Pub Date : 1999-10-17DOI: 10.1109/SFFCS.1999.814593
M. Dyer, A. Frieze, M. Jerrum
We prove two results concerning approximate counting of independent sets in graphs with constant maximum degree /spl Delta/. The first result implies that the Monte-Carlo Markov chain technique is likely to fail if /spl Delta//spl ges/6. The second shows that no fully polynomial randomized approximation scheme can exist for /spl Delta//spl ges/25, unless P=NP under randomized reductions.
{"title":"On counting independent sets in sparse graphs","authors":"M. Dyer, A. Frieze, M. Jerrum","doi":"10.1109/SFFCS.1999.814593","DOIUrl":"https://doi.org/10.1109/SFFCS.1999.814593","url":null,"abstract":"We prove two results concerning approximate counting of independent sets in graphs with constant maximum degree /spl Delta/. The first result implies that the Monte-Carlo Markov chain technique is likely to fail if /spl Delta//spl ges/6. The second shows that no fully polynomial randomized approximation scheme can exist for /spl Delta//spl ges/25, unless P=NP under randomized reductions.","PeriodicalId":385047,"journal":{"name":"40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1999-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124871506","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1999-10-17DOI: 10.1109/SFFCS.1999.814614
Maria Luisa Bonet, Nicola Galesi
The paper is concerned with the complexity of proofs and of searching for proofs in two propositional proof systems: Resolution and Polynomial Calculus (PC). For the former system we show that the recently proposed algorithm of E. Ben-Sasson and A. Wigderson (1999) for searching for proofs cannot give better than weakly exponential performance. This is a consequence of showing optimality of their general relationship, referred to as size-width trade-off. We moreover obtain the optimality of the size width trade-off for the widely used restrictions of resolution: regular, Davis-Putnam, negative, positive and linear. As for the second system, we show that the direct translation to polynomials of a CNF formula having short resolution proofs, cannot be refuted in PC with degree less than /spl Omega/ (log n). A consequence of this is that the simulation of resolution by PC of M. Clegg, J. Edmonds and R. Impagliazzo (1996) cannot be improved to better than quasipolynomial in the case where we start with small resolution proofs. We conjecture that the simulation of M. Clegg et al. is optimal.
{"title":"A study of proof search algorithms for resolution and polynomial calculus","authors":"Maria Luisa Bonet, Nicola Galesi","doi":"10.1109/SFFCS.1999.814614","DOIUrl":"https://doi.org/10.1109/SFFCS.1999.814614","url":null,"abstract":"The paper is concerned with the complexity of proofs and of searching for proofs in two propositional proof systems: Resolution and Polynomial Calculus (PC). For the former system we show that the recently proposed algorithm of E. Ben-Sasson and A. Wigderson (1999) for searching for proofs cannot give better than weakly exponential performance. This is a consequence of showing optimality of their general relationship, referred to as size-width trade-off. We moreover obtain the optimality of the size width trade-off for the widely used restrictions of resolution: regular, Davis-Putnam, negative, positive and linear. As for the second system, we show that the direct translation to polynomials of a CNF formula having short resolution proofs, cannot be refuted in PC with degree less than /spl Omega/ (log n). A consequence of this is that the simulation of resolution by PC of M. Clegg, J. Edmonds and R. Impagliazzo (1996) cannot be improved to better than quasipolynomial in the case where we start with small resolution proofs. We conjecture that the simulation of M. Clegg et al. is optimal.","PeriodicalId":385047,"journal":{"name":"40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1999-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132410758","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1999-10-17DOI: 10.1109/SFFCS.1999.814631
J. Kleinberg, Y. Rabani, É. Tardos
We consider the issue of network routing subject to explicit fairness conditions. The optimization of fairness criteria interacts in a complex fashion with the optimization of network utilization and throughput; in this work, we undertake an investigation of this relationship through the framework of approximation algorithms. In this work we consider the problem of selecting paths for routing so as to provide a bandwidth allocation that is as fair as possible (in the max-min sense). We obtain the first approximation algorithms for this basic optimization problem, for single-source unsplittable routings in an arbitrary directed graph. Special cases of our model include several fundamental load balancing problems, endowing them with a natural fairness criterion to which our approach can be applied. Our results form an interesting counterpart to the work of Megiddo (1974), who considered max-min fairness for single-source fractional flow. The optimization problems in our setting become NP-complete, and require the development of new techniques for relating fractional relaxations of routing to the equilibrium constraints imposed by the fairness criterion.
{"title":"Fairness in routing and load balancing","authors":"J. Kleinberg, Y. Rabani, É. Tardos","doi":"10.1109/SFFCS.1999.814631","DOIUrl":"https://doi.org/10.1109/SFFCS.1999.814631","url":null,"abstract":"We consider the issue of network routing subject to explicit fairness conditions. The optimization of fairness criteria interacts in a complex fashion with the optimization of network utilization and throughput; in this work, we undertake an investigation of this relationship through the framework of approximation algorithms. In this work we consider the problem of selecting paths for routing so as to provide a bandwidth allocation that is as fair as possible (in the max-min sense). We obtain the first approximation algorithms for this basic optimization problem, for single-source unsplittable routings in an arbitrary directed graph. Special cases of our model include several fundamental load balancing problems, endowing them with a natural fairness criterion to which our approach can be applied. Our results form an interesting counterpart to the work of Megiddo (1974), who considered max-min fairness for single-source fractional flow. The optimization problems in our setting become NP-complete, and require the development of new techniques for relating fractional relaxations of routing to the equilibrium constraints imposed by the fairness criterion.","PeriodicalId":385047,"journal":{"name":"40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1999-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114305535","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1999-10-17DOI: 10.1109/SFFCS.1999.814586
U. Feige
Selection tasks generalize some well studied problems, such as collective coin flipping and leader election. We present new selection protocols in the full information model, and new negative results. In particular when there are (1+/spl delta/)n/2 good players, we show a protocol that chooses a good leader with probability /spl Omega/(/spl delta//sup 1.65/), and show that every leader election protocol has success probability O(/spl delta//sup 1-/spl epsiv//), for every /spl epsiv/>0. Previously known protocols for this problem have success probability that is exponentially small in 1//spl delta/, and no nontrivial upper bounds on the success probability were known.
{"title":"Noncryptographic selection protocols","authors":"U. Feige","doi":"10.1109/SFFCS.1999.814586","DOIUrl":"https://doi.org/10.1109/SFFCS.1999.814586","url":null,"abstract":"Selection tasks generalize some well studied problems, such as collective coin flipping and leader election. We present new selection protocols in the full information model, and new negative results. In particular when there are (1+/spl delta/)n/2 good players, we show a protocol that chooses a good leader with probability /spl Omega/(/spl delta//sup 1.65/), and show that every leader election protocol has success probability O(/spl delta//sup 1-/spl epsiv//), for every /spl epsiv/>0. Previously known protocols for this problem have success probability that is exponentially small in 1//spl delta/, and no nontrivial upper bounds on the success probability were known.","PeriodicalId":385047,"journal":{"name":"40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1999-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134497540","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1999-10-17DOI: 10.1109/SFFCS.1999.814634
R. Kolpakov, G. Kucherov
A repetition in a word w is a subword with the period of at most half of the subword length. We study maximal repetitions occurring in w, that is those for which any extended subword of w has a bigger period. The set of such repetitions represents in a compact way all repetitions in w. We first prove a combinatorial result asserting that the sum of exponents of all maximal repetitions of a word of length n is bounded by a linear function in n. This implies, in particular that there is only a linear number of maximal repetitions in a word. This allows us to construct a linear-time algorithm for finding all maximal repetitions. Some consequences and applications of these results are discussed, as well as related works.
{"title":"Finding maximal repetitions in a word in linear time","authors":"R. Kolpakov, G. Kucherov","doi":"10.1109/SFFCS.1999.814634","DOIUrl":"https://doi.org/10.1109/SFFCS.1999.814634","url":null,"abstract":"A repetition in a word w is a subword with the period of at most half of the subword length. We study maximal repetitions occurring in w, that is those for which any extended subword of w has a bigger period. The set of such repetitions represents in a compact way all repetitions in w. We first prove a combinatorial result asserting that the sum of exponents of all maximal repetitions of a word of length n is bounded by a linear function in n. This implies, in particular that there is only a linear number of maximal repetitions in a word. This allows us to construct a linear-time algorithm for finding all maximal repetitions. Some consequences and applications of these results are discussed, as well as related works.","PeriodicalId":385047,"journal":{"name":"40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1999-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133956463","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1999-10-17DOI: 10.1109/SFFCS.1999.814600
Matteo Frigo, C. Leiserson, H. Prokop, S. Ramachandran
This paper presents asymptotically optimal algorithms for rectangular matrix transpose, FFT, and sorting on computers with multiple levels of caching. Unlike previous optimal algorithms, these algorithms are cache oblivious: no variables dependent on hardware parameters, such as cache size and cache-line length, need to be tuned to achieve optimality. Nevertheless, these algorithms use an optimal amount of work and move data optimally among multiple levels of cache. For a cache with size Z and cache-line length L where Z=/spl Omega/(L/sup 2/) the number of cache misses for an m/spl times/n matrix transpose is /spl Theta/(1+mn/L). The number of cache misses for either an n-point FFT or the sorting of n numbers is /spl Theta/(1+(n/L)(1+log/sub Z/n)). We also give an /spl Theta/(mnp)-work algorithm to multiply an m/spl times/n matrix by an n/spl times/p matrix that incurs /spl Theta/(1+(mn+np+mp)/L+mnp/L/spl radic/Z) cache faults. We introduce an "ideal-cache" model to analyze our algorithms. We prove that an optimal cache-oblivious algorithm designed for two levels of memory is also optimal for multiple levels and that the assumption of optimal replacement in the ideal-cache model. Can be simulated efficiently by LRU replacement. We also provide preliminary empirical results on the effectiveness of cache-oblivious algorithms in practice.
{"title":"Cache-oblivious algorithms","authors":"Matteo Frigo, C. Leiserson, H. Prokop, S. Ramachandran","doi":"10.1109/SFFCS.1999.814600","DOIUrl":"https://doi.org/10.1109/SFFCS.1999.814600","url":null,"abstract":"This paper presents asymptotically optimal algorithms for rectangular matrix transpose, FFT, and sorting on computers with multiple levels of caching. Unlike previous optimal algorithms, these algorithms are cache oblivious: no variables dependent on hardware parameters, such as cache size and cache-line length, need to be tuned to achieve optimality. Nevertheless, these algorithms use an optimal amount of work and move data optimally among multiple levels of cache. For a cache with size Z and cache-line length L where Z=/spl Omega/(L/sup 2/) the number of cache misses for an m/spl times/n matrix transpose is /spl Theta/(1+mn/L). The number of cache misses for either an n-point FFT or the sorting of n numbers is /spl Theta/(1+(n/L)(1+log/sub Z/n)). We also give an /spl Theta/(mnp)-work algorithm to multiply an m/spl times/n matrix by an n/spl times/p matrix that incurs /spl Theta/(1+(mn+np+mp)/L+mnp/L/spl radic/Z) cache faults. We introduce an \"ideal-cache\" model to analyze our algorithms. We prove that an optimal cache-oblivious algorithm designed for two levels of memory is also optimal for multiple levels and that the assumption of optimal replacement in the ideal-cache model. Can be simulated efficiently by LRU replacement. We also provide preliminary empirical results on the effectiveness of cache-oblivious algorithms in practice.","PeriodicalId":385047,"journal":{"name":"40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1999-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128813170","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1999-10-17DOI: 10.1109/SFFCS.1999.814635
Avi Shoshan, Uri Zwick
We show that the all pairs shortest paths (APSP) problem for undirected graphs with integer edge weights taken from the range {1, 2, ..., M} can be solved using only a logarithmic number of distance products of matrices with elements in the range (1, 2, ..., M). As a result, we get an algorithm for the APSP problem in such graphs that runs in O~(Mn/sup /spl omega//) time, where n is the number of vertices in the input graph, M is the largest edge weight in the graph, and /spl omega/<2.376 is the exponent of matrix multiplication. This improves, and also simplifies, an O~(M/sup (/spl omega/+1)/2/n/sup /spl omega//) time algorithm of Galil and Margalit (1997).
{"title":"All pairs shortest paths in undirected graphs with integer weights","authors":"Avi Shoshan, Uri Zwick","doi":"10.1109/SFFCS.1999.814635","DOIUrl":"https://doi.org/10.1109/SFFCS.1999.814635","url":null,"abstract":"We show that the all pairs shortest paths (APSP) problem for undirected graphs with integer edge weights taken from the range {1, 2, ..., M} can be solved using only a logarithmic number of distance products of matrices with elements in the range (1, 2, ..., M). As a result, we get an algorithm for the APSP problem in such graphs that runs in O~(Mn/sup /spl omega//) time, where n is the number of vertices in the input graph, M is the largest edge weight in the graph, and /spl omega/<2.376 is the exponent of matrix multiplication. This improves, and also simplifies, an O~(M/sup (/spl omega/+1)/2/n/sup /spl omega//) time algorithm of Galil and Margalit (1997).","PeriodicalId":385047,"journal":{"name":"40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1999-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123352021","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1999-10-17DOI: 10.1109/SFFCS.1999.814605
John Watrous
We define a class of stochastic processes based on evolutions and measurements of quantum systems, and consider the complexity of predicting their long term behavior. It is shown that a very general class of decision problems regarding these stochastic processes can be efficiently solved classically in the space-bounded case. The following corollaries are implied by our main result for any space-constructible space bound s satisfying s(n)=/spl Omega/(log n): (i) any space O(s) uniform family of quantum circuit acting on s qubits and consisting of unitary gates and measurement gates defined in a typical way by matrices of algebraic numbers can be simulated by an unbounded error space O(s) ordinary (i.e., fair-coin flipping) probabilistic Turing machine, and hence by space O(s) uniform classical (deterministic) circuits of depth O(s/sup 2/) and size 2/sup 0/(s); (2) any quantum Turing machine running in space s, having arbitrary algebraic transition amplitudes, allowing unrestricted measurements during its computation, and having no restrictions on running time can be simulated by a space O(s) ordinary probabilistic Turing machine in the unbounded error setting. We also obtain the following classical result: any unbounded error probabilistic Turing machine running in space s that allows algebraic probabilities and algebraic cut-point can be simulated by a space O(s) ordinarily probabilistic Turing machine with cut-point 1/2. Our technique for handling algebraic numbers in the above simulations may be of independent interest. It is shown that any real algebraic number can be accurately approximated by a ratio of GapL functions.
{"title":"On quantum and classical space-bounded processes with algebraic transition amplitudes","authors":"John Watrous","doi":"10.1109/SFFCS.1999.814605","DOIUrl":"https://doi.org/10.1109/SFFCS.1999.814605","url":null,"abstract":"We define a class of stochastic processes based on evolutions and measurements of quantum systems, and consider the complexity of predicting their long term behavior. It is shown that a very general class of decision problems regarding these stochastic processes can be efficiently solved classically in the space-bounded case. The following corollaries are implied by our main result for any space-constructible space bound s satisfying s(n)=/spl Omega/(log n): (i) any space O(s) uniform family of quantum circuit acting on s qubits and consisting of unitary gates and measurement gates defined in a typical way by matrices of algebraic numbers can be simulated by an unbounded error space O(s) ordinary (i.e., fair-coin flipping) probabilistic Turing machine, and hence by space O(s) uniform classical (deterministic) circuits of depth O(s/sup 2/) and size 2/sup 0/(s); (2) any quantum Turing machine running in space s, having arbitrary algebraic transition amplitudes, allowing unrestricted measurements during its computation, and having no restrictions on running time can be simulated by a space O(s) ordinary probabilistic Turing machine in the unbounded error setting. We also obtain the following classical result: any unbounded error probabilistic Turing machine running in space s that allows algebraic probabilities and algebraic cut-point can be simulated by a space O(s) ordinarily probabilistic Turing machine with cut-point 1/2. Our technique for handling algebraic numbers in the above simulations may be of independent interest. It is shown that any real algebraic number can be accurately approximated by a ratio of GapL functions.","PeriodicalId":385047,"journal":{"name":"40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1999-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132091205","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1999-10-17DOI: 10.1109/SFFCS.1999.814577
Eric Vigoda
We consider the problem of sampling uniformly from the set of proper k-colorings of a graph with maximum degree /spl Delta/. Our main result is the design Markov chain that converges in O(nk log n) time to the desired distribution when k>11/6 /spl Delta/.
{"title":"Improved bounds for sampling colorings","authors":"Eric Vigoda","doi":"10.1109/SFFCS.1999.814577","DOIUrl":"https://doi.org/10.1109/SFFCS.1999.814577","url":null,"abstract":"We consider the problem of sampling uniformly from the set of proper k-colorings of a graph with maximum degree /spl Delta/. Our main result is the design Markov chain that converges in O(nk log n) time to the desired distribution when k>11/6 /spl Delta/.","PeriodicalId":385047,"journal":{"name":"40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1999-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123999063","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1999-10-17DOI: 10.1109/SFFCS.1999.814576
M. Blaser
We prove a lower bound of 5/2n/sup 2/-3n for the rank of n/spl times/n-matrix multiplication over an arbitrary field. Similar bounds hold for the rank of the multiplication in noncommutative division algebras and for the multiplication of upper triangular matrices.
{"title":"A 5/2n/sup 2/-lower bound for the rank of n/spl times/n-matrix multiplication over arbitrary fields","authors":"M. Blaser","doi":"10.1109/SFFCS.1999.814576","DOIUrl":"https://doi.org/10.1109/SFFCS.1999.814576","url":null,"abstract":"We prove a lower bound of 5/2n/sup 2/-3n for the rank of n/spl times/n-matrix multiplication over an arbitrary field. Similar bounds hold for the rank of the multiplication in noncommutative division algebras and for the multiplication of upper triangular matrices.","PeriodicalId":385047,"journal":{"name":"40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1999-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121672836","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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