Skip to main content
SpringerLink
Log in

Heuristically Ordered Search in State Graphs

  • Chapter
  • First Online:
A Guided Tour of Artificial Intelligence Research

  • 921 Accesses

Abstract

Heuristic search in state graphs is one of the oldest topics of artificial intelligence. Initially, the theoretical and practical interest of this research area has been demonstrated by application to solving games such as sliding puzzles. We first present the definitions and properties of the widely used algorithms A* and IDA*, able to find solutions of minimum length, when the length of a solution is simply defined as the sum of the costs of its components. The gradual construction of a minimal solution is guided by evaluation functions that can take into account some pieces of knowledge coming from empirical data; when these functions satisfy some relations, the discovery of a minimal solution is ensured. Several relaxations of these algorithms that produce solutions of ‘almost minimal’ length, or which are less computationally demanding, are discussed. Some extensions, based on other notions of length of a solution and of evaluation functions, are also recalled.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 149.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    The and/or graphs are special cases of hyper-graphs (Berge 1970).

  2. 2.

    Heuristic comes from Greek heuriskein which conveys the ideas of finding, and of what is useful for discovery.

  3. 3.

    Representable by a simple graph rather than an hypergraph (Berge 1970).

  4. 4.

    Other puzzles provide a field of experimentation and demonstration, e.g., the Rubiks cube. Classic problems of graph theory or operational research have been revisited: optimal itinerary calculus, travelling salesman problem. Among other application fields, let us mention: automatic theorem-proving (Kowalski 1970; Chang and Lee 1973; Minker et al. 1973; Dubois et al. 1987), image processing (Martelli 1976), path finding in robotics (Chatila 1982; Gouzènes 1984; Lirov 1987), automatic generation of natural language (Farreny et al. 1984), spelling correctors (Pérennou et al. 1986), breakdown diagnosis (Gonella 1989), dietetics (Buisson 2008).

  5. 5.

    If we associate 0 to the empty cell and if we represent each state by the sequence of the numbers of the 16 cells (for instance: from top to bottom and from left to right) then a problem can be solved if and only if the representation of the state goal is an even permutation of the representation of the initial state.

  6. 6.

    Two nodes of a graph G satisfy the relationship of connectivity if and only if they are connected by at least a chain. It is an equivalence relationship that allows us to distinguish equivalence classes in the set of the nodes of G. If there is only one equivalence class, G is a connected graph. The subgraph of consisting of all nodes of an equivalence class and all edges of G that connect two nodes of the class, is called the connected component of G.

  7. 7.

    For instance, the Moore-Dijkstra algorithm (Dijkstra 1959; Gondran and Minoux 1979).

  8. 8.

    See also Nilsson (1971, 1980, 1998).

  9. 9.

    We reason here in the setting of directed graphs. The model can be applied to undirected graphs: substitute edge to arc, chain to path. The state graph of (\(n^2-1\))-puzzles can be represented by an undirected graph as well than by a directed graph (each edge of the undirected graph corresponds to 2 opposite arcs of the directed graph).

  10. 10.

    In (\(n^2-1\))-puzzles, the cost of an arc (or edge) is commonly considered equal to 1.

  11. 11.

    In (\(n^2-1\))-puzzles, commonly the objective is to minimize the number of moves from s to t.

  12. 12.

    It is supposed that each state has only a finite number of sons, e.g., 4 at most for the (\(n^2-1\))-puzzles.

  13. 13.

    And compared to its generalization called uniform-cost algorithm. Among the nodes, never developed before, the breadth first algorithm favors the development of those nodes reachable from s by a path of which the number of arcs is minimal; the uniform-cost algorithm prefers the nodes reachable from s by a path which the sum of arc costs is minimal. While the breadth first algorithm guarantees to find a path from s to t whose number of arcs is minimal (if such a path exists), the uniform-cost algorithm guarantees to find a path from s to t whose sum of arc costs is minimal (if such a path exists).

  14. 14.

    We shall see that \(GS_d\) is an arborescence, so C will be the single path joining s to t in \(GS_d\).

  15. 15.

    That is to say: coming from experience or observation (without previous theory).

  16. 16.

    For any x, \(GR_x\) is an arborescence with root s, so it exists a single path from s to n in \(GR_x\).

  17. 17.

    Alternatively, we may wish to exploit knowledge about the current state n, not reducible to a function of only n (for example also depending on the progress of the ongoing search).

  18. 18.

    If it exists; else we let: \(h^*(n) = \infty \).

  19. 19.

    We have introduced \(h^*\) before, but in the specific context of the state graphs of (\(n^2 -1\))-puzzles.

  20. 20.

    A better name might be: comparison theorem.

  21. 21.

    The processor, supposed unique, allocates disjointed periods of operation to the two \(A^*\).

  22. 22.

    The first idea is to give control, at each development cycle, to the algorithm which owns a node with better evaluation in its front.

  23. 23.

    Pohl supposes that arc costs are positive, and consequently the functions sh and th.

  24. 24.

    Respecting the 5 conditions introduced in the statement of the admissibility of \(A^*\) algorithms.

  25. 25.

    This algorithm can be denoted \(A^*_\varepsilon \); when \(\varepsilon = 0\), we recognize algorithm \(A^*\).

  26. 26.

    Denoted \(A_\varepsilon \) by the authors; we prefer: \(A_\varepsilon ^\alpha \); when \(\alpha = 0\), the algorithm is \(A_\varepsilon ^*\) ; it is \(A^*\) if \(\alpha = \varepsilon = 0\).

  27. 27.

    Listed in Korf (1985a). Denote them: K4, 1 to K4, 100. The goal node is standard: empty cell to the top on left, then tiles 1 to 15 from left to right and from top to bottom: see state t in Fig. 3.

  28. 28.

    It is supposed that for any state, the number of sons is finite. To each son of a state is assigned a rank of generation.

  29. 29.

    On a Sun Sparc Station.

  30. 30.

    We could resort to \(IDA^*\), with P or better: \(P + LC + C\).

  31. 31.

    This technique uses \(h = \max (h_{1-5}, h_{6-10}, h_{11-15})\) rather than \(h = \underline{h} = \underline{h}_{1-5} + \underline{h}_{6-10} + \underline{h}_{11-15}\).

  32. 32.

    As well as several of the mentioned algorithms (B, C, D . . .) and others not evocated so far, such as: \(BF^*\) (Pearl 1990), \(A^{**}\) (Dechter and Pearl 1985, 1988), SDW (Köll and Kaindl 1992).

  33. 33.

    A path is elementary if it does not include two occurrences of the same node.

  34. 34.

    Path whose final node is identical to the initial node.

References

  • Bagchi A, Mahanti A (1983) Search algorithms under different kinds of heuristics–a comparative study. JACM: J ACM 30(1):1–21

    Article  MathSciNet  MATH  Google Scholar 

  • Barr A, Feigenbaum E, Cohen P (1982) The handbook of artificial intelligence. Morgan Kaufman, Los Altos CA

    MATH  Google Scholar 

  • Berge C (1970) Graphes et Hypergraphes. Dunod, Paris

    MATH  Google Scholar 

  • Brungger A (1997) Solving hard combinatorial optimization problems in parallel : two case studies. PhD thesis, ETH Zurich

    Google Scholar 

  • Buisson J-C (2008) Nutri-educ, a nutrition software application for balancing meals, using fuzzy arithmetic and heuristic search algorithms. Artif Intell Med 42(3):213–227

    Article  MathSciNet  Google Scholar 

  • Chakrabarti PP, Ghose S, Acharya A, de Sarkar SC (1989) Heuristic search in restricted memory. Artif Intell 41(2):197–222

    Article  MathSciNet  MATH  Google Scholar 

  • Chang CL, Lee RCT (1973) Symbolic logic and mechanical theorem proving. Academic Press, New York

    MATH  Google Scholar 

  • Charniak, McDermott (1985) Introduction to artificial intelligence. Addison-Wesley, Reading

    MATH  Google Scholar 

  • Chatila R (1982) Path planning and environment learning in a mobile robot system. In: Proceedings ECAI-82, Orsay, pp 211–215

    Google Scholar 

  • Culberson JC, Schaeffer J (1998) Pattern databases. Comput Intell 14(4):318–334

    Article  MathSciNet  Google Scholar 

  • de Champeaux D, Sint L (1977) An improved bidirectional heuristic search algorithm. JACM 24(2):177–191

    Google Scholar 

  • Dechter R, Pearl J (1985) Generalized best first search strategies and the optimality of A*. JACM 32(3):505–536

    Google Scholar 

  • Dechter R, Pearl J (1988) The optimality of A\(^*\). In: Kanal L, Kumar V (eds) Search in artificial intelligence. Springer, Berlin, pp 166–199

    Chapter  Google Scholar 

  • Delahaye JP (2011) Le Rubik’s Cube : pas plus de 20 mouvements!. Pour la Science 400:98–103

    Google Scholar 

  • Dijkstra EW (1959) A note on two problems in connexion with graphs. Numer Math 1:269–271

    Article  MathSciNet  MATH  Google Scholar 

  • Dubois D, Lang J, Prade H (1987) Theorem proving under uncertainty. A possibility theory-based approach. In: Proceedings of the 10th international joint conference on artificial intelligence, Milan, Italy, pp 984–986

    Google Scholar 

  • Eckerle J, Schuierer S (1995) Efficient memory-limited graph search. Lect Notes Comput Sci 981:101–112

    Article  Google Scholar 

  • Farreny H (1987) Exercices programmés d’intelligence artificielle. Méthode + programmes. Masson

    Google Scholar 

  • Farreny H (1995) Recherche Heuristiquement Ordonnée dans les graphes d’états: algorithmes et propriétés. Manuels informatiques Masson, Masson

    Google Scholar 

  • Farreny H (1997a) Recherche heuristiquement ordonnée: Généralisations compatibles avec la complétude et l’admissibilité. Technique et Science Informatiques 16(7):925–953

    Google Scholar 

  • Farreny H (1997b) Recherche Heuristiquement Ordonnée dans les graphes d’états : élargissement des cas admissibilité et sous-admissibilité. Revue d’Intelligence Artificielle 11(4):407–448

    Google Scholar 

  • Farreny H (1999) Completeness and admissibility for general heuristic search algorithms – a theoretical study: basic concepts and proofs. J Heuristics 5(3):353–376

    Google Scholar 

  • Farreny H, Ghallab M (1987) Éléments d’intelligence artificielle. Hermès

    Google Scholar 

  • Farreny H, Piquet-Gauthier S, Prade H (1984) Méthode de recherche ordonnée pour la désignation d’objets en génération de phrases. In: Proceedings of 6th international congress cybernetics and systems, pp 647–652

    Google Scholar 

  • Felner A, Korf RE, Hanan S (2004) Additive pattern database heuristics. J Artif Intell Res (JAIR) 22:279–318

    Article  MathSciNet  MATH  Google Scholar 

  • Gardner M (1979) Jeux mathématiques du “Scientific american”. CEDIC

    Google Scholar 

  • Ghallab M, Allard DG (1983) \(A_{\varepsilon }\) - an efficient near admissible heuristic search algorithm. In: Bundy A (ed) Proceedings of the 8th international joint conference on artificial intelligence, Karlsruhe, FRG. William Kaufmann, pp 789–791

    Google Scholar 

  • Gondran M, Minoux M (1979) Graphes et algorithmes. Eyrolles, Paris

    MATH  Google Scholar 

  • Gonella R (1989) Diagnostic de pannes sur avions : mise en œuvre d’un raisonnement révisable. PhD thesis, ENSAE, Toulouse, France

    Google Scholar 

  • Gouzènes L (1984) Strategies for solving collision-free trajectories problems for mobile and manipulator robots. Int J Robot Res 3(4):51–65

    Google Scholar 

  • Hansson O, Mayer A, Yung M (1992) Criticizing solutions to relaxed models yields powerful admissible heuristics. Inf Sci 63(3):207–227

    Article  Google Scholar 

  • Harris LR (1974) The heuristic search under conditions of error. Artif Intell 5(3):217–234

    Article  MathSciNet  MATH  Google Scholar 

  • Hart PE, Nilsson NJ, Raphael B (1968) A formal basis for the heuristic determination of minimum cost paths. IEEE Trans Syst Sci Cybern SSC–4(2):100–107

    Article  Google Scholar 

  • Ibaraki T (1978) Branch-and-bound procedure and state-space representation of combinatorial optimization problems. Inf Control 36(1):1–27

    Article  MathSciNet  MATH  Google Scholar 

  • Johnson WW, Storey WE (1879) Notes on the “15” puzzle. Am J Math 2:397–404

    Article  MathSciNet  Google Scholar 

  • Kaindl H, Khorsand A (1994) Memory-bounded bidirectional search. In: Proceedings of the twelfth national conference on artificial intelligence (AAAI-94). AAAI Press, Seattle, Washington, pp 1359–1364

    Google Scholar 

  • Kanal LN, Kumar V (eds) (1988) Search in artificial intelligence. Springer, Berlin

    Google Scholar 

  • Köll AL, Kaindl H (1992) In: Neumann B (ed) A new approach to dynamic weighting. Wiley, Vienna, Austria, pp 16–17

    Google Scholar 

  • Korf RE (1985a) Depth-first iterative deepening: an optimal admissible tree search. Artif Intell 27(1):97–109

    Article  MathSciNet  MATH  Google Scholar 

  • Korf RE (1985b) Iterative-deepening A*: an optimal admissible tree search. In: Proceedings of the ninth international joint conference on artificial intelligence (IJCAI-85), Los Angeles, California. Morgan Kaufmann, pp 1034–1036

    Google Scholar 

  • Korf RE (1993) Linear-space best-first search. Artif Intell 62(1):41–78

    Article  MathSciNet  MATH  Google Scholar 

  • Korf RE (1997) Finding optimal solutions to Rubik’s Cube using pattern databases. In: Proceedings of the 14th national conference on artificial intelligence (AAAI-96), pp 700–705

    Google Scholar 

  • Korf RE (2000) Recent progress in the design and analysis of admissible heuristic functions. In: Proceedings of the 17th national conference on artificial intelligence (AAAI 2000), Austin, Texas, USA. AAAI / MIT Press, pp 1165–1170

    Google Scholar 

  • Korf RE, Felner A (2002) Disjoint pattern database heuristics. AIJ Artif Intell 134(1–2):9–22

    Article  MATH  Google Scholar 

  • Korf RE, Taylor LA (1996) Finding optimal solutions to the twenty-four puzzle. In: Proceedings of the 14th national conference on artificial intelligence (AAAI-96), Portland, Oregon, USA. AAAI Press / The MIT Press, pp 1202–1207

    Google Scholar 

  • Kowalski R (1970) Search strategies for theorem proving. In: Meltzer B, Michie D (eds) Machine intelligence, vol 5. American Elsevier, pp 181–201

    Google Scholar 

  • Kumar V, Kanal LN (1988) The CDP: a unifying formulation for heuristic search, dynamic programming, and branch-and-bound. In: Kanal LN, Kumar V (eds) Search in artificial intelligence. Springer, Berlin, pp 1–27

    MATH  Google Scholar 

  • Laurière J-L (1986) Intelligence Artificielle: résolution de problèmes par l’homme et la machine. Editions Eyrolles, Paris

    Google Scholar 

  • Lirov Y (1987) A locally optimal fast obstacle-avoiding path-planning algorithm. Math Model 9(1):63–68

    Article  MathSciNet  Google Scholar 

  • Manzini G (1995) BIDA*: an improved perimeter search algorithm. Artif Intell 75(2):347–360

    Article  MathSciNet  MATH  Google Scholar 

  • Martelli A (1976) Application of heuristic search methods to edge and contour detection. Commun ACM 19(2):73–83

    Article  MATH  Google Scholar 

  • Martelli A (1977) On the complexity of admissible search algorithms. Artif Intell 8(1):1–13

    Article  MathSciNet  MATH  Google Scholar 

  • Mero L (1984) A heuristic search algorithm with modifiable estimate. Artif Intell 23(1):13–27

    Article  MathSciNet  MATH  Google Scholar 

  • Minker J, Fishman DH, McSkimin JR (1973) The Q* algorithm-A search strategy for a deductive question-answering system. Artif Intell 4(3–4):225–243

    Article  MATH  Google Scholar 

  • Nilsson NJ (1971) Problem solving methods in artificial intelligence. McGraw-Hill, New York

    Google Scholar 

  • Nilsson NJ (1980) Principles of Artificial Intelligence. Tioga, Palo Alto. Principes d’Intelligence Artificielle, Cepadues 1988

    Google Scholar 

  • Nilsson NJ (1998) Artificial intelligence: a new synthesis. Morgan Kaufmann Publishers, San Francisco

    MATH  Google Scholar 

  • Pearl J (1984) Heuristics - intelligent search strategies for computer problem solving. Addison-Wesley series in artificial intelligence. Addison-Wesley. Heuristiques - Stratégies de recherche intelligentes pour la résolution de problèmes par ordinateur, Cepadues 1990

    Google Scholar 

  • Pearl J, Kim JH (1982) Studies in semi-admissible heuristics. IEEE Trans Pattern Anal Mach Intell 4(4):392–399

    Article  MATH  Google Scholar 

  • Pérennou G, Lahens F, Daubèze P, De Calmes M (1986) Rôle et structure du lexique dans le correcteur vortex. In: Séminaire GRECO/GALF Lexiques et traitement automatique des langages. Université Paul Sabatier, Toulouse

    Google Scholar 

  • Pohl I (1971) Bi-directional search. In: Meltzer B, Michie D (eds) Machine intelligence, 6th edn. Edinburgh University Press, Edinburgh, Scotland, pp 127–140

    Google Scholar 

  • Rao NV, Kumar V, Korf RE (1991) Depth-first versus best-first search. In: Dean TL, McKeown K (eds) Proceedings of the 9th national conference on artificial intelligence, Anaheim, CA, USA, 14–19 July 1991, vol 1. AAAI Press / The MIT Press, pp 434–440

    Google Scholar 

  • Ratner D, Warmuth M (1990) The \((n^{2} - 1)\)-puzzle and related relocation problems. JSCOMP J Symb Comput 10:111–137

    Article  MathSciNet  MATH  Google Scholar 

  • Reinefeld A (1993) Complete solution of the eight-puzzle and the benefit of node ordering in \(IDA^*\). In: Proceedings of the 11th international joint conference on artificial intelligence. Chambéry, France, pp 248–253

    Google Scholar 

  • Rich E (1983) Artificial intelligence. McGraw-Hill, New York

    Google Scholar 

  • Rokicki T, Kociemba H, Davidson M, Dethridge J (2010) God’s number is 20. http://www.cube20.org/

  • Russell S (1992) Efficient memory-bounded search methods. In: Neumann B (ed) Proceedings of the 10th European conference on artificial intelligence, Vienna, Austria. Wiley, pp 1–5

    Google Scholar 

  • Russell SJ, Norvig P (2003) Artificial intelligence: a modern approach, 2nd edn. Prentice Hall, Upper Saddle River, NJ

    MATH  Google Scholar 

  • Sen AK, Bagchi A (1989) Fast recursive formulations for best-first search that allow controlled use of memory. In: Sridharan NS (ed) Proceedings of the 11th international joint conference on artificial intelligence, Detroit, MI, USA. Morgan Kaufmann, pp 297–302

    Google Scholar 

  • Shapiro S (1987) Encyclopedia of AI. Wiley-Interscience, Hoboken

    Google Scholar 

  • Shirai Y, Tsujii J (1984) Artificial intelligence: concepts, techniques, and applications. Wiley series in computing. Wiley, New Jersey

    Google Scholar 

  • Winston PH (1984) Artificial intelligence. Addison-Wesley, Reading

    MATH  Google Scholar 

  • Yager RR (1986) Paths of least resistance in possibilistic production systems. Fuzzy Sets Syst 19(2):121–132

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Previously with IRIT-CNRS, Université Paul Sabatier (now retired), Toulouse, France

    Henri Farreny

Authors
  1. Henri Farreny
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Henri Farreny .

Editor information

Editors and Affiliations

  1. CRIL-CNRS, Université d’Artois and Institut Universitaire de France, Lens, France

    Pierre Marquis

  2. Aix Marseille Université, Université de Toulon, CNRS, LIS, Marseille, France

    Odile Papini

  3. IRIT, CNRS and Université Paul Sabatier, Toulouse, France

    Henri Prade

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Farreny, H. (2020). Heuristically Ordered Search in State Graphs. In: Marquis, P., Papini, O., Prade, H. (eds) A Guided Tour of Artificial Intelligence Research. Springer, Cham. https://doi.org/10.1007/978-3-030-06167-8_1

Download citation

Publish with us

Policies and ethics

Search

Navigation