering on the set of vertices reachable from a given vertex. This led us to conjecture that if we ....

S. A. Cook and C. W. Rackoff, "Space Lower Bounds for Maze Threadability of Restricted Machines," SIAM J. Comput., 9(3):636-652, Aug 1980.

....necessary for Go with the winners to work well. Are they also necessary for other local optimization algorithms One approach to answering this positively is to prove a lower bound for optimization in a restricted model that captures the notion of local search , such as a modified JAG model [CR80]. 7 Acknowledgements We would like to thank T. C. Hu for raising the general issue of analyzing search algorithms, and Christos Papadimitriou and Umesh Vazirani for helpful discussions. ....

S. A. Cook and C. W. Rackoff. Space lower bounds for maze threadability on restricted machines. SIAM Journal on Computing, 9(4):635-- 652, 1980.

....have been used to study the time space tradeoff of the element distinctness problem by Borodin et al. BFMadH 87] and Yao [Yao88] See Borodin [Bor93] for a recent survey on time space tradeoffs for various problems on various models. For the st connectivity problem, Cook and Rackoff [CR80] introduce a natural structured model called the Jumping Automaton for Graphs (JAG) Informally, a JAG has a set of pebbles which have the dual purpose of marking the nodes in the input graph and gathering information about the input. The pebbles can move around in the graph by walking along the ....

....powerful enough to express all known deterministic algorithms for graph connectivity problems with at most a polynomial factor blow up in time and logarithmic factor blow up in space. For example, depth and breadth first search can both be translated directly to JAG algorithms. Cook and Rackoff [CR80] show how to simulate Savitch s algorithm using effectively the same time and space resources. There has been much more success in proving lower bounds on this structured model. First, Cook and Rackoff [CR80] prove a space lower bound of Omega Gamma 44 2 n= log log n) which almost matches the ....

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S. A. Cook and C. W. Rackoff. Space lower bounds for maze threadability on restricted machines. SIAM Journal on Computing, 9(3):636--652, August 1980.

.... because it can try all possible choices of midpoints, it can use time exponential in its space bound, or n 2(logn) There are two other algorithms that solve s t connectivity deterministically with essentially the same time and space performance as Savitch s algorithm: Cook and Rackoff [CR80] present an algorithm similar to Savitch s that solves stcon on their more restrictive Jumping Automata on Graphs (JAG) model. Nisan [Nis90] uses pseudorandom generators to derandomize the random walk algorithm of Aleliunas et al. AKL 79] for more on the random walk algorithm, see Section ....

....space Omega# log n) Any nondeterministic Turing machine that recognizes directed s t nonconnectivity using the adjacency matrix or edge list encoding uses space Omega# log n) 16 Some tighter lower bounds are known for s t connectivity on restricted models of computation. Cook and Rackoff [CR80] show an Omega# log 2 n= log log n) space lower bound for stcon on their JAG model, closely matching Savitch s upper bound. Berman and Simon [BS83] extend this result to give a similar lower bound on a randomized version of the JAG. Beame et al. BBR 90] give time space lower bounds for ....

S. A. Cook and C. W. Rackoff. Space lower bounds for maze threadability on restricted machines. SIAM Journal on Computing, 9(3):636--652, August 1980.

....is the s t connectivity problem in directed graphs, see [3, 4, 14] and references therein. Given a directed graph, the problem is to decide whether there exists a path from a distinguished node s to a distinguished node t . Most of the results are developed 3 in the JAG model by Cook and Rackoff [10]. The best time space tradeoffs currently known [4, 14] only imply a polynomial lower bound on the computation time if no upper bounds are imposed in the space used by the computation. Given the current knowledge of the s t connectivity problem it seems unlikely that one can prove ....

S.A. Cook and C.W. Rackoff. Space lower bounds for maze threadability on restricted machines. SIAM Journal on Computing, 9:636--652, 1980.

....Over ordered structures, FO DTC) captures L and (FO TC) captures NL. On the other hand, in the absence of ordering, FO TC) is strictly more powerful than (FO DTC) GM92] An apparently quite different structured model of logspace machines is the Jumping Automaton on Graphs (JAG) [CR80]. We show that the JAG model is intimately related to these logics on locally ordered structures. We argue that the usual JAG model is unreasonably weak and should be replaced, wherever possible, by the two way JAG model, which we define. Furthermore, the language (FO DTC) over two way ....

....(FO DTC COUNT) in which counting quantifiers are present. Attempts to extend this proof to separate the languages with ordering and thus separate L from NL remain unsuccessful. An apparently quite different structured model of logspace machines is the Jumping Automaton on Graphs (JAG) [CR80]. It is known that the JAG model is not powerful enough to search all graphs. This may be considered as some evidence that L 6= NL. Unfortunately, the same proof shows that the JAG is not powerful enough to search all trees, a problem that is easily seen to be in L. Thus, the JAG model, like the ....

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S. A. Cook and C. W. Rackoff, "Space Lower Bounds for Maze Threadability of Restricted Machines," SIAM J. Comput. , 9(3):636-652, Aug 1980.

....the most interesting open problem. There is no fundamental reason why our upper bound is the best possible. We thus hope that this work will spark interest in proving a time space tradeoff for USTCON, even in a restricted model of space bounded computation such as the JAGs of Cook and Rackoff [5]. For a restricted version of the JAG model, Beame et al. 2] have shown that space p implies time #.n 2 = p log n for bounded degree graphs. Acknowledgement We are very grateful to Lyle Ramshaw for a thorough reading of the manuscript and many useful comments and corrections. ....

S. A. Cook and C. W. Rackoff. Space lower bounds for maze threadability on restricted machines. SIAM Journal on Computing, 9(3):636--652, 1980.

....is, the simultaneous time and space requirements of algorithms for directed connectivity. No nontrivial lower bounds are known for general models of computation (such as Turing machines) on either the space, or on the simultaneous space and time required to solve stcon, although Cook and Rackoff [3] and Tompa [8] have obtained lower bounds for restricted models. This paper presents new upper bounds for the problem. The standard algorithms for connectivity, breadth and depth first search, run in optimal time Theta(m n) and use Theta(n log n) space. At the other extreme, Savitch s ....

S. A. Cook and C. W. Rackoff. Space lower bounds for maze threadability on restricted machines. SIAM Journal on Computing, 9(3):636--652, Aug. 1980.

....Portions of this work were performed while the author was at the University of Toronto, Canada. TIME SPACE LOWER BOUNDS 1191 graph s encoding. A natural structured model for the problem of st connectivity is the jumping automaton for graphs, or JAG, introduced by Cook and Racko# [11]. A JAG moves a set of pebbles on the graph. There are two basic operations moving a pebble along a directed edge in the graph and jumping a pebble from its current location to the vertex occupied by another pebble. Although the JAG model is structured, it is not weak. In particular, it is ....

....most known deterministic algorithms for graph connectivity can be implemented on it. Poon [18] introduces the more powerful node named JAG (NNJAG) an extension of the JAG model where the computation is allowed to depend on the names of the nodes on which the pebbles are located. Cook and Racko# [11] prove a lower bound of #(log 2 n log log n) on the space required for a JAG to compute directed st connectivity (stcon) Berman and Simon [8] extend this result to randomized JAGs, and Poon [18] extends it to a probabilistic version of the NNJAG. Tompa [23] shows lower bounds on the product of ....

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<F4.668e+05> S. A. Cook and C. W.<F3.811e+05> Rackoff,<F3.365e+05> Space lower bounds for maze threadability on restricted<F3.811e+05> machines, SIAM J. Comput., 9 (1980), pp. 636--652.

....natural to consider structured computational models [12] whose basic operations are based on the structure of the input, as opposed to being based on the bits in the input s encoding. A natural structured model for stcon is the jumping automaton for graphs, or JAG, introduced by Cook and Racko# [13]. A JAG moves a set of pebbles on the graph. There are two basic operations moving a pebble along a directed edge in the graph and jumping a pebble from its current location to the node occupied by another pebble. Although the JAG model is structured, it is powerful enough to simulate most known ....

....by another pebble. Although the JAG model is structured, it is powerful enough to simulate most known algorithms for stcon and related problems. For example, depth first and breadth first search, random walks [1] and the algorithms of Savitch and Barnes et al. can all be simulated on a JAG (see [13, 27]) To our kowledge, all known deterministic or probabilistic algorithms for directed stcon are implementable on a JAG. However, it is not clear how a nondeterministic JAG can simulate Immerman s and Szelepcsenyi s O(log n) space algorithm for directed st nonconnectivity (stcon) 19, 29] This ....

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<F3.748e+05> S. A. Cook and C. W.<F3.815e+05> Rackoff,<F3.419e+05> Space lower bounds for maze threadability on restricted<F3.815e+05> machines, SIAM J. Comput., 9 (1980), pp. 636--652.

....to date, has produced few non trivial lower bounds on general models of computation. This has led researchers to pursue lower bounds for various weaker models where the behavior of computation is more tenable than on Turing machines. There has been some success, for example, in models such as JAGs [CR80, Poo93], but for more general models things are pretty bleak. The situation is the same in Descriptive Complexity, where the familiar classes of computational complexity have been characterized as exactly the properties expressible in corresponding logical languages over ordered structures (see, e.g. ....

S. A. Cook and C. W. Rackoff. Space lower bounds for maze threadability of restricted machines. SIAM J. Comput., 9 (3):636--652, 1980.

....that is, one whose basic move is based on the adjacencies of the graph, as opposed to one whose basic move is based on the bits in the graph s encoding. An appropriate structured model for proving such a tradeoff is some variant of the JAG ( jumping automaton for graphs ) of Cook and Rackoff [24]. Such an automaton has a set of states, and a limited supply of pebbles that it can move from vertex to adjacent vertex ( walk ) or directly to a vertex containing another pebble ( jump ) The purpose of its pebbles is to mark certain vertices temporarily, so that they are recognizable when some ....

....when some other pebble reaches them. The pebbles represent vertex names that a structured algorithm might record in its workspace. Walking represents replacing a vertex name by some adjacent vertex found in the input. Jumping represents copying a previously recorded vertex name. Rabin (see [24]) Savitch [45] Blum and Sakoda [13] Blum and Kozen [12] Hemmerling [30] and others have considered similar models; see Hemmerling s monograph for an extensive bibliography (going back over a century) emphasizing results for labyrinths graphs embedded in two or three dimensional ....

[Article contains additional citation context not shown here]

S. A. Cook and C. W. Rackoff. Space lower bounds for maze threadability on restricted machines. SIAM Journal on Computing, 9(3):636--652, Aug. 1980.

....is the s t connectivity problem in directed graphs, see [3, 4, 14] and references therein. Given a directed graph, the problem is to decide whether there exists a path from a distinguished node s to a distinguished node t. Most of the results are developed in the JAG model by Cook and Rackoff [10]. The best time space tradeoffs currently known [4, 14] only imply a polynomial lower bound on the computation time if no upper bounds are imposed in the space used by the computation. Given the current knowledge of the s t connectivity problem it seems unlikely that one can prove ....

S.A. Cook and C.W. Rackoff, Space lower bounds for maze threadability on restricted machines, SIAM J. Comput., 9 (1980), pp. 636--652.

....Closure operator [I87] FO DTC ) captures L and (FO TC ) captures NL. On the other hand, in the absence of ordering, FO TC) is strictly more powerful than (FO DTC) GM92] An apparently quite different structured model of logspace machines is the Jumping Automaton on Graphs (JAG) [CR80]. We show that the JAG model is intimately related to these logics on one way locally ordered (1LO) structures. We argue that the usual JAG model is unreasonably weak and should be replaced, wherever possible, by the two way JAG model, which we define. Furthermore, the language (FO DTC 2LO) ....

....(FO DTC COUNT) in which counting quantifiers are present. Attempts to extend this proof to separate the languages with ordering and thus separate L from NL remain unsuccessful. An apparently quite different structured model of logspace machines is the Jumping Automaton on Graphs (JAG) [CR80]. It is known that the JAG model is not powerful enough to search all graphs. This may be considered as some evidence that L 6= NL. Unfortunately, the same proof shows that the JAG is not powerful enough to search all trees, a problem that is easily seen to be in L. Thus, the JAG model, like the ....

[Article contains additional citation context not shown here]

S. A. Cook and C. W. Rackoff, "Space Lower Bounds for Maze Threadability of Restricted Machines," SIAM J. Comput. , 9(3):636-652, Aug 1980.

....the simultaneous time and space requirements of algorithms for directed connectivity. No nontrivial lower bounds are known for general models of computation (such as Turing machines) on either the space, or on the simultaneous space and time required to solve stcon, although Cook and Rackoff [3] and Tompa [8] have obtained lower bounds for restricted models. This paper presents new upper bounds for the problem. The standard algorithms for connectivity, breadth and depth first search, run in optimal time 2(m n) and use 2(n log n) space. At the other extreme, Savitch s Theorem [7] ....

S. A. Cook and C. W. Rackoff. Space lower bounds for maze threadability on restricted machines. SIAM Journal on Computing, 9(3):636--652, Aug. 1980.

....Thus it is natural to consider a structured model [Bo82] whose basic operations are based on the structure of the graph, as opposed to being based on the bits in the graph s encoding. A natural structured model is the JAG ( jumping automaton for graphs ) introduced by Cook and Rackoff [CR80]. It has a set of states and a limited supply of labeled pebbles that it can either move from a node to an adjacent node ( walk ) or move directly to a node containing another pebble ( jump ) The pebbles represent node names that a structured algorithm might record in its workspace and are ....

....it is general enough so that most known algorithms for graph connectivity can be implemented on it. For example, a JAG can perform depth first or breadth first search. It avoids cycling by leaving a pebble on each node when it first visits it. This uses O (n log n) space. Cook and Rackoff [CR80] show that the JAG model is powerful enough to execute an adaptation of Savitch s algorithm [Sa70] for directed st connectivity using only O i log 2 n j space. Poon [Po93a] shows that Barnes et al. BBRS92] sub linear space, polynomial time algorithm for directed s t connectivity runs on a ....

[Article contains additional citation context not shown here]

S. A. Cook and C. W. Rackoff. Space lower bounds for maze threadability on restricted machines. SIAM Journal on Computing, 9(3):636--652, August 1980.

....limited space. Aleliunas et al. 1] proved the existence of polynomial length UTSs, yielding polynomial time and logarithmic space (non uniform) deterministic algorithms for undirected graph traversal. Beame et al. 4] used variants of the JAG ( jumping automaton for graphs ) of Cook and Rackoff [7] to study time space tradeoffs for undirected graph traversal. Good lower bounds on the length of UTSs provide a prerequisite to proving time space tradeoffs for these computational models. The main subject of our study is reflecting sequences on labeled chains, which derives from UTSs on labeled ....

S.A. Cook and C.W. Rackoff. Space lower bounds for maze threadability on restricted machines. SIAM Journal on Computing, 9(3):636--652, 1980.

....to consider structured computational models [Bor82] whose basic operations are based on the structure of the input, as opposed to being based on the bits in the input s encoding. A natural structured model for stcon is the jumping automaton for graphs , or JAG, introduced by Cook and Rackoff [CR80] A JAG moves a set of pebbles on the graph. There are two basic operations moving a pebble along a directed edge in the graph, and jumping a pebble from its current location to the node occupied by another pebble. Although the JAG model is structured, it is powerful enough to simulate most ....

....pebble. Although the JAG model is structured, it is powerful enough to simulate most known algorithms for stcon and related problems. For example, depth first and breadth first search, random walks [AKL 79] and the algorithms of Savitch and Barnes et al. can all be simulated on a JAG, see [CR80] and [Poo96] To the authors knowledge, all known deterministic or probabilistic algorithms for directed stcon are implementable on a JAG. However, it is not clear how a nondeterministic JAG can simulate Immerman s and Szelepcs enyi s O(log n) space algorithm for directed st nonconnectivity ....

[Article contains additional citation context not shown here]

S. A. Cook and C. W. Rackoff. Space lower bounds for maze threadability on restricted machines. SIAM Journal on Computing, 9(3):636--652, August 1980.

....ordering the parity of the number of vertices in a graph is not expressible in (FO TC) In [EI95] we introduced local orderings in graphs as an intermediate step between ordered and unordered graphs. We showed that the language (FO DTC 1LO) extends the Jumping Automata on Graphs (JAG) model [CR80] to a more robust complexity class that This research was supported by NSF grant CCR 9207797. still permits interesting lower bounds on graph reachability. On the other hand, we showed that the language (FO TC 1LO) is strong enough to express a total ordering on the set of vertices reachable ....

....in addition to the one way local ordering, F , on the outgoing edges. There is no assumption about consistency between F and H . In [EI95] it was shown that (FO DTC 1LO) is related to, but strictly more powerful than a well known structured model for space bounded computation called JAGs [CR80] It was there also shown that with (FO TC 1LO) one can express a total ordering on all vertices reachable from a particular vertex. It was then conjectured that (FO TC 1LO COUNT) NL. It will follow from our lower bound in section 4 that this conjecture is false. However, we shall prove ....

S. A. Cook and C. W. Rackoff. Space lower bounds for maze threadability of restricted machines. SIAM J. Comput., 9 (3):636--652, 1980.

....step is to explore time space tradeoffs for the problem: the simultaneous time and space requirements of algorithms for ustcon. Even for this simpler intermediate problem, no nontrivial lower bounds are known for general models of computation (such as Turing machines) although Cook and Rackoff [11] and Beame, et al. 6] have obtained lower bounds for restricted models. This paper presents new upper bounds for the problem. For probabilistic algorithms, much is known about the simultaneous time and space requirements for ustcon. The random walk result of Aleliunas, Karp, Lipton, Lov asz, ....

....applicable, still time optimal, and still require space 2(n log n) in the worst case. Savitch s Theorem [22] provides a deterministic 2(log 2 n) space algorithm for stcon, but it requires time exponential in its space bound: n 2(logn) A similar but more subtle algorithm by Cook and Rackoff [11] for their more restricted JAG model has essentially the same performance. No sublinear space, polynomial time algorithm is known for stcon, and there is evidence suggesting that none is possible. Specifically, Tompa [24] has shown that certain natural approaches to solving stcon admit no such ....

S. A. Cook and C. W. Rackoff. Space lower bounds for maze threadability on restricted machines. SIAM Journal on Computing, 9(3):636--652, Aug. 1980.

.... [RS87] RS93] Rabin proposed the idea of dropping pebbles to mark nodes [Rab67] This suggestion led to work exploring the searching capabilities of a finite automaton supplied with pebbles (e.g. BS77] BK78] Sav72] Cook and Rackoff generalized the idea of pebbles to jumping automata [CR80] However, most previous work has concentrated on learning undirected graphs or graphs with distinguishable nodes. The power behind the two robot model lies in the robots abilities to recognize each other and to move independently. Nonetheless, it is not obvious how to harness this power. If the ....

Stephen A. Cook and Charles W. Rackoff. Space lower bounds for maze threadability on restricted machines. SIAM Journal on Computing, 9:636--652, 1980.

....model or the RAM is beyond the reach of current techniques. Hence, restricted models of computation are considered. One model that has been a successful tool for understanding the complexity of graph connectivity is the jumping automaton for graphs (JAG) model introduced by Cook and Rackoff [CR80]. This model restricts the manner in which it is allowed to access the input and in the type of information about the input that it is allowed to access. In addition, some of its workspace is structured to contain only a certain type of information. As well, its basic operations are based on the ....

....Figure 1.2: The JAG model for undirected graphs v 1 v 3 v 4 v 2 deg = 3 v 4 v 1 v 3 3 Ajacency List Current State names of p nodes deg = 2 deg = 4 deg = 2 : v v v v , v v 1 5 3 4 9 1 v 2 v Figure 1. 3: The structured allocation of the workspace that the JAG models The JAG [CR80] is a finite automaton with p distinguishable pebbles and q states. The space charged to the model is defined to be S = p log 2 n log 2 q. This is because it requires log 2 n bits to store which of the n nodes a pebble is on and log 2 q bits to record the current state. The input to this JAG ....

[Article contains additional citation context not shown here]

S. A. Cook and C. W. Rackoff. Space lower bounds for maze threadability on restricted machines. SIAM Journal on Computing, 9(3):636--652, August 1980.

....s t connectivity is the problem of detecting whether there is a path from a distinguished vertex s to a distinguished vertex t in a directed graph. We prove time space lower bounds of ST = Omega Gamma n 2 = log n) and S 1=2 T = Omega Gamma mn 1=2 ) for Cook and Rackoff s JAG model [8], where n is the number of vertices and m the number of edges in the input graph, and S is the space and T the time used by the JAG. We also prove a timespace lower bound of S 1=3 T = Omega Gamma m 2=3 n 2=3 ) on the more powerful node named JAG model of Poon [13] These bounds approach the ....

....a structured model [4] whose basic operations are based on the structure of the graph, as opposed to being based on the bits in the graph s encoding. A natural structured model for the problem of s t connectivity is the jumping automaton for graphs , or JAG, introduced by Cook and Rackoff [8]. A JAG moves a set of pebbles on the graph. There are two basic operations moving a pebble along a directed edge in the graph, and jumping a pebble from its current location to the vertex occupied by another pebble. Although the JAG model is structured, it is not weak. In particular, it is ....

[Article contains additional citation context not shown here]

S. A. Cook and C. W. Rackoff. Space lower bounds for maze threadability on restricted machines. SIAM Journal on Computing, 9(3):636--652, Aug. 1980.

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S. Cook and C. Racko. Space lower bounds for maze threadability on restricted machines. SIAM Journal on Computing, 9(3):636-652, 1980.

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