Baruch Schieber and Uzi Vishkin. On finding lowest common ancestors: Simplification and parallelization. SIAM Journal on Computing, 17(6):1253--1262, December 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....decomposition, in O(d) time we can determine in which subtree of the computation tree to continue the search for the pair (a; b) Details will be given in the full paper. We also augment the fair split tree with any data structure for answering lowest common ancestor queries in O(logn) time (e.g. [14] is more than sufficient) These augmentations can all be computed within the same asymptotic time bound as the construction of the well separated pair decomposition. To answer a pair query fa; bg in O(log n) time, we begin by finding the lowest common ancestor of a and b in the fair split tree. ....

B. Schieber and U. Vishkin. On finding lowest common ancestors: Simplifications and parallelization. SIAM J. Comput. 17 (1988), pp. 1253--1262.

....of ancestor labeling schemes appears in [73] Additional results on labeling schemes (like reachability in planar digraphs) appear in [23,118] Another example for a piece of non numeric information that may be required in rooted trees is the least common ancestor of v and w. Standard solutions [66,111] can answer such queries in constant time with a suitable preprocessing of the tree, but cannot be applied in a localized computation setting, as they require some accesses to a global table of O(n) items. In [100] it is shown that the identifier of the least common ancestor can be found using a ....

B. Schieber, U. and Vishkin. On finding lowest common ancestors: simplification and parallelization. SIAM Journal on Computing, 17(6):1253-- 1262, Dec. 1988.

....time bound for the algorithm is derived as follows: We can solve the TSP on the metric space induced by G[E] in time O(n) We then root the tree G[E] at an arbitrary vertex. With preprocessing time, the least common ancestor of any pair of vertices can be found in constant time (see [HT84, SV88] Thus, we can implement ALG TSP in such a way that the invocations of Step 7 take total time R ) This means that ALG TSP R ) The balancing in ALG LA T can be accomplished in time R ) Completion of the graph by computing all pairs shortest paths can be done in E n log n) ....

B. Schieber and U. Vishkin, On finding lowest common ancestors: Simplification and parallelization, SIAM Journal on Computing 17 (1988), no. 6, 1253--1262.

....improving the time complexity of each broadcast. This is done by dividing the tree into paths, and broadcasting over each path in one unit of time. In the next section we give details of our tree decomposition algorithm. Other methods of tree decomposition appear in different contexts, e.g. in [HT84, SV88, T83]. 3.1 The Topology Broadcast Algorithm First, we give a brief description of the overall algorithm. Prior to the t execution of the broadcast, node i has obtained some information about the topology of the network. By assumption, the node is always aware of its local topology. Information on ....

B. Schieber and U. Vishkin, On finding lowest common ancestors: simplification and parallelization, SIAM Journal on Computing, Vol. 17, No. 6, December 1988, pp. 1253-1262.

....n Gamma i, S[n] if n Gamma i i) has length to its complement. We build a single suffix tree containing all suffixes of S and S in O(n) time. Further, we augment this tree with the capability to perform leastcommon ancestor (LCA) queries in constant time after linear preprocessing time [8, 16]. This LCA data structure enables us to return the length of the longest prefix match of two given suffixes in constant time. To find the end most possible fold, we can search for the longest prefix match of S [n Gamma i 2] where the jth fold attempt takes place at i = j Gamma 1) 2 if ....

B. Schieber and U. Vishkin. On finding lowest common ancestors: Simplification and parallelization. SIAM Journal on Computing, 17(6):1253--1262, 1988.

.... solvable with the suffix tree are also solvable with binary search on the suffix array by a slowdown of log z factor However, we can easily observe that a naive algorithm for the substring traversal problem requires O(z 2) time and O(z log z) time without and with the constant time lcp computation [11]. We present a simple and efficient algorithm that simulates the traversal of the suffix tree for a given text in O(z) time with the suffix array combined with an additional information called the height array [10] This algorithm traverses a virtual suffix tree using the left to right scan of ....

....Lemma 10 For each 1 i 1, the followi9 properties hold: The leaf The lea ode lca is represeted by lca = i, Hgt[i] Proof: The proof is immediate from the definitions of SA and Hgt. 1 In general, constant time decision of the relation requires the constant time lea com putation [11]. In our special case, however, it is decidable by simply examining the string depth of the nodes. Lemma 11 I the i th stage of Traversal with Array of Fig. if (L,H) top(S) ad (L, H,i) i, Hgt[i] be the pair represetig lca, the: L,H) rcsp) L,H,i) iff Proof: Suppose that the ....

B. Schieber and U. Vishkin, On finding lowest common ancestors: simplifications an paral- lelization, SIAM J. Cornpti!7, 17, 1253 1262, 1988.

....is regarded as a onedimensional string. For each internal node u of Isuffix tree IST (A) we define the min index of u to be the minimum position (i; j) in OE I such that l ij is a descendant of u. The least common ancestor of two nodes v and w in a tree is denoted by lca(v; w) By the results of [12, 18] the computation of lca of two nodes can be done in constant time after linear time preprocessing on a tree. The Ilength of the longest common Iprefix of two Istrings Iff and Ifi is denoted by Ilcp(Iff; Ifi) For all nodes u; v in Isuffix tree IST (A) the following property is satisfied between ....

B. Schieber and U. Vishkin, On finding lowest common ancestors: simplification and parallelization, SIAM J. Comput. 17, (1988), 1253--1262.

....of ear P i are contained on an ear with label smaller than i. Let us analyze the complexity of Algorithm 2.1. Step 1 requires the computation of a spanning tree T and its preorder numbering with respect to the root r [CV86] Step 2a requires the computation of least common ancestors in T [SV88]. Step 2b requires sorting of integers in the range [0: n Gamma 1] C88] Step 3a requires the computation of the minimum value in each adjacency list [KR90] Step 3b can be performed efficiently in parallel by the following simple method using the Euler tour technique on trees [TV84] Note ....

B. Schieber, U. Vishkin, "On finding lowest common ancestors: simplification and parallelization," Proc. 3rd Aegean Workshop on Computing, SpringerVerlag LNCS 319, 1988, pp. 111-123.

....time ifexists, where m = Q . Search) We implemented the above abstract data type by a data structure called a virtual su#x tree, which is just the su#x array SA coupled with the height array Hgt[12] the inverse array Rank of SA[11] and the ran min query for constant time lcp information[6, 14].Here, Hgt is the array that stores the the length of the longest common prefixes of adjacent su#xes in SA [12] Although a reconstructible su#x dictionary can be implemented by the su#x tree [10] combined with markingtechnique, the virtual su#x tree has advantages over the su#x tree when ....

....are important. For Traverse operation, we developed a linear time algorithm [7] that simulates bottom up traversal of the su#x tree when SA and Hgt aregiven, while well known simulation of the su#x tree by binary search takes O(n 2 ) time and O(n log n) time without and with lcpinfo. [14]) resp. We also developed a linear time algorithm for building the array Hgt from SA and T [7] For Reconq operation, we use the inverse array Rank and the sorting of ranks for update of Pos array, and also use constant time lcp information for update of Hgt array in claimed worst case time ....

B.Schieber and U.Vishkin, On finding lowest common ancestors: simplifications an parallelization, SIAM J. Computing, 17, 1253--1262, 1988.

....The length of the longest common prefix of two strings ff, fi is denoted by lcp(ff; fi) The least common ancestor of two nodes u, v is denoted by lca(u; v) Then the following property is satisfied between lcp and lca: lcp(L(u) L(v) jL(lca(u; v) j for all nodes u; v in T S . By the results of [HT84, SV88] the computation of lca of two nodes can be done in constant time after linear time preprocessing on a tree. 3 Construction algorithm Let the odd tree T o be the suffix tree of all suffixes beginning at odd positions, and the even tree T e be the suffix tree of all suffixes beginning at even ....

B. Schieber and U. Vishkin, On finding lowest common ancestors: simplification and parallelization, SIAM J. Comput. 17, (1988), 1253--1262.

....query. Since the number of nodes in A is of constant size, the second phase can in constant time construct the spheres S i and S o . To check the third condition, we assume that lowest common ancestor preprocessing has been performed on the tree (such preprocessing can be performed in O(n) time [71]) Then jAj jAj lowest common ancestor queries can verify that no node in A is an ancestor of another node in A (and hence no node in A is a descendant of another node in A) The second condition can be checked in constant time if we augment the fair split tree to contain the size of the subtree ....

....we construct and perform operations on binary trees. For consistency, if a node has only a single child, we consider the child to be the left child of the parent node. Thus, the inorder traversal of the trees we consider is well defined. First we define the lowest common ancestor (LCA) problem [71]. Given a rooted tree, we wish to answer queries on the tree of the following form. Given two nodes u and v in the tree, find the node w with greatest depth that is an ancestor of both u and v. Next we define the range maxima queries problem. We are given a list (a 1 ; a 2 ; a s ) of numbers ....

[Article contains additional citation context not shown here]

Schieber, B., and Vishkin, U., "On Finding Lowest Common Ancestors: Simplification and Parallelization," SIAM Journal Computing, vol. 17, 1988, 12531262.

....T LCAN(N,d,u) Note that a T LCAN where u = 1 is a limiting case of a CB LCAN; TLCAN (N,d,1) is equivalent to CB LCAN(N,d,1) CB LCANs and T LCANs are based on a tree structure. The least common ancestor of two nodes in a tree is the deepest node which counts both nodes among its descendants [7, 16]. Analogously, two PEs communicating using an LCAN need only utilize switches as high as one of their least common ancestor switches. A least common ancestor (LCA) switch of two PEs is a switch in level i, such that: 1) it connects to both PEs through switches in levels 0 through (i 0 1) and 2) ....

B. Schieber and U. Vishkin, On finding lowest common ancestors: simplification and parallelization, SIAM J. Comput., Vol. 17, No. 6, December 1988, pp. 1253-1262.

....a Cartesian tree for Am 1;n = a m 1 ; a n ) The tree for an empty sub array is the empty tree. CHAPTER 3. APPLICATIONS OF ANSVP 70 The preprocessing procedure starts with the construction of the Cartesian tree for A, and then proceeds with answering queries for lowest common ancestors [89, 14] in trees. The definition of the Cartesian tree implies that MIN(i; j) is the lowest common ancestor of a i and a j in the Cartesian tree. Therefore, each range minimum query can be answered in constant time by answering the corresponding query for lowest common ancestor in the Cartesian tree. ....

....left and right children of w, respectively. CHAPTER 3. APPLICATIONS OF ANSVP 71 Let l be the index of the leftmost leaf in the subtree of u. Then, MIN(i; j) minfS v (l i) P u ( j l 1)g. The lowest common ancestor of any two given vertices in T can be found using the inorder numbering of T [89]. Range minimum search has been used to solve the parallel triconnectivity [82] and pattern matching with scaling problems [7] The first problem deals with determining in parallel the 3 vertex connectivity for graphs and further decomposing graphs into triconnected components. The second problem ....

SHIEBER, B., AND VISHKIN, U. On Finding Lowest Common Ancestors: Simplification and Parallelization. SIAM Journal on Computing 17 (1988), 1253-- 1262.

....the following result is achieved. Theorem 2.1. For two prev encoded pstrings of length n and m, respectively, O (D (n m) time and O (n m) space are sufficient to determine the p edit distance D. These bounds can be improved by using p suffix tree [1, 9] and lowest common ancestor (LCA) [5, 11] techniques, though the resulting algorithm is not practical. Theorem 2.2. For two pstrings of length n and m, respectively, O( n m) log j Sigmaj log j Pij) D 2 ) time and O (n m) space are sufficient to determine the p edit distance D. The above algorithms do not report a minimal edit ....

....m, a pstring text T of length n, and an integer k, O( k log j Sigmaj log j Pij) n m) time and O( n m) space are sufficient to report all final positions of subpstrings of T that are within p edit distance k of P . This theorem depends on using p suffix trees [1, 9] and LCA techniques [5, 11]. In practice, the LCA techniques may require excessive overhead. A more practical approach would precompute all longest pmatches over a threshold length using Dup and use direct comparison to find shorter pmatches. Acknowledgment The author would like to thank Raffaele Giancarlo for helpful ....

B. Schieber and U. Vishkin. On finding lowest common ancestors: Simplification and parallelization. SIAM J. Computing, 17:1253--1262, 1988.

....remain small (10 ) Although there is an O(logN) computational cost per route lookup, algorithmic routing is still desirable for serial simulators such as ns 2, where large scale simulations may not be performed at all with a limited amount of memory. Besides, with elegant data structure [21], this lookup can be done in O(1) time. The last two bullets imply that the degree of distortion scales well to topology size and it is likely that most communication is done through routes that are only slightly different. For simulation scenarios that have single or few senders, multiple k ary ....

B. Schieber and U. Vishkin. On Finding Lowest Common Ancestors: Simplification and Parallelization. SIAM J. Comput., 17(6):1253--1262, 1988.

....(80 ) relative differences remain small (10 ) Although there is an O(logN) computational cost per route lookup, algorithmic routing is still desirable for sequential simulators where large scale simulations may not run at all with a limited amount of memory. With elegant tree addressing scheme [25], this lookup can be done in O(1) time. The last two bullets imply that the degree of distortion scales well to topology size and it is likely that most communication is done through routes that are only slightly different. For simulation scenarios that have single or few senders, multiple k ary ....

B. Schieber and U. Vishkin, "On Finding Lowest Common Ancestors: Simplification and Parallelization," SIAM J. Comput., vol. 17, no. 6, pp. 1253--1262, 1988.

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Baruch Schieber and Uzi Vishkin. On finding lowest common ancestors: Simplification and parallelization. SIAM Journal on Computing, 17(6):1253--1262, December 1988.

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B. Schieber and U. Vishkin. On Finding lowest common ancestors:simplifications and parallelization. SIAM Journal on Computing, 17:1253-62, 1988.

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B. Schieber and U. Vishkin, On finding lowest common ancestors: simplifications and parallelization, SIAM J. Comput., 17:1253-62, 1988.

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B. Schieber and U. Vishkin. On finding lowest common ancestors: Simplification and parallelization. SIAM Journal on Computing, 17:1253--1262, 1988.

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Baruch Schieber and Uzi Vishkin. On finding lowest common ancestors: Simplification and parallelization. SIAM Journal on Computing, 17(6):1253--1262, December 1988.

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B. Schieber and U. Vishkin, On finding lowest common ancestors: simplification and parallelization, SIAM J. Comput. 17(1988), No. 6, 1253-1262.

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B. Schieber and U. Vishkin. On finding lowest common ancestors: Simplification and parallelization. SIAM J. on Computing, 17(6):1253--1262, 1988.

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Schieber, B., U. Vishkin. 1988. On finding lowest common ancestors: simplification and parallelization. SIAM Journal on Computing 17, 1253-1262.

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B. Schieber and U.Vishkin, On finding lowest common ancestors: simplification and parallelization, SIAM J. Cornput. 17 (1988), pp. 11253-1262.

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