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M-tree

M-trees are tree data structures that are similar to R-trees and B-trees. It is constructed using a metric and relies on the triangle inequality for efficient range and k-NN queries. While M-trees can perform well in many conditions, the tree can also have large overlap and there is no clear strategy on how to best avoid overlap. In addition, it can only be used for distance functions that satisfy the triangle inequality, while many advanced dissimilarity functions used in information retrieval do not satisfy this.[1]

Contents

Overview

2D M-Tree visualized using ELKI. The tree has a single level of leaf nodes. Due to a suboptimal split heuristic, there is a large overlap.

As in any Tree-based data structure, the M-Tree is composed of Nodes and Leaves. In each node there is a data object that identifies it uniquely and a pointer to a sub-tree where its children reside. Every leaf has several data objects. For each node there is a radius r that defines a Ball in the desired metric space. Thus, every node n and leaf l residing in a particular node N is at most distance r from N, and every node n and leaf l with node parent N keep the distance from it.

M-Tree construction

Components

An M-Tree has these components and sub-components:

  1. Root.
    1. Routing object O.
    2. Radius Or.
    3. A set of Nodes or (exclusive) Leafs.
  2. Node.
    1. Routing object O.
    2. Radius Or.
    3. Distance from this node to its parent Op.
    4. A set of Nodes or (exclusive) Leafs.
  3. Leaf.
    1. Routing object O.
    2. Radius Or.
    3. Distance from this leaf to its parent Op.
    4. Objects.
  4. Objects.
    1. Feature value (usually a d-dimensional vector).

Insert

The main idea is first to find a leaf node N where the new object O belongs. If N is not full then just attach it to N. If N is full then invoke a method to split N. The algorithm is as follows:

Algorithm Insert  Input: Node N  of M-Tree MT, Entry O_{n}  Output: A new instance of MT containing all entries in original MT plus O_{n}
  N_{e} ← Entries of node N  if N is not a leaf then  {   /*Look for entries that the new object fits into*/   let N_{in} be entries such that d(O_{r}, O_{n}) <= r(O_{r})   if N_{in} is not empty then   {  /*If there are one or more entry, then look for an entry such that is closer to the new object*/  let be O_{r}^{*} \in N_{in} such that d(O_{r}^{*}, O_{n}) is minimum   }   else   {  /*If there are no such entry, then look for an entry with minimal distance from its edge to the new object*/  let be O_{r}^{*} \in N_{in} such that d(O_{r}^{*}, O_{n}) - r(O_{r}) is minimum  /*Upgrade the new radii of the entry*/  r(O_{r}^{*}) = d(O_{r}^{*}, O_{n})   }   /*Continue inserting in the next level*/   return insert(T(O_{r}^{*}), O_{n});  else  {   /*If the node has capacity then just insert the new object*/   if N is not full then   {  store(N, O_{n})   }   /*The node is at full capacity, then it is needed to do a new split in this level*/   else   {  split(N, O_{n}) }  }
  • "←" is a shorthand for "changes to". For instance, "largest ← item" means that the value of largest changes to the value of item.
  • "return" terminates the algorithm and outputs the value that follows.

Split

If the split method arrives to the root of the tree, then it choose two routing objects from N, and creates two new nodes containing all the objects in original N, and store them into the new root. If split methods arrives to a node N that is not the root of the tree, the method choose two new routing objects from N, re-arrange every routing object in N in two new nodes N_{1} and N_{2}, and store this new nodes in the parent node N_{p} of original N. The split must be repeated if N_{p} has not enough capacity to store N_{2}. The algorithm is as follow:

Algorithm Split  Input: Node N  of M-Tree MT, Entry O_{n}  Output: A new instance of MT containing a new partition.
  /*The new routing objects are now all those in the node plus the new routing object*/  let be NN entries of N \cup O  if N is not the root then  { /*Get the parent node and the parent routing object*/ let O_{p} be the parent routing object of N let N_{p} be the parent node of N  }  /*This node will contain part of the objects of the node to be split*/  Create a new node N'  /*Promote two routing objects from the node to be split, to be new routing objects*/  Promote(N, O_{p1}, O_{p2})  /*Choose which objects from the node being split will act as new routing objects*/  Partition(N, O_{p1}, O_{p2}, N_{1}, N_{2})  /*Store entries in each new routing object*/  Store N_{1}'s entries in N and N_{2}'s entries in N'  if N is the current root then  {  /*Create a new node and set it as new root and store the new routing objects*/  Create a new root node N_{p}  Store O_{p1} and O_{p2} in N_{p}  }  else  {  /*Now use the parent rouing object to store one of the new objects*/  Replace entry O_{p} with entry O_{p1} in N_{p}  if N_{p} is no full then  {  /*The second routinb object is stored in the parent only if it has free capacity*/  Store O_{p2} in N_{p}  }  else  {   /*If there is no free capacity then split the level up*/   split(N_{p}, O_{p2})  }  }
  • "←" is a shorthand for "changes to". For instance, "largest ← item" means that the value of largest changes to the value of item.
  • "return" terminates the algorithm and outputs the value that follows.

M-Tree Queries

Range queries

Range queries

k-NN queries

See also

References

  1. ^ Ciaccia, Paolo; Patella, Marco; Zezula, Pavel (1997). "M-tree An Efficient Access Method for Similarity Search in Metric Spaces". Proceedings of the 23rd VLDB Conference Athens, Greece, 1997. IBM Almaden Research Center: Very Large Databases Endowment Inc.. pp. 426–435. p426. http://www.vldb.org/conf/1997/P426.PD F. Retrieved 2010-09-07.
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