Optimal parallel verification of minimum spanning trees in logarithmic time

We present the first optimal parallel algorithms for the verification and sensitivity analysis of minimum spanning trees. Our algorithms are deterministic and run inO(logn) time and require linear-work in the CREW PRAM model. These algorithms are used as a subroutine in the linear-work randomized algorithm for finding minimum spanning trees of Cole, Klein, and Tarjan. Research partially supported by a National Science Foundation Graduate Fellowship and by DIMACS (Center for Discrete Mathematics and Theoretical Computer Science), a National Science Foundation Science and Technology Center, Grant No. NSF-STC88-09648. Research at Princeton University was partially supported by the National Science Foundation, Grant No. CCR-8920505, the Office of Naval Research, Contract No. N00014-91-J-1463, and by DIMACS (Center for Discrete Mathematics and Theoretical Computer Science), a National Science Foundation Science and Technology Center, Grant No. NSF-STC88-09648.     

Planar geometric location problems

We present anO(n ² log³ n) algorithm for the two-center problem, in which we are given a setS ofn points in the plane and wish to find two closed disks whose union containsS so that the larger of the two radii is as small as possible. We also give anO(n ² log⁵ n) algorithm for solving the two-line-center problem, where we want to find two strips that coverS whose maximum width is as small as possible. The best previous solutions of both problems requireO(n ³) time.Pankaj Agarwal has been supported by DIMACS (Center for Discrete Mathematics and Theoretical Computer Science), an NSF Science and Technology Center, under Grant STC-88-09648. Micha Sharir has been supported by the Office of Naval Research under Grants N00014-89-J-3042 and N00014-90-J-1284, by the National Science Foundation under Grant CCR-89-01484, by DIMACS, and by grants from the US-Israeli Binational Science Foundation, the Fund for Basic Research administered by the Israeli Academy of Sciences, and the G.I.F., the German-Israeli Foundation for Scientific Research and Development. A preliminary version of this paper has appeared inProceedings of the Second Annual ACM-SIAM Symposium on Discrete Algorithms, 1991, pp. 449–458.     

A faster parallel algorithm for a matrix searching problem

We give an improved parallel algorithm for the problem of computing the tube minima of a totally monotonen ×n ×n matrix, an important matrix searching problem that was formalized by Aggarwal and Park and has many applications. Our algorithm runs inO(log logn) time withO(n²/log logn) processors in theCRCW-PRAM model, whereas the previous best ran inO((log logn)²) time withO(n²/(log logn)² processors, also in theCRCW-PRAM model. Thus we improve the speed without any deterioration in thetime ×processors product. Our improved bound immediately translates into improvedCRCW-PRAM bounds for the numerous applications of this problem, including string editing, construction of Huffmann codes and other coding trees, and many other combinatorial and geometric problems.This research was supported by the Office of Naval Research under Grants N00014-84-K-0502 and N00014-86-K-0689, the Air Force Office of Scientific Research under Grant AFOSR-90-0107, the National Science Foundation under Grant DCR-8451393, and the National Library of Medicine under Grant R01-LM05118. Part of the research was done while the author was at Princeton University, visiting the DIMACS center.     

Parallel algorithms for arrangements

We give the first efficient parallel algorithms for solving the arrangement problem. We give a deterministic algorithm for the CREW PRAM which runs in nearly optimal bounds ofO (logn log^* n) time andn ²/logn processors. We generalize this to obtain anO (logn log^* n)-time algorithm usingn ^d /logn processors for solving the problem ind dimensions. We also give a randomized algorithm for the EREW PRAM that constructs an arrangement ofn lines on-line, in which each insertion is done in optimalO (logn) time usingn/logn processors. Our algorithms develop new parallel data structures and new methods for traversing an arrangement.This work was supported by the National Science Foundation, under Grants CCR-8657562 and CCR-8858799, NSF/DARPA under Grant CCR-8907960, and Digital Equipment Corporation. A preliminary version of this paper appeared at the Second Annual ACM Symposium on Parallel Algorithms and Architectures [3].     

Ray shooting in polygons using geodesic triangulations

LetP be a simple polygon withn vertices. We present a simple decomposition scheme that partitions the interior ofP intoO(n) so-called geodesic triangles, so that any line segment interior toP crosses at most 2 logn of these triangles. This decomposition can be used to preprocessP in a very simple manner, so that any ray-shooting query can be answered in timeO(logn). The data structure requiresO(n) storage andO(n logn) preprocessing time. By using more sophisticated techniques, we can reduce the preprocessing time toO(n). We also extend our general technique to the case of ray shooting amidstk polygonal obstacles with a total ofn edges, so that a query can be answered inO( logn) time.Work by Bernard Chazelle has been supported by NSF Grant CCR-87-00917. Work by Herbert Edelsbrunner has been supported by NSF Grant CCR-89-21421. Work by Micha Sharir has been supported by ONR Grants N00014-89-J-3042 and N00014-90-J-1284, by NSF Grant CCR-89-01484, and by grants from the U.S.-Israeli Binational Science Foundation, the Fund for Basic Research administered by the Israeli Academy of Sciences, and the G.I.F., the German-Israeli Foundation for Scientific Research and Development.     

Incremental topological flipping works for regular triangulations

A set ofn weighted points in general position in ^d defines a unique regular triangulation. This paper proves that if the points are added one by one, then flipping in a topological order will succeed in constructing this triangulation. If, in addition, the points are added in a random sequence and the history of the flips is used for locating the next point, then the algorithm takes expected time at mostO(nlogn+n ^[d/2]). Under the assumption that the points and weights are independently and identically distributed, the expected running time is between proportional to and a factor logn more than the expected size of the regular triangulation. The expectation is over choosing the points and over independent coin-flips performed by the algorithm.The research of both authors was supported by the National Science Foundation under Grant CCR-8921421 and the research by the first author was also supported under the Alan T. Waterman award, Grant CCR-9118874. Any opinions, findings, conclusions, or recommendations expressed in this publication are those of the authors and do not necessarily reflect the view of the National Science Foundation.     

An improved technique for output-sensitive hidden surface removal

We derive a new output-sensitive algorithm for hidden surface removal in a collection ofn triangles, viewed from a pointz such that they can be ordered in an acyclic fashion according to their nearness toz. Ifk is the combinatorial complexity of the outputvisibility map, then we obtain a sophisticated randomized algorithm that runs in (randomized) timeO(n^4/3 log^2.89 n +k ^3/5 n ^{4/5 +} for any > 0. The method is based on a new technique for tracing the visible contours using ray shooting.Work by the first author was partially supported by the ESPRIT II Basic Research Actions Program of the EC, under Contract No. 3075 (project ALCOM). Work by the second author has been supported by Office of Naval Research Grant N00014-87-K-0129, by National Science Foundation Grant CCR-89-01484, and by grants from the U.S.-Israeli Binational Science Foundation, the NCRD-the Israeli National Council for Research and Development-and the Fund for Basic Research in Electronics, Computers, and Communication administered by the Israeli Academy of Sciences. A preliminary version of this paper appeared as part of the conference proceedings paper [17].     

Quasi-optimal upper bounds for simplex range searching and new zone theorems

This paper presents quasi-optimal upper bounds for simplex range searching. The problem is to preprocess a setP ofn points in ^d so that, given any query simplexq, the points inP q can be counted or reported efficiently. Ifm units of storage are available (n <m <n ^d ), then we show that it is possible to answer any query inO(n ¹⁺/m ^1/d ) query time afterO(m ¹⁺) preprocessing. This bound, which holds on a RAM or a pointer machine, is almost tight. We also show how to achieveO(logn) query time at the expense ofO(n ^d+) storage for any fixed > 0. To fine-tune our results in the reporting case we also establish new zone theorems for arrangements and merged arrangements of planes in 3-space, which are of independent interest.A preliminary version of this paper has appeared in theProceedings of the Sixth Annual ACM Symposium on Computational Geometry, June 1990, pp. 23–33. Work on this paper by Bernard Chazelle has been supported by NSF Grant CCR-87-00917 and NSF Grant CCR-90-02352. Work on this paper by Micha Sharir has been supported by Office of Naval Research Grant N00014-87-K-0129, by National Science Foundation Grants DCR-83-20085 and CCR-8901484, and by grants from the U.S.-Israeli Binational Science Foundation, the NCRD—the Israeli National Council for Research and Development, and the Fund for Basic Research administered by the Israeli Academy of Sciences. Work by Emo Welzl has been supported by Deutsche Forschungsgemeinschaft Grant We 1265/1–2. Micha Sharir and Emo Welzl have also been supported by a grant from the German-Israeli Binational Science Foundation. Last but not least, all authors thank DIMACS, an NSF Science and Technology Center, for additional support under Grant STC-88-09648.     

Lazy structure sharing for query optimization

We studylazy structure sharing as a tool for optimizing equivalence testing on complex data types. We investigate a number of strategies for implementing lazy structure sharing and provide upper and lower bounds on their performance (how quickly they effect ideal configurations of our data structure). In most cases when the strategies are applied to a restricted case of the problem, the bounds provide nontrivial improvements over the naïve linear-time equivalence-testing strategy that employs no optimization. Only one strategy, however, which employs path compression, seems promising for the most general case of the problem.Work completed while at Princeton University and supported by a Fannie and John Hertz Foundation Fellowship, National Science Foundation Grant No. CCR-8920505, and the Center for Discrete Mathematics and Theoretical Computer Science (DIMACS) under NSF-STC-91-19999.Work completed while at Princeton University and DIMACS and supported by DIMACS under NSF-STC-91-19999.Research at Princeton University partially supported by the National Science Foundation, Grant No. CCR-8920505, the Office of Naval Research, Contract No. N00014-91-J-1463, and by DIMACS under NSF-STC-91-19999.     

Generalized Voronoi diagrams for a ladder: II. Efficient construction of the diagram

We present a collection of algorithms, all running in timeO(n ² logn (n) ^o((n)3)) for some fixed integers(where (n) is the inverse Ackermann's function), for constructing a skeleton representation of a suitably generalized Voronoi diagram for a ladder moving in a two-dimensional space bounded by polygonal barriers consisting ofn line segments. This diagram, which is a two-dimensional subcomplex of the dimensional configuration space of the ladder, is introduced and analyzed in a companion paper by the present authors. The construction of the diagram described in this paper yields a motion-planning algorithm for the ladder which runs within the same time bound given above.Work on this paper has been supported in part by Office of Naval Research Grant N00014-82-K-0381, and by grants from the Digital Equipment Corporation, the Sloan Foundation, the System Development Foundation, the IBM corporation, and by National Science Foundation CER Grant No. DCR-8320085. Work by the second author has also been supported in part by a grant from the US-Israeli Binational Science Foundation.     

Efficient algorithms for minimum-cost flow problems with piecewise-linear convex costs

We present two efficient algorithms for the minimum-cost flow problem in which arc costs are piecewise-linear and convex. Our algorithms are based on novel algorithms of Orlin, which were developed for the case of linear arc costs. Our first algorithm uses the Edmonds-Karp scaling technique. Its complexity isO(M logU(m+n logM)) for a network withn vertices,m arcs, M linear cost segments, and an upper boundU on the supplies and the capacities. The second algorithm is a strongly polynomial version of the first, and it uses Tardos's idea of contraction. Its complexity isO(M logM(m+n logM)). Both algorithms improve by a factor of at leastM/m the complexity of directly applying existing algorithms to a transformed network in which arc costs are linear.The final stage of this work was performed while Ron Shamir was a visitor at DIMACS (Center for Discrete Mathematics and Theoretical Computer Science), Rutgers University. Supported in part by National Science Foundation Grant NSF-STC88-09648, and by Air Force Grants AFOSR-89-0512 and AFOSR-90-0008.     

A better performance guarantee for approximate graph coloring

Approximate graph coloring takes as input a graph and returns a legal coloring which is not necessarily optimal. We improve the performance guarantee, or worst-case ratio between the number of colors used and the minimum number of colors possible, toO(n(log logn)³/(logn)³), anO(logn/log logn) factor better than the previous best-known result.The work of the first author was supported by Air Force Grant AFOSR-86-0078 and NSF PYI Grant 8657527-CCR. The work of the second author was supported by a National Science Foundation Graduate Fellowship.     

A mildly exponential approximation algorithm for the permanent

A new approximation algorithm for the permanent of ann ×n 0,1-matrix is presented. The algorithm is shown to have worst-case time complexity exp(O(n ^1/2 log² n)). Asymptotically, this represents a considerable improvement over the best existing algorithm, which has worst-case time complexity exp((n)).Supported by SERC Grant GR/F 90363; work done in part while visiting DIMACS (Center for Discrete Mathematics and Computer Science).Supported by an NSF PYI grant, with matching equipment grant from the AT&T Foundation; work done in part while visiting DIMACS.     

An optimal algorithm for shortest paths on weighted interval and circular-arc graphs,with applications

We give the first linear-time algorithm for computing single-source shortest paths in a weighted interval or circular-arc graph, when we are given the model of that graph, i.e., the actual weighted intervals or circular-arcsand the sorted list of the interval endpoints. Our algorithm solves this problem optimally inO(n) time, wheren is the number of intervals or circular-arcs in a graph. An immediate consequence of our result is anO(qn + n logn)-time algorithm for the minimum-weight circle-cover problem, whereq is the minimum number of arcs crossing any point on the circle; then logn term in this time complexity is from a preprocessing sorting step when the sorted list of endpoints is not given as part of the input. The previously best time bounds were0(n logn) for this shortest paths problem, andO(qn logn) for the minimum-weight circle-cover problem. Thus we improve the bounds of both problems. More importantly, the techniques we give hold the promise of achieving similar (logn)-factor improvements in other problems on such graphs.The research of M. J. Atallah was supported in part by the Leonardo Fibonacci Institute, Trento, Italy, by the Air Force Office of Scientific Research under Contract AFOSR-90-0107, and by the National Science Foundation under Grant CCR-9202807. D. Z. Chen's research was supported in part by the Leonardo Fibonacci Institute, Trento, Italy. The research of D. T. Lee was supported in part by the Leonardo Fibonacci Institute, Trento, Italy, by the National Science Foundation, and the Office of Naval Research under Grants CCR-8901815, CCR-9309743, and N00014-93-1-0272.     

Randomized competitive algorithms for the list update problem

We prove upper and lower bounds on the competitiveness of randomized algorithms for the list update problem of Sleator and Tarjan. We give a simple and elegant randomized algorithm that is more competitive than the best previous randomized algorithm due to Irani. Our algorithm uses randomness only during an initialization phase, and from then on runs completely deterministically. It is the first randomized competitive algorithm with this property to beat the deterministic lower bound. We generalize our approach to a model in which access costs are fixed but update costs are scaled by an arbitrary constantd. We prove lower bounds for deterministic list update algorithms and for randomized algorithms against oblivious and adaptive on-line adversaries. In particular, we show that for this problem adaptive on-line and adaptive off-line adversaries are equally powerful.A preliminary version of these results appeared in a joint paper with S. Irani in theProceedings of the 2nd Symposium on Discrete Algorithms, 1991 [17].This research was partially supported by NSF Grants CCR-8808949 and CCR-8958528.This research was partially supported by NSF Grant CCR-9009753.This research was supported in part by the National Science Foundation under Grant CCR-8658139, by DIMACS, a National Science Foundation Science and Technology center, Grant No. NSF-STC88-09648.     

An efficient algorithm for maxdominance, with applications

Given a planar setS ofn points,maxdominance problems consist of computing, for everyp S, some function of the maxima of the subset ofS that is dominated byp. A number of geometric and graph-theoretic problems can be formulated as maxdominance problems, including the problem of computing a minimum independent dominating set in a permutation graph, the related problem of finding the shortest maximal increasing subsequence, the problem of enumerating restricted empty rectangles, and the related problem of computing the largest empty rectangle. We give an algorithm for optimally solving a class of maxdominance problems. A straightforward application of our algorithm yields improved time bounds for the above-mentioned problems. The techniques used in the algorithm are of independent interest, and include a linear-time tree computation that is likely to arise in other contexts.The research of this author was supported by the Office of Naval Research under Grants N00014-84-K-0502 and N00014-86-K-0689, and the National Science Foundation under Grant DCR-8451393, with matching funds from AT&T.This author's research was supported by the National Science Foundation under Grant DCR-8506361.     

Space-bounded simulation of multitape turing machines

A new proof of a theorem of Hopcroft, Paul, and Valiant is presented: Every deterministic multitape Turing machine of time complexityT(n) can be simulated by a deterministic Turing machine of space complexityT(n)/logT(n). The proof includes an overlap argument.Supported by National Science Foundation Grant No. MCS-78-04343.Supported by National Science Foundation Grant No. MCS-77-19754 and the Fannie and John Hertz Foundation.     

Efficient parallel algorithms forr-dominating set andp-center problems on trees

We develop efficient parallel algorithms for ther-dominating set and thep-center problems on trees. On a concurrent-read exclusive-write PRAM, our algorithm for ther-dominating set problem runs inO(logn log logn) time withn processors. The algorithm for thep-center problem runs inO(log² n log logn) time withn processors.Xin He was supported in part by an Ohio State University Presidential Fellowship, and by the Office of Research and Graduate Studies of Ohio State University. Yaacov Yesha was supported in part by the National Science Foundation under Grant No. DCR-8606366.     

A linear-time algorithm for finding an ambitus

We devise a linear-time algorithm for finding an ambitus ín an undirected graph. An ambitus is a cycle in a graph containing two distinguished vertices such that certain different groups of bridges (calledB ^itp-,B ^itQ-, andB ^itPQ-bridges) satisfy the property that a bridge in one group does not interlace with any bridge in the other groups. Thus, an ambitus allows the graph to be cut into pieces, where, in each piece, certain graph properties may be investigated independently and recursively, and then the pieces can be pasted together to yield information about these graph properties in the original graph. In order to achieve a good time-complexity for such an algorithm employing the divide-and-conquer paradigm, it is necessary to find an ambitus quickly. We also show that, using ambitus, linear-time algorithms can be devised for abiding-path-finding and nonseparating-induced-cycle-finding problems.The research of B. Mishra was supported in part by National Science Foundation Grants DMS-8703458 and CCR-9002819. R. E. Tarjan's research at Princeton University was partially supported by DIMACS, a National Science Foundation Science and Technology Center, Grant No. NSF-STC88-09648, and by National Science Foundation Grant CCR-8929505.     

Expected parallel time and sequential space complexity of graph and digraph problems

This paper determines upper bounds on the expected time complexity for a variety of parallel algorithms for undirected and directed random graph problems. For connectivity, biconnectivity, transitive closure, minimum spanning trees, and all pairs minimum cost paths, we prove the expected time to beO(log logn) for the CRCW PRAM (this parallel RAM machine allows resolution of write conflicts) andO(logn · log logn) for the CREW PRAM (which allows simultaneous reads but not simultaneous writes). We also show that the problem of graph isomorphism has expected parallel timeO(log logn) for the CRCW PRAM andO(logn) for the CREW PRAM. Most of these results follow because of upper bounds on the mean depth of a graph, derived in this paper, for more general graphs than was known before.For undirected connectivity especially, we present a new probabilistic algorithm which runs on a randomized input and has an expected running time ofO(log logn) on the CRCW PRAM, withO(n) expected number of processors only.Our results also improve known upper bounds on the expected space required for sequential graph algorithms. For example, we show that the problems of finding connected components, transitive closure, minimum spanning trees, and minimum cost paths have expected sequential spaceO(logn · log logn) on a deterministic Turing Machine. We use a simulation of the CRCW PRAM to get these expected sequential space bounds.This research was supported by National Science Foundation Grant DCR-85-03251 and Office of Naval Research Contract N00014-80-C-0647.This research was partially supported by the National Science Foundation Grants MCS-83-00630, DCR-8503497, by the Greek Ministry of Research and Technology, and by the ESPRIT Basic Research Actions Project ALCOM.