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]

Overview

2D M-Tree visualized using ELKI. Due to the axis scales, the spheres appear ellipsoidal. Every blue sphere (leaf) is contained in a red sphere (directory nodes). Leaves overlap, but not too much.

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:

Non-leaf nodes
1. A set of routing objects N_RO.
2. Pointer to Node's parent object O_p.
Leaf nodes
1. A set of objects N_O.
2. Pointer to Node's parent object O_p.
Routing Object
1. (Feature value of) routing object O_r.
2. Covering radius r(O_r).
3. Pointer to covering tree T(O_r).
4. Distance of O_r from its parent object d(O_r,P(O_r))
Object
1. (Feature value of the) object O_j.
2. Object identifier oid(O_j).
3. Distance of O_j from its parent object d(O_j,P(O_j))

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  $O_{n}$

   $N_{e} \gets N$ 's routing objects or objects
  if  $N$  is not a leaf then
  {
       /* Look for entries that the new object fits into */
       let  $N_{in}$  be routing objects from  $N_{e}$ 's set of routing objects  $N_{RO}$  such that  $d(O_{r}, O_{n}) \le r(O_{r})$ 
       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 */
           $O_{r}^{*} = \min_{O_{r}\in N_{in}} d(O_{r}, O_{n})$ 
       }
       else
       {
          /* If there are no such entry, then look for an object with minimal distance from */ 
          /* its covering radius's edge to the new object */
           $O_{r}^{*} = \min_{O_{r}\in N_{in}} d(O_{r}, O_{n}) - r(O_{r})$ 
          /* 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 these 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 */
  Create new objects  $O_{p1}$  and  $O_{p2}$ .
  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 routing 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 Query

A range query is where a minimum similarity/maximum distance value is speciﬁed. For a given query object Q ∈ D and a maximum search distance r(Q), the range query range(Q, r(Q)) selects all the indexed objects Oj such that d(Oj, Q) ≤ r(Q).^[2]

Algorithm RangeSearch starts from the root node and recursively traverses all the paths which cannot be excluded from leading to qualifying objects.

Algorithm RangeSearch
Input: Node  $N$  of M-Tree MT,   $Q$ : query object,  $r(Q)$ : search radius

Output: all the DB objects such that  $d(Oj, Q) \le r(Q)$

{ 
  let  $O_{p}$  be the parent object of node  $N$ ;

  if  $N$  is not a leaf then { 
    for each entry( $O_{r}$ ) in  $N$  do {
          if  $| d(O_{p}, Q) - d(O_{r}, O_{p}) | \le r(Q) + r(O_{r})$  then { 
            Compute  $d(O_{r}, Q)$ ;
            if  $d(O_{r}, Q) \le r(Q) +r(O_{r})$  then
              RangeSearch(*ptr( $T(O_{r}$ )), $Q$ , $r(Q)$ ); 
          }
    }
  }
  else { 
    for each entry( $O_{j}$ ) in  $N$  do {
          if  $| d(O_{p}, Q) - d(O_{j}, O_{p}) | \le r(Q)$  then { 
            Compute  $d(O_{j}, Q)$ ;
            if  $d(O_{j}, Q)$  ≤  $r(Q)$  then
              add  $oid(O_{j})$  to the result;
          }
    }
  }
}

"←" 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.

$oid(O_{j})$ is the identiﬁer of the object which resides on a separate data ﬁle.

$T(O_{r})$ is a sub-tree – the covering tree of $O_{r}$

k-NN queries

K Nearest Neighbor (k-NN) query takes the cardinality of the input set as an input parameter. For a given query object Q ∈ D and an integer k ≥ 1, the k-NN query NN(Q, k) selects the k indexed objects which have the shortest distance from Q, according to the distance function d. ^[2]

References

↑ Ciaccia, Paolo; Patella, Marco; Zezula, Pavel (1997). "M-tree An Efficient Access Method for Similarity Search in Metric Spaces" (PDF). Proceedings of the 23rd VLDB Conference Athens, Greece, 1997. IBM Almaden Research Center: Very Large Databases Endowment Inc. pp. 426–435. p426. Retrieved 2010-09-07.
1 2 P. Ciaccia, M. Patella, F. Rabitti, P. Zezula. "Indexing Metric Spaces with M-tree" (PDF). Department of Computer Science and Engineering. University of Bologna. p. 3. Retrieved 19 November 2013.

Tree data structures

Search trees (dynamic sets/associative arrays)	2–3 2–3–4 AA (a,b) AVL B B+ B* B^x (Optimal) Binary search Dancing HTree Interval Order statistic (Left-leaning) Red-black Scapegoat Splay T Treap UB Weight-balanced

Heaps	Binary Binomial Fibonacci Leftist Pairing Skew Van Emde Boas

Tries	Ctrie C-trie (compressed ADT) Hash Radix Suffix Ternary search X-fast Y-fast

Spatial data partitioning trees	BK BSP Cartesian Hilbert R k-d (implicit k-d) M Metric MVP Octree Priority R Quad R R+ R* Segment VP X

Other trees	Cover Exponential Fenwick Finger Fractal tree index Fusion Hash calendar iDistance K-ary Left-child right-sibling Link/cut Log-structured merge Merkle PQ Range SPQR Top

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