DOCSLIB.ORG

Multiway Range Trees: Scalable IP Lookup with Fast Updates

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Multiway Range Trees: Scalable IP Lookup with Fast Updates Multiway Range Trees: Scalable IP Lookup with Fast Updates Subhash Suri George Varghese Priyank Ramesh Warkhede Department of Computer Science Washington University St. Louis, MO 63130. Abstract—In this paper, we introduce a new IP lookup scheme with A. Our Contribution log n n worst-case search and update time of O , where is the number of prefixes in the forwarding table. Our scheme is based on a new data structure, a multiway range tree. While existing lookup schemes are good We introduce a new IP lookup scheme whose worst-case log n n for IPv4, they do not scale well in both lookup speed and update costs search and update time is O , where is the number when addresses grow longer as in the IPv6 proposal. Thus our lookup of prefixes in the forwarding table. Our scheme is based on a scheme is the first lookup scheme to offer fast lookups and updates for new data structure, a multiway range tree, which achieves the IPv6 while remaining competitive for IPv4. optimal lookup time of binary search, but can also be updated The possibility of a global Internet with addressable appli- fast when a prefix is added or deleted. (By contrast, ordinary ances has motivated a proposal to move from the older In- binary search [5] is a precomputation-based scheme whose n ternet Ipv4 routing protocol with 32 bit addresses to a new worst-case update time is .) protocol, IPv6 with 128 bit addresses. While there is uncer- Our main contribution is to modify binary search for tainty about IPv6, any future Internet protocol will require at prefixes to provide fast update times. In standard binary least 64 bit, and possibly even 128 bit addresses as in IPv6. search [5], each prefix maps to an address range. A set of n Thus for this paper we simply use IPv6 and 128 bits to refer n prefixes partition the address line into at most inter- to the next generation IP protocol and its address length. vals. We build a tree, whose leaves correspond to interval Our paper deals with address lookup, a key bottleneck endpoints. All packet headers mapped to an interval have the log n for Internet routers. Our paper describes the first IP lookup same longest prefix match. Search time is O . scheme that scales well as address lengths increase as in IPv6, The problem is update: a prefix range can be split into n while providing fast search times and fast update times. intervals, which need to be updated when the prefix is There are four requirements for a good lookup scheme: added or deleted. Thus our first idea is associating an address v search time, memory, scalability, and update time. Search span with every tree node v ; the span of is the maximal time is important for lookup to not be a bottleneck. Schemes range of addresses into which the final search can fall after that are memory-efficient lead to faster search because com- reaching v . Thus the root span is the entire address range. pact data structures fit in fast but expensive Static RAM. Scal- We then associate with every node v the set of prefixes that ability in both number of prefixes and address length is im- contain the span of v . However, this results in storing redun- portant because prefix databases are growing, and a switch to dant prefixes. IPv6 will increase address prefix lengths. Finally, schemes To remove this redundancy, we store a prefix with a node v with fast update time are desirable because i) instabilities in if and only if the same prefix is not stored at the parent w of w backbone protocols [4] can cause rapid insertion and deletion the node. Since the span of v is a subrange of the span of , of prefixes. ii) multicast forwarding support requires route any prefix that contains the span of w , will contain the span entries to be added as part of the forwarding process and not of v . Using this compression rule we can prove that only a by a separate routing protocol iii) Any lookup scheme that small number of prefixes need be precomputed and stored; log n does not support fast incremental updates will require two thus updates change information only at O nodes. By copies of the lookup structure in fast memory, one for up- increasing the arity of the tree, we can reduce the height of log n d d dates and one for lookup. the tree to O , for any integer . We pick so that the d The main drawback of existing lookup schemes is their entire node can fit in one cache line. For instance, a 256 bit lack of both scalability and fast update. Current IP lookup cache line and arity-14 yields a tree height of about 5 or 6 for schemes come in two categories: those that use precomputa- the largest IPv4 databases. tion, and those that do not. Schemes like “binary search on Our third contribution is to extend multiway range trees prefix lengths” [9], “Lulea compressed tries” [1], or “binary to IPv6. On a 32-bit machine, a single 128-bit address will search on intervals” [5] perform precomputation to speed up require 4 memory accesses. Thus, an IP lookup scheme de- search; however, adding a prefix can cause the data struc- signed for IPv4 could slow down by a factor of 4 when used ture to be rebuilt. Thus, precomputation-based schemes have for IPv6. In practice, since we are assuming wide memories n a worst-case update time of . On the other hand, trie- of at least 128 bits, we can handle such wide prefixes in one based schemes (e.g., [8], [7]) do not use precomputation; memory access. Unfortunately, the use of 128 bit addresses however, their search time grows linearly with the prefix result in the use of a small branching factor and hence large length. tree heights. To avoid this problem, we generalize our scheme to the case of k -word prefixes, giving a worst-case search and with 32K additional 32-bit prefixes described by the sequence k log n n updates times of O , where is the total number . d k xy of prefixes present in the database. Notice that the factor of Thus all IP addresses of the form where the is additive. LSB of y will have as their best matching pre- The rest of this paper is organized as follows. Section I fix. The binary search method [5] handles this by setting match p describes basic binary search. Section II describes our main for all the 32K prefixes of length 32. data structure, the multiway range tree. Section III briefly Now consider adding the prefix Since this new pre- xy describes how to handle updates efficiently. Section IV de- fix is now a better match for prefixes of the form , match p scribes our extension to long addresses. Section V describes we have to update for 32K values of our experiments and measurements. Section VI states our p. Thus update can take linear time. We avoid this problem conclusions. using the following ideas. I. IP LOOKUP BY BINARY OR MULTIWAY SEARCH A. Address Spans We first review binary search [5] prefix matching. Suppose Our key idea is to associate address spans with nodes and r r r m keys of the tree; intuitively, the address span of a node is the m the database consists of prefixes . Each pre- b e fix r matches a contiguous interval of addresses. For range of addresses that can be reached through the node. For instance, the interval of prefix r begins at point the multiway tree in Figure 1, first consider a leaf node, such z x x and ends at . View the IP address as . For each key in it, we define the span of to be space as a line in which each prefix maps to a contiguous in- the range from its predecessor key to the key itself. (When terval, and each destination address is a point. Prefix intervals the predecessor key doesn’t exist, we use the artificial guard are disjoint but one prefix can completely contain another. key .) To ensure that spans are disjoint, each span in- The longest matching prefix problem reduces to determining cludes the right endpoint of its range, but not the left end- c spanaa the shortest interval containing a query point. point. Thus, the spans of a b and are , ba b spancb c span , . p p p n m n Let , where , denote the distinct end- points of all the prefix intervals, sorted in ascending order. The span of a leaf node is defined to be the union of the spanz c match p p spans of all the keys in it. Thus, we have . i With each key i , we store two prefixes, and h p match p matc The span of a non-leaf node is defined as the union of the i i . The first, , is the longest prefix that match p p spans of all the descendants of the node. Thus, for in- i begins or ends at i . The second, , is the longest spanv i spanw i o p p stance, , and , and i prefix whose interval includes the range i , where spanuo p p . (In fact, it is easy to check that nodes at i i is the predecessor key of . the same level in the tree form a partition of the total address Given a destination address q , we perform binary search span defined by all the input prefixes.) We now describe how n q in log time to determine the successor of , which is the to associate prefixes with nodes of the tree.
Load more

Recommended publications
  • A Call-By-Need Lambda Calculus
    A CallByNeed Lamb da Calculus Zena M Ariola Matthias Fel leisen Computer Information Science Department Department of Computer Science University of Oregon Rice University Eugene Oregon Houston Texas John Maraist and Martin Odersky Philip Wad ler Institut fur Programmstrukturen Department of Computing Science Universitat Karlsruhe UniversityofGlasgow Karlsruhe Germany Glasgow Scotland Abstract arguments value when it is rst evaluated More pre cisely lazy languages only reduce an argument if the The mismatchbetween the op erational semantics of the value of the corresp onding formal parameter is needed lamb da calculus and the actual b ehavior of implemen for the evaluation of the pro cedure b o dy Moreover af tations is a ma jor obstacle for compiler writers They ter reducing the argument the evaluator will remember cannot explain the b ehavior of their evaluator in terms the resulting value for future references to that formal of source level syntax and they cannot easily com parameter This technique of evaluating pro cedure pa pare distinct implementations of dierent lazy strate rameters is called cal lbyneed or lazy evaluation gies In this pap er we derive an equational characteri A simple observation justies callbyneed the re zation of callbyneed and prove it correct with resp ect sult of reducing an expression if any is indistinguish to the original lamb da calculus The theory is a strictly able from the expression itself in all p ossible contexts smaller theory than the lamb da calculus Immediate Some implementations attempt
    [Show full text]
  • NU-Prolog Reference Manual
    NU-Prolog Reference Manual Version 1.5.24 edited by James A. Thom & Justin Zobel Technical Report 86/10 Machine Intelligence Project Department of Computer Science University of Melbourne (Revised November 1990) Copyright 1986, 1987, 1988, Machine Intelligence Project, The University of Melbourne. All rights reserved. The Technical Report edition of this publication is given away free subject to the condition that it shall not, by the way of trade or otherwise, be lent, resold, hired out, or otherwise circulated, without the prior consent of the Machine Intelligence Project at the University of Melbourne, in any form or cover other than that in which it is published and without a similar condition including this condition being imposed on the subsequent user. The Machine Intelligence Project makes no representations or warranties with respect to the contents of this document and specifically disclaims any implied warranties of merchantability or fitness for any particular purpose. Furthermore, the Machine Intelligence Project reserves the right to revise this document and to make changes from time to time in its content without being obligated to notify any person or body of such revisions or changes. The usual codicil that no part of this publication may be reproduced, stored in a retrieval system, used for wrapping fish, or transmitted, in any form or by any means, electronic, mechanical, photocopying, paper dart, recording, or otherwise, without someone's express permission, is not imposed. Enquiries relating to the acquisition of NU-Prolog should be made to NU-Prolog Distribution Manager Machine Intelligence Project Department of Computer Science The University of Melbourne Parkville, Victoria 3052 Australia Telephone: (03) 344 5229.
    [Show full text]
  • Variable-Arity Generic Interfaces
    Variable-Arity Generic Interfaces T. Stephen Strickland, Richard Cobbe, and Matthias Felleisen College of Computer and Information Science Northeastern University Boston, MA 02115 [email protected] Abstract. Many programming languages provide variable-arity func- tions. Such functions consume a fixed number of required arguments plus an unspecified number of \rest arguments." The C++ standardiza- tion committee has recently lifted this flexibility from terms to types with the adoption of a proposal for variable-arity templates. In this paper we propose an extension of Java with variable-arity interfaces. We present some programming examples that can benefit from variable-arity generic interfaces in Java; a type-safe model of such a language; and a reduction to the core language. 1 Introduction In April of 2007 the C++ standardization committee adopted Gregor and J¨arvi's proposal [1] for variable-length type arguments in class templates, which pre- sented the idea and its implementation but did not include a formal model or soundness proof. As a result, the relevant constructs will appear in the upcom- ing C++09 draft. Demand for this feature is not limited to C++, however. David Hall submitted a request for variable-arity type parameters for classes to Sun in 2005 [2]. For an illustration of the idea, consider the remarks on first-class functions in Scala [3] from the language's homepage. There it says that \every function is a value. Scala provides a lightweight syntax for defining anonymous functions, it supports higher-order functions, it allows functions to be nested, and supports currying." To achieve this integration of objects and closures, Scala's standard library pre-defines ten interfaces (traits) for function types, which we show here in Java-like syntax: interface Function0<Result> { Result apply(); } interface Function1<Arg1,Result> { Result apply(Arg1 a1); } interface Function2<Arg1,Arg2,Result> { Result apply(Arg1 a1, Arg2 a2); } ..
    [Show full text]
  • Small Hitting-Sets for Tiny Arithmetic Circuits Or: How to Turn Bad Designs Into Good
    Electronic Colloquium on Computational Complexity, Report No. 35 (2017) Small hitting-sets for tiny arithmetic circuits or: How to turn bad designs into good Manindra Agrawal ∗ Michael Forbes† Sumanta Ghosh ‡ Nitin Saxena § Abstract Research in the last decade has shown that to prove lower bounds or to derandomize polynomial identity testing (PIT) for general arithmetic circuits it suffices to solve these questions for restricted circuits. In this work, we study the smallest possibly restricted class of circuits, in particular depth-4 circuits, which would yield such results for general circuits (that is, the complexity class VP). We show that if we can design poly(s)-time hitting-sets for Σ ∧a ΣΠO(log s) circuits of size s, where a = ω(1) is arbitrarily small and the number of variables, or arity n, is O(log s), then we can derandomize blackbox PIT for general circuits in quasipolynomial time. Further, this establishes that either E6⊆#P/poly or that VP6=VNP. We call the former model tiny diagonal depth-4. Note that these are merely polynomials with arity O(log s) and degree ω(log s). In fact, we show that one only needs a poly(s)-time hitting-set against individual-degree a′ = ω(1) polynomials that are computable by a size-s arity-(log s) ΣΠΣ circuit (note: Π fanin may be s). Alternatively, we claim that, to understand VP one only needs to find hitting-sets, for depth-3, that have a small parameterized complexity. Another tiny family of interest is when we restrict the arity n = ω(1) to be arbitrarily small.
    [Show full text]
  • Making a Faster Curry with Extensional Types
    Making a Faster Curry with Extensional Types Paul Downen Simon Peyton Jones Zachary Sullivan Microsoft Research Zena M. Ariola Cambridge, UK University of Oregon [email protected] Eugene, Oregon, USA [email protected] [email protected] [email protected] Abstract 1 Introduction Curried functions apparently take one argument at a time, Consider these two function definitions: which is slow. So optimizing compilers for higher-order lan- guages invariably have some mechanism for working around f1 = λx: let z = h x x in λy:e y z currying by passing several arguments at once, as many as f = λx:λy: let z = h x x in e y z the function can handle, which is known as its arity. But 2 such mechanisms are often ad-hoc, and do not work at all in higher-order functions. We show how extensional, call- It is highly desirable for an optimizing compiler to η ex- by-name functions have the correct behavior for directly pand f1 into f2. The function f1 takes only a single argu- expressing the arity of curried functions. And these exten- ment before returning a heap-allocated function closure; sional functions can stand side-by-side with functions native then that closure must subsequently be called by passing the to practical programming languages, which do not use call- second argument. In contrast, f2 can take both arguments by-name evaluation. Integrating call-by-name with other at once, without constructing an intermediate closure, and evaluation strategies in the same intermediate language ex- this can make a huge difference to run-time performance in presses the arity of a function in its type and gives a princi- practice [Marlow and Peyton Jones 2004].
    [Show full text]
  • Lambda-Calculus Types and Models
    Jean-Louis Krivine LAMBDA-CALCULUS TYPES AND MODELS Translated from french by René Cori To my daughter Contents Introduction5 1 Substitution and beta-conversion7 Simple substitution ..............................8 Alpha-equivalence and substitution ..................... 12 Beta-conversion ................................ 18 Eta-conversion ................................. 24 2 Representation of recursive functions 29 Head normal forms .............................. 29 Representable functions ............................ 31 Fixed point combinators ........................... 34 The second fixed point theorem ....................... 37 3 Intersection type systems 41 System D­ ................................... 41 System D .................................... 50 Typings for normal terms ........................... 54 4 Normalization and standardization 61 Typings for normalizable terms ........................ 61 Strong normalization ............................. 68 ¯I-reduction ................................. 70 The ¸I-calculus ................................ 72 ¯´-reduction ................................. 74 The finite developments theorem ....................... 77 The standardization theorem ......................... 81 5 The Böhm theorem 87 3 4 CONTENTS 6 Combinatory logic 95 Combinatory algebras ............................. 95 Extensionality axioms ............................. 98 Curry’s equations ............................... 101 Translation of ¸-calculus ........................... 105 7 Models of lambda-calculus 111 Functional
    [Show full text]
  • Decompositions of Functions Based on Arity
    DECOMPOSITIONS OF FUNCTIONS BASED ON ARITY GAP MIGUEL COUCEIRO, ERKKO LEHTONEN, AND TAMAS´ WALDHAUSER1 Abstract. We study the arity gap of functions of several variables defined on an arbitrary set A and valued in another set B. The arity gap of such a function is the minimum decrease in the number of essential variables when variables are identified. We establish a complete classification of functions according to their arity gap, extending existing results for finite functions. This classification is refined when the codomain B has a group structure, by providing unique decompositions into sums of functions of a prescribed form. As an application of the unique decompositions, in the case of finite sets we count, for each n and p, the number of n-ary functions that depend on all of their variables and have arity gap p. 1. Introduction Essential variables of functions have been investigated in multiple-valued logic and computer science, especially, concerning the distribution of values of functions whose variables are all essential (see, e.g., [9, 16, 22]), the process of substituting constants for variables (see, e.g., [2, 3, 14, 16, 18]), and the process of substituting variables for variables (see, e.g., [5, 10, 16, 21]). The latter line of study goes back to the 1963 paper by Salomaa [16] who consid- ered the following problem: How does identification of variables affect the number of essential variables of a given function? The minimum decrease in the number of essential variables of a function f : An → B (n ≥ 2) which depends on all of its variables is called the arity gap of f.
    [Show full text]
  • An Introduction to Prolog
    Appendix A An Introduction to Prolog A.1 A Short Background Prolog was designed in the 1970s by Alain Colmerauer and a team of researchers with the idea – new at that time – that it was possible to use logic to represent knowledge and to write programs. More precisely, Prolog uses a subset of predicate logic and draws its structure from theoretical works of earlier logicians such as Herbrand (1930) and Robinson (1965) on the automation of theorem proving. Prolog was originally intended for the writing of natural language processing applications. Because of its conciseness and simplicity, it became popular well beyond this domain and now has adepts in areas such as: • Formal logic and associated forms of programming • Reasoning modeling • Database programming • Planning, and so on. This chapter is a short review of Prolog. In-depth tutorials include: in English, Bratko (2012), Clocksin and Mellish (2003), Covington et al. (1997), and Sterling and Shapiro (1994); in French, Giannesini et al. (1985); and in German, Baumann (1991). Boizumault (1988, 1993) contain a didactical implementation of Prolog in Lisp. Prolog foundations rest on first-order logic. Apt (1997), Burke and Foxley (1996), Delahaye (1986), and Lloyd (1987) examine theoretical links between this part of logic and Prolog. Colmerauer started his work at the University of Montréal, and a first version of the language was implemented at the University of Marseilles in 1972. Colmerauer and Roussel (1996) tell the story of the birth of Prolog, including their try-and-fail experimentation to select tractable algorithms from the mass of results provided by research in logic.
    [Show full text]
  • Reducing the Arity in Unbiased Black-Box Complexity
    Reducing the Arity in Unbiased Black-Box Complexity Benjamin Doerr and Carola Winzen∗ Max-Planck-Institut f¨ur Informatik, 66123 Saarbr¨ucken, Germany Abstract We show that for all 1 < k ≤ log n the k-ary unbiased black-box complexity of the n- dimensional OneMax function class is O(n/k). This indicates that the power of higher arity operators is much stronger than what the previous O(n/ log k) bound by Doerr et al. (Faster black-box algorithms through higher arity operators, Proc. of FOGA 2011, pp. 163–172, ACM, 2011) suggests. The key to this result is an encoding strategy, which might be of independent interest. We show that, using k-ary unbiased variation operators only, we may simulate an unrestricted memory of size O(2k) bits. 1 Introduction Black-box complexity theory tries to give a theory-driven answer to the question how difficult a problem is to be solved by general purpose optimization approaches (“black-box algorithms”). The recently introduced notion of unbiased black-box complexity in addition allows a distinction regarding the arity of the variation operators employed (see the theory track best-paper award winner by Lehre and Witt [LW10]). The only result so far indicating that there exists a hierarchy of the unbiased black-box models with respect to their arity (that is, the only result indicating that for any k ∈ N the (k + 1)-ary operators are strictly more powerful than k-ary ones) is the result of Doerr, Johannsen, K¨otzing, Lehre, Wagner, and Winzen showing that the k-ary unbiased black-box complexity of the n-dimensional OneMaxn function class is O(n/ log k), 1 < k ≤ n.
    [Show full text]
  • Additive Decomposition Schemes for Polynomial Functions Over Fields Miguel Couceiro, Erkko Lehtonen, Tamas Waldhauser
    Additive decomposition schemes for polynomial functions over fields Miguel Couceiro, Erkko Lehtonen, Tamas Waldhauser To cite this version: Miguel Couceiro, Erkko Lehtonen, Tamas Waldhauser. Additive decomposition schemes for polyno- mial functions over fields. Novi Sad Journal of Mathematics, Institut za matematiku, 2014, 44(2), pp.89-105. hal-01090554 HAL Id: hal-01090554 https://hal.archives-ouvertes.fr/hal-01090554 Submitted on 18 Feb 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. This paper appeared in Novi Sad J. Math. 44 (2014), 89{105. ADDITIVE DECOMPOSITION SCHEMES FOR POLYNOMIAL FUNCTIONS OVER FIELDS MIGUEL COUCEIRO, ERKKO LEHTONEN, AND TAMAS´ WALDHAUSER Abstract. The authors' previous results on the arity gap of functions of several vari- ables are refined by considering polynomial functions over arbitrary fields. We explicitly describe the polynomial functions with arity gap at least 3, as well as the polynomial functions with arity gap equal to 2 for fields of characteristic 0 or 2. These descriptions are given in the form of decomposition schemes of polynomial functions. Similar descrip- tions are given for arbitrary finite fields. However, we show that these descriptions do not extend to infinite fields of odd characteristic.
    [Show full text]
  • Modern Perl, Fourth Edition
    Prepared exclusively for none ofyourbusiness Prepared exclusively for none ofyourbusiness Early Praise for Modern Perl, Fourth Edition A dozen years ago I was sure I knew what Perl looked like: unreadable and obscure. chromatic showed me beautiful, structured expressive code then. He’s the right guy to teach Modern Perl. He was writing it before it existed. ➤ Daniel Steinberg President, DimSumThinking, Inc. A tour de force of idiomatic code, Modern Perl teaches you not just “how” but also “why.” ➤ David Farrell Editor, PerlTricks.com If I had to pick a single book to teach Perl 5, this is the one I’d choose. As I read it, I was reminded of the first time I read K&R. It will teach everything that one needs to know to write Perl 5 well. ➤ David Golden Member, Perl 5 Porters, Autopragmatic, LLC I’m about to teach a new hire Perl using the first edition of Modern Perl. I’d much rather use the updated copy! ➤ Belden Lyman Principal Software Engineer, MediaMath It’s not the Perl book you deserve. It’s the Perl book you need. ➤ Gizmo Mathboy Co-founder, Greater Lafayette Open Source Symposium (GLOSSY) Prepared exclusively for none ofyourbusiness We've left this page blank to make the page numbers the same in the electronic and paper books. We tried just leaving it out, but then people wrote us to ask about the missing pages. Anyway, Eddy the Gerbil wanted to say “hello.” Prepared exclusively for none ofyourbusiness Modern Perl, Fourth Edition chromatic The Pragmatic Bookshelf Dallas, Texas • Raleigh, North Carolina Prepared exclusively for none ofyourbusiness Many of the designations used by manufacturers and sellers to distinguish their products are claimed as trademarks.
    [Show full text]
  • Every Permutation CSP of Arity 3 Is Approximation Resistant
    Every Permutation CSP of arity 3 is Approximation Resistant Moses Charikar∗ Venkatesan Guruswamiy Rajsekar Manokaranz Dept. of Computer Science Dept. of Computer Science & Engg. Dept. of Computer Science Princeton University University of Washington Princeton University Princeton, NJ 08540. Seattle, WA 98195. Princeton, NJ 08540. [email protected] [email protected] [email protected] Abstract—A permutation constraint satisfaction problem Max 3LIN, Unique Games, etc. are examples of CSPs. A (permCSP) of arity k is specified by a subset Λ ⊆ Sk of CSP of arity k over domain [q] = f0; 1; : : : ; q − 1g (called permutations on f1; 2; : : : ; kg. An instance of such a permCSP a q-ary kCSP) is specified by a predicate P :[q]k ! f0; 1g. consists of a set of variables V and a collection of constraints each of which is an ordered k-tuple of V . The objective is An instance of such a CSP consists of a set of variables to find a global ordering σ of the variables that maximizes V and a collection C of constraints each of which applies the number of constraint tuples whose ordering (under σ) P to a k-tuple of literals (which are variables or their follows a permutation in Λ. This is just the natural extension “negations,” i.e., translates modulo q). The objective is to of constraint satisfaction problems over finite domains (such find an assignment A : V ! [q] of values to the variables as Boolean CSPs) to the world of ordering problems. The simplest permCSP corresponds to the case when Λ that maximizes the number of constraints of C that are consists of the identity permutation on two variables.
    [Show full text]