In a number of practical situations, data have structure and the relations among its component parts need to be coded with suitable data models. Trees are usually utilized for representing data for which hierarchical relations can be defined. This is the case in a number of fields like image analysis, natural language processing, protein structure, or music retrieval, to name a few. In those cases, procedures for comparing trees are very relevant. An approximate tree edit distance algorithm has been introduced for working with trees labeled only at the leaves. In this paper, it has been applied to handwritten character recognition, providing accuracies comparable to those by the most comprehensive search method, being as efficient as the fastest.This work is supported by the Spanish Ministry projects DRIMS (TIN2009-14247-C02), and Consolider Ingenio 2010 (MIPRCV, CSD2007-00018), partially supported by EU ERDF and the Pascal Network of Excellence

A. Habrard

B. DasGupta

C.-M. Lee

D. Rizo

D.F. Robinson

G. Valiente

H. Samet

K. Zhang

K.-C. Tai

P. Bille

R.A. Finkel

R.B. Russell

S.M. Selkow

T. Jiang

Calvo-Zaragoza, Jorge

Rizo, David

Iñesta, José M.

RUa Reposity University of Alicante

A distance for partially labeled treesNo Author GivenNo Institute GivenAbstract. Trees are a powerful data structure for representing datafor which hierarchical relations can be defined. It has been applied in anumber of fields like image analysis, natural language processing, proteinstructure, or music retrieval, to name a few. Procedures for comparingtrees are very relevant in many tasks where tree representations are in-volved. The computation of these measures is usually time consumingand different authors have proposed algorithms that are able to com-pute them in a reasonable time, by means of approximated versions ofthe similarity measure. Other methods require that the trees are fullylabeled for the distance to be computed. The measure utilized in thispaper is able to deal with trees labeled only at the leaves that runs inO(|T1| × |T2|) time. Experiments and comparative results are provided.Key words: Tree edit distance, approximate distances, qtrees1 IntroductionThe computation of a measure of the similarity between two trees is a subject ofinterest in very different areas where trees are suitable structures for data coding.Trees are able to code hierarchical relations in their structure in a natural wayand they have been utilized in many tasks, like text document analysis [10],protein structure [14], image representation and coding [3], or music coding andretrieval [12], to name just a few.Different approaches have been posed in order to perform this comparison.Some of them impose a restriction on how the comparison is performed, whileothers establish valid mappings. Some methods pay more attention on the treestructure, and others pay it on the content of the nodes and the leaves. Most ofthem are designed to work with fully labeled trees.The method proposed in this paper is designed to work with partially labeledtrees, more precisely with those labeled only at the leaves. This fact, focus moreon the coded content and the relations within its context. One of the fields wherethis situation is relevant is music comparison and retrieval. Trees have been usedfor this task and a number of representation and comparison schemes have beenapplied based on tree edit distances [12] or probabilistic similarity schemes [5].In any case, the computation of these measures is usually a time consumingtask and different authors have proposed algorithms that are able to computethem in a reasonable time [16], through approximated versions of the similaritymeasure. In this paper, a new algorithm is presented, able to deal with treeslabeled only at the leaves that runs in O(|T1| × |T2|) time.2 Tree comparison methodsIn general, trees can be divided in orderered and not ordered trees, and in evo-lutionary and not evolutionary trees. In this work only ordered trees are con-sidered. Regarding the evolutionary trees, they are often used to conceptuallyrepresent the evolutionary relationship of species or organisms in biology, evolu-tion of works in linguistics, statistical classifications, or even tracking computerviruses. An evolutionary tree can be defined as a tree with distinct labels atleaves. Several algorithms have been given to solve the comparison of evolution-ary trees [13, 2, 9]. In this work we deal with trees that have not distinct labels,and thus they are not evolutionary trees, so those algorithms are not applicablefor our problem.A number of similarity measures for ordered non-evolutionary trees have beendefined in the literature [1]. Some of them measure the sequence of operationsneeded to transform one tree in another one, others look for the longest commonpath from the root to a tree node. There are methods that allow wildcards in thematching process in the so-called variable-length doesn’t care (VLDC) distance.Several taxonomies have been proposed. The interested reader can look up ahierarchy of tree edit distances in [8] and [18].Table 1: Some tree edit and alignment distance algorithms and their time complexities.Tai [17] O(|T | × |T ′| × depth(T )2 × depth(T ′)2)Shasha & Zhang [19] O(|T | × |T ′| × min{depth(T ), | leaves(T )|} × min{depth(T ′), | leaves(T ′)|})Jiang [7] O(|T | × |T ′| × (rank(T ) + rank(T ′))2)Selkow [16] O(|T | × |T ′|)Valiente [18] O(|T | × |T ′| × log (|T | + |T ′|))Some of the most relevant tree edit and alignment distances have been com-piled in Table 1 together with their time complexities. These measures are theones utilized for comparison in this paper, except that of Tai due to its very highcomplexity.The classical edit distance between two trees d(T, T ′) is computed usingan edit script e = e1 · · · en that is a sequence of edit operations allowing thetransformation of a tree T to a tree T ′. The cost of an edit script C(e) is thesum of the costs of the edit operations involved in the script: C(e) =∑ni=1 c(ei).Given S(T, T ′), the set of all the scripts that enable the emission of T ′ given T ,the edit distance between T and T ′ is defined by: d(T, T ′) = mine∈S(T,T ′) C(e).The edit operations allowed over two trees are any of the following:– relabel the label l of a node v of T by the label l′ of another node w of T ′,denoted by (v, w) (Fig. 1a).– deletion of a non-root node v from T , denoted by (v, λ), consists of deletingit, making the children of v become the children of its parent node, in theposition that was occupied by v, preserving this way the left to right orderingof leaves (Fig. 1b).– insertion of a non-root node w in T ′, denoted by denoted by (λ,w). Given asequence wi · · ·wj of subtrees of a common parent w, the insertion of nodew′ makes those wi · · ·wj subtrees children of w′, and w′ child of w (Fig. 1c).Note that the operation (λ, λ) is not allowed.a bT T1 2(a) Substitute operationT T1 2aab(b) Delete operationaT T1 2ab(c) Insert operationFig. 1: Tree edit operations (from [6]).The edit cost of each operation, c(ei) is given based on that of the edit costof the symbols for the labels, c : Σ × Σ → R that depends on the particularapplication. Therefore, c(ei) denotes the cost of applying the edit operation(v, w).The approaches by Selkow [16], Valiente [18], and Jiang [7] are restrictedversion of this general methodology.3 Proposed tree comparison algorithmThe edit distances presented above are designed to work with fully labeled trees.In order to apply those algorithms to trees labeled only at the leaves, the non-labeled inner nodes can be assigned a special label “empty”. However, it isexpected they don’t work as well as they do with fully labeled trees.In order to overcome this situation two approaches are possible. The firstone consists of labeling all nodes using any bottom-up propagation scheme usingspecific knowledge of the application domain. The main drawback of that optionis that any intermediate process will condition the resulting trees, with a loss ofgenerality. The second approach is to design a distance function able to comparepartially labeled trees.The partially labeled tree comparison algorithm (sp) is based on the assump-tion that the similarity value between a labeled leaf and a non-labeled innernode should be the average of the chances of finding that leaf in the descendantsof that inner node. Fig. 2a shows the simplest case of having two leaf trees:sp(TA, TB) = δ(a, b), where δ(a, b) = 1 ⇐⇒ a = b, and 0 elsewhere. For com-paring the trees shown in Fig. 2b, the chances of finding the label a in TB arecomputed as sp(TA, TB) = (δ(a, x) + δ(a, y))/2. If, instead of a label, y anothertree is placed there, the function should be computed recursively. Finally, whennone of the trees is composed by a single leaf (Fig. 2c), the similarity function iscomputed like an edit distance between sequences wx and yz, where each symbolis a tree.This measuring method omits the accounting of the insertion or deletion ofnodes and only measures the chance of finding matching labels, giving moreimportance to the information hierarchically contained in the tree than to thetree structure.(a) (b) (c)Fig. 2: Similarity function sp representative cases.This method is designed for working with partially labeled trees, but wecan slightly adapt the original idea to work with fully labeled trees. The caseof comparing a leaf to a non-leaf tree (Fig. 3a) is computed as sp(TA, TB) =(δ(a, b) + δ(a, x) + δ(a, y))/3. And in the same way as in the case of non-labelednodes, the similarity sp(TA, TB) between two fully labelled trees (Fig. 3b) iscomputed as the edit distance between the sequences wx and yz, where eachsymbol is a tree, plus now the similarity between the labels a and b.(a) (b)Fig. 3: Similarity function sp working on fully labelled trees.The algorithmic details of this method can be found at [11]. In that paper it isalso shown that the time complexity of the proposed algorithm is O(|T1|× |T2|),like that fastest of the methods cited in Table 1.4 Experiments and resultsThe experiments are devised to show that the proposed tree distance is able toprovide good results in a reasonable time when compared to other classical treecomparison algorithms. For that, we need to select an application where the datacan be described in terms of trees.In general, images can be coded as QuadTrees [15]. In a QuadTree, each innernode has four children. Each leaf represents a region in the image, labeled withany of its properties. If the considered property in the region covered by thatleaf is not homogeneous enough (the deviation of their pixel values are over athreshold), the leaf is converted to a inner node and 4 leaves are created, splittingthe region into 4 sub-regions. This procedure is repeated recursively until al theregions are homogeneous. At this point the tree is built, containing labels onlyat the leaves.In particular, for binary images, the alphabet for the labels is Σ = {0, 1}.That is the case for the images of the isolated hand-written character imagesthat have been used and coded as QuadTrees (Fig. 4) for our experiments. Theproblem to solve is to identify a character or digit from a set made by differentwriters.It should be pointed out that this evaluation is not oriented to outperform thestate of the art in hand-written OCR, but to check whether the proposed tree editalgorithm is useful for solving tree-encoded structure comparisons efficiently. Inorder to this, its performance has been compared to those of other tree distances,not to other handwritten OCR methods.(a) Original image (b) Image encoded as QuadTreeFig. 4: Encoding image as a QuadTree.Corpora. The NIST image gallery [4] was utilized. To code the character imageswith QuadTrees the usual procedure has been followed, splitting recursively theimage in four parts until they reach a level where all the pixels in each new parthave the same value (foreground or background). That value will be placed onlyin the leaves.Two corpora have been used in our experimental setup. The first consists ofa set of 19,540 digits, the same number for each one. The second corpus consistsof a set of 50,400 letter-and-digits images, with the same number for each classagain.Character classification accuracy The experiments have been performedusing the following scheme. The corpora have been split into 10 folds. Then,given each fold, for each class, a prototype has been taken as a query, beingcompared to all prototypes in the fold, obtaining a list L of prototypes sortedby similarity value, Lk being the k-th position in the list.The accuracy of the system has been computed using the mean reciprocalrank (MRR). Let q be a query, p a prototype, and let the function c(p) returnthe class of the prototype p, the reciprocal rank RR for a query is computed as:RR(q) =1argmink{c(Lk(q)) = c(q)}(1)The denominator should be read as the minimum position in the list, k, so thatthe predicted class matches that of the query.The mean reciprocal rank is the mean of all RR(q) for all the queries.Running times were measured in milliseconds, taking into account only thetest phase, leaving aside the construction of the representations that may bedone off-line. All experiments were performed using a Sun machine with 8 GbRAM and 8 Intel(R) Xeon(R) CPU X5355 running at 2.66GHz, with a SUSELinux with kernel version 2.6.4.1 ResultsThe achieved results have been shown in the Fig. 5. For each tree comparisonmethod the MRR has been extracted to be used as y-axis in the bars. Theprocessing time for each query has also been used as the other y-axis.The results plotted in Fig. 5a denote that the proposed algorithm achievesa success rate comparable to the best algorithms but the processing time isalmost as efficient as the faster method. When working with alphanumeric cor-pus (Fig. 5b), the proposed algorithm behaves in a similar level of performanceregarding the other methods. 0 0.2 0.4 0.6 0.8 1Proposed Selkow Align Shasha 1 10 100 1000MRRms (log)MRR Time(a) Digits corpus 0 0.2 0.4 0.6 0.8 1Proposed Selkow Align Shasha 1 10 100 1000MRRms (log)MRR Time(b) Alphanumeric corpusFig. 5: Success rates of proposed method compared to classical tree edit distances.Thus, it seems that the proposed algorithm is able to compare hand-writtencharacter images encoded as QuadTree better than the other classical tree com-parison algorithms in terms of trade-off between time and success rate.5 ConclusionsAn approximate tree edit distance algorithm has been introduced for workingwith trees labeled only at the leaves. In order to assess its performance, it hasbeen applied to handwritten character recognition using the well-known NISTdatabase. Our aim was not to outperform the state of the art but to show thatthe proposed tree distance is able to provide good results in a reasonable timewhen compared to other tree distances.The performance has been compared to classical tree similarity algorithmsfrom the literature. The results show that, in the used context, the proposedalgorithm achieved accuracies comparable to those by the most comprehensivesearch method and it is as efficient as the fastest.References1. Philip Bille. A survey on tree edit distance and related problems. Theorical Com-puter Science, 337(1-3):217–239, 2005.2. B. DasGupta, X. He, T. Jiang, M. Li, J. Tromp, and L. Zhang. On distancesbetween phylogenetic trees. In SODA ’97: Proceedings of the eighth annual ACM-SIAM symposium on Discrete algorithms, pages 427–436, Philadelphia, PA, USA,1997. Society for Industrial and Applied Mathematics.3. Raphael A. Finkel and Jon Louis Bentley. Quad trees: A data structure for retrievalon composite keys. Acta Inf., 4:1–9, 1974.4. M.D. Garris and R.A. Wilkinson. Nist special database 3: Handwritten segmentedcharacters. NIST, Gaithersburg, Md.5. Amaury Habrard, Jose´ M.. In˜esta, David Rizo, and Marc Sebban. Melody recogni-tion with learned edit distances. Lecture Notes in Computer Science, 5342:86–96,2008.6. Carsten Isert. The editing distance between trees, 1999.7. Tao Jiang, Lusheng Wang, and Kaizhong Zhang. Alignment of trees – an alterna-tive to tree edit. Theoretical Computer Science, 143(1):137 – 148, 1995.8. Tetsuji Kuboyama, Kilho Shin, and Tetsuhiro Miyahara. A hierarchy of tree editdistance measures. Theoretical Computer Science and its Applications, 2005.9. Chuan-Min Lee, Ling-Ju Hung, Maw-Shang Chang, Chia-Ben Shen, and Chuan-Yi Tang. An improved algorithm for the maximum agreement subtree problem.Information Processing Letters, 94(5):211 – 216, 2005.10. Mitchell Marcus, Grace Kim, Mary Ann Marcinkiewicz, Robert Macintyre, AnnBies, Mark Ferguson, Karen Katz, and Britta Schasberger. The penn treebank: An-notating predicate argument structure. In In ARPA Human Language TechnologyWorkshop, pages 114–119, 1994.11. Authors ommited for blind review. New partially labelled tree similarity measure:a case study. Lecture Notes in Computer Science, 6218:296–305, august 2010.12. David Rizo, Kjell Lemstro¨m, and Jose´ M. In˜esta. Tree representation in combinedpolyphonic music comparison. Computer Music Modeling and Retrieval. Genesis ofMeaning in Sound and Music. Lecture Notes in Computer Science, 5493/2009:177–195, 2009.13. D. F. Robinson and L. R. Foulds. Comparison of phylogenetic trees. MathematicalBiosciences, 53(1-2):131–147, February 1981.14. Robert B. Russell and Geoffrey J. Barton. Multiple protein sequence alignmentfrom tertiary structure comparison: Assignment of global and residue confidencelevels. Proteins: Structure, Function, and Bioinformatics, 14:309–323, 2004.15. Hanan Samet. The quadtree and related hierarchical data structures. ACM Com-put. Surv., 16(2):187–260, 1984.16. Stanley M. Selkow. The tree-to-tree editing problem. Information ProcessingLetters, 6(6):184–186, 1977.17. Kuo-Chung Tai. The tree-to-tree correction problem. J. ACM, 26(3):422–433,1979.18. G. Valiente. An efficient bottom-up distance between trees. In International Sym-posium on String Processing and Information Retrieval, pages 212–219, Los Alami-tos, CA, USA, 2001. IEEE Computer Society.19. K. Zhang and D. Shasha. Simple fast algorithms for the editing distance betweentrees and related problems. SIAM J. Comput., 18(6):1245–1262, 1989.

A distance for partially labeled trees

Abstract

Similar works

Full text

Available Versions

RUa Reposity University of Alicante

Crossref