Pub Date : 2018-12-17DOI: 10.1109/SFCS.2001.959912
Peter Burgisser
The existence of string functions, which are not polynomial time computable, but whose graph is checkable in polynomial time, is a basic assumption in cryptography. We prove that in the framework of algebraic complexity, there are no such families of polynomial functions of p-bounded degree overfields of characteristic zero. The proof relies on a polynomial upper bound on the approximative complexity of a factor g of a polynomial f in terms of the (approximative) complexity of f and the degree of the factor g. This extends a result by E. Kaltofen (1986). The concept of approximative complexity allows us to cope with the case that a factor has an exponential multiplicity, by using a perturbation argument. Our result extends to randomized (two-sided error) decision complexity.
{"title":"The complexity of factors of multivariate polynomials","authors":"Peter Burgisser","doi":"10.1109/SFCS.2001.959912","DOIUrl":"https://doi.org/10.1109/SFCS.2001.959912","url":null,"abstract":"The existence of string functions, which are not polynomial time computable, but whose graph is checkable in polynomial time, is a basic assumption in cryptography. We prove that in the framework of algebraic complexity, there are no such families of polynomial functions of p-bounded degree overfields of characteristic zero. The proof relies on a polynomial upper bound on the approximative complexity of a factor g of a polynomial f in terms of the (approximative) complexity of f and the degree of the factor g. This extends a result by E. Kaltofen (1986). The concept of approximative complexity allows us to cope with the case that a factor has an exponential multiplicity, by using a perturbation argument. Our result extends to randomized (two-sided error) decision complexity.","PeriodicalId":378126,"journal":{"name":"Proceedings 2001 IEEE International Conference on Cluster Computing","volume":"71 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123161057","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 : 2001-10-14DOI: 10.1109/SFCS.2001.959898
M. Thorup
It is shown that a planar digraph can be preprocessed in near-linear time, producing a near-linear space distance oracle that can answer reachability queries in constant time. The oracle can be distributed as an O(log n) space label for each vertex and then we can determine if one vertex can reach another considering their two labels only. The approach generalizes to approximate distances in weighted planar digraphs where we can then get a (1+/spl epsi/) approximation distance in O(log log /spl Delta/+1//spl epsi/) time where /spl Delta/ is the longest finite distance in the graph and weights are assumed to be non-negative integers. Our scheme can be extended to find and route along the short dipaths. Our technique is based on a novel dipath decomposition of planar digraphs that instead of using the standard separator with O(/spl radic/n) vertices, in effect finds a separator using a constant number of dipaths.
{"title":"Compact oracles for reachability and approximate distances in planar digraphs","authors":"M. Thorup","doi":"10.1109/SFCS.2001.959898","DOIUrl":"https://doi.org/10.1109/SFCS.2001.959898","url":null,"abstract":"It is shown that a planar digraph can be preprocessed in near-linear time, producing a near-linear space distance oracle that can answer reachability queries in constant time. The oracle can be distributed as an O(log n) space label for each vertex and then we can determine if one vertex can reach another considering their two labels only. The approach generalizes to approximate distances in weighted planar digraphs where we can then get a (1+/spl epsi/) approximation distance in O(log log /spl Delta/+1//spl epsi/) time where /spl Delta/ is the longest finite distance in the graph and weights are assumed to be non-negative integers. Our scheme can be extended to find and route along the short dipaths. Our technique is based on a novel dipath decomposition of planar digraphs that instead of using the standard separator with O(/spl radic/n) vertices, in effect finds a separator using a constant number of dipaths.","PeriodicalId":378126,"journal":{"name":"Proceedings 2001 IEEE International Conference on Cluster Computing","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2001-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115556040","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 : 2001-10-14DOI: 10.1109/SFCS.2001.959881
Sariel Har-Peled, Kasturi R. Varadarajan
Shape fitting is a fundamental optimization problem in computer science. The authors present a general and unified technique for solving a certain family of such problems. Given a point set P in R/sup d/, this technique can be used to /spl epsi/-approximate: (i) the min-width annulus and shell that contains P, (ii) minimum width cylindrical shell containing P, (iii) diameter, width, minimum volume bounding box of P, and (iv) all the previous measures for the case the points are moving. The running time of the resulting algorithms is O(n + 1//spl epsi//sup c/), where c is a constant that depends on the problem at hand. Our new general technique enables us to solve those problems without resorting to a careful and painful case by case analysis, as was previously done for those problems. Furthermore, for several of those problems our results are considerably simpler and faster than what was previously known. In particular, for the minimum width cylindrical shell problem, our solution is the first algorithm whose running time is subquadratic in n. (In fact we get running time linear in n.).
{"title":"Approximate shape fitting via linearization","authors":"Sariel Har-Peled, Kasturi R. Varadarajan","doi":"10.1109/SFCS.2001.959881","DOIUrl":"https://doi.org/10.1109/SFCS.2001.959881","url":null,"abstract":"Shape fitting is a fundamental optimization problem in computer science. The authors present a general and unified technique for solving a certain family of such problems. Given a point set P in R/sup d/, this technique can be used to /spl epsi/-approximate: (i) the min-width annulus and shell that contains P, (ii) minimum width cylindrical shell containing P, (iii) diameter, width, minimum volume bounding box of P, and (iv) all the previous measures for the case the points are moving. The running time of the resulting algorithms is O(n + 1//spl epsi//sup c/), where c is a constant that depends on the problem at hand. Our new general technique enables us to solve those problems without resorting to a careful and painful case by case analysis, as was previously done for those problems. Furthermore, for several of those problems our results are considerably simpler and faster than what was previously known. In particular, for the minimum width cylindrical shell problem, our solution is the first algorithm whose running time is subquadratic in n. (In fact we get running time linear in n.).","PeriodicalId":378126,"journal":{"name":"Proceedings 2001 IEEE International Conference on Cluster Computing","volume":"235 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2001-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121867785","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 : 2001-10-14DOI: 10.1109/SFCS.2001.959934
M. Dyer, A. Frieze
We consider the problem of generating a random q-colouring of a graph G=(V, E). We consider the simple Glauber Dynamics chain. We show that if the maximum degree /spl Delta/>c/sub l/ ln n and the girth g>c/sub 2/ ln ln n (n=|V|), then this chain mixes rapidly provided C/sub 1/, C/sub 2/ are sufficiently large, q/A>/spl beta/, where /spl beta//spl ap/1.763 is the root of /spl beta/=e/sup 1//spl beta//. For this class of graphs, this beats the 11/spl Delta//6 bound of E. Vigoda (1999) for general graphs. We extend the result to random graphs.
{"title":"Randomly colouring graphs with lower bounds on girth and maximum degree","authors":"M. Dyer, A. Frieze","doi":"10.1109/SFCS.2001.959934","DOIUrl":"https://doi.org/10.1109/SFCS.2001.959934","url":null,"abstract":"We consider the problem of generating a random q-colouring of a graph G=(V, E). We consider the simple Glauber Dynamics chain. We show that if the maximum degree /spl Delta/>c/sub l/ ln n and the girth g>c/sub 2/ ln ln n (n=|V|), then this chain mixes rapidly provided C/sub 1/, C/sub 2/ are sufficiently large, q/A>/spl beta/, where /spl beta//spl ap/1.763 is the root of /spl beta/=e/sup 1//spl beta//. For this class of graphs, this beats the 11/spl Delta//6 bound of E. Vigoda (1999) for general graphs. We extend the result to random graphs.","PeriodicalId":378126,"journal":{"name":"Proceedings 2001 IEEE International Conference on Cluster Computing","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2001-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123843185","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 : 2001-10-14DOI: 10.1109/SFCS.2001.959896
Stefan S. Dantchev, Søren Riis
We prove exponential lower bounds on the resolution proofs of some tautologies, based on rectangular grid graphs. More specifically, we show a 2/sup /spl Omega/(n)/ lower bound for any resolution proof of the mutilated chessboard problem on a 2n/spl times/2n chessboard as well as for the Tseitin tautology (G. Tseitin, 1968) based on the n/spl times/n rectangular grid graph. The former result answers a 35 year old conjecture by J. McCarthy (1964).
{"title":"\"Planar\" tautologies hard for resolution","authors":"Stefan S. Dantchev, Søren Riis","doi":"10.1109/SFCS.2001.959896","DOIUrl":"https://doi.org/10.1109/SFCS.2001.959896","url":null,"abstract":"We prove exponential lower bounds on the resolution proofs of some tautologies, based on rectangular grid graphs. More specifically, we show a 2/sup /spl Omega/(n)/ lower bound for any resolution proof of the mutilated chessboard problem on a 2n/spl times/2n chessboard as well as for the Tseitin tautology (G. Tseitin, 1968) based on the n/spl times/n rectangular grid graph. The former result answers a 35 year old conjecture by J. McCarthy (1964).","PeriodicalId":378126,"journal":{"name":"Proceedings 2001 IEEE International Conference on Cluster Computing","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2001-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114358825","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 : 2001-10-14DOI: 10.1109/SFCS.2001.959939
N. Alon, A. Lubotzky, A. Wigderson
We consider the standard semi-direct product A/spl times/B of finite groups A, B. We show that with certain choices of generators for these three groups, the Cayley graph of A/spl times/B is (essentially) the zigzag product of the Cayley graphs of A and B. Thus, using the results of O. Reingold et al. (2000), the new Cayley graph is an expander if and only if its two components are. We develop some general ways of using this construction to obtain large constant-degree expanding Cayley graphs from small ones. A. Lubotzky and B. Weiss (1993) asked whether expansion is a group property; namely, is being an expander for (a Cayley graph of) a group G depend solely on G and not on the choice of generators. We use the above construction to answer the question in the negative, by showing an infinite family of groups A/sub i//spl times/B/sub i/ which are expanders with one choice of a (constant-size) set of generators and are not with another such choice. It is interesting to note that this problem is still open, though for "natural" families of groups like the symmetric groups S/sub n/ or the simple groups PSL(2, p).
{"title":"Semi-direct product in groups and zig-zag product in graphs: connections and applications","authors":"N. Alon, A. Lubotzky, A. Wigderson","doi":"10.1109/SFCS.2001.959939","DOIUrl":"https://doi.org/10.1109/SFCS.2001.959939","url":null,"abstract":"We consider the standard semi-direct product A/spl times/B of finite groups A, B. We show that with certain choices of generators for these three groups, the Cayley graph of A/spl times/B is (essentially) the zigzag product of the Cayley graphs of A and B. Thus, using the results of O. Reingold et al. (2000), the new Cayley graph is an expander if and only if its two components are. We develop some general ways of using this construction to obtain large constant-degree expanding Cayley graphs from small ones. A. Lubotzky and B. Weiss (1993) asked whether expansion is a group property; namely, is being an expander for (a Cayley graph of) a group G depend solely on G and not on the choice of generators. We use the above construction to answer the question in the negative, by showing an infinite family of groups A/sub i//spl times/B/sub i/ which are expanders with one choice of a (constant-size) set of generators and are not with another such choice. It is interesting to note that this problem is still open, though for \"natural\" families of groups like the symmetric groups S/sub n/ or the simple groups PSL(2, p).","PeriodicalId":378126,"journal":{"name":"Proceedings 2001 IEEE International Conference on Cluster Computing","volume":"232 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2001-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123361709","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 : 2001-10-14DOI: 10.1109/SFCS.2001.959923
T. Roughgarden
We consider a directed network in which every edge possesses a latency function specifying the time needed to traverse the edge given its congestion. Selfish, noncooperative agents constitute the network traffic and wish to travel from a source s to a sink t as quickly as possible. Since the route chosen by one network user affects the congestion (and hence the latency) experienced by others, we model the problem as a noncooperative game. Assuming each agent controls only a negligible portion of the overall traffic, Nash equilibria in this noncooperative game correspond to s-t flows in which all flow paths have equal latency. We give optimal inapproximability results and approximation algorithms for several network design problems of this type. For example, we prove that for networks with n nodes and continuous, nondecreasing latency functions, there is no approximation algorithm for this problem with approximation ratio less than n/2 (unless P = NP). We also prove this hardness result to be best possible by exhibiting an n/2-approximation algorithm. For networks in which the latency of each edge is a linear function of the congestion, we prove that there is no (4/3 - /spl epsi/)-approximation algorithm for the problem (for any /spl epsi/ > 0, unless P = NP); the existence of a 4/3-approximation algorithm follows easily from existing work, proving this hardness result sharp.
{"title":"Designing networks for selfish users is hard","authors":"T. Roughgarden","doi":"10.1109/SFCS.2001.959923","DOIUrl":"https://doi.org/10.1109/SFCS.2001.959923","url":null,"abstract":"We consider a directed network in which every edge possesses a latency function specifying the time needed to traverse the edge given its congestion. Selfish, noncooperative agents constitute the network traffic and wish to travel from a source s to a sink t as quickly as possible. Since the route chosen by one network user affects the congestion (and hence the latency) experienced by others, we model the problem as a noncooperative game. Assuming each agent controls only a negligible portion of the overall traffic, Nash equilibria in this noncooperative game correspond to s-t flows in which all flow paths have equal latency. We give optimal inapproximability results and approximation algorithms for several network design problems of this type. For example, we prove that for networks with n nodes and continuous, nondecreasing latency functions, there is no approximation algorithm for this problem with approximation ratio less than n/2 (unless P = NP). We also prove this hardness result to be best possible by exhibiting an n/2-approximation algorithm. For networks in which the latency of each edge is a linear function of the congestion, we prove that there is no (4/3 - /spl epsi/)-approximation algorithm for the problem (for any /spl epsi/ > 0, unless P = NP); the existence of a 4/3-approximation algorithm follows easily from existing work, proving this hardness result sharp.","PeriodicalId":378126,"journal":{"name":"Proceedings 2001 IEEE International Conference on Cluster Computing","volume":"26 3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2001-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116628548","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 : 2001-10-14DOI: 10.1109/SFCS.2001.959886
B. Barak, Oded Goldreich, S. Goldwasser, Yehuda Lindell
Resettably-sound proofs and arguments maintain soundness even when the prover can reset the verifier to use the same random coins in repeated executions of the protocol. We show that resettably-sound zero-knowledge arguments for NP exist if collision-free hash functions exist. In contrast, resettably-sound zero-knowledge proofs are possible only for languages in P/poly. We present two applications of resettably-sound zero-knowledge arguments. First, we construct resettable zero-knowledge arguments of knowledge for NP, using a natural relaxation of the definition of arguments (and proofs) of knowledge. We note that, under the standard definition of proof of knowledge, it is impossible to obtain resettable zero-knowledge arguments of knowledge for languages outside BPP. Second, we construct a constant-round resettable zero-knowledge argument for NP in the public-key model, under the assumption that collision-free hash functions exist. This improves upon the sub-exponential hardness assumption required by previous constructions. We emphasize that our results use non-black-box zero-knowledge simulations. Indeed, we show that some of the results are impossible to achieve using black-box simulations. In particular, only languages in BPP have resettably-sound arguments that are zero-knowledge with respect to black-box simulation.
{"title":"Resettably-sound zero-knowledge and its applications","authors":"B. Barak, Oded Goldreich, S. Goldwasser, Yehuda Lindell","doi":"10.1109/SFCS.2001.959886","DOIUrl":"https://doi.org/10.1109/SFCS.2001.959886","url":null,"abstract":"Resettably-sound proofs and arguments maintain soundness even when the prover can reset the verifier to use the same random coins in repeated executions of the protocol. We show that resettably-sound zero-knowledge arguments for NP exist if collision-free hash functions exist. In contrast, resettably-sound zero-knowledge proofs are possible only for languages in P/poly. We present two applications of resettably-sound zero-knowledge arguments. First, we construct resettable zero-knowledge arguments of knowledge for NP, using a natural relaxation of the definition of arguments (and proofs) of knowledge. We note that, under the standard definition of proof of knowledge, it is impossible to obtain resettable zero-knowledge arguments of knowledge for languages outside BPP. Second, we construct a constant-round resettable zero-knowledge argument for NP in the public-key model, under the assumption that collision-free hash functions exist. This improves upon the sub-exponential hardness assumption required by previous constructions. We emphasize that our results use non-black-box zero-knowledge simulations. Indeed, we show that some of the results are impossible to achieve using black-box simulations. In particular, only languages in BPP have resettably-sound arguments that are zero-knowledge with respect to black-box simulation.","PeriodicalId":378126,"journal":{"name":"Proceedings 2001 IEEE International Conference on Cluster Computing","volume":"110 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2001-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131640662","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 : 2001-10-14DOI: 10.1109/SFCS.2001.959885
B. Barak
The simulation paradigm is central to cryptography. A simulator is an algorithm that tries to simulate the interaction of the adversary with an honest party, without knowing the private input of this honest party. Almost all known simulators use the adversary's algorithm as a black-box. We present the first constructions of non-black-box simulators. Using these new non-black-box techniques, we obtain several results that were previously proven to be impossible to obtain using black-box simulators. Specifically, assuming the existence of collision resistent hash functions, we construct a new zero-knowledge argument system for NP that satisfies the following properties: 1. This system has a constant number of rounds with negligible soundness error. 2. It remains zero knowledge even when composed concurrently n times, where n is the security parameter. Simultaneously obtaining 1 and 2 has been recently proven to be impossible to achieve using black-box simulators. 3. It is an Arthur-Merlin (public coins) protocol. Simultaneously obtaining 1 and 3 was known to be impossible to achieve with a black-box simulator. 4. It has a simulator that runs in strict polynomial time, rather than in expected polynomial time. All previously known constant-round, negligible-error zero-knowledge arguments utilized expected polynomial-time simulators.
{"title":"How to go beyond the black-box simulation barrier","authors":"B. Barak","doi":"10.1109/SFCS.2001.959885","DOIUrl":"https://doi.org/10.1109/SFCS.2001.959885","url":null,"abstract":"The simulation paradigm is central to cryptography. A simulator is an algorithm that tries to simulate the interaction of the adversary with an honest party, without knowing the private input of this honest party. Almost all known simulators use the adversary's algorithm as a black-box. We present the first constructions of non-black-box simulators. Using these new non-black-box techniques, we obtain several results that were previously proven to be impossible to obtain using black-box simulators. Specifically, assuming the existence of collision resistent hash functions, we construct a new zero-knowledge argument system for NP that satisfies the following properties: 1. This system has a constant number of rounds with negligible soundness error. 2. It remains zero knowledge even when composed concurrently n times, where n is the security parameter. Simultaneously obtaining 1 and 2 has been recently proven to be impossible to achieve using black-box simulators. 3. It is an Arthur-Merlin (public coins) protocol. Simultaneously obtaining 1 and 3 was known to be impossible to achieve with a black-box simulator. 4. It has a simulator that runs in strict polynomial time, rather than in expected polynomial time. All previously known constant-round, negligible-error zero-knowledge arguments utilized expected polynomial-time simulators.","PeriodicalId":378126,"journal":{"name":"Proceedings 2001 IEEE International Conference on Cluster Computing","volume":"82 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2001-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121688935","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 : 2001-10-14DOI: 10.1109/SFCS.2001.959897
Jittat Fakcharoenphol, Satish Rao
The authors present an O(n log/sup 3/ n) time algorithm for finding shortest paths in a planar graph with real weights. This can be compared to the best previous strongly polynomial time algorithm developed by R. Lipton et al., (1978 )which ran in O(n/sup 3/2/) time, and the best polynomial algorithm developed by M. Henzinger et al. (1994) which ran in O/spl tilde/(n/sup 4/3/) time. We also present significantly improved algorithms for query and dynamic versions of the shortest path problems.
{"title":"Planar graphs, negative weight edges, shortest paths, and near linear time","authors":"Jittat Fakcharoenphol, Satish Rao","doi":"10.1109/SFCS.2001.959897","DOIUrl":"https://doi.org/10.1109/SFCS.2001.959897","url":null,"abstract":"The authors present an O(n log/sup 3/ n) time algorithm for finding shortest paths in a planar graph with real weights. This can be compared to the best previous strongly polynomial time algorithm developed by R. Lipton et al., (1978 )which ran in O(n/sup 3/2/) time, and the best polynomial algorithm developed by M. Henzinger et al. (1994) which ran in O/spl tilde/(n/sup 4/3/) time. We also present significantly improved algorithms for query and dynamic versions of the shortest path problems.","PeriodicalId":378126,"journal":{"name":"Proceedings 2001 IEEE International Conference on Cluster Computing","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2001-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125521029","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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