623:
can be determined to be true or false for a given implicit graph. Rivest and
Vuillemin proved that any deterministic algorithm for any nontrivial graph property must test a quadratic number of pairs of vertices. The full Aanderaa–Karp–Rosenberg conjecture is that any deterministic algorithm for a monotonic graph property (one that remains true if more edges are added to a graph with the property) must in some cases test every possible pair of vertices. Several cases of the conjecture have been proven to be true—for instance, it is known to be true for graphs with a prime number of vertices—but the full conjecture remains open. Variants of the problem for randomized algorithms and quantum algorithms have also been studied.
82:, one may define an implicit graph in which each vertex represents one of the possible states of the cube, and each edge represents a move from one state to another. It is straightforward to generate the neighbors of any vertex by trying all possible moves in the puzzle and determining the states reached by each of these moves; however, an implicit representation is necessary, as the state space of Rubik's Cube is too large to allow an algorithm to list all of its states.
615:
all graphs in a family. Because of this difference, every graph has an implicit representation. For instance, the rule could be to look up the pair of vertices in a separate adjacency matrix. However, an algorithm that is given as input an implicit graph of this type must operate on it only through the implicit adjacency test, without reference to how the test is implemented.
437:-vertex graphs are together enough to guarantee the existence of an adjacency labeling scheme; this question, which Spinrad restated as a conjecture, remains open. Among the families of graphs which satisfy the conditions of the conjecture and for which there is no known adjacency labeling scheme are the family of disk graphs and line segment intersection graphs.
465:
of polynomial size, the graph consisting of all possible vertex identifiers. Conversely, if an induced universal graph of this type can be constructed, then the identities of its vertices may be used as labels in an adjacency labeling scheme. For this application of implicit graph representations, it
308:
and each vertex of the graph is represented by the numbers of the two endpoints of its corresponding interval. With this representation, one may check whether two vertices are adjacent by comparing the numbers that represent them and verifying that these numbers define overlapping intervals. The same
622:
is the question of whether a graph belongs to a given family of graphs; the answer must remain invariant under any relabeling of the vertices. In this context, the question to be determined is how many pairs of vertices must be tested for adjacency, in the worst case, before the property of interest
614:
concerns implicit graphs given as a set of labeled vertices with a black-box rule for determining whether any two vertices are adjacent. This definition differs from an adjacency labeling scheme in that the rule may be specific to a particular graph rather than being a generic rule that applies to
347:
of a partial order is the minimum number of linear orders whose intersection is the given partial order. If a partial order has bounded order dimension, then an adjacency labeling scheme for the vertices in its comparability graph may be defined by labeling each vertex with its position in each of
172:
in order to prove separations between complexity classes that are stronger than the known separations for non-relativized models. For instance, Childs et al. used neighborhood representations of implicit graphs to define a graph traversal problem that can be solved in polynomial time on a
266:-tuple of its own number and the numbers of its neighbors. Two vertices are adjacent when the first numbers in their identifiers appear later in the other vertex's identifier. More generally, the same approach can be used to provide an implicit representation for graphs with bounded
533:
vertices. In particular, planar graphs have arboricity at most three, so they have universal graphs with a nearly-cubic number of vertices. This bound was improved by
Gavoille and Labourel who showed that planar graphs and minor-closed graph families have a labeling scheme with
325:. However, a geometric intersection graph representation does not always imply the existence of an adjacency labeling scheme, because it may require more than a logarithmic number of bits to specify each geometric object. For instance, representing a graph as a
426:-vertex graphs but does not have an adjacency labeling scheme, because one could transform any given graph into a larger graph in this family by adding a new isolated vertex for each edge, without changing its labelability. Kannan et al. asked whether having a
466:
is important that the labels use as few bits as possible, because the number of bits in the labels translates directly into the number of vertices in the induced universal graph. Alstrup and Rauhe showed that any tree has an adjacency labeling scheme with
348:
the defining linear orders, and determining that two vertices are adjacent if each corresponding pair of numbers in their labels has the same order relation as each other pair. In particular, this allows for an adjacency labeling scheme for the
419:, do not have adjacency labeling schemes. However, even families of graphs in which the number of graphs in the family is small might not have an adjacency labeling scheme; for instance, the family of graphs with fewer edges than vertices has
157:(the class of problems that may be characterized by reachability in implicit graphs with polynomial-length bitstrings). In this complexity-theoretic context, the vertices of an implicit graph may represent the states of a
232:: it is straightforward to check whether two vertices are adjacent but finding the neighbors of any vertex may involve looping through all vertices and testing which ones are neighbors.
1148:
Curtis, Andrew R.; Izurieta, Clemente; Joeris, Benson; Lundberg, Scott; McConnell, Ross M. (2010), "An implicit representation of chordal comparability graphs in linear time",
1393:
869:
Childs, Andrew M.; Cleve, Richard; Deotto, Enrico; Farhi, Edward; Gutmann, Sam; Spielman, Daniel A. (2003), "Exponential algorithmic speedup by a quantum walk",
1414:
1310:
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576:
bits per label. The bound for planar graphs was improved again by Bonamy, Gavoille, and
Piliczuk who showed that planar graphs have a labelling scheme with
611:
161:, and the edges may represent possible state transitions, but implicit graphs may also be used to represent many other types of combinatorial structure.
384:
Not all graph families have local structures. For some families, a simple counting argument proves that adjacency labeling schemes do not exist: only
1186:
378:
747:
93:
have been defined in connection with implicit graphs, defined as above by a rule or algorithm for listing the neighbors of a vertex. For instance,
755:
105:
algorithm for listing the neighbors of any vertex) and a vertex of odd degree in the graph, and must find a second vertex of odd degree. By the
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from graphs that are very far from being acyclic. In contrast, such a fast time is not possible in neighborhood-based implicit graph models,
958:
626:
Bender and Ron have shown that, in the same model used for the evasiveness conjecture, it is possible in only constant time to distinguish
137:
of one vertex to another in an implicit graph may also be used to characterize space-bounded nondeterministic complexity classes including
1309:; Esperet, Louis; Joret, Gwenaël; Gavoille, Cyril; Micek, Piotr; Morin, Pat (2020), "Adjacency Labelling for Planar Graphs (and Beyond)",
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1353:
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870:
427:
78:, in that it provides easy access to the neighbors of each vertex. For instance, in searching for a solution to a puzzle such as
304:. It may be given an adjacency labeling scheme in which the points that are endpoints of line segments are numbered from 1 to 2
158:
86:
1482:
279:
70:
which are described in terms of graphs. In this context, an implicit graph may be defined as a set of rules to define all
1409:
43:
71:
399:
bits may be used to represent an entire graph, so a representation of this type can only exist when the number of
343:
has a vertex for each set element and an edge between two set elements that are related by the partial order. The
318:
271:
224:) that takes as input two vertex identifiers and determines whether or not they are the endpoints of an edge in
141:(the class of problems that may be characterized by reachability in implicit directed graphs whose vertices are
731:
The standard 3Ă—3Ă—3 Rubik's Cube contains 4.3252 Ă— 10 states, and is too large to search exhaustively.
126:
97:
is the class of problems in which one is given as input an undirected implicit graph (in which vertices are
1358:
803:
743:
627:
1271:
Bonamy, Marthe; Gavoille, Cyril; Pilipczuk, Michał (2020), "Shorter
Labeling Schemes for Planar Graphs",
1215:
1477:
772:
340:
47:
1103:
Ma, Tze Heng; Spinrad, Jeremy P. (1991), "Cycle-free partial orders and chordal comparability graphs",
1363:
1079:
416:
372:
352:
336:
55:
50:
or edges are not represented as explicit objects in a computer's memory, but rather are determined
165:, another complexity class, captures the complexity of finding local optima in an implicit graph.
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1207:
1130:
902:
876:
841:
829:
700:
675:
515:
480:
293:
1387:
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809:
586:
bits per label. Finally
Dujmović et al showed that planar graphs have a labelling scheme with
118:
106:
1043:
799:
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344:
326:
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31:
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for any specified vertex. This type of implicit graph representation is analogous to an
1348:
1240:
994:
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309:
approach works for other geometric intersection graphs including the graphs of bounded
289:
169:
122:
75:
768:
719:
1471:
1445:, Lecture Notes in Comput. Sci., vol. 1853, Berlin: Springer, pp. 809–820,
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953:
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349:
1380:
748:"On the complexity of the parity argument and other inefficient proofs of existence"
704:
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275:
134:
906:
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855:
1241:"Shorter Implicit Representation for Planar Graphs and Bounded Treewidth Graphs"
1195:
Proceedings of the 43rd Annual IEEE Symposium on
Foundations of Computer Science
114:
1344:
1203:
1162:
999:"Planar orientations with low out-degree and compaction of adjacency matrices"
492:
267:
185:
In the context of efficient representations of graphs, J. H. Muller defined a
1450:
949:
688:
554:
329:
may require exponentially many bits for the coordinates of the disk centers.
301:
51:
1351:(1975), "A generalization and proof of the Aanderaa-Rosenberg conjecture",
1187:"Small induced-universal graphs and compact implicit graph representations"
872:
Proceedings of the Thirty-Fifth Annual ACM Symposium on Theory of
Computing
411:. Graph families that have larger numbers of graphs than this, such as the
1372:
890:
113:, but the problems that can be defined in this way may not necessarily be
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881:
673:
Korf, Richard E. (2008), "Linear-time disk-based implicit graph search",
310:
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but that requires exponential time to solve on any classical computer.
154:
1412:; Sturtevant, Dean (1983), "A topological approach to evasiveness",
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1441:(2000), "Testing acyclicity of directed graphs in sublinear time",
1321:
1283:
808:, Graduate Texts in Computer Science, Springer-Verlag, p. 48,
228:. That is, this type of implicit representation is analogous to an
846:
1274:
Proceedings of the 2020 ACM-SIAM Symposium on
Discrete Algorithms
1418:, Los Alamitos, CA, USA: IEEE Computer Society, pp. 31–33,
1249:
Proceedings of the 15th annual
European Symposium on Algorithms
1042:
Spinrad, Jeremy P. (2003), "2. Implicit graph representation",
1312:
61st IEEE Annual
Symposium on Foundations of Computer Science
832:(2009), "Equilibria, fixed points, and complexity classes",
355:, which come from partial orders of dimension at most four.
491:
bits per label, from which it follows that any graph with
1357:, Albuquerque, New Mexico, United States, pp. 6–11,
235:
Graph families with adjacency labeling schemes include:
168:
Implicit graph models have also been used as a form of
720:"Minimizing disk I/O in two-bit breadth-first search"
66:
The notion of an implicit graph is common in various
109:, such a vertex exists; finding one is a problem in
1443:Automata, languages and programming (Geneva, 2000)
216:, together with an algorithm (that may depend on
153:(the analogous class for undirected graphs), and
1072:Sphere and dot product representations of graphs
727:Proc. 23rd AAAI Conf. on Artificial Intelligence
317:, and subfamilies of these families such as the
927:, Ph.D. thesis, Georgia Institute of Technology
129:because it contains the problem of computing a
117:, as it is unknown whether PPA = NP.
1354:Proc. 7th ACM Symposium on Theory of Computing
1048:, American Mathematical Soc., pp. 17–30,
956:(1992), "Implicit representation of graphs",
596:bits per label giving a universal graph with
441:Labeling schemes and induced universal graphs
8:
1415:Symposium on Foundations of Computer Science
553:bits per label, and that graphs of bounded
449:has an adjacency labeling scheme, then the
258:and let the identifier for a vertex be the
220:but is independent of the individual graph
1392:: CS1 maint: location missing publisher (
1239:Arnaud, Labourel; Gavoille, Cyril (2007),
1037:
1035:
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1031:
526:bits per label and a universal graph with
250:neighbors, one may number the vertices of
121:is an analogous class defined on implicit
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918:
916:
1185:Alstrup, Stephen; Rauhe, Theis (2002),
756:Journal of Computer and System Sciences
665:
379:(more unsolved problems in mathematics)
1385:
1069:Kang, Ross J.; MĂĽller, Tobias (2011),
800:"Exercise 3.7 (Everything is a Graph)"
943:
941:
939:
937:
935:
54:from some other input, for example a
7:
959:SIAM Journal on Discrete Mathematics
332:Low-dimensional comparability graphs
201:of graphs to be an assignment of an
428:forbidden subgraph characterization
403:-vertex graphs in the given family
612:Aanderaa–Karp–Rosenberg conjecture
212:-bit identifier to each vertex of
25:
875:, New York: ACM, pp. 59–68,
925:Local structure in graph classes
375:have an implicit representation?
125:that has attracted attention in
1045:Efficient Graph Representations
366:Unsolved problem in mathematics
159:nondeterministic Turing machine
87:computational complexity theory
1:
769:10.1016/S0022-0000(05)80063-7
360:The implicit graph conjecture
36:implicit graph representation
27:Algorithmically defined graph
1331:10.1007/978-3-540-75520-3_52
1293:10.1007/978-3-540-75520-3_52
1258:10.1007/978-3-540-75520-3_52
1150:Discrete Applied Mathematics
1021:10.1016/0304-3975(91)90020-3
1007:Theoretical Computer Science
923:Muller, John Harold (1988),
856:10.1016/j.cosrev.2009.03.004
557:have a labeling scheme with
101:-bit binary strings, with a
62:Neighborhood representations
373:hereditary family of graphs
1499:
371:Does every slowly-growing
319:distance-hereditary graphs
181:Adjacency labeling schemes
1204:10.1109/SFCS.2002.1181882
1163:10.1016/j.dam.2010.01.005
718:Korf, Richard E. (2008),
280:minor-closed graph family
191:adjacency labeling scheme
133:. The problem of testing
18:Implicit graph conjecture
1451:10.1007/3-540-45022-X_68
652:, an implicit model for
642:, an implicit model for
834:Computer Science Review
744:Papadimitriou, Christos
689:10.1145/1455248.1455250
683:(6), Article 26, 40pp,
628:directed acyclic graphs
127:algorithmic game theory
805:Descriptive Complexity
457:may be represented as
278:and the graphs in any
1483:Graph data structures
1373:10.1145/800116.803747
891:10.1145/780542.780552
341:partially ordered set
239:Bounded degree graphs
1437:Bender, Michael A.;
1315:, pp. 577–588,
1277:, pp. 446–462,
1252:, pp. 582–593,
729:, pp. 317–324,
461:of a common induced
430:and having at most
417:triangle-free graphs
353:comparability graphs
1424:10.1109/SFCS.1983.4
830:Yannakakis, Mihalis
337:comparability graph
285:Intersection graphs
242:If every vertex in
56:computable function
1198:, pp. 53–62,
1119:10.1007/BF00385814
676:Journal of the ACM
498:has a scheme with
453:-vertex graphs in
445:If a graph family
294:intersection graph
197:in a given family
149:-bit bitstrings),
91:complexity classes
1345:Rivest, Ronald L.
948:Kannan, Sampath;
815:978-0-387-98600-5
459:induced subgraphs
107:handshaking lemma
68:search algorithms
16:(Redirected from
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38:(or more simply
32:graph algorithms
30:In the study of
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640:Black box group
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1014:(2): 243–266,
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966:(4): 596–603,
954:Rudich, Steven
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796:Immerman, Neil
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763:(3): 498–532,
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1055:0-8218-2815-0
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287:
284:
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276:planar graphs
273:
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1478:Graph theory
1442:
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1408:Kahn, Jeff;
1403:
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1080:the original
1071:
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963:
957:
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773:the original
760:
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726:
713:
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135:reachability
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80:Rubik's Cube
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606:Evasiveness
588:(1+o(1))log
407:is at most
270:or bounded
115:NP-complete
1472:Categories
1322:2003.04280
1284:1908.03341
1225:2011-07-13
1089:2011-07-12
950:Naor, Moni
782:2011-07-12
661:References
656:algorithms
646:algorithms
602:vertices.
493:arboricity
272:degeneracy
268:arboricity
254:from 1 to
89:, several
1439:Ron, Dana
1359:CiteSeerX
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847:0802.2831
570:(log log
555:treewidth
547:(log log
302:real line
72:neighbors
1388:citation
1381:16220596
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798:(1999),
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705:13969607
634:See also
323:cographs
313:and the
311:boxicity
48:vertices
1459:1795937
1212:1820524
1172:2602811
1127:1129614
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