Polynomial-time metrics for attributed trees

We address the problem of comparing attributed trees and propose four novel distance measures centered around the notion of a maximal similarity common subtree. The proposed measures are general and defined on trees endowed with either symbolic or continuous-valued attributes and can be applied to rooted as well as unrooted trees. We prove that our measures satisfythe metric constraints and provide a polynomial-time algorithm to compute them. This is a remarkable and attractive property, since the computation of traditional edit-distance-based metrics is, in general, NP-complete, at least in the unordered case. We experimentally validate the usefulness of our metrics on shape matching tasks and compare them with (an approximation of) edit-distance

Using airborne lidar to discern age classes of cottonwood trees in a riparian area

Airborne lidar (light detecting and ranging) is a useful tool for probing the structure of forest canopies. Such information is not readily available from other remote sensing methods and is essential for modern forest inventories. In this study, small-footprint lidar data were used to estimate biophysical properties of young, mature, and old cottonwood trees in the San Pedro River basin near Benson, Arizona. The lidar data were acquired in June 2004, using Optech's 1233 ALTM during flyovers conducted at an altitude of 600 m. Canopy height, crown diameter, stem dbh, canopy cover, and mean intensity of return laser pulses from the canopy surface were estimated for the cottonwood trees from the data. Linear regression models were used to develop equations relating lidar-derived tree characteristics with corresponding field acquired data for each age class of cottonwoods. The lidar estimates show a good degree of correlation with ground-based measurements. This study also shows that other parameters of young, mature, and old cottonwood trees such as height and canopy cover, when derived from lidar, are significantly different (P < 0.05). Additionally, mean crown diameters of mature and young trees are not statistically different at the study site (P = 0.31). The results illustrate the potential of airborne lidar data to differentiate different age classes of cottonwood trees for riparian areas quickly and quantitatively. Copyright © 2006 by the Society of American Foresters

Decomposition of Trees and Paths via Correlation

We study the problem of decomposing (clustering) a tree with respect to costs attributed to pairs of nodes, so as to minimize the sum of costs for those pairs of nodes that are in the same component (cluster). For the general case and for the special case of the tree being a star, we show that the problem is NP-hard. For the special case of the tree being a path, this problem is known to be polynomial time solvable. We characterize several classes of facets of the combinatorial polytope associated with a formulation of this clustering problem in terms of lifted multicuts. In particular, our results yield a complete totally dual integral (TDI) description of the lifted multicut polytope for paths, which establishes a connection to the combinatorial properties of alternative formulations such as set partitioning.Comment: v2 is a complete revisio

Faster Algorithms for the Maximum Common Subtree Isomorphism Problem

The maximum common subtree isomorphism problem asks for the largest possible isomorphism between subtrees of two given input trees. This problem is a natural restriction of the maximum common subgraph problem, which is ${\sf NP}$-hard in general graphs. Confining to trees renders polynomial time algorithms possible and is of fundamental importance for approaches on more general graph classes. Various variants of this problem in trees have been intensively studied. We consider the general case, where trees are neither rooted nor ordered and the isomorphism is maximum w.r.t. a weight function on the mapped vertices and edges. For trees of order $n$ and maximum degree $\Delta$ our algorithm achieves a running time of $\mathcal{O}(n^2\Delta)$ by exploiting the structure of the matching instances arising as subproblems. Thus our algorithm outperforms the best previously known approaches. No faster algorithm is possible for trees of bounded degree and for trees of unbounded degree we show that a further reduction of the running time would directly improve the best known approach to the assignment problem. Combining a polynomial-delay algorithm for the enumeration of all maximum common subtree isomorphisms with central ideas of our new algorithm leads to an improvement of its running time from $\mathcal{O}(n^6+Tn^2)$ to $\mathcal{O}(n^3+Tn\Delta)$, where $n$ is the order of the larger tree, $T$ is the number of different solutions, and $\Delta$ is the minimum of the maximum degrees of the input trees. Our theoretical results are supplemented by an experimental evaluation on synthetic and real-world instances