Tree edit distance

Here, we briefly review tree edit distance and edit distance mapping (see also Figure 1) for rooted, labelled and unordered trees [12, 15, 16].

Let T be a rooted unordered tree. We assume that each node v has a label ℓ(v) over an alphabet Σ. r(T), V(T), and E(T) denote the root of T, the set of nodes in T, and the set of edges in T, respectively. For a node v ∈ V(T), anc(v) denotes the set of ancestors of v. In the following, n denotes the number of nodes in a larger input tree (i.e., n = max{|V(T₁)|, |V(T₂)|}).

An edit operation on a tree T is either a deletion, an insertion, or a substitution, where each operation is defined as follows (see also Figure 1):

Deletion: Delete a non-root node v in T with parent u, making the children of v become children of u. The children are inserted in the place of v into the set of the children of u.

Insertion: Inverse of delete. Insert a node v as a child of u in T, making v the parent of some of the children of u.

Substitution: Change the label of a node v in T.

For each of edit operations, the cost is defined as follows:

γ(a, b): cost of substituting a node with label a to label b,

γ(a, ∈): cost of deleting a node labeled with a,

γ(∈, a): cost of inserting a node labeled with a.

The edit distance dist(T₁, T₂) between two unordered trees T₁ and T₂ is defined as the cost of the minimum cost sequence of edit operations that transforms T₁ to T₂. In this paper, we adopt the following standard assumption so that dist(T₁, T₂) becomes a distance metric [12, 15]:

where Σ′ = ΣU {∈}. We call T₂ a subtree of T₁ if T₂ is obtained from T₁ only by deletion operations. It should be noted that this definition of subtree is different from a subgraph of a tree.

Let I₁ and I₂ be the sets of nodes in V(T₁) and V(T₂) not appearing in M, respectively. Then, the following relation holds [12, 15]:

Here we define a score function f(u, v) for (u, v) ∈ V(T₁) × V(T₂) by

f(u, v) = γ(ℓ(u), ∈) + γ(∈, ℓ(v)) - γ(ℓ(u), ℓ(v)).

It is seen that f(u, v) = f(v, u) ≥ 0 holds. It should also be noted that under the unit cost model (i.e., γ(a, b) = 1 for all a ≠ b), f(v, v) = 2 and f(u, v) = 1 hold for ℓ(u) ≠ ℓ(v). Let score(M) be the score of a mapping M defined by score(M) = ∑_{( u,v )∈ M} f(u, v). Let M _OPT be the mapping with the maximum score. Then, we can see from the definition that the following property holds [17]:

(1)

assuming that the root of T₁ corresponds to the root of T₂ in M _OPT , where this assumption can be removed if we add dummy nodes having the same label to T₁ and T₂ as the new roots.

Reduction to maximum clique

Let G(V, E) be an undirected graph. Then, a subgraph G′(V′, E′) of G(V, E) is called a clique if it is a complete subgraph (i.e., {{v _i ,v _j } | v _i , v _j ∈ V′, v _i ≠ v _j } = E′). The maximum clique problem is to find a maximum clique (i.e., a clique with the maximum number of vertices) in a given undirected graph. Though the maximum clique problem is known to be NP-hard, several practical algorithms have been developed [20–23]. In some cases, weighted versions of the maximum clique problem are utilized. Among such variants, we consider the case that weights are given to vertices. Let w(v) denote the weight of a vertex v in G(V, E). Then, a weighted version of the maximum clique problem is to find a clique G′(V′, E′) which maximizes ∑ _v∈V′ w(v). In this paper, we call this variant the maximum vertex weighted clique problem, whereas the maximum clique problem denotes the original one.

Our proposed method is based on a simple reduction from the edit distance problem for unordered trees to the maximum clique problem. Based on Eq. (1), for calculating the tree edit distance, it is enough to find a mapping M which maximizes ∑_{(u, v)∈M}f(u, v). In order to find such a mapping, we construct an undirected graph G(V,E) from two input trees T₁ and T₂ by

V = {(u, v) | u ∈ V(T₁), u ≠ r (T₁), v ∈ V(T₂), v ≠ r(T₂)},

E = {{(u₁,v₁), (u₂,v₂ )} | u₁ ≠ u₂, v₁ ≠ v₂, u₁∈ anc(u₂) iff v₁ ∈ anc(v₂), u₂ ∈ anc(u₁) iff v₂ ∈ anc(v₁)},

where the first two conditions and the last two conditions in the definition of E correspond to conditions (i) and (ii) for the edit distance mapping, respectively. We can see that there is a one-to-one correspondence between the set of cliques and the set of edit distance mappings, where we let r(T₁) correspond to r(T₂) (because the root cannot be deleted or inserted). Here, we assign weight w(x) to each vertex x = (u, v) ∈ V by w(x) = f(u, v). Then, we can see from Eq. (1) that the tree edit distance can be obtained by finding a maximum vertex weighted clique.

It is to be noted that if we consider the case of γ(a, ∈) = γ(∈, a) = 1, γ(a, a) = 0 for all a ∈ Σ, and γ(a, b) = 2 for all a ≠ b, we have f(v, v) = 2 and f(u, v) = 0 for ℓ(u) ≠ ℓ(v), and thus we can use a non-weighted version of maximum clique algorithms (see Figure 2). In such a case, the resulting mapping gives a largest common subtree (a tree with the largest number of nodes which is a subtree of both T₁ and T₂) [17].

Maximum clique algorithms

In this study, we use algorithms for both the maximum clique problem and the maximum vertex weighted clique problem. For both problems, Tomita and his collaborators have been developing several algorithms. Recent studies on comparison with other existing algorithms suggest that their algorithms are fastest in most cases[22]. Based on several preliminary experiments, we chose MCQ and MWCQ as algorithms for the maximum clique problem and the maximum vertex weighted clique problem, respectively, where MWCQ is basically an extended version of MCQ. Details of MCQ and MWCQ are given in [21] and [20], respectively.

Though there are some theoretical studies on related algorithms [23], the worst case time complexities of MCQ and MWCQ are left open. Therefore, we cannot discuss the time complexity of our proposed method, whereas it is straight-forward to see that the graph obtained by reduction from input trees has O(|V(T₁)| × |V(T₂)|) nodes and O(|V(T₁)|² × |V(T₂)|²) edges.

Results on efficiency

Though MCQ-based method is very fast, it makes extensive use of identity of node labels (node pairs without non-identical labels are ignored and thus the number of remaining nodes in G(V, E) becomes very small). On the other hand, MWCQ-based method takes all node pairs between T₁ and T₂ into account and thus is not very fast. Compared with an existing method [3], MCQ-based method is much faster but solves an easier problem (it seems from their results that their method can be applied to comparison of trees with up to 90 nodes (sum of two input trees) though CPU time is not shown in [3]). On the other hand, it seems from Table 1 that MWCQ-based method has a similar performance with that in [3] though [3] solves non-labeled cases whereas we solved labeled cases. However, MWCQ-based method has a merit: it can handle general editing cost functions whereas the method in [3] can only handle the unit editing cost.

Results on similar structure search

Though the ordered and unordered tree edit distances are widely-accepted (dis)similarity measures on trees, we performed computational experiments in order to examine how it is useful for similarity search for glycans. We used a dataset compiled by Yamanishi et al. [9] on four properties on glycans, where we used 355 structures among 356 glycan structures listed in their list since we could not obtain one structure. Though this dataset is prepared for evaluating machine learning methods, we applied it to evaluation of search methods. We compared the following four similarity search methods: global glycan alignment and local glycan alignment implemented in the KCaM glycan search tool (version of Sept. 2004 with the default parameters) [6], unit cost ordered tree edit distance, and unit cost unordered tree edit distance (i.e., MWCQ-based method). Glycan alignment scores were introduced for efficient comparison of glycan structures. Though it is based on tree edit distance, the deletion (and corresponding insertion) operation is simplified so that only one child and its descendants can survive if a node is deleted. Therefore, there is a possibility that similar structures are missed by glycan alignment.

We evaluated the performance of similarity search using the AUC score [27]. In order to apply the AUC score, we need positive and negative samples. For that purpose, each pair of sequences in the dataset is regarded as a positive sample if the distance (resp., alignment score) is smaller (resp., greater) than a given threshold. Otherwise, it is regarded as a negative sample. Each positive sample is classified into either true positive or false positive according to whether sequences in the pair belong to the same class. Similarly, each negative sample is classified into either false negative or true negative according to whether sequences in the pair belong to the same class. Then, true positive rate and false positive rate are defined as the ratio of the number of true positive samples to the number of true positive and false negative samples and the ratio of the number of false positive samples to the number of false positive and true negative samples, respectively. The ROC (Receiver Operating Characteristic) curve is a graphical plot of the true positive rate vs. the false positive rate obtained by varying the threshold. The AUC (Area Under Curve) score is defined as the area under the ROC curve: AUC scores of 1 and 0.5 correspond to complete classification and random classification, respectively. The resulting ROC curves are shown in Figure 3 and Figure 4, and the resulting AUC scores are shown in Table 3. It should be mentioned that we could not obtain meaningful AUC scores for plasma and serum datasets (i.e., AUC scores for these data sets were less than 0.65 though the local alignment method produced better results). Since it seems that these data sets are not appropriate for simple search, Table 3 lists AUC scores only for leukemia and erythrocyte datasets. It is seen from the table that the tree edit distance measures are better than the alignment scores for leukemia data but are worse for erythrocyte data. It is also seen that the AUC scores for ordered tree edit distance are very close to the AUC scores for unordered tree edit distance. Though we cannot conclude that the unordered tree edit distance is better than other similarity measures for glycan search, it is comparative to other measures.

In order to see more differences between ordered tree edit distance and unordered tree edit distance, we computed ordered tree edit distances when the order of children of every node was reversed in one of the input trees. The results are also shown in Table 3 (denoted as “reversed ordered tree”), Figure 3, and Figure 4 (denoted as ‘reverse’). For the result on the erythrocyte dataset, it is seen that the difference between ordered tree edit distance and unordered tree edit distance becomes larger (i.e., the difference increases from 0.004 to 0.008) though it is still small. Though we do not clearly understand the reason of this small difference, it might be because a single path in each glycan structure is relevant for the features studied in this paper.

The total CPU time for computing the distances (or scores) between all pairs of glycans in the dataset is also shown for each method in Table 3. Though the proposed clique-based method took more CPU time than other methods, the differences were not very large. It should be mentioned that we used a clique-based method for computing ordered tree edit distance for simplicity of implementation and thus CPU time on ordered tree edit distance would be much larger here than that by an efficient dynamic programming-based algorithm [14], but that is not relevant because CPU time for unordered tree edit distance is fast enough in Table 3.

Here, we briefly explain the methodological differences among measures. Figure 5 illustrates the difference between unordered tree edit distance and ordered tree edit distance. As shown in Figure 5, suppose that there exist two trees T₁ and T₂ with roots r₁ and r₂. Suppose further that r₁ has two subtrees A and B, and r₂ has two subtrees B′ and A′ in these orders, where A and A′ (B and B′, respectively) are similar to each other, and A is larger than B. If two trees are regarded as ordered, ordered tree edit mapping takes only matching between A and A′ into account. Otherwise, unordered tree mapping takes matchings between A and A′, and between B and B′ into account. Figure 6 illustrates the difference of the deletion operation between tree edit and glycan alignment. In tree edit, all children of the deleted node u become children of the parent v of u. However, in glycan alignment, only one child can be a child of v and the other children are deleted along with their descendants, where the surviving child is chosen so that the resulting score is maximized. It is seen from these figures that the tree edit distance for unordered trees provides the most flexible matching.