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193
Fractional cascading: I. A data structuring technique
 Algorithmica
, 1986
"... Abstract. In computational geometry many search problems and range queries can be solved by performing an iterative search for the same key in separate ordered lists. In this paper we show that, if these ordered lists can be put in a onetoone correspondence with the nodes of a graph of degree d so ..."
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Cited by 174 (6 self)
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Abstract. In computational geometry many search problems and range queries can be solved by performing an iterative search for the same key in separate ordered lists. In this paper we show that, if these ordered lists can be put in a onetoone correspondence with the nodes of a graph of degree d so that the iterative search always proceeds along edges of that graph, then we can do much better than the obvious sequence of binary searches. Without expanding the storage by more than a constant factor, we can build a datastructure, called a fractional cascading structure, in which all original searches after the first can be carried out at only log d extra cost per search. Several results related to the dynamization of this structure are also presented. A companion paper gives numerous applications of this technique to geometric problems.
Modular Decomposition and Transitive Orientation
, 1999
"... A module of an undirected graph is a set X of nodes such for each node x not in X, either every member of X is adjacent to x, or no member of X is adjacent to x. There is a canonical linearspace representation for the modules of a graph, called the modular decomposition. Closely related to modular ..."
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Cited by 111 (12 self)
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A module of an undirected graph is a set X of nodes such for each node x not in X, either every member of X is adjacent to x, or no member of X is adjacent to x. There is a canonical linearspace representation for the modules of a graph, called the modular decomposition. Closely related to modular decomposition is the transitive orientation problem, which is the problem of assigning a direction to each edge of a graph so that the resulting digraph is transitive. A graph is a comparability graph if such an assignment is possible. We give O(n +m) algorithms for modular decomposition and transitive orientation, where n and m are the number of vertices and edges of the graph. This gives linear time bounds for recognizing permutation graphs, maximum clique and minimum vertex coloring on comparability graphs, and other combinatorial problems on comparability graphs and their complements.
Internet Packet Filter Management and Rectangle Geometry
, 2001
"... We consider rule sets for internet packet routing and filtering, where each rule consists of a range of source addresses, a range of destination addresses, a priority, and an action. A given packet should be handled by the action from the maximum priority rule that matches its source and destination ..."
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Cited by 87 (1 self)
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We consider rule sets for internet packet routing and filtering, where each rule consists of a range of source addresses, a range of destination addresses, a priority, and an action. A given packet should be handled by the action from the maximum priority rule that matches its source and destination. We describe new data structures for quickly finding the rule matching an incoming packet, in nearlinear space, and a new algorithm for determining whether a rule set contains any conflicts, in time O(n 3/2 ). 1 Introduction The working of the current Internet and its posited evolution depend on efficient packet filtering mechanisms: databases of rules, maintained at various parts of the network, which use patterns to filter out sets of IP packets and specify actions to be performed on those sets. Typical filter patterns are based on packet header information such as the source or destination IP addresses. The actions to be performed depend on where the packet filtering is performed i...
Topologically Sweeping Visibility Complexes via Pseudotriangulations
, 1996
"... This paper describes a new algorithm for constructing the set of free bitangents of a collection of n disjoint convex obstacles of constant complexity. The algorithm runs in time O(n log n + k), where k is the output size, and uses O(n) space. While earlier algorithms achieve the same optimal run ..."
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Cited by 85 (9 self)
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This paper describes a new algorithm for constructing the set of free bitangents of a collection of n disjoint convex obstacles of constant complexity. The algorithm runs in time O(n log n + k), where k is the output size, and uses O(n) space. While earlier algorithms achieve the same optimal running time, this is the first optimal algorithm that uses only linear space. The visibility graph or the visibility complex can be computed in the same time and space. The only complicated data structure used by the algorithm is a splittable queue, which can be implemented easily using redblack trees. The algorithm is conceptually very simple, and should therefore be easy to implement and quite fast in practice. The algorithm relies on greedy pseudotriangulations, which are subgraphs of the visibility graph with many nice combinatorial properties. These properties, and thus the correctness of the algorithm, are partially derived from properties of a certain partial order on the faces of th...
Membership in constant time and almostminimum space
 SIAM Journal on Computing
, 1999
"... Abstract. This paper deals with the problem of storing a subset of elements from the bounded universeM = {0,...,M−1} so that membership queries can be performed efficiently. In particular, we introduce a data structure to represent a subset of N elements of M in a number of bits close to the informa ..."
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Cited by 73 (3 self)
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Abstract. This paper deals with the problem of storing a subset of elements from the bounded universeM = {0,...,M−1} so that membership queries can be performed efficiently. In particular, we introduce a data structure to represent a subset of N elements of M in a number of bits close to the informationtheoretic minimum, B = lg
Marked Ancestor Problems
, 1998
"... Consider a rooted tree whose nodes can be marked or unmarked. Given a node, we want to find its nearest marked ancestor. This generalises the wellknown predecessor problem, where the tree is a path. ..."
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Cited by 62 (5 self)
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Consider a rooted tree whose nodes can be marked or unmarked. Given a node, we want to find its nearest marked ancestor. This generalises the wellknown predecessor problem, where the tree is a path.
Verification and Sensitivity Analysis Of Minimum Spanning Trees In Linear Time
 SIAM J. COMPUT
, 1992
"... Komlos has devised a way to use a linear number of binary comparisons to test whether a given spanning tree of a graph with edge costs is a minimum spanning tree. The total computational work required by his method is much larger than linear, however. We describe a lineartime algorithm for verifyi ..."
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Cited by 59 (1 self)
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Komlos has devised a way to use a linear number of binary comparisons to test whether a given spanning tree of a graph with edge costs is a minimum spanning tree. The total computational work required by his method is much larger than linear, however. We describe a lineartime algorithm for verifying a minimum spanning tree. Our algorithm combines the result of Komlos with a preprocessing and table lookup method for small subproblems and with a previously known almostlineartime algorithm. Additionally, we present an optimal deterministic algorithm and a lineartime randomized algorithm for sensitivity analysis of minimum spanning trees.
A THEORY OF ALTERNATING PATHS AND BLOSSOMS FOR PROVING CORRECTNESS OF THE O(√VE) GENERAL GRAPH MAXIMUM MATCHING ALGORITHM
, 1994
"... ..."
Undirected Single Source Shortest Paths in Linear Time
 J. Assoc. Comput. Mach
, 1997
"... The single source shortest paths problem (SSSP) is one of the classic problems in algorithmic graph theory: given a weighted graph G with a source vertex s, find the shortest path from s to all other vertices in the graph. Since 1959 all theoretical developments in SSSP have been based on Dijkstra& ..."
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Cited by 56 (3 self)
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The single source shortest paths problem (SSSP) is one of the classic problems in algorithmic graph theory: given a weighted graph G with a source vertex s, find the shortest path from s to all other vertices in the graph. Since 1959 all theoretical developments in SSSP have been based on Dijkstra's algorithm, visiting the vertices in order of increasing distance from s. Thus, any implementation of Dijkstra 's algorithm sorts the vertices according to their distances from s. However, we do not know how to sort in linear time. Here, a deterministic linear time and linear space algorithm is presented for the undirected single source shortest paths problem with integer weights. The algorithm avoids the sorting bottleneck by building a hierechical bucketing structure, identifying vertex pairs that may be visited in any order. 1 Introduction Let G = (V; E), jV j = n, jEj = m, be an undirected connected graph with an integer edge weight function ` : E ! N and a distinguished source vertex...
Greedy matchings
"... Suppose that each member of a set of n applicants ranks a subset of a set of m posts in strict order of preference. A matching is a set of (post, applicant) pairs such that each applicant and each post appears in at most one pair. A greedy matching is one in which the maximum possible number of appl ..."
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Cited by 54 (16 self)
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Suppose that each member of a set of n applicants ranks a subset of a set of m posts in strict order of preference. A matching is a set of (post, applicant) pairs such that each applicant and each post appears in at most one pair. A greedy matching is one in which the maximum possible number of applicants are matched to their rst choice post, and subject to that condition, the maximum possible number are matched to their second choice post, and so on. This is a relevant concept in any practical matching situation where the preferences are on only one side of the market. A greedy matching can be found by a transformation to the classical problem of maximum weight bipartite matching. However an exponentially decreasing sequence of weights must be assigned to the entries in each preference list, and this adversely a ects the complexity of the algorithm (and its performance in practice). Here, we describe a