D. Sleator, R. Tarjan, W. Thurston, Short encodings of evolving structures, SIAM J. Discrete Math., 5:428-450, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....(NNI) Waterman and Smith (1978) 0 0 0 1 2 3 4 1 2 3 4 1 2 3 4 Figure 15: Rotation In the link L n as we have defined it, each maximal simplex corresponds to a binary tree, and two maximal simplices share a codimension 1 face if and only if the corresponding trees di#er by a rotation move. In Sleator et al. 1992) it is shown that the maximal rotation distance between two trees on n leaves is O(n log n) while the maximal rotation distance between two trees contained in the same associahedron is exactly 2n 6 (see Sleator et al. 1988) These results give an indication of the size of our space of trees. ....

D. D. Sleator, R. E. Tarjan, and W. P. Thurston. Short encodings of evolving structures. SIAM J. Discrete Mathematics, 5(3):428--450, 1992.

....are defined on trees or other data structures. The huge literature on this ranges from pattern matching and cognition to search strategies on internet and computational biology. As an example we mention nearest neighbor interchange distance between evolutionary trees in computational biology, [24, 21]. A priori it is not immediate what is the most appropriate universal symmetric informational distance between two strings, that is, the minimal quantity of information sufficient to translate between x and y, generating either string effectively from the other. We give evidence that such notions ....

D. Sleator, R. Tarjan, W. Thurston, Short encodings of evolving structures, SIAM J. Discrete Math., 5:428--450, 1992.

....here, and will be given in a separate comprehensive study of diameter properties of G n [24] Lemma 4 The number of elements within distance i from any given binary coupling tree in G n is less than or equal to Gamma n 2i i Delta 4 i . This can be shown using so called short encodings [25]; for a detailed proof see [24] 15 Theorem 5 The diameter of G n satisfies d(G n ) 1 4 log(n ) 1 4 n log(n=e) 10) Herein (and in what follows) log = log 2 is the logarithm in basis 2, and e is the basis of the natural logarithm. To prove the theorem, let ffi = d(G n ) then by ....

D.D. Sleator, R.E. Tarjan and W.P. Thurston, Short encodings of evolving structures, SIAM J. Disc. Math. 5 (1992) 428-450. 20

....criteria [24] or from di#erent genes [15, 16, 17, 18, 14] Comparing these trees to find their similarities (e.g. agreement or consensus) and dissimilarities, i.e. distance, is thus an important issue in computational molecular biology. The nearest neighbor interchange (nni) distance [29, 28, 34, 3, 6, 2, 19, 20, 23, 33, 22, 21, 26] is a natural distance metric that has been extensively studied. Despite its many appealing aspects such as simplicity and sensitivity to tree topologies, computing this distance has remained very challenging, and many algorithmic and complexity issues about computing this distance have ....

....algorithms concerning the nni distance on both unweighted and weighted phylogenies. We finally settle almost all questions regarding the nni distance. We show that computing the nni distance is NP complete (cf. 2) The proof is quite involved and it uses the lower and upper bounds in [3, 33, 26] for sorting on a degree 3 tree by nni operations. The problem is also shown to be NP complete for unlabeled trees, answering another open question in [3] cf. 3. We will give an e#cient approximation algorithm for computing the nni distance on weighted phylogenies with a performance ratio of 4 ....

D. Sleator, R. Tarjan and W. Thurston, Short encodings of evolving structures, SIAM J. Discr. Math., 5(1992), 428--450.

.... is necessarily bounded by n lg(n) O(n) Section 5 shows how an Omega Gamma n lg(n) lower bound for the diameter can be obtained from an upper bound for the number of trees within a certain distance of any given tree, for which the technique of short encodings introduced by Sleator et al. in [13] can be used. We conclude that the diameter d(G n ) grows like n lg(n) 2 Binary coupling trees and the graph G n Following Knuth [9, Section 2.3] we say that a tree is a finite set of one or more nodes such that (a) there is one specially designated node called the root of the tree and (b) the ....

....the diameter of G n , one might wonder whether a better lower bound, i.e. of the order n lg(n) can be established. This is indeed the case, as was already indicated in [10] In the rest of this section we will show how to obtain an Omega Gamma n lg(n) lower bound for the diameter of G n . In [13] Sleator et al. provide a tool for deriving an upper bound for the number of combinatorial objects within m transformations from a given object. They take advantage of the fact that, for many series of transformations, one can interchange the order of the transformations, without affecting the ....

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D. D. Sleator, R. E. Tarjan and W. P. Thurston, Short encodings of evolving structures, SIAM J. Disc. Math. 5 (1982) 428--450.

....[18] or from different genes [12] in the study of molecular evolution. Comparing these trees to find their similarities (e.g. agreement or consensus) and dissimilarities, i.e. distance, is thus an important issue in computational molecular biology. The nearest neighbor interchange (nni) distance [26, 24, 32, 4, 5, 3, 16, 17, 19, 30, 20, 21, 23] and the subtree transfer distance [12, 13, 15] are two major distance metrics that have been proposed and extensively studied for different reasons. Despite their many appealing aspects such as simplicity and sensitivity to tree topologies, computing these distances has remained very challenging. ....

....the nni distance and linear cost subtree transfer distance on both unweighted and weighted phylogenies. We finally settle almost all questions regarding the nni distance. We show that computing the nni distance is NP complete. The proof is quite nontrivial and it uses the lower and upper bounds [4, 30, 23] for sorting on a tree by nni operations in an essential way. The problem is also shown to be NP complete for unlabeled trees, answering another open question in [4] We will give an efficient O(log n) approximation algorithm for computing the nni distance on weighted phylogenies, where n is the ....

[Article contains additional citation context not shown here]

D. Sleator, R. Tarjan, W. Thurston, Short encodings of evolving structures, SIAM J. Discr. Math., 5(1992), 428--450.

....[18] or from different genes [12] in the study of molecular evolution. Comparing these trees to find their similarities (e.g. agreement or consensus) and dissimilarities, i.e. distance, is thus an important issue in computational molecular biology. The nearest neighbor interchange (nni) distance [25, 24, 32, 4, 5, 3, 16, 17, 19, 29, 20, 21, 23] and the subtree transfer distance [12, 13, 15] are two major distance metrics that have been proposed and extensively studied for different reasons. Despite their many appealing aspects such as simplicity and sensitivity to tree topologies, computing these distances has remained very challenging. ....

....the nni distance and linear cost subtree transfer distance on both unweighted and weighted phylogenies. We finally settle almost all questions regarding the nni distance. We show that computing the nni distance is NP complete. The proof is quite nontrivial and it uses the lower and upper bounds [4, 29, 23] for sorting on a 3 tree by nni operations in an essential way. The problem is also shown to be NP complete for unlabeled trees, answering another open question in [4] We will give an efficient O(logn) approximation algorithm for computing the nni distance on weighted phylogenies, where n is ....

[Article contains additional citation context not shown here]

D. Sleator, R. Tarjan, W. Thurston, Short encodings of evolving structures, SIAM J. Discr. Math., 5(1992), 428--450.

....are defined on trees or other data structures. The huge literature on this ranges from pattern matching and cognition to search strategies on internet and computational biology. As an example we mention nearest neighbor interchange distance between evolutionary trees in computational biology, [24, 21]. A priori it is not immediate what is the most appropriate universal symmetric informational distance between two strings, that is, the minimal quantity of information sufficient to translate between x and y, generating either string effectively from the other. We give evidence that such notions ....

D. Sleator, R. Tarjan, W. Thurston, Short encodings of evolving structures, SIAM J. Discrete Math., 5:428--450, 1992.

....optimality criteria [24] or from different genes [15, 16, 17, 18, 14] Comparing these trees to find their similarities (e.g. agreement or consensus) and dissimilarities, i.e. distance, is thus an important issue in computational molecular biology. The nearest neighbor interchange (nni) distance [29, 28, 34, 3, 6, 2, 19, 20, 23, 33, 22, 21, 26] is a natural distance metric that has been extensively studied. Despite its many appealing aspects such as simplicity and sensitivity to tree topologies, computing this distance has remained very challenging, and many algorithmic and complexity issues about computing this distance have remained ....

....algorithms concerning the nni distance on both unweighted and weighted phylogenies. We finally settle almost all questions regarding the nni distance. We show that computing the nni distance is NP complete (cf. x 2) The proof is quite involved and it uses the lower and upper bounds in [3, 33, 26] for sorting on a degree 3 tree by nni operations. The problem is also shown to be NP complete for unlabeled trees, answering another open question in [3] cf. x 3. We will give an efficient approximation algorithm for computing the nni distance on weighted phylogenies with a performance ratio ....

D. Sleator, R. Tarjan and W. Thurston, Short encodings of evolving structures, SIAM J. Discr. Math., 5(1992), 428--450.

....computing an optimal subtreetransfer sequence when the D st (T 1 ; T 2 ) is small, say at most d. An n O(d) algorithm for this problem is trivial. With careful inspection, one can derive an algorithm that runs in O(n O(1) Delta d O(d 2 ) time. It turns out that by using the results in [23, 18], we can improve this asymptotically to O(n Delta 2 21d=2 ) time. Definition 1 Let T 1 and T 2 be the two trees being compared. An edge e 1 2 T 1 is good if there is another edge e 2 2 T 2 such that e 1 and e 2 partition the leaf labels of T 1 and T 2 identically; e 1 is bad otherwise. The ....

....edges have zero weights. Assume also the two (unweighted or weighted) trees involved in the distance calculation share no good edge pairs (Definition 1 or Definition 2, as appropriate) In the unweighted case, it is known that there are two trees which are at a distance of Omega Gamma n log n) [23]. However, in the weighted case, our factor 2 approximation algorithm and the lower bounds in Lemma 6 imply that any two trees are at a distance of at most O(1) Several open questions still remain and may be worth persuing further: 1. Is the linear cost subtree transfer problem NP hard when the ....

D. Sleator, R. Tarjan, W. Thurston, Short encodings of evolving structures, SIAM J. Discr. Math., 5(1992), 428--450.

....computing an optimal subtree transfer sequence when the D st (T 1 ; T 2 ) is small, say at most d. An n O(d) algorithm for this problem is trivial. With careful inspection, one can derive an algorithm that runs in O(n O(1) Delta d O(d 2 ) time. It turns out that by using the results in [22, 16], we could improve this asymptotically to O(n 2 log n n Delta 2 23d=2 ) time. Definition 1 Let T 1 and T 2 be the two trees being compared. An edge e 1 2 T 1 is good if there is another edge e 2 2 T 2 such that e 1 and e 2 partition the leaf labels of T 1 and T 2 identically; e 1 is bad ....

D. Sleator, R. Tarjan, W. Thurston, Short encodings of evolving structures, SIAM J. Discr. Math., 5(1992), 428--450.

....internal edge (u; v) exchange B C or B D. internal edge. The nni distance, D nni (T 1 ; T 2 ) between two trees T 1 and T 2 is defined as the minimum number of nni operations required to transform one tree into the other. Although the distance has been studied extensively in the literature [37, 35, 47, 6, 10, 5, 25, 26, 29, 42, 30, 31, 33], the computational complexity of computing it has puzzled the research community for nearly 25 years until recently [7] An nni operation can also be viewed as moving a subtree past a neighboring internal node. A more general operation is to transfer a subtree from one place to another arbitrary ....

....and a faulty NP completeness proof [47, 5, 25, 26, 29, 30, 33] 9 K. Culik II and D. Wood [6] improved later by [33] proved that n log n O(n) nni moves are sufficient to transform a tree of n leaves to any other tree with the same set of leaves. D. Sleator, R. Tarjan, and W. Thurston [42] proved an Omega Gamma n log n) lower bound for most pair of trees. A restricted version of the nni operation, known as the tree rotation operation (discussed in Section 6) was considered in [41] and a trivial approximation algorithm with approximation ratio of 2 was given. But given two ....

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D. Sleator, R. Tarjan, W. Thurston, Short encodings of evolving structures, SIAM J. Discr. Math., 5, pp. 428--450, 1992.

....number of nni operations required to transform one tree into the other. K. Culik II and D. Wood [13] improved later by [51] proved that n log n O(n) nni moves are sufficient to transform a tree of n leaves to any other tree with the same set of leaves. D. Sleator, R. Tarjan, and W. Thurston [69] proved an Omega Gamma n log n) lower bound for most pair of trees. Although the distance has been studied extensively in the literature u v D A u v C A B B C B D B C B D u v C D A Figure 2: The two possible nni operations on an internal edge (u; v) exchange B C or B D. 60, 56, 83, 13, ....

....W. Thurston [69] proved an Omega Gamma n log n) lower bound for most pair of trees. Although the distance has been studied extensively in the literature u v D A u v C A B B C B D B C B D u v C D A Figure 2: The two possible nni operations on an internal edge (u; v) exchange B C or B D. [60, 56, 83, 13, 17, 40, 41, 43, 69, 51], the computational complexity of computing it has puzzled the research community for nearly 25 years until recently when the authors in [14] showed this problem to be NP hard (an erroneous proof of the NP hardness of the nni distance between unlabeled trees was published in [43] Since computing ....

D. Sleator, R. Tarjan, W. Thurston. Short encodings of evolving structures, SIAM J. Discr. Math., 5 (1992), pp. 428-450.

....the diameter, can be replaced by n moves, each only costing a number of steps equal to the diameter. The lower bound follows from the following: Lemma 1. The number of trees within distance m from any given tree is at most 3 n Gamma2 2 4m . Proof. Analogous to Theorem 5. 1 of the elegant work [12], which states an 3 2n Gamma4 2 3n upper bound on the number of plane triangulations on n vertices within m flips of a given triangulation. Their result is proven by giving a short encoding of derivations on graphs. Their graphs and our trees (at least the internal nodes) share the property ....

....can use the same encoding, with an extra bit per operation to specify which of the two nni s corresponding to an internal edge is used. That is why our bound uses 2 4m instead of 2 3m . The 3 n Gamma2 part corresponds to the initial 3 valued labels, one for each internal node of the tree. In [12], an initial label was needed for each of the 2n Gamma 4 vertices in the dual graph of a triangulation. It turns out that by a completely different, and rather unexpected approach, one can improve the upper bound in 4 to the upper bound in 5. Lemma 2. Delta(G) n log n O(n) Proof. This ....

D. Sleator, R. Tarjan, W. Thurston, Short encodings of evolving structures, SIAM Journal on Discrete Mathematics 5, 428--450, 1992.

....use Kolmogorov complexity to do the average case analysis via four examples, and exhibits a surprising property of the celebrated associated universal distribution. The four examples are: average case analysis of Heapsort [17, 15] average nni distance between two binary rooted leave labeled trees [20], compact routing in computer networks [3] average case analysis of an adder algorithm [4] The property is that the average case complexity of any algorithm whatsoever equals its worst case complexity if the inputs are distributed according to the Universal Distribution [14] We provide the ....

....moves to convert (i) to (ii) K. Culik II and D. Wood [6] improved by [13] proved that n log n O(n) nni moves are sufficient to transform a tree of n leaves to any other tree with the same set of leaves. But the question is, is this the best upper bound D. Sleator, R. Tarjan, and W. Thurston [20] proved an Omega Gamma n log n) lower bound for most pairs of trees, essentially using the incompressibility method. Note, they proved their results for a more general graph transformation system. The idea behind the proof is simple. Consider T 1 and T 2 such that C(T 1 jT 2 ) i) Reptilian ....

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D. Sleator, R. Tarjan, and W. Thurston, Short encodings of evolving structures, SIAM J. Discr. Math., 5(1992), 428-450.

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D. Sleator, R. Tarjan, W. Thurston, Short encodings of evolving structures, SIAM J. Discrete Math., 5:428-450, 1992.

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