Even, S. and Y. Shiloach. 1981. An on-line edge deletion problem. Journal of the Association of Computing Machinery 28:1--4.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the labels assigned to the pixels to reflect the change in the components. Henceforth, this problem will be referred to as the MCC (Maintaining Connected Components) problem. Maintaining connected components problem in graphs that are dynamically chang ing has been dealt with by many researchers [2, 5, 27]. Eppstein [4] gave a lower bound of ( for any algorithm maintaining connected components in array rep resentation. For dynamically changing inputs, the traditional approach is to repeat the computation for each change. This approach is called the batch approach [17] In most cases, dynamically ....

....standard BFS is implemented as queue. In the interleaved breadth first search(IBFS) we have a pair of nodes which serve as seed for two separate BFS processes respectively. The two BFS processes visit the nodes starting from the seeds, in an interleaved fashion. This technique is discussed in in [5]. Here we present a modification of the technique to suit our purpose. Suppose for region 1, we get a set Sa of neighboring nodes in all the directions. Out of the nodes in Sa, our objective is to find those that are connected and those that are disconnected. For each component except the largest ....

[Article contains additional citation context not shown here]

Even. S., and Shiloach. Y. An On-Line Edge Deletion Problem. J. A CM, 28(1):1 4, January 1981.

....structures is O(n ) 2 Overview of our algorithms Our algorithms use an ecient data structure for maintaining the set of vertices lying within distance d from a given vertex under deletion of edges. This objective can be achieved if we can maintain BFS tree up to depth d. Even and Shiloach [4] had given an algorithm for maintaining a BFS tree under deletion of edges in an undirected graph; it was later shown by King and Henzinger [6] that the algorithm can be easily extended to directed graph as well while maintaining the same amortized update time of O(d) per edge deletion. Based on ....

....operation updates the BFS tree T and returns the set X of vertices that earlier lay within distance d from v but have increased their distance from v (due to deletion of edge e) Amortized cost of this operation will be O(d) in addition to O(jXj) time required to report the set X. Lemma 2. 1 [6, 4] For a vertex v and distance d n, there exist a data structure T d for maintaining BFS tree from a vertex v under deletion of edges such that all the four operations above can be performed within the stated time bounds. We will denote the data structure T as out tree of depth d rooted at v. ....

S. Even and Y. Shiloach. An on-line edge-deletion problem. Journal of association for computing machinery, 28:1-4, 1981.

....Thorup. A lower bound of Omega Gamma 20 n= log log n) for fully dynamic connectivity was proved independently by Fredman and Henzinger [6] and Miltersen, Subramanian, Vitter, and Tamassia [14] The decremental connectivity problem has been considered of independent interest by Even and Shiloach [3], but their bounds have been surpassed by the above mentioned fully dynamic algorithms. Recently, the following deterministic bound was obtained by Henzinger, King, and Warnow [10] starting from a graph with n nodes and m edges, suppose that the edges are deleted in batches (with no queries in ....

.... with n isolated vertices, we can maintain a spanning forest on line during m insertions and deletions in total time O(m log n) We shall refer to the algorithm of Theorem 3 as the HTK algorithm, as it is based on the algorithm of Henzinger and King [9] First we have Lemma 4 (Even Shiloach [3]) If a spanning forest is maintained during edge deletions, at an extra total cost of O(n log n) for the deletions, we can support connectivity queries in constant time. Thus, we can restrict our attention to the maintenance of a spanning tree. 4 Description A: The components of a data ....

S. Even and Y. Shiloach. An On-Line Edge-Deletion Problem. Journal of ACM, 28(1) 1981, 1--4.

....to find all such components, or simply just one. The batch deletion problem may be done by brute force. After each batch, simply compute connectivity. If there were b batches of deletions, this would result in bm running time, regardless of the value of b0. A 1981 algorithm by Even and Shiloach [3] considered the dynamic graph problem for in dividual deletions only. Over the course of deleting every edge from a graph, their algorithm spends O(m log n) time processing individual deletions each of which results in a new component and O(mn) time processing the remaining deletions. Thus, if ....

....edges for connectivity, we can determine exactly which pairs of newly deleted edges have become separated. For each such newly separated pair, we proceed to search from each endpoint, alternating between the two searches, until one component is completely visited (this technique derives from [3]) It is not hard to see that the searches for components cost in total O(m log n) time during the course of the algorithm. If we use the best deterministic fully dynamic algorithm for connectivity, then the m queries and O(m) updates cost O(mx ) time; while if we use the randomized fully dynamic ....

S. Even and Y. Siftloach, An on-line edge deletion problem, J. ACM, vol. 28, No. 1 (1981), pp. 1-4.

....paper we consider on line decremental connectivity in trees. Let T be a tree with n nodes from which edges are deleted interspersed with m on line connectivity queries. Two nodes are connected if and only if none of the edges on the unique path between them have been deleted. Even and Shiloach [1] gave an O(n log n m) algorithm to solve this problem. In this paper we give an O(n m) algorithm for the same problem. In section 2 we combine standard tree techniques and the result due to Even and Shiloach, in order to reduce the problem to connectivity in trees of small size. The reduced ....

S. Even and Y. Shiloach. An on-line edge-deletion problem. Journal of the Association for Computing Machinery, 28:1-4, 1981.

....leads to a running time of O( P i m 2 i ) We present an algorithm with time O( p l P i m 1:5 i ) For constant l our algorithm takes time O( P i m 1:5 i ) Lock Step Search Both improved algorithms use lock step search. Lock step search was introduced by Even and Shiloach [2] to maintain the connected components of an undirected graph under edge deletions. After the deletion of the edge (u; v) they start a (depth first) search for the resulting component in parallel at u and at v: the 3 algorithm executes one step of the search at u and one step of the search at v ....

S. Even and Y. Shiloach, "An On-Line Edge-Deletion Problem", J. ACM 28 (1981), 1--4.

....i is maintained, where Count i [x; y] is the number of nodes v such that x 2 In i (v) and y 2 Out i (v) Now, E i is maintained so as to contain edges (x; y) such that Count i Gamma1 [x; y] 0. Trees In i (v) and Out i (v) are maintained for any node v by means of deletions only data structures [6] which are rebuilt from scratch after each v centered insertion of edges. Our data structures for polynomials over Boolean matrices P i play the same role as King s forests F i of In i and Out i trees and of counters Count i . 30 While King s data structures require O(n 3 Delta log n) ....

S. Even and Y. Shiloach. An on-line edge-deletion problem. Journal of the ACM, 28:1--4, 1981.

....y Work partially supported by NSF Grant CCR 9014605. z On leave from Universit a di Roma, 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 ....

S. Even and Y. Shiloach. An on-line edge deletion problem. J. Assoc. Comput. Mach., 28:1--4, 1981.

....S i:x2S i out(x; n=2 i ) and In(x) S i:x2S i in(x; n=2 i ) 2. For each u 2 V the sets out(u; log 2 n) and in(u; log 2 n) are maintained. The out(x; k) and in(x; k) sets are maintained in a decremental environment using a (modi cation of a) technique proposed by Even and Shiloach [16] for undirected graphs. Each set is called a BFS structure, since it implicitly maintains a spanning tree for the descendants of x as it would have been computed by a BFS algorithm. The idea is to maintain for each vertex j that belongs to such a structure T a set Up(j) that stores the incoming ....

S. Even and Y. Shiloach. An on-Line edge deletion problem. Journal of the ACM, 28:1-4, 1981.

....received. Similarly, each partial solution (as well as the final one) may need to be returned as soon as it is available [32, 40, 56] It is helpful to note here that, when no deadlines are imposed, computations for which inputs arrive while the algorithm is in progress are referred to as on line [27, 33, 35, 36], incremental [21, 22, 49, 59] dynamic [11, 12, 67] and updating [20, 23, 28, 38, 53, 54, 62, 66] It is also important to note that our definition, while striving to be as general as possible, is particularly suitable for our purposes in this paper. Many other definitions exist; see, for ....

S. Even and Y. Shiloach, An on-line edge deletion problem, Journal of the ACM , 28, 1982, 1--4.

....information around a spanning forest. However, finding a generalization that worked was rather delicate, particularly for biconnectivity, where we needed to make a careful recycling of information, leading to the first polylogarithmic algorithm for this problem. The reader is referred to [5, 7, 8, 10, 11, 22] for discussions of other problems that get improved bounds by our new fully dynamic graph algorithms. 47 Chapter 6 Connectivity In this chapter, we present a simple O(log 2 n) time deterministic fully dynamic algorithm for graph connectivity. First we give a high level description, ignoring ....

S. Even and Y. Shiloach. An on-line edge-deletion problem. Journal of the ACM, 28(1):1--4, January 1981.

....kind of tree encoding, and applied it to the reverse problem of the above; namely, that of answering m connectivity queries to a forest F over n nodes as the edges are deleted. Again, they achieve a linear time algorithm, but this time, the best previous algorithm is that of Even and Shiloach [2] with running time O(m n log n) Like the standard union nd algorithm, Even and Shiloach s algorithm is very simple, likely with very small constants, but log is a suciently fast growing function, that there is a hope of beating it for n not too large. E mail:stephen itu.dk. The IT ....

.... Department of Computer Science, University of Copenhagen, Universitetsparken 1, 2100 Copenhagen, Denmark z E mail:mthorup research.att.com AT T Labs Research For this paper, we implemented and compared the ASS algorithm of Alstrup et al. 1] and the ESalgorithm of Even and Shiloach [2]. Quite interestingly, the ASS algorithm started winning already around 1000 nodes. This emphasizes the importance of being very careful dismissing creative usage of word operations as being just a theoretical game, without any practical relevance. The operations are there in standard programming ....

S. Even and Y. Shiloach. An on-line edge-deletion problem. JACM, 28:1-4, 1981.

....found on level i then a tree of F i which was split by the deletion remains split. We use the following: Claim 2. 1 O( P w(T 1 ) summed over all smaller components T 1 which split from a tree T on any given level during all Replace operations is O(w(T ) log n) The proof of the claim follows [5]. The first time a smaller component T 1 of a tree T is searched, it can have weight no greater than w(T ) 2. Between two successive times that L f (i) v) contributes to the weight of a smaller component T 1 and that component splits off, the weight of a smaller component T 1 containing v ....

S. Even and Y. Shiloach, "An On-Line Edge-Deletion Problem", J. ACM 28 (1981), 1--4.

....in the worst case. We present an algorithm that is similar to that of Xie and Beerel but takes (n log n) steps in the worst case. Our algorithm depends critically on lockstep search, the idea of performing backward and forward searches in the graph simultaneously. Lockstep search is used in [ES81] to update the connected components in an undirected graph after an edge has been deleted, and in [HT96] to decide Streett emptiness. Although these papers contain a complexity analysis that is similar to ours, they use lockstep search in a di erent fashion: to recompute (strongly) connected ....

S. Even and Y. Shiloach. An on-line edge-deletion problem. Journal of the Association for Computing Machinery, 28(1):1-4, January 1981.

....received. Similarly, each partial solution (as well as the final one) may need to be returned as soon as it is available [31, 39, 55] It is helpful to note here that, when no deadlines are imposed, computations for which inputs arrive while the algorithm is in progress are referred to as on line [26, 32, 34, 35], incremental [20, 21, 48, 58] dynamic [10, 11, 66] and updating [19, 22, 27, 37, 52, 53, 61, 65] It is also important to note that our definition, while striving to be as general as possible, is particularly suitable for our purposes in this paper. Many other definitions exist; see, for ....

S. Even and Y. Shiloach, An on-line edge deletion problem, Journal of the ACM , 28, 1982, 1--4.

....of batches which do not result in a new component. The batch deletion problem can be solved by computing connectivity from scratch after each batch. If there were b batches of deletions, this would result in bm running time, regardless of the value of b 0 . A 1981 algorithm by Even and Shiloach [14] considered the dynamic graph problem for individual deletions only. Over the course of deleting every edge from a graph, their algorithm spends O(m log n) time processing individual deletions each of which results in a new component and O(mn) time processing the remaining deletions. Thus, if ....

....all edges in that batch from our data structure one by one. After each edge (a; b) is deleted, we query the pair a; b. If a and b are no longer connected, we search from both a and b, alternating between the two searches, until one component is completely visited (this technique derives from [14]) We relabel the nodes of the completely visited component. The length of the search is thus proportional to the number of edges in the component with the fewer number of edges. Since each edge is in the component with the fewer edges at most log m times over the course of the algorithm, each ....

S. Even and Y. Shiloach, An on-line edge deletion problem, J. ACM, Vol. 28, No. 1 (1981), pp. 1-4.

....the decremental connectivity problem for trees: given an undirected tree T = V; E) all jEj = n 1 edges are dynamically deleted while m intermixed connectivity queries have to be answered. This problem has been investigated in several papers, see e.g. the references in [7] Shiloach and Even [1] have presented an algorithm that solves the problem in O(n log n m) time. We shall present a O(n m) algorithm for the same problem. In section 2 we will review the O(n log n) algorithm, in section 3 we combine this algorithm with standard tree techniques to reduce the problem to connectivity ....

....of Even Shiloach s algorithm in section 6. Finally in section 7 we briefly discuss applications of the algorithm. 2 Shiloach and Even s algorithm Given a tree of size n, a O(n log n m) algorithm for solving the on line decremental connectivity problem was described by Even and Shiloach [1]. It can be summarised thus: Algorithm 1 (Even Shiloach) Preprocessing. Each node in the tree is assigned an identical number, indicating full initial connectivity. Deletion. When an edge is removed from some tree, the two resulting trees are simultaneously traversed until the smaller tree S ....

[Article contains additional citation context not shown here]

S. Even and Y. Shiloach. An on-line edge-deletion problem. Journal of the Association for computing Machinery, 28:1--4, 1981.

....Thorup. A lower bound of Omega Gamma 20 n= log log n) for fully dynamic connectivity was proved independently by Fredman and Henzinger [6] and Miltersen, Subramanian, Vitter, and Tamassia [14] The decremental connectivity problem has been considered of independent interest by Even and Shiloach [3], but their bounds have been surpassed by the above mentioned fully dynamic algorithms. Recently, the following deterministic bound was obtained by Henzinger, King, and Warnow [10] starting from a graph with n nodes and m edges, suppose that the edges are deleted in batches (with no queries in ....

.... with n isolated vertices, we can maintain a spanning forest on line during m insertions and deletions in total time O(m log 2 n) We shall refer to the algorithm of Theorem 3 as the HTK algorithm, as it is based on the algorithm of Henzinger and King [9] First we have Lemma 4 (Even Shiloach [3]) If a spanning forest is maintained during edge deletions, at an extra total cost of O(n log n) for the deletions, we can support connectivity queries in constant time. Thus, we can restrict our attention to the maintenance of a spanning tree. Description A: The components of a data structure ....

S. Even and Y. Shiloach. An On-Line Edge-Deletion Problem. Journal of ACM, 28(1) 1981, 1--4.

.... Similarly, each partial solution (as well as the final one) may need to be returned as soon as it is available [21, 28, 37] It is helpful to note here that, when no deadlines are imposed, computations for which inputs arrive while the algorithm is in progress are referred to as on line [18, 22, 23, 24], incremental [14, 15, 32, 39] dynamic [7, 8, 44] and updating [13, 16, 19, 26, 34, 35, 41, 43] 2.2 Real time optimization The first example of a computation for which a parallel solution is consistently better than a sequential one was provided by real time optimization. The realtime ....

S. Even and Y. Shiloach, An on-line edge deletion problem, Journal of the ACM , 28, 1982, 1--4.

....update. Our technique relies on some of the same intuition as was used in Henzinger and King [6] in their randomized algorithm. Our deterministic algorithm is, however, much simpler, and in contrast to their algorithm, it generalizes to the minimum spanning tree problem. The reader is referred to [1, 3, 4, 6] for discussions of problems that get improved by our improvements for the fully dynamic connectivity and minimum spanning tree problems. 2 Connectivity In this section, we present an O(log 2 n) time deterministic fully dynamic algorithm for graph connectivity. First we give a high level ....

Shimon Even and Yossi Shiloach. An on-line edge-deletion problem. Journal of the ACM, 28(1):1--4, January 1981.

....paper we consider on line decremental connectivity in trees. Let T be a tree with n nodes from which edges are deleted interspersed with m on line connectivity queries. Two nodes are connected if and only if none of the edges on the unique path between them have been deleted. Even and Shiloach [1] gave an O(n log n m) algorithm to solve this problem. In this paper we give an O(n m) algorithm for the same problem. In section 2 we combine standard tree techniques and the result due to Even and Shiloach, in order to reduce the problem to connectivity in trees of small size. The reduced ....

S. Even and Y. Shiloach. An on-line edge-deletion problem. Journal of the Association for Computing Machinery, 28:1--4, 1981.

....of batches which do not result in a new component. The batch deletion problem can be solved by computing connectivity from scratch after each batch. If there were b batches of deletions, this would result in bm running time, regardless of the value of b 0 . A 1981 algorithm by Even and Shiloach [6] considered the dynamic graph problem for individual deletions only. Over the course of deleting every edge from a graph, their algorithm spends O(m log n) time processing individual deletions each of which results in a new component and O(mn) time processing the remaining deletions. Thus, if ....

....we delete all edges in that batch from our data structure one by one. After each edge is deleted, we query the edge. If its endpoints are no longer connected, we search from each endpoint, alternating between the two searches, until one component is completely visited (this technique derives from [6]) We relabel its nodes. The length of the search is thus proportional to the number of edges incident to the component with the fewer number of edges. Since each edge is incident to the component with the fewer number of edges at most log m times over the course of the algorithm, each edge is ....

S. Even and Y. Shiloach, An on-line edge deletion problem, J. ACM, Vol. 28, No. 1 (1981), pp. 1-4.

....leads to a running time of O( P i m 2 i ) We present an algorithm with time O( p l P i m 1:5 i ) For constant l our algorithm takes time O( P i m 1:5 i ) Lock Step Search Both improved algorithms use lock step search. Lock step search was introduced by Even and Shiloach [2] to maintain the connected components of an undirected graph under edge deletions. After the deletion of the edge (u; v) they start a (depth first) search for the resulting component in parallel at u and at v: the algorithm executes one step of the search at u and one step of the search at v ....

S. Even and Y. Shiloach, "An On-Line Edge-Deletion Problem", J. ACM 28 (1981), 1--4.

....vertex v in a microtree, a bit string that encodes the vertices on the path from v to the root of its microtree. They show how to answer connectivity queries using a constant number of bitwise operations on these bit strings and applying the Even and Shiloach decremental connectivity algorithm [11] to the upper part of T . The Alstrup, Secher, and Spork algorithm [4] runs on a pointer machine: since the connectivity queries return yes no answers, they need not index tables to recover the answers. In contrast, while their method can be extended to solve the off line LCAs problem in linear ....

S. Even and Y. Shiloach. An on-line edge deletion problem. J. ACM, 28(1):1--4, 1981.

....for maintaining a bfs tree of a directed graph in any dynamic or semi dynamic setting. On the other hand, the additional topological constraints of a bfs tree with respect to a sssp tree implies that the problem of dynamically maintain a bfs tree could be harder than maintaining a sssp tree. In [5] an algorithm is given for maintaining the bfs structure of an undirected graph under edge deletions in O(n) amortized time, where a bfs structure is the union of all possible shortest paths from a fixed vertex. Note that their approach does not allow to maintain a bfs tree within the same time ....

S. Even and Y. Shiloach, An On-Line Edge Deletion Problem, Journal of the ACM, 28 (1981), 1--4.

....of computing a solution. The newly arrived data must be incorporated in the solution at hand. The final solution is to be returned by a certain deadline. Real time computations form a subclass of a larger class of problems known variably as on line, incremental , dynamic, and updating computations [7, 12, 13, 14, 17, 18, 19, 20, 21, 22, 24, 25, 30, 31, 32, 36, 38, 40, 41]. What distinguishes a real time problem from problems in the larger class is the presence of deadlines by which the input is to be processed, by which the output is to be produced, and so on. 3 Modern Cryptography The purpose of contemporary cryptography is the protection of digital ....

S. Even and Y. Shiloach, An on-line edge deletion problem, Journal of the ACM , 28, 1982, 1--4.

.... operation [11] using the dynamic trees of Sleator and Tarjan [22, 23] However, in both cases 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 ....

S. Even, and Y. Shiloach, "An on-line edge deletion problem", J. Assoc. Comput. Mach. 28 (1981), 1--4.

....Award. y Department of Computer Science, University of Victoria, Victoria, BC. Email: val csr.uvic.ca. This research was supported by an NSERC Grant. 1 Throughout the paper the logarithms are base 2. Previous Work. In recent years a lot of work has been done in fully dynamic algorithms (see [1, 3, 4, 6, 7, 8, 10, 11, 12, 15, 16, 18] for connectivity related work in undirected graphs) There is also a large body of work for restricted classes of graphs and for insertions only algorithms. Currently the best time bounds for fully dynamic algorithms in undirected n node graphs are: O( p n) per update for a minimum spanning ....

S. Even and Y. Shiloach, "An On-Line Edge-Deletion Problem", J. ACM 28 (1981), 1--4.

.... w(T 1 ) log ng ( m 0 in =k) c) log n k 0 ) O( m 0 in =k) c) log n k 0 ) Claim 1 O( P w(T 1 ) summed over all smaller components T 1 which split from a tree T on any given level during all Replace operations is O(w(T ) log n) The proof of the claim is not hard and follows [5]. The details are omitted here. There are at most k edges per level (except for level 0, which has at most k nontree edges) Each L j (v) consists of edges from c levels. Since level 0 tree edges do not belong to any list L j (v) the maximum weight of a tree w(T ) is ck. Thus the total cost ....

S. Even and Y. Shiloach, "An On-Line Edge-Deletion Problem", J. ACM 28 (1981), 1--4.

....deal of interest in what are known as incremental, dynamic, or on line algorithms. The idea is to develop algorithms that can adjust their answers efficiently in response to changes in the input data. Domains for such algorithms have included: ffl graph theoretic algorithms: connectivity [ES81, Har83, 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 ....

....the hard direction that is more efficient, or simpler, than a fully dynamic incremental algorithm. For example, the method of Even and Shiloach for updating connected 5 List maintenance problems are a notable exception; see, e.g. ST85] 94 components requires O(n) per deletion (amortized) ES81] By working in only one direction, a partially dynamic algorithm can throw away structure as it goes, whereas a fully dynamic algorithm typically needs to maintain complicated history information. Finally, we note that there has been quite a bit of work on persistent algorithms, which can be ....

S. Even and Y. Shiloach. An on-line edge deletion problem. Journal of the ACM, 28(1):1--4, January 1981.

....slowly increasing function [116] Therefore, we have amortized O(ff(m; n) time per operation. The fully dynamic data structure of [48] takes O( p m) time per operation. This is the best result handling edge deletions. Other semi dynamic techniques supporting only edge deletions are shown in [41], where a data structure is presented supporting constant time queries and O(q mn) time to perform q edge deletions; and in [89] where q edge deletions are performed in O(mg m log m) time for a graph of genus g. These results have been extended to the maintenance of 2, 3, and 4 connectivity. ....

S. Even and Y. Shiloach, "An On-Line Edge Deletion Problem," J. ACM 28 (1981), 1--4.

....is computed. Subsequently, more data arrive at regular intervals. Each new datum received must be incorporated into the solution. Real time computation is sometimes known as on line computation, by contrast with off line computation in which all the required data are available at the outset [14, 16, 17, 21, 28]. The adjectives updating , incremental , and dynamic are also often used to refer to algorithms that receive and process new data [4, 9, 10, 11] Examples of real time computations include sorting a sequence of numbers, computing the convex hull of a set of points in the plane, and finding the ....

S. Even and Y. Shiloach, An on-line edge deletion problem, Journal of the ACM , 28, 1982, 1--4.

....weight of each tree, whether an edge 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 ....

S. Even and Y. Shiloach. An on-line edge deletion problem. J. ACM, 28:1--4, 1981.

....to use the previously found solutions as much as possible, rather than recomputing everything from scratch. Dynamization problems in graph theory are not only of practical interest, but of theoretical as well, and have already been examined and discussed by various authors, such as Even Shiloach [4] and Ibaraki Katoh [8] transitive closures) La Poutr e van Leeuwen [11] transitive closures and reductions) Frederickson [6] minimum spanning trees) and Goto Sangiovanni Vincentelli [7] Rohnert [14] Even Gazit [3] and Ausiello et al. 2] least cost paths) Our problem can be formalized ....

S.Even and Y.Shiloach. An On-line Edge-Deletion Problem. JACM 28-1 (1981) 1-4.

....inherently exponential. Algorithm C2 avoids this problem. 2. In Algorithm C2, if we look at each component machine F separately, then this is on line maintenance of the strongly connected components of dynamic graphs. Algorithm C2 is a quadratic algorithm. For undirected graphs, one can do better [ES81, Fr85, Ha82]. However, for directed graphs, it is a challenging problem. On the other hand, our problem is not really on line . We know which edges to delete at the very beginning, and further deletions are due to the mutual constraints of the machines involved and are done iteratively. It would be an ....

S. Even and Y. Shiloach, "An On-line Edge Deletion Problem," J. Assoc. Comput. Mach, Vol. 28, pp. 1-4, 1981.

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Even, S. and Y. Shiloach. 1981. An on-line edge deletion problem. Journal of the Association of Computing Machinery 28:1--4.

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S. Even and Y. Shiloach. An on-line edge-deletion problem. Journal of the ACM, 28(1):1--4, 1981.

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S. Even and Y. Shiloach. An on-line edge-deletion problem. Journal of the ACM, 28(1):1--4, 1981.

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S. Even and Y. Shiloach. An on-line edge-deletion problem. J. of the Asso. For Comp., 28:1--4, March 1981.

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S. Even and Y. Shiloach. An on-line edge-deletion problem. Journal of the ACM, 28(1):1--4, 1981.

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Even, S. and Shiloach, Y., "An on-line edge-deletion problem," J. ACM 28(1) pp. 1-4 (January 1981).

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S. Even and Y. Shiloach. An on-line edge-deletion problem. Journal of the ACM, 28:1--4, 1981.

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S. Even and Y. Shiloach. An on-line edge-deletion problem. Journal of the Association for Computing Machinery, 28(1):1-4, January 1981.

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S. Even and Y. Shiloach, An on-line edge deletion problem, Journal of the ACM , 28, 1982, 1--4.

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Shimon Even and Yossi Shiloach. An on-line edge-deletion problem. Journal of the ACM, 28(1):1--4, January 1981.

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S. Even and Y. Shiloach, "An On-Line EdgeDeletion Problem", J. ACM 28 (1981), I 4.

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S. Even and Y. Shiloach, "An On-Line Edge-Deletion Problem", J. ACM 28 (1981), 1--4.

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S. Even and Y. Shiloach, "An On-Line Edge-Deletion Problem", J. ACM 28, 1981, pp. 1--4.

No context found.

S. Even and Y. Shiloach, "An On-Line Edge-Deletion Problem", J. ACM 28, 1981, pp. 1--4.

No context found.

S. Even and Y. Shiloach, "An On-Line Edge-Deletion Problem", J. ACM 28 (1981), 1--4.

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