K. Mehlhorn. Graph Algorithms and NP-Completeness, volume 2 of Data Structures and Algorithms. Springer-Verlag, Heidelberg, West Germany, 1984.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of the tree, and node pointer is a pointer to a node of the tree. The actual value of height depends on which kind of tree is used, e.g. standard binary trees or AVL trees. 6.2.2. Weight Balanced Trees. So called weight balanced trees have been introduced in [18] and are treated in detail in [19] and in [20] Definition 6.4. We define: 1) Let T be a binary tree with left subtree Tt and right subtree Tr. Then is called the root balance of T. Here IT] denotes the number of leaves of tree T. 2) Tree T is of bounded balance c if for every subtree T of T: c p(T ) 1 c (3) BB[c] is ....

....Deletemin take time O(logN) in BB[c] trees. Here N is the number of leaves in the BB[c] tree. Some of the above operations can move the root balance of some nodes on the path of search outside the permissible range [c, 1 c] This can be repaired by single and double rotations (for details see [19] and [20] BB[c] trees are binary trees with bounded height. In fact it is proved in [19] that ldN 1 height(T) ld(1 c) 1, where N is the number of leaves in the BB[c] tree T. A template for the above operations is shown in Figure 7, where floor(x) is ne node pointer : ....

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K. Mehlhorn. Sorting and Searching, volume 1 of Data Structures and Algorithms. SpringerVerlag, Berlin, 1984.

....the set of edges E, where E is defined by (v,w) E w= L(v) for some i [1, el. With these definitions the following lemma is trivially true. Lemma 2.1. A discrete loop completes if and only if the corresponding loop digraph is acyclic. Each acyclic digraph can be topologically sorted (cf. [15]) i.e. we can find a mapping ord: V 0, 1, N, N 1) such that for all edges (v, w) E we have ord(v) ord(w) Since we are only interested in completing discrete loops, we restrict ourselves to discrete loops that result in topologically sorted loop digraphs. This is certainly the case if ....

....loops. Besides the loop digraph corresponding to a certain loop is very important in this section to prove properties of discrete loops. Note, however, that the vertex 0 can be avoided since the underlying loop is monotonically increasing. In order to calculate u we can use an algorithm given in [15] which determines the longest path in topologically sorted digraphs. The path is supposed to start at node s. for k in 1. 1 loop c(s) O; for k in s. N loop for i in 1, e loop c(Yi(k) max c(Yi(k) end loop; A similar procedure can be used to determine the shortest path in . for k ....

K. Mehlhorn. Graph Algorithms and NP-Completeness, volume 2 of Data Structures and Algorithms. Springer-Verlag, Berlin, 1984.

....N(V;E; cap; s; t; w) then f is called an integer ow i for all edges (a; b) 2 E, f(a; b) is an integer. De nition 11 (integer ow network) If N(V;E; cap; s; t; w) is a weighted transport network, then N is called an integer ow network i for all edges (a; b) 2 E, cap(a; b) is an integer. In [10] the following theorem is proved: Theorem 1. If N(V;E; cap; s; t; w) is an integer ow network then there is a maximal ow minimal weight ow f for N such that f is an integer ow. 3 Distances between sets of points In this section we discuss some existing distance measures between sets of ....

....and size W (B) Proof: The weights and capacities of the graph of the minimal weight maximal ow problem associated with this metric can be computed in #A:#B time. These numbers are all integers. The minimal weight maximal ow problem can be solved in polyniomial time in size W (A) and size W (B) [10]. This proves the theorem. 2 c Example 2. Assume one has to choose among a number of 7 element sets the one most representative for a 100 element set B. Without weights, only the best 7 elements of B will determine the outcome. Using a weight 14 (or 15) for the elements in the ....

K. Mehlhorn. Graph algorithms and NP-completeness, volume 2 of Data structures and algorithms. Springer, 1984.

....in construction of xed parameter tractable algorithms [DF99, Nie98] we employ the method of reduction to a problem kernel 2 to obtain recognition algorithms such that an exponential function on k contributes only an additive term to the overall complexity. This builds on previous work [Meh84, BG93]; our results can be seen as generalizations of the algorithms designed for the vertex cover problem [BFR98, NR99, CKJ99] Fast algorithms of this type have been generated for other problems [DF95, FPT95, KST99, DFS99, CCDF97, MR99] We make use of our upper bound on the size of obstructions to ....

Kurt Mehlhorn. Graph Algorithms and NP-Completeness, volume 2 of Data Structures and Algorithms. Springer Verlag, Berlin, 1984.

....advances in construction of xed parameter tractable algorithms [DF99, Nie98] we employ the method of reduction to a problem kernel to obtain recognition algorithms such that an exponential function on k contributes only an additive term to the overall complexity. This builds on previous work [Meh84, BG93]; our results can be seen as generalizations of the algorithms designed for the vertex cover problem [BFR98, NR99, CKJ99] Fast algorithms of this type have been generated for other problems [DF95, FPT95, KST99, DFS99, CCDF97, MR99] We make use of our upper bound on the size of obstructions to ....

Kurt Mehlhorn. Graph Algorithms and NP-Completeness, volume 2 of Data Structures and Algorithms. Springer Verlag, Berlin, 1984.

....cap; s; t; w) then f is called an integer flow iff for all edges (a; b) 2 E, f(a; b) is an integer. Definition 11 (integer flow network) If N(V; E; cap; s; t; w) is a weighted transport network, then N is called an integer flow network iff for all edges (a; b) 2 E, cap(a; b) is an integer. In [10] the following theorem is proved: Theorem 1 If N(V; E; cap; s; t; w) is an integer flow network then there is a maximal flow minimal weight flow f for N such that f is an integer flow. 3 Distances between sets of points In this section we discuss some existing distance measures between sets of ....

....Since M max x2A;y2B d(x; y) d 0m (A; B) can be written in the form d 0m (A; B) min r2MaxMatch(A;B) X (x;y)2r d 0 (x; y) j#A Gamma #Bj: M 2 To compute it, one has to solve a minimal weight maximal matching problem. This can be done in a time bounded by a polynomial in #A and #B [10]. Hence, d m (A; B) can be computed in time bounded by a polynomial in A, B and T . 2 5 Normalised matching metric. Instance based learning systems such as RIBL [7] and clustering algorithms (e.g. agglomerative clustering algorithms using distances) make use of normalised similarity measures, ....

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K. Mehlhorn. Graph algorithms and NP-completeness, volume 2 of Data structures and algorithms. Springer, 1984.

....by Ben Amram [3] Among the above proofs all but La Poutr e s assume m n. However Tarjan s proof was extended for all m by Banachowski [2] and the rest extend without any essential change. In this paper m is not restricted. The path compression algorithm has worst case time (log n) Mehlhorn [14] asked if it was possible to get o(log n) worst case time. Blum [6] answered this positively by giving an O(log n= log log n) algorithm, and proved that this is optimal for a class of pointer algorithms; Fredman and Saks [8] gave a matching lower bound for the cell probe model, and Ben Amram [3] ....

K. Mehlhorn. Sorting and Searching, volume 1 of Data Structures and Algorithms. SpringerVerlag, 1984.

....These results hold even for the relaxed version. Our results also demonstrate that a binary tree scheme with the same complexities can be designed. This is an improvement over the existing results. 1 Introduction We focus on the type of multi way trees usually referred to as (a; b) trees [8, 15], and in particular, we adopt the relaxed (a; b) trees [11, 12] In the context Supported in part by the Danish Natural Sciences Research Council (SNF) and in part by the IST Programme of the EU under contract number IST 1999 14186 (ALCOM FT) y Department of Mathematics and Computer Science, ....

Kurt Mehlhorn. Sorting and Searching, volume 1 of Data Structures and Algorithms. Springer-Verlag, 1986.

....Remark 4.1. Let M denote the path from f to some f 0 2 F 0 , f 0 2 R (f) with maximum weight W (f) P g D(g) where g runs through all vertices on M. Then W (f) is equal to S(f) Remark 4.2. Using G(f) the quantity S(f) can be computed off line at compile time in O(kV k kEk) time (cf. e.g. [9]) Definition 4.3. Let p be a monotonical recursive procedure. We define N : F F to be a function such that N (f) fmax , where fmax is such that D(fmax ) max f2R(f) D(f ) and recdep(f max ) recdep(f) Gamma 1. Definition 4.4. We call a monotonical recursive procedure p locally ....

K. Mehlhorn. Graph Algorithms and NPCompleteness, volume 2 of Data Structures and Algorithms. Springer-Verlag, Berlin, 1984.

....the number of internal nodes on this layer, which is 2 h 2 i = 2 #upd 2 i on layer i. a; b) Trees In this section, we consider (a; b) trees [2, 5] i.e. multi way search trees satisfying that the number of children of each node (except maybe the root) is between a and b for b 2a. In [5, 6], it is shown that rebalancing is exponentially decreasing in (a; b) trees. Here, we demonstrate how to obtain comparable constants to [5] using Theorem 1. To represent intermediate states between operations, we assume that nodes are capable of storing b 1 pointers. We de ne the set of local ....

Kurt Mehlhorn. Sorting and Searching, volume 1 of Data Structures and Algorithms. Springer-Verlag, rst edition, 1984.

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K. Mehlhorn. Sorting and Searching, volume 1 of Data Structures and Algorithms. Springer-Verlag, Heidelberg, West Germany, 1984.

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K. Mehlhorn. Graph Algorithms and NP-Completeness, volume 2 of Data Structures and Algorithms. Springer-Verlag, Heidelberg, West Germany, 1984.

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K. Mehlhorn. Graph Algorithms and NP-Completeness, volume 2 of Data Structures and Algorithms. Springer, 1984.

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Kurt Mehlhorn. Graph Algorithms and NP-Completeness, volume 2 of Data Structures and Algorithms. Springer Verlag, Berlin, 1984.

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Kurt Mehlhorn. Graph Algorithms and NP-Completeness, volume 2 of Data Structures and Algorithms. Springer Verlag, Berlin, 1984.

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K. Mehlhorn. Sorting and Searching, volume 1 of Data Structures and Algorithms. Springer Verlag, Berlin, 1984.

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Kurt Mehlhorn. Graph Algorithms and NP-Completeness, volume 2 of Data Structures and Algorithms. Springer-Verlag, 1984.

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K. Mehlhorn. Sorting and Searching, volume 1 of Data Structures and Algorithms. Springer-Verlag, 1994. 168

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K. Mehlhorn. Sorting and Searching, volume 1 of Data Structures and Algorithms. Springer-Verlag, 1994.

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K. Mehlhorn. Sorting and Searching, volume 1 of Data Structures and Algorithms. Springer-Verlag, 1994.

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K. Mehlhorn. Sorting and Searching, volume 1 of Data Structures and Algorithms. Springer-Verlag, Heidelberg, West Germany, 1984.

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K. Mehlhorn. Sorting and Searching, volume 1 of Data Structures and Algorithms. Springer-Verlag, 1994.

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K. Mehlhorn. Sorting and Searching, volume 1 of Data Structures and Algorithms. Springer-Verlag, 1994.

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K. Mehlhorn. Sorting and Searching, volume 1 of Data Structures and Algorithms. Springer-Verlag, 1994.

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K. Mehlhorn. Sorting and Searching, volume 1 of Data Structures and Algorithms. Springer-Verlag, 1994.

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