BK-tree

This picture depicts the BK-tree for the set W of words {"book", "books", "cake", "boo", "boon", "cook", "cape", "cart"} obtained by using the Levenshtein distance • each node u is labeled by a string of w_u \in W; • each arc (u,v) is labeled by d_{uv} = d(w_u,w_v) where w_u denotes the word assigned to u. The BK-tree is built so that: • for all node u of the BK-tree, the weight assigned to its egress arcs are distinct; • for all arc e=(u,v) labeled by k, each descendant v' of v satisfies the following equation: d(w_u, w_{v'}) = k: • Example 1: Consider the arc from "book" to "books". The distance between "book" and any word in {"books", "boo", "boon", "cook"} is equal to 1; • Example 2: Consider the arc from "books" to "boo". The distance between "books" and any word in {"boo", "boon", "cook"} is equal to 2. == Insertion ==

The insertion primitive is used to populate a BK-tree t according to a discrete metric d. Input: • t: the BK-tree; • d_{uv} denotes the weight assigned to an arc (u, v); • w_u denotes word assigned to a node u; • d: the discrete metric used by t (e.g. the Levenshtein distance); • w: the element to be inserted into t; Output: • The node of t corresponding to w Algorithm: • If the t is empty: • Create a root node r in t • w_r \leftarrow w • Return r • Set u to the root of t • While u exists: • k \leftarrow d(w_u, w) • If k = 0: • Return u • Find v the child of u such that d_{uv} = k • If v is not found: • Create the node v • w_v \leftarrow w • Create the arc (u, v) • d_{uv} \leftarrow k • Return v • u \leftarrow v == Lookup ==

Given a searched element w, the lookup primitive traverses the BK-tree to find the closest element of w. The key idea is to restrict the exploration of t to nodes that can only improve the best candidate found so far by taking advantage of the BK-tree organization and of the triangle inequality (cut-off criterion). Input: • t: the BK-tree; • d: the corresponding discrete metric (e.g. the Levenshtein distance); • w: the searched element; • d_{max}: the maximum distance allowed between the best match and w, defaults to +\infty; Output: • w_{best}: the closest element to w stored in t and according to d or \perp if not found; Algorithm: • If t is empty: • Return \perp • Create S a set of nodes to process, and insert the root of t into S. • (w_{best}, d_{best}) \leftarrow (\perp, d_{max}) • While S \ne \emptyset: • Pop an arbitrary node u from S • d_u \leftarrow d(w, w_u) • If d_u : • (w_{best}, d_{best}) \leftarrow (w_u, d_u) • For each egress-arc (u, v): • If |d_{uv} - d_u| : (cut-off criterion) • Insert v into S. • Return w_{best} == Example of the lookup algorithm ==

Consider the example 8-node B-K Tree shown above and set w="cool". S is initialized to contain the root of the tree, which is subsequently popped as the first value of u with w_u="book". Further d_u=2 since the distance from "book" to "cool" is 2, and d_{best}=2 as this is the best (i.e. smallest) distance found thus far. Next each outgoing arc from the root is considered in turn: the arc from "book" to "books" has weight 1, and since |1-2|=1 is less than d_{best}=2, the node containing "books" is inserted into S for further processing. The next arc, from "book" to "cake," has weight 4, and since |4-2|=2 is not less than d_{best}=2, the node containing "cake" is not inserted into S. Therefore, the subtree rooted at "cake" will be pruned from the search, as the word closest to "cool" cannot appear in that subtree. To see why this pruning is correct, notice that a candidate word c appearing in "cake"s subtree having distance less than 2 to "cool" would violate the triangle inequality: the triangle inequality requires that for this set of three numbers (as sides of a triangle), no two can sum to less than the third, but here the distance from "cool" to "book" (which is 2) plus the distance from "cool" to c (which is less than 2) cannot reach or exceed the distance from "book" to "cake" (which is 4). Therefore, it is safe to disregard the entire subtree rooted at "cake". Next the node containing "books" is popped from S and now d_u=3, the distance from "cool" to "books." As d_u > d_{best}, d_{best} remains set at 2 and the single outgoing arc from the node containing "books" is considered. Next, the node containing "boo" is popped from S and d_u=2, the distance from "cool" to "boo." This again does not improve upon d_{best} = 2. Each outgoing arc from "boo" is now considered; the arc from "boo" to "boon" has weight 1, and since |2-1|=1 , "boon" is added to S. Similarly, since |2-2|=0 , "cook" is also added to S. Finally each of the two last elements in S are considered in arbitrary order: suppose the node containing "cook" is popped first, improving d_{best} to distance 1, then the node containing "boon" is popped last, which has distance 2 from "cool" and therefore does not improve the best result. Finally, "cook" is returned as the answer w_{best} with d_{best}=1. == Time Complexity ==

The efficiency of BK-trees depends strongly on the structure of the tree and the distribution of distances between stored elements. In the average case, both insertion and lookup operations take \Theta(\log n) time, assuming the tree remains relatively balanced and the distance metric distributes elements evenly. In the worst case, when the data or distance function causes the tree to become highly unbalanced (for example, when many elements are at similar distances), both insertion and lookup can degrade to \Theta(n). In practical applications, the actual performance depends on the choice of distance metric (e.g., the Levenshtein distance) and on the allowed search radius during approximate matching. == See also ==