G. F. Italiano. Amortized efficiency of a path retrieval data structure. Theoretical Computer Science, 48(2-- 3):273--281, 1986.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....worst case analysis. Furthermore we consider an intermediate model between worst case analysis and average case analysis: the semi random adversary introduced in [3] 1 Introduction Significant progress has been recently made in the design of algorithms and data structures for dynamic graphs [1, 5, 6, 8, 11, 12, 13, 16, 17, 18, 19, 20, 21, 24]. These data structures support insertions and deletions of edges and or nodes in a graph, in addition to several types of queries. The goal is to compute the new solution in the modified graph without having to recompute it from scratch. Usually, the sequence of insertions deletions of edges is ....

.... of O(ff) for undirected graphs Then, for an arbitrary sequence of insertions and connectivity (for undirected graphs) or reachability (for directed graphs) queries between a pair of vertices, the update amortized time for directed graphs is O(n) instead of O(ff(n; n) for undirected graphs [16, 19, 23]. If we consider deletions of edges there are solutions for special classes of graphs such as directed acyclic graphs [17] The fully dynamic problem has also been studied [11, 19, 21] but, to the best of our knowledge, no fully dynamic data structure exists for general directed graphs that, in ....

G.F. Italiano, Amortized efficiency of a path retrieval data structure, Theoret. Comp. Sci., 48, 273-281, 1986. 10

....number of nodes) Moreover, at any moment, the data structure can answer the following type of query in O(1) time: given two nodes in the graph, are these nodes 2 or 3 edge connected. I Introduction Recently there has been a growing interest in dynamic or on line graph algorithms (see e.g. [3, 8, 9, 10, 18]) A graph algorithm is called dynamic or on line if it maintains some information related to a graph while the graph is being changed (e.g. by inserting or deleting a node or an edge) A dynamic algorithm exploits a suitable data representation for a graph and uses information of the old graph to ....

....dynamic algorithm does not need to compute a new solution for the new graph from scratch, i.e. by using the new graph as input only, and a better performance may be expected compared to an algorithm that simply ecomputes . Dynamic algorithms are known for e.g. computing transitive closures (cf. [8, 9, 10], or cf. 17] for planar graphs) minimal spanning trees (cf. 3] incremental planarity testing (cf. 2] and maintaining shortest paths (cf. 18] One sometimes uses the term on line algorithm when only insertions (of nodes or edges) are allowed. This research was partially supported by ....

G.F. Italiano, Amortized efficiency of a path retrieval data structure, Theoretical Computer Science, 48, (1986), pp. 273-281.

....consider the problems of reachability from a start vertex and of maintaining transitive closure. Since one of the purposes of this document is to serve as a collection of optimization hints for compiler writers, some of the presented solutions and techniques are already known in the literature [3, 4, 6, 7, 2] and are described here just for the sake of completeness. We present original contributions from the algorithmic point of view in Section4, with the dynamic maintenance of topological ordering; in Section6, with a parallel implementation of an algorithm for the dynamic transitive closure. ....

....O(IGReach(s)l) Dynamic Reachability A very efficient dynamic solution for this problem is possible in case of either sequences of edges insertion, or sequences of deletions in a day. The solution proposed here is similar to the one proposed in [7] and has the same time and space complexity of [3, 4, 9]. The performances of the algorithms are the following: O(m) time for any sequence of edge insertions, that is costant amortized time per edge insertion starting from an empty graph; O(m) time for any sequence of edge deletions, that is costant amortized time per edge deletion finishing ....

G. F. Italiano. Amortized efficiency of a path retrieval data structure. Theoret. Cornput. Sci., 48:273 281, 1986.

....transitive closure datastructure. Note that Palsberg et al. use the same in [7] Worthy of note is the fact that it maintains an adjacency matrix M such that M [x; y] 1 if and only if there is a path from x to y. Furthermore, O(n 2 ) insert operations can be performed in O(n 3 ) time[6]. To maintain the transitive closed property of the adjacency matrix, insertion of an edge may result in the addition of many other edges. An insert operation returns a list R of all new edges added. HG a graph for storing [x; y] edges. This is implemented as an adjacency matrix. Q a set of (x; ....

....we note that P strictly increases on each iteration. Since P is bounded by 2n 2 , we know that the iteration step is executed at most 2n 2 times. Insertion of O(n 2 ) edges into ITC can be done in O(n 3 ) time, since the insertion operation for ITC has linear amortised time complexity [6]. We note that R returned by the insertion operation is a list of new edges added to ITC, and moreover the number of edges in ITC is bounded by n 2 so over the whole computation, step A5 appends at most n 2 elements to Q, taking O(n 2 ) time. The matrix PE has at most jEj 2 elements in its ....

G. F. Italiano. Amortized efficiency of a path retrieval data structure. Theoretical Computer Science, 48:273--281, 1986.

.... e Sistemistica, Universit a di Roma La Sapienza , Via Salaria 113 00198 Roma, Italy, ffrigioni,ioffreda,nanni,pasqualog dis.uniroma1.it 1 1 Introduction A lot of efforts have been done in the last years in order to devise efficient algorithms for dynamic graph problems (e.g. see [6, 9, 13, 14, 15, 16, 18, 20, 23, 24, 25, 26, 30, 31, 32]) motivated by theoretical as well as practical applications. In the literature, the most used dynamic model is the following: we are given a graph G and we want to answer queries on a property P of G, while the graph is changing due to insertions and deletions of edges. For instance, if the ....

....from scratch. These 2 considerations state that it is very hard to devise efficient fully dynamic solutions for the single source shortest paths problem in the standard cost models, and this holds also for other important dynamic graph problems, as, for example for transitive closure (see e.g. [24, 25]) 1.2 The output complexity model For the reasons given above, some researchers have tried to investigate the possibility of defining other cost models for the computational analysis of dynamic graph problems [4, 20, 30] One of them is the output complexity model, first considered by ....

G. F. Italiano. Amortized efficiency of a path retrieval data structure. Theoretical Computer Science, 48:273--281, 1986.

....T at most once. L4. A transaction T may begin by locking any SCC. Subsequently, 1 One can use incremental graph algorithms. Such algorithms can dynamically maintain certain kinds of information about a graph in the face of updates to the graph without recomputing the information from scratch (Italiano 1986; Italiano 1988) More information about the specific algorithms is given in Chapter 3, Section 3.3.2. 20 L5. All the nodes of an SCC are locked together in one step provided all the entry points of that SCC in the present state of G have been locked by T in the past and T is now holding a lock ....

Italiano, G. F. 1986. Amortized Efficiency of a Path Retrieval Data Structure. Theoretical Computer Science, 48(2,3):273--281.

....with a one sided error, its techniques are quite different from the algorithm presented here. Other related work includes partially dynamic algorithms. The best result for updates allowing edge insertions only is O(n) amortized time per inserted edge, and O(1) time per query by Italiano (1986)[12], and by La Poutre and van Leeuwen (1987) 16] This improved upon Ibaraki and Katoh s (1983) 11] algorithm with a total cost of O(n 3 ) for an arbitrary number of insertions. There is also Yellin s (1993) 19] algorithm, with a total cost of O(m Delta) for any number of insertions, where m is ....

G. F. Italiano, "Amortized efficiency of a path retrieval data structure", Theoretical Computer Science 48, 1986, pp. 273-281.

....is also Monte Carlo, its techniques are quite different from the algorithm presented here. Other related work includes partially dynamic algorithms. The best result for updates allowing only edge insertions is O(n) amortized time per inserted edge and O(1) time per query by Italiano (1986)[13], and by La Poutre and van Leeuwen (1987) 16] This improved upon Ibaraki and Katoh s (1983) 11] algorithm with running time O(n 3 ) for an arbitrary number of insertions. Also there is Yellin s (1993) 18] algorithm with cost O(m Delta) for m insertions, where m is the number of edges ....

G. F. Italiano, "Amortized efficiency of a path retrieval data structure", Theoretical Computer Science 48, 1986, pp. 273-281.

....base. The exact improvement will depend on the environment in which the knowledge 7 One can use incremental graph algorithms. Such algorithms can dynamically maintain certain kinds of information about a graph in the face of updates to the graph without recomputing the information from scratch (Italiano 1986; Italiano 1988) base will be used. We plan to explore this issue in our future research. From now on, we will assume that the knowledge base is always connected and has a single root. 3.3.2 Acquiring Locks The rules presented in this section are the core of the DDG policy as they specify how ....

Italiano, G. F. 1986. Amortized Efficiency of a Path Retrieval Data Structure. Theoretical Computer Science, 48(2,3):273--281.

....these values is shown in Figure 2.7. The use of constraints is optional, but they are usually beneficial for large designs, because the more constrained a design is, the quicker a good solution can be found. A sample constraints specification is shown in Figure 2.8. drl myLib f ALU 32 452 [12,28,15] [12,30,16] Multiplier 32 671 [34,82,61] g Figure 2.6. Sample datapath resource library (DRL) constraints name f max area = val max delay = val resource name = val . resource name = val g Figure 2.7. Input format for constraints. 16 constraints myCon f max area = 923 max delay = ....

....algorithm, it is possible to update the reachability of vertices more efficiently. A dynamic transitive closure algorithm developed by Cicerone [39] which is a generalization of another algorithm proposed by La Poutr e and van Leeuwen [37] is used. Other similar algorithms include Italiano s [28, 29] algorithm and Yellin s algorithm [44] The algorithm proposed by Cicerone uses a counting technique to solve the problem. Information on edges existing in the graph is maintained explicitly in an adjacency matrix. As resource edges are added or deleted, the adjacency matrix is updated to reflect ....

Italiano, G. F. Amortized efficiency of a path retrieval data structure. In Theoretical Computer Science (1986), pp. 48:273--281.

....materialized views upon updates. It is also closely related to the problem of partially evaluating definite logic programs [27] Finally, when restricted to standard transitive closure programs, our task can be viewed as solving the incremental transitive closure computation problem for graphs [10, 14, 20, 21]. More detailed comparison will be given in Section 6. In general, all these optimization approaches store extra information to reduce the time required for subsequent computations. In our case, we store the answer to the query in one database state (and possibly additional derived facts) to ....

....iterations, but at the price of using recursive algorithms. Incremental evaluation of arbitrary Datalog [16] An algorithm is given in [16] for transforming an arbitrary Datalog query into an incremental query for arbitrary updates, but which is not in general nonrecursive. Graph algorithms [21, 20, 26, 24]. Graph algorithms for on line evaluation of transitive closure of graphs are given in [21, 20] and a method to optimize transitive queries by using subtrees in graphs constructed in previous evaluations is presented in [24] The main difference is that they use more elaborate data structures and ....

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G.F. Italiano. Amortized efficiency of a path retrieval data structure. Theoretical Computer Science, 48:273-281, 1986.

....materialized views upon updates. It is also closely related to the problem of partially evaluating definite logic programs [26] Finally, when restricted to standard transitive closure programs, our task can be viewed as solving the incremental transitive closure computation problem for graphs [10, 14, 19, 20]. More detailed comparison will be given in Section 6. In general, all these optimization approaches store extra information to reduce the time required for subsequent computations. In our case, we store the answer to the query in one database state (and possibly additional derived facts) to ....

....for two classes of graphs (including the acyclic graphs) Incremental evaluation of arbitrary Datalog [15] An algorithm is given in [15] for transforming an arbitrary Datalog query into an incremental query for arbitrary updates, but which is not in general nonrecursive. Graph algorithms [20, 19, 25, 23]. Graph algorithms for on line evaluation of transitive closure of graphs are given in [20, 19] and a method to optimize transitive queries by using subtrees in graphs constructed in previous evaluations is presented in [23] The main difference is that they use more elaborate data structures and ....

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G.F. Italiano. Amortized efficiency of a path retrieval data structure. Theoretical Computer Science, 48:273-281, 1986.

....Algorithms 132 an edge (u; v) 2 E iff there is a u v path in G is called the transitive closure of G; G need not be explicitely stored. Let m denote jE j. We have implemented three algorithms for dynamic transitive closure in this release of the library, namely Italiano s algorithm [15, 16], Yellin s algorithm [25] and the algorithm of Cicerone et.al. 3] which is a generalization of another algorithm proposed by La Poutr e and van Leeuwen [18] All of these algorithms are partially dynamic: the incremental versions apply to any digraph, while the decremental ones apply only on ....

....one) Italiano s and Yellin s data structures support both types: a Boolean path query in O(1) time and a find path query in O( time, where is the number of edges of the reported path. The data structure of Cicerone et.al. 3] supports only Boolean path queries in O(1) time. The algorithms in [3, 15, 16] require O(n 2 ) space. In the following, let G 0 denote the initial digraph having n vertices and m 0 edges. Italiano s algorithm is initialized in O(n 2 nm 0 ) time, while Cicerone et.al. s algorithm is initialized in O(n 2 ) time. The Boolean path version of Yellin s algorithm is ....

[Article contains additional citation context not shown here]

G. F. Italiano. Amortized efficiency of a path retrieval data structure. Theoretical Computer Science 48:273--281, 1986.

.... Che84] spanning trees [SP73, CH78, FS84, Fre85] spanning forests [Wes89] shortest paths [Rod68, Che76, GSV78, Fuj81, CC82, Gaz83, EG85, AMSN89, AIMSN90, Ita91] biconnected components [Sac86, WT92, BT90] triconnected components [Ita91, BT90] transitive closure [IK83, Ita86, Ita88, LPv88, YS88, Yel91] planar graphs [Tam88, TP90, BT89, EIT 92, PT88] ffl computational geometry [Ov81, CBT 92] ffl data bases [ABJ89] ffl syntax directed editors and grammars [Rep82, RTD83, Rep88, ACR 87] ffl data flow analysis [Ryd83, RP88, RC86, CR88, Mar89, Bur90, ....

....can be extended to handle both additions and deletions, but it would appear that in this case the worst case performance could be quite bad, and the amortized performance no better since it would be possible to get in a situation where the worst case was repeated an arbitrary number of times. In [Ita86] Italiano introduced an algorithm that can process a sequence of n edge additions and searchpath operations in O(n) amortized time per operation. Hence a sequence of q = O(m) edge additions requires O(nm) time, an improvement over Ibaraki and Katoh s method when m n 2 . The operation ....

G. F. Italiano. Amortized efficiency of a path retrieval data structure. Theoretical Computer Science, 48:273--281, 1986. 112

....per edge deletion in dense digraphs. The data structure uses O(n 2 ) space. Semi dynamic algorithms were presented which improve this result. A data structure for arbitrary digraphs that support edge insertions in amortized O(n) time, queries in O(1) time, and use O(n 2 ) space are given in [64]. An extension of this result to finding regular paths for a regular language is given in [14] A semi dynamic algorithm for acyclic digraphs that support edge deletions in in amortized O(n) time, queries in O(1) time, and use O(n 2 ) space is presented in [65] La Poutr e and van Leeuwen [85] ....

G.F. Italiano, "Amortized Efficiency of a Path Retrieval Data Structure," Theoretical Computer Science 48 (1986), 273--281.

....path from u to v) However, in the dynamic realm the problem seems much harder. The best previously known dynamic results for general digraphs are semi dynamic data structures with O(n 2 ) space, constant query time, and O(n) update time. Data structures that support insertions only are given in [12,50,84]. Data structures that are restricted to acyclic digraphs and support only deletions can be found in [12,51] We can obtain better results if we restrict ourselves to specific classes of graphs. For series parallel digraphs, Italiano, Marchetti Spaccamela, and Nanni [52] gave an O(n) space data ....

G.F. Italiano, "Amortized efficiency of a path retrieval data structure," Theoretical Computer Science 48 (1986), 273--281.

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G. F. Italiano. Amortized efficiency of a path retrieval data structure. Theoretical Computer Science, 48(2-- 3):273--281, 1986.

....(currently 2:38 [2] Previous Work. For the incremental version of the problem, the first algorithm was proposed by Ibaraki and Katoh [11] in 1983: its running time was O(n 3 ) over any sequence of insertions. This bound was later improved to O(n) amortized time per insertion by Italiano [12] and also by La Poutr e and van Leeuwen [17] Yellin [19] gave an O(m ffi max ) algorithm for m edge insertions, where m is the number of edges in the final transitive closure and ffi max is the maximum out degree of the final graph. All these algorithms maintain explicitly the transitive ....

G. F. Italiano. Amortized efficiency of a path retrieval data structure. Theoretical Computer Science, 48(2--3):273--281, 1986.

....Italy. 1 Introduction In the last decade there has been a growing interest in dynamic problems on graphs. In particular, much attention has been devoted to the dynamic maintenance of connected components [14, 16, 43] and higher connectivity [8, 10, 18, 19, 20, 21, 32, 35, 53] transitive closure [29, 30, 31, 37, 47, 55], planarity [7, 8, 46] shortest paths [2, 5, 13, 39, 44] and minimum spanning trees [10, 11, 16] In these problems one would like to answer queries on graphs that are undergoing a sequence of updates, such as insertions and deletions of edges and vertices. The goal of a dynamic graph algorithm ....

G. F. Italiano. Amortized efficiency of a path retrieval data structure. Theoret. Comput. Sci., 48:273--281, 1986.

....(currently 2:736 [2] Previous Work. For the incremental version of the problem, the first algorithm was proposed by Ibaraki and Katoh [7] in 1983: its running time was O(n 3 ) over any sequence of insertions. This bound was later improved to O(n) amortized time per insertion by Italiano [8] and also by La Poutre and van Leeuwen [13] Yellin [15] gave an O(m ffi max ) algorithm for m edge insertions, where m is the number of edges in the final transitive closure and ffi max is the maximum out degree of the final graph. All these algorithms maintain explicitly the transitive ....

G. F. Italiano. Amortized efficiency of a path retrieval data structure. Theoretical Computer Science, 48(2-- 3):273--281, 1986.

....Grant CDA 9024735. z University of Venice Ca Foscari , Venice, Italy. Supported in part by the ESPRIT LTR Project no. 20244 (ALCOM IT) and by a Research Grant from University of Venice Ca Foscari . Most of the efficient data structures available for directed graphs are partially dynamic [2, 13, 29, 30, 31, 37, 39, 43, 53], and only preliminary results are available for fully dynamic problems [25] For this reason, an alternative viewpoint that has been proposed is to measure the complexity of a dynamic algorithm as a function of the output change [17, 40] The main dynamic problems considered on directed graphs ....

G. F. Italiano. Amortized efficiency of a path retrieval data structure. Theor. Comput. Sci., 48:273--281, 1986.

.... no better bound than O( p m ) is known for the corresponding fully dynamic problems [11] Moreover, despite intensive research on dynamic problems on graphs (such as dynamic maintenance of connectivity [7, 8, 10, 11, 14, 20, 22, 29, 30] 2 and 3 connectivity [7, 12, 29, 30] transitive closure [3, 4, 15, 16, 17, 18, 19, 31], planar graphs [6, 7, 19, 25] shortest paths [2, 9, 21, 24, 31] and minimum spanning trees [5, 8, 11, 24] there are very few graphtheoretic problems for which a fully dynamic non trivial algorithm is known. As mentioned in [30] the fully dynamic maintenance of the connected components of a ....

G. F. Italiano, "Amortized efficiency of a path retrieval data structure", Theoret. Comput. Sci. 48 (1986), 273--281.

....e is currently a spanning edge, and if so, which tree it belongs to. Dynamic problems on graphs have been extensively studied. Several algorithms have been proposed for maintaining fundamental structural information about dynamic graphs, such as connectivity [9, 10, 15, 24, 26] transitive closure [17, 18, 19, 20, 21, 34, 23], and shortest paths [1, 8, 25, 28, 34] Dynamic planar graphs arise in communication networks, graphics, and VLSI design, and they occur in algorithms that build planar subdivisions such as Voronoi diagrams. Algorithms have been proposed for maintaining the embedding of a planar graph [29] and ....

G. F. Italiano. Amortized efficiency of a path retrieval data structure. Theoret. Comput. Sci., 48:273--281, 1986.

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G. F. Italiano. Amortized efficiency of a path retrieval data structure. Theoretical Computer Science, 48:273--281, 1986.

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G. F. Italiano. Amortized efficiency of a path retrieval data structure. Theoretical Computer Science, 48:273--281, 1986.

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