Abstract
Background
A multilabeled tree, or MULtree, is a phylogenetic tree where two or more leaves share a label, e.g., a species name. A MULtree can imply multiple conflicting phylogenetic relationships for the same set of taxa, but can also contain conflictfree information that is of interest and yet is not obvious.
Results
We define the information content of a MULtree T as the set of all conflictfree quartet topologies implied by T, and define the maximal reduced form of T as the smallest tree that can be obtained from T by pruning leaves and contracting edges while retaining the same information content. We show that any two MULtrees with the same information content exhibit the same reduced form. This introduces an equivalence relation among MULtrees with potential applications to comparing MULtrees. We present an efficient algorithm to reduce a MULtree to its maximally reduced form and evaluate its performance on empirical datasets in terms of both quality of the reduced tree and the degree of data reduction achieved.
Conclusions
Our measure of conflictfree information content based on quartets is simple and topologically appealing. In the experiments, the maximally reduced form is often much smaller than the original tree, yet retains most of the taxa. The reduction algorithm is quadratic in the number of leaves and its complexity is unaffected by the multiplicity of leaf labels or the degree of the nodes.
Keywords:
Phylogenetic trees; Evolutionary trees; Multilabeled trees; Reduction; Singlylabeled treesBackground
Multilabeled trees, also known as MULtrees, are phylogenetic trees that can have more than one leaf with the same label [15] (Figure 1). MULtrees arise naturally and frequently in data sets containing multiple gene sequences for the same species [6], but they can also arise in biogeographical studies or cospeciation studies where leaves represent individual taxa yet are labeled with their areas [7] or hosts [8].
Figure 1. A MULtree. Numbers in parenthesis next to labels indicate the multiplicity of the respective labels and are not part of the labels themselves.
MULtrees, unlike singlylabeled trees, can contain conflicting specieslevel phylogenetic information due to biological processes such as whole genome duplications [9] or incomplete lineage sorting [10], to artifactual processes such as inferential error, or, frequently, an unknown combination of several factors. However, they can also contain substantial amounts of conflictfree information.
Here we provide a way to extract this information; specifically, we have the following results.
• We introduce a new quartetbased measure of the information content of a MULtree, defined as the set of conflictfree quartets that the tree displays (see MULTrees and information content on page 3).
• We introduce the concept of the maximallyreduced form (MRF) of a MULtree T, the smallest tree with the same information content as T (see Maximally reduced MULTrees on page 4), and show that any two MULtrees with the same information content have the same MRF (Theorem 3).
• We present a simple algorithm to construct the MRF of a MULtree (see The reduction algorithm on page 7). Its running time is quadratic in the number of leaves and does not depend on the multiplicity of the leaf labels or the degrees of the internal nodes.
• We present computational experience with an implementation of our MRF algorithm (see Results and discussion on page 8). In our test data, the MRF is often significantly smaller than the original tree, while retaining most of the taxa.
We now give the intuition behind our notion of information content, deferring the formal definitions of this and other concepts to the next section. Quartets (i.e., sets of four species) are a natural starting point, since they are the smallest subsets from which we can draw meaningful topological information. A singlylabeled tree implies exactly one topology on any quartet. More precisely, each edge e in a singlylabeled tree implies a bipartition (A,B) of the leaf set, where each part is the set of leaves on one of the two sides of e. From (A,B), we derive a collection of bipartitions abcd of quartets, such that {a,b} ⊆ A and {c,d} ⊆ B. Clearly, if one edge in a singlylabeled tree implies some bipartition q = abcd of {a,b,c,d}, then there can be no other edge that implies a bipartition, such as acbd, that is in conflict with q. Indeed, the quartet topologies implied by a singlylabeled tree uniquely identify it [11].
The situation for MULtrees is more complicated, as illustrated in Figure 1. Here, the presence of two copies of labels b and c — b(1) and b(2), and, c(1) and c(2) — leads to two conflicting topologies on the quartet {b,c,d,e}. Edge (u,v) implies the bipartition bcde, corresponding to the labels {b(1),c(1),d,e}, while edge (v,w) implies bdce corresponding to the leaves {b(2),c(2),d,e}. On the other hand, the quartet topology afbc, implied by edge (t,u), has no conflict with any other topology that the tree exhibits on {a,b,c,f}. We show that the set of all such conflictfree quartet topologies is compatible (Theorem 1). That is, for every MULtree T there exists at least one singlylabeled tree that displays all the conflictfree quartets of T — and possibly some other quartets as well. Motivated by this, we only view conflictfree quartet topologies as informative, and define the information content of a MULtree as the set of all conflictfree quartet topologies it implies.
We should note that conflicting quartets may well provide valuable information, whether about paralogy, deep coalescence, or mistaken annotations. In some cases, specieslevel phylogenetic information can be recovered from conflicted quartets through application of, e.g., genetree speciestree reconciliation (generally an NPhard problem [12]). However, this is not feasible when the underlying cause of multiplicity is unknown or when conducting largescale analyses. Our definition of information content is deliberately designed to make no assumptions about the cause of conflict. It is also conservative with respect to species relationships, i.e., it does not introduce quartets not originally supported by the data. Further, knowing the information content of a MULtree allows us to easily identify its conflicting quartets as well.
A MULtree may have leaves that can be pruned and edges that can be contracted without altering the tree’s information content, i.e., without adding or removing conflictfree quartets. For example, in Figure 1, every quartet topology that edge (v,w) implies is either in conflict with some other topology (e.g., for set {b,c,d,e}) or is already implied by some other edge (e.g., afce is also implied by (t,u)). Thus, (v,w) can be contracted without altering the information content. In fact, the information content remains unchanged if we also contract (u,v) and remove the leaves labeled b(1) and c(1). We define the MRF of a MULtree T as the tree that results from applying informationpreserving edge contraction and leaf pruning operations repeatedly to T, until it is no longer possible to do so. The MRF of the tree in Figure 1 is shown in Figure 2. In this case, the MRF is singlylabeled; however, this is not true in general (see An example on page 8). If the MRF is itself a MULtree, it is not possible to reduce the original to a singlylabeled tree without either adding at least one quartet that did not exist conflictfree in T or by losing one or more conflictfree quartets.
Since any two MULtrees with the same information content have the same MRF, rather than comparing MULtrees directly, we can instead compare their MRFs. This is appealing mathematically, because it focuses on conflictfree information content, and also computationally, since an MRF can be much smaller than the original MULtree. Indeed, on our test data, the MRF was frequently singlylabeled. This reduction in input size is especially significant if the MULtree is an input to an algorithm whose running time is exponential in the label multiplicity, such as Ganapathy et al.’s algorithm to compute the contractandrefine distance between two area cladograms [7] or Huber et al.’s algorithm to determine if a collection of “multisplits” can be displayed by a MULtree [13].
For our experiments, we also implemented a postprocessing step, which converts the MRF to a singlylabeled tree, rendering it available for analyses that require singlylabeled trees, including supermatrix [14,15] and supertree methods [1619]. On the trees in our data set, the combined taxon loss between the MRF computation and the postprocessing was much lower than it would have been had we simply removed all duplicate taxa from the original trees.
Previous work on MULtrees has concentrated on finding ways to reduce MULtrees to singlylabeled trees (typically in order to provide inputs to supertree methods) [5], and to develop metrics and algorithms to compare MULtrees [7,2022]. In contrast to our approach — which is purely topologybased and is agnostic with respect to the cause of label multiplicity — the assumption underlying much of the literature on MULtrees is that taxon multiplicity results from gene duplication. Thus, methods to obtain singlylabeled trees from MULtrees usually work by pruning subtrees at putative duplication nodes. Although the proposed algorithms are polynomial, they are unsatisfactory in various ways. For example, in [5] if the subtrees are neither identical nor compatible, then the subtree with smaller information content is pruned, which seems to discard too much information. Further, the algorithm is only efficient for binary rooted trees. In [20] subtrees are pruned arbitrarily, while in [21] at each putative duplication node a separate analysis is done for each possible pruned subtree. Although the latter approach is better than pruning arbitrarily, in the worst case it can end up analyzing exponentially many subtrees.
MULTrees and information content
A MULtree is a triple (T,M,ψ), where (i) T is an unrooted tree^{a} with leaf set
An edge (u,v) in T is internal if neither u nor v belong to
Let (u,v) be an edge in T and T^{′} be the result of deleting (u,v) from T. Then
A (resolved) quartet in a MULtree T is a bipartition abcd of a set of labels {a,b,c,d} such that there is an edge (u,v) in T with
The information content of an edge (u,v) of a MULtree T, denoted Δ(u,v), is the set of quartets resolved by (u,v). An edge (u,v) in tree T is informative if Δ(u,v) > 0; (u,v) is maximally informative if there is no other edge (u^{′},v^{′}) in T with Δ(u,v) ⊂ Δ(u^{′},v^{′}). The information content of T, denoted
The next result shows that the quartets in
Theorem 1
For every MULtree T, there is a singly labeled tree T^{′}such that
Proof
Repeat the following step until T has no multiplyoccurring labels. Pick any multiplyoccurring label ℓ in T, select an arbitrary leaf labeled by ℓ, and relabel every other leaf labeled by ℓ, by a new, unique, label. The resulting tree T^{′} is singly labeled, and all labels of T are also present in T^{′}. Consider a quartet abcd in T, that is resolved by edge (u,v). Assume that
Note that there are examples where the containment indicated by the above result is proper.
To conclude this section, we give some results that are useful for the MULtree reduction
algorithm (see The reduction algorithm, beginning on page 7). In the next lemmas, (u,v) and (w,x) denote two edges in tree T that lie on the path
Figure 3. Supportive illustration for the proof of Lemma 1.
Lemma 1.
If
Proof
Refer to Figure
3. Since
Together with Lemma 1, the next result allows us to check whether the information
content of an edge is a subset of that of another based solely on the cardinalities
of the
Lemma 2
Δ(u,v) ⊆ Δ(w,x) if and only if
Proof
(Only if) Suppose Δ(u,v) ⊆ Δ(w,x); therefore,
(If) Suppose
Lemma 3
Suppose Δ(u,v) ⊆ Δ(w,x). Then, for any edge (y,z) onP_{u,x}such thatvis closer toythan toz, Δ(u,v) ⊆ Δ(y,z) ⊆ Δ(w,x).
Proof
By Lemma 2, since Δ(u,v) ⊆ Δ(w,x), we have
Maximally reduced MULTrees
Our goal is to provide a way to reduce a MULtree T as much as possible, while preserving its information content. Our reduction algorithm uses the following operations. Prune (v): Delete leaf v from T. If, as a result, v’s neighbor u becomes a degreetwo node, connect the former two neighbors of u by an edge and delete u. Contract (e): Delete an internal edge e and identify its endpoints.
A leaf v in T is prunable if the tree that results from pruning v has the same information content as T. An internal edge e in T is contractible if the tree that results from contracting e has the same information content as T. T is maximally reduced if it has no prunable leaf and no contractible internal edge.
Theorem 2
Every internal edge in a maximally reduced tree T resolves a quartet that is resolved by no other edge.
Proof
We rely on two facts. First, every internal node in the tree has degree at least three. Second, every internal edge in the tree resolves a quartet; otherwise, the edge would be contractible and the tree would not be maximally reduced.
Consider any edge (u,v) in the tree. To prove that (u,v) resolves a quartet not resolved by any other edge, we need to show that there exists a quartet abcd of the form shown in Figure 4. First, we describe how to select leaves a and b. Consider the following cases:
1. u has at least two neighbors i and j, apart from v, that are internal nodes. Then, we select any
Figure 4. Supportive illustration for the proof of Theorem 2. Quartet abcd is resolved only by edge (u,v). Here,
2. u has only one neighbor i ≠ v that is an internal node. Then, at least one of u’s neighboring leaves must participate in a quartet that (u,v) resolves. Without such a leaf, (u,v) would resolve the same set of quartets as (u,i), so one of these two edges would be contractible, contradicting the assumption that
the tree is maximally reduced. We select this leaf as b and we select any
3. All neighbors of u, except v, are leaves. Then, at least two of its neighbors must participate in a quartet, because (u,v) must resolve a quartet. We select the two neighbors as a and b.
In every case, we can select the desired leaves a and b. By a similar argument, we can also select the desired c and d. This proves the existence of the desired quartet abcd. Therefore, each internal edge of T uniquely resolve a quartet. □
The next result shows that the set of quartets resolved by a maximally reduced tree uniquely identifies the tree.
Theorem 3
Let T and T^{′}be two maximally reduced trees such that
The maximally reduced form (MRF) of a MULtree T is the tree that results from repeatedly pruning prunable leaves and contracting contractible edges from T until this is no longer possible. Theorem 3 shows that we can indeed talk about “the” MRF of T. Before proving Theorem 3, we mention some of its consequences.
Corollary 1
Every MULtree has a unique MRF.
Corollary 2
Any two MULtrees with the same information content have the same MRF.
Corollary 3
If a maximally reduced MULtreeTis not singlylabeled, there does not exist a singlylabeled treeT^{′}such that
Note that Corollary 3 does not contradict Theorem 1. If the MULtree in Theorem 1
is maximally reduced and not singlylabeled, the containment is proper; i.e.,
Figure 5. A maximally reduced MULtree.
Corollary 4
The relation “sharing a common MRF” is an equivalence relation on the set of MULtrees.
The last result implies that MULtrees can be partitioned into equivalence classes, where each class consists of the set of all trees with the same information content. Thus, instead of comparing MULtrees directly, we can compare their maximally reduced forms.
We now proceed to the proof of Theorem 3. We need two lemmas.
Lemma 4
There is a bijectionϕbetween the respective sets of internal edges of T and T^{′}with the following property. Let (u,v) be an internal edge in T and let (u^{′},v^{′}) = ϕ(u,v). Then,
Proof
Consider an edge (u,v) in T. By Theorem 2, (u,v) must resolve a quartet abcd not resolved by any other edge as shown in Figure
4. We claim that this quartet must be resolved uniquely by an edge (u^{′},v^{′}) in T^{′}. Suppose not. Using arguments similar to those in the proof of Lemma 3, we can show
that all edges that resolve abcd in T^{′} form a path (u^{′},x^{′},…,w^{′},v^{′}), where possibly x^{′} = w^{′}, as shown in Figure
6. Here,
Figure 6. Supportive illustration for the proof of Lemma 4.
Since (w^{′},v^{′}) resolves a quartet not resolved by any other edge, by Theorem 2 there exists a label
ℓ as shown, where
Now consider any label
We have shown that there is a onetoone mapping ϕ from edges in T to edges of T^{′} such that Δ(e) = Δ(ϕ(e)). To complete the proof, we show that ϕ is onto. Suppose that for some edge e^{′} in T^{′} there is no edge e in T such that ϕ(e) = e^{′}. But then e^{′} must resolve a quartet not resolved by any other edge in T^{′}. This quartet cannot be in
Let ϕ be the bijection between the edge sets of T and T^{′} from the preceding lemma.
Lemma 5
Let (u,v) and (v,x) be any two neighboring internal edges inT, and let (p,q) = ϕ(u,v) and (r,s) = ϕ(v,x) be the corresponding edges in T^{′}such that
Proof
Since (u,v) and (v,x) are neighbors, and each resolves a quartet that is not resolved by the other,
and
Let (z,w) = ϕ^{−1}(t,r) be the edge in T corresponding to (t,r). Irrespective of the position of (z,w) in T, (1) and (2) cannot be simultaneously true with respect to edges (u,v), (v,x) and (z,w) in T. Therefore, q = r, which proves the desired result. □
Proof of Theorem 3
Lemmas 4 and 5 show that T and T^{′} are isomorphic with respect to their internal edges. It remains to show a onetoone
correspondence between their leaf sets. For this, we match up the leaves attached
at every pendant node in T and T^{′}. We start with pendant nodes to which only one internal edge is attached. For example,
consider an internal edge (u,v) in T such that v is a pendant node and
Identifying contractible edges and prunable leaves
In preparation for the MULtree reduction algorithm of the next section, we give some results that help to identify contractible edges and prunable leaves.
The setting for the next result is the same as for Lemmas 2 and 3: (u,v) and (w,x) are two edges in tree T that lie on the path
Lemma 6
Suppose Δ(u,v) ⊆ Δ(w,x) then
1. every internal edge on a subtree branching out from P_{u,x}is contractible, and
2. if Δ(u,v) = Δ(w,x), every leaf on a subtree branching out from P_{u,x}is prunable. Thus, the entire subtree can be deleted without changing the information content of the tree.
Proof
Refer to Figure 7. □
1. Consider any edge (A,B) in a subtree branching out of P_{u,x}, as shown. We claim that
Figure 7. Supportive illustration for the proof of Lemma 6.
To prove the claim, observe first that, by definition,
Now,
2. Suppose Δ(u,v) = Δ(w,x). By an argument similar to the one used in the proof of Lemma 3, we can show that
any edge (y,z) on the path P_{v,w} = (v…w) (see Figure
7) satisfies
Now, let S be the tree obtained after pruning leaf c.
(a)
(b)
Hence, c is prunable. □
Lemma 7
Suppose that T is a MULtree where no pendant node is adjacent to two or more leaves with the same label. Let ℓ be any multiplyoccurring label in T and let T^{′}be the minimal subtree of T that spans all the leaves labeled by ℓ. Then, any leaf in T labeled ℓ attached to a pendant node of degree at least three in T^{′}is prunable.
Proof
Refer to Figure 8. Consider any pendant node v of degree at least three in T^{′} attached to a leaf labeled ℓ. Clearly deleting the leaf does not change the information content of any edge in T_{u} or T_{y}. Now consider an edge (w,x) in T^{′} as shown. Note that ℓ ∈ C^{wx}, so ℓ does not contribute to Δ(w,x). After deleting the leaf, we still have ℓ ∈ C^{wx}, so Δ(w,x) remains unchanged. Therefore, the leaf is prunable. □
Figure 8. Supportive illustration for the proof of Lemma 7. The leaves attached to pendant nodes u, v, and y are labeled by ℓ, and the subtrees indicated by T_{u} and T_{y} do not contain a leaf labeled with ℓ. Nodes u and y have degree two in T^{′}, while v has degree three.
The reduction algorithm
We now describe a O(n^{2}) algorithm to compute the MRF of an nleaf MULtree T. In the previous section, the MRF was defined as the tree obtained by applying informationpreserving pruning and contraction operations to T, in any order, until it is no longer possible. For efficiency, however, the sequence in which these steps are performed is important. Our algorithm has three distinct phases: a preprocessing step, redundant edge contraction, and pruning of redundant leaves. We describe these next and then give an example.
Preprocessing
For every edge (u,v) in T, we compute
Edge contraction and subtree pruning
Next, we repeatedly find pairs of adjacent edges (u,v) and (v,w) such that Δ(u,v) ⊆ Δ(v,w) or viceversa, and contract the less informative of the two. By Lemmas 1 and 2,
we can compare Δ(u,v) and Δ(v,w) in constant time using the precomputed values of
Pruning redundant leaves
The tree that is left at this point has no contractible edges; however, it can still
have prunable leaves. We first prune any leaf with a label ℓ that does not participate in any resolved quartet. Such an ℓ has the property that for every edge (u,v),
Next, we consider sets of leaves with the same label ℓ that share a common neighboring pendant node. Such leaves can be found in linear time. For each such set, we delete all but one element. Let T be the tree that results from removing such leaves. Now, the only prunable leaves with a given label ℓ that might remain are leaves attached to different pendant nodes. By Lemma 7, we can identify and prune such leaves by performing the following steps.
1. For each label ℓ, consider the subgraph on the leaves labeled by ℓ.
2. In this subgraph, delete any leaf not attached to a degree 2 pendant node as it is a redundant leaf.
This takes O(n) time per label and O(n^{2}) time total. The space used is O(n). Hence, the overall time and space complexities are O(n^{2}) time and O(n), respectively.
The resulting tree has no contractible edges nor prunable leaves. Therefore, it is the MRF of the orginal MULtree.
An example
We illustrate the reduction of the unrooted MULtree shown in Figure 9(a) to its MRF.
1. In the preprocessing step, we find that
Figure 9. Reduction to the MRF. Supportive illustrations for the example on page 8.
2. Since Δ(u,v) ⊂ Δ(v,w), contract (u,v). The result is shown in Figure 9(c).
3. Since Δ(v,w) = Δ(w,y), delete the subtree branching out at w from the path from v to y and contract (v,w). The result is shown in Figure 9(d).
4. Prune taxon 6, which does not participate in any quartet, and all duplicate taxa at the pendant nodes. The result, shown in Figure 9(e), is the MRF of the original tree.
Results and discussion
We implemented our MULtree reduction algorithm, as well as a second step that restricts the MRF to the set of labels that appear only once, which yields a singlylabeled tree. We tested our twostep program on a set of 110,842 MULtrees obtained from the PhyLoTA database [6] ( http://phylota.net/ webcite; GenBank eukaryotic nucleotide sequences, release 184, June 2011), which included a broad range of labelset sizes, from 4 to 1500 taxa.
There were 8,741 trees (7.8%) with essentially no information content; these lost all resolution either when reduced to their MRFs, or in the second step. The remaining trees fell into two categories. Trees in set A had a singlylabeled MRF; 65,709 trees (59.3%) were of this kind. Trees in set B were reduced to singlylabeled trees in the second step; 36,392 trees (32.8%) were of this kind. Reducing a tree to its MRF (step 1), led to an average taxon loss of 0.83% of the taxa in the input MULtree. The total taxon loss after the second step (reducing the MRFs in set B to singlylabeled trees), averaged 12.81%. This taxon loss is not trivial, but it is far less than the 41.27% average loss from the alternative, naïve, approach in which all MULtaxa (taxa that label more than one leaf) are removed at the outset. Note that, by the definition of MRFs, taxa removed in the first step do not contribute to the information content, since all nonconflicting quartets are preserved. On the other hand, taxa removed in the second step do alter the information content, because each such taxon participated in some nonconflicting quartet. Information content, in this case, will be lost but new information is never introduced, so the algorithm can be considered conservative.
Taxon loss is sensitive to the number of total taxa and, especially, MULtaxa, as demonstrated in Figure 10. The grey function shows the percentage of MULtaxa in the original input trees, which is the taxon loss if we had restricted the input MULtrees to the set of singlylabeled leaves. The black function shows the percentage of MULtaxa lost after steps 1 and 2 of our reduction procedure.
Figure 10. Experimental results: Taxon loss in the second step.
In addition to the issue of taxon loss, we investigated the effect of our reduction on edge loss, i.e., the level of resolution within the resulting singlylabeled tree. Input MULtrees were binary and therefore had more nodes than twice the number of taxa (Figure 11, solid line), whereas a binary tree on singly labeled taxa would have approximately as many nodes as twice the number of taxa (Figure 11, dashed line). We found that, although there was some edge loss, the number of nodes in the reduced singlylabeled trees (Figure 11, dotted line) corresponded well to the total possible, indicating low levels of edge loss. Note that each point on the dotted or solid lines represents an average over all trees with the same number of taxa.
Figure 11. Experimental results: Quality of reduced singlylabeled trees.
We have integrated our reduction algorithm into STBase (available at http://stbase.org/ webcite), a phylogenetic tree search engine that takes a userprovided list of species names and finds matches with a precomputed collection of phylogenetic trees, more than half of which are MULtrees, assembled from GenBank sequence data. The trees returned are ranked by a tree quality criterion that takes into account overlap with the query set, support values for the branches, and degree of resolution. We have added functionality to provide reduced singlylabeled trees as well as the MULtrees based on the full leaf set and the label sets from the reduced singlylabeled trees are used in downstream supermatrix construction.
Conclusions
We introduced an efficient algorithm to reduce a multilabeled MULtree to a maximally reduced form with the same information content, defined as the set of nonconflicting quartets it resolves. We showed that the information content of a MULtree uniquely identifies the MULtree’s maximally reduced form. This has potential application in comparing MULtrees by significantly reducing the number of comparisons as well as in extracting specieslevel information efficiently and conservatively from large sets of trees, irrespective of the underlying cause of multiple labels. Our algorithm can easily be adapted to work for rooted trees.
Further work investigating the relationship of the MRF to the original tree under various biological circumstances is also underway. We might expect, for example, that wellsampled nuclear gene families reduce to very small MRF trees, and that annotation errors in chloroplast gene sequences (in which we expect little gene duplication), result in relatively large MRF trees. Comparing the MRF to the original MULtree may well provide a method for efficiently assessing and segregating data sets with respect to the causes of multiple labels.
It would be interesting to compare our results with some of the other approaches for reducing MULtrees to singlylabeled trees (e.g., [5]) or, indeed, to evaluate if our method can benefit from being used in conjunction with such approaches.
Endnote
^{a}The results presented here can be extended to rooted trees, using triplets instead of quartets, exploiting the wellknown bijection between rooted and unrooted trees ( [23], p. 20).
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
MMM and DFB conceived the problem. AD, DFB and MMM designed the experiments and drafted the manuscript. AD designed and implemented the algorithms, and implemented the experiments. DFB coordinated the project. All authors read and approved the final manuscript.
Acknowledgements
This work was supported in part by National Science Foundation grant DEB0829674. We thank Mike Sanderson for helping to motivate this work, for many discussions about the problem formulation, and for our ongoing collaboration in the STBase project. Sylvain Guillemot listened to numerous early versions of our proofs and offered many insightful comments.
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