A New Compressed Suffix Tree Supporting Fast Search and its Construction Algorithm Using Optimal Working Space Dong Kyue Kim 1 andHeejin Park 2 1 School of Electrical and Computer Engineering, Pusan National Univ. 2 College of Information and Communications, Hanyang Univ.

Contents Preliminaries Previous results Our contribution Conclusion

Suffix Tree The suffix tree (ST) of a text T A compacted trie for all the suffixes of T. An example for accagat#. agat# c accagat#at# gat#t# a g c g cagat# ccagat# t # # t c a We assume that # is the lexicographically smallest special symbol.

Suffix Array The suffix array (SA) of a text T pos array lcp array

Suffix Array pos The pos array of T stores the starting positions of the lexicographically sorted suffixes of T. 18# 21a c c a g a t # 34a g a t # 46a t # 53c a g a t # 62c c a g a t # 75g a t # 87t # The suffix array (SA) of a text T pos array lcp array T = accagat#

Suffix Array pos lcp The pos array of T stores the starting positions of the lexicographically sorted suffixes of T. The lcp array of T stores the length of the longest common prefix of every adjacent suffixes in the pos array. For example, lcp[3] stores 1 that is the length of the longest common prefix of accagat# and agat#. 18# 210a c c a g a t # 341a g a t # 461a t # 530c a g a t # 621c c a g a t # 750g a t # 870t # The suffix array (SA) of a text T pos array lcp array T = accagat#

Storing Suffix Trees in Arrays Suffix trees can be stored in arrays if it is used as a static data structure.  If a suffix tree is used as a static data structure, they can be implemented using arrays instead of using nodes and pointers in a similar way a complete binary tree is stored in an array. Array-based data structures storing suffix trees Enhanced suffix arrays (ESA) Linearlized suffix trees (LST)

Enhanced Suffix Array Enhanced suffix array developed by Abouelhoda et al. [SPIRE ’02, WABI ’02, JDA ’04] a pos array + an lcp array + a child table The child table is an array implementation of the suffix tree topology whose node branching is implemented by the linked list. Pattern search takes O(m| Σ|) time.  m: pattern length, |Σ|: size of alphabets

Linearlized Suffix Tree Linearlized suffix tree An improvement on ESA developed by Kim et al. [SPIRE ’04] a pos array + an lcp array + a new child table The new child table is an array implementation of the suffix tree topology whose node branching is implemented by the complete binary tree. Pattern search takes O(m log | Σ|) time.  m: pattern length, |Σ|: size of alphabets

Compressed Full-text Indices Compressed full-text indices  Occupy O(n log|Σ|)-bit space.  All full-text indices (ST, SA, ESA, LST) we just introduced occupy O(n)-word space. Compressed suffix array (CSA)  Succinct representation of pos array. Compressed suffix tree (CST)  Succinct representation of a pos array, an lcp array, and a suffix tree topology.

Previous Results Munro et al. [1998], Sadakane[2002]  A succinct representation of a suffix tree topology Grossi and Vitter [2000]  A succinct representation of a pos array Sadakane [2002]  A succinct representation of an lcp array These data structures require O(n log|Σ|)-bit space, however, when they were introduced, the working space is more than O(n log|Σ|) bits.

Previous Results Hon et al.[2002][2003] developed O(n log|Σ|)-bit working space algorithms for constructing CSTs and CSAs that run in O(n log ε n) time. Their construction algorithm for CSTs can construct CSTs supporting O(n log ε n |Σ|)-time pattern search. However, it cannot construct CSTs supporting O(n log ε n log|Σ|)-time pattern search.

Our Contribution We first present a new CST supporting O(n log ε n log|Σ|)- time pattern search. Then, we present an algorithm for constructing the new CST running in optimal O(n log|Σ|)-bit working space and O(n log ε n) time.

New Compressed Suffix Tree Our new compressed suffix tree is a succinct representation of the linearlized suffix tree (LST).  a succinct representation of a pos array,  a succinct representation of an lcp array, and  a succinct representation of a child table, which stores a suffix tree topology.

New Compressed Suffix Tree Succinct representation of a pos array and an lcp array are the same as before.  a succinct representation of a pos array (Grossi & Vitter)  a succinct representation of an lcp array (Sadakane) Succinct representation of a child table, which stores a suffix tree topology, is a new one.

Previous Compressed Suffix Tree Topology Previous succinct representation of a suffix tree is a Parentheses representation. In this representation, every node is represented by a pair of parentheses. A pair of parentheses of a node encloses its children’s parentheses ( () (() () ()) (() ()) () ())

Previous Compressed Suffix Tree Topology ( () (() () ()) (() ()) () ()) In this representation, parent-child relationship is stored implicitly. To find a child of a node, a range-minima query is required.

New compressed tree topology Our succinct representation differs from the previous one in that we store the parent-child relationship explicitly rather than implicitly. Range-minima query is not required.

Child Table We first describe a child table and then the succinct representation of a child table, i.e., the compressed child table. A child table stores an lcp-interval tree that is a modification of a suffix tree.  We first show how to modify a suffix tree to an lcp- interval tree.  Then, how to store an lcp-interval tree into a child table.

Child Table suffix tree  lcp-interval tree  child table agat#accagat#at# gat# t# cagat# ccagat# # The suffix tree for accagat# agat#accagat# at# gat# t# cagat# ccagat# # The suffix tree for accagat# whose node branching is a complete binary tree

Child Table suffix tree  lcp-interval tree  child table agat# accagat# at# gat# t# cagat# ccagat# [1] # agat#accagat# at# gat# t# cagat# ccagat# # [1..8] [1..6][7..8] [1..4] [5..6] [2..4] [2..3] Each node in the suffix tree is replaced by the interval in the pos array which stores the suffixes in the subtree rooted at the node. [2][3] [5] [6] [4] [7] [8] lcp-interval tree

Child Table suffix tree  lcp-interval tree  child table lcp-interval tree [1..8] [1..6][7..8] [1..4] [5..6] [2..4] [2..3] [1] [2] [3] [8] [7] [4] [5] [6] child table Each interval [i..j] have only to store the first index of its right child, denoted by child (i,j), so that it can compute its two children.  Interval [1..8] have only to store 7 to compute its two children [1..6] and [7..8].  Interval [1..6] stores 5 to compute its two children [1..4] and [5..6].

Child Table suffix tree  lcp-interval tree  child table lcp-interval tree [1..8] [1..6][7..8] [1..4] [5..6] [2..4] [2..3] [1] [2] [3] [8] [7] [4] [5] [6] child table Where is child (i,j) stored? We store child (i,j) in cldtab [i] or cldtab [j].  If [i..j] is a right child, child (i,j) is stored in cldtab [i].  If [i..j] is a left child, child (i,j) is stored in cldtab [j].  Interval [7..8] is a right child so child (7,8) = 8 is stored in cldtab [7].  Interval [1..6] is a left child so child (1,6) = 5 is stored in cldtab [6].

Compressed Child Table [1..8] [1..6][7..8] [1..4] [5..6] [2..4] [2..3] [1] [2] [3] [8] [7] [4] [5] [6] child table difference child table diff sign child table  difference child table  compressed child table Difference child table  diff array  sign array

Compressed Child Table [1..8] [1..6][7..8] [1..4] [5..6] [2..4] [2..3] [1] [2] [3] [8] [7] [4] [5] [6] child table difference child table diff sign child table  difference child table  compressed child table Difference child table  diff array  sign array In a diff array, instead of storing child (i,j), we store min{j- child (i,j), child (i,j)-i}.  For an interval [1..4] whose child(1,4) = 2, we compute 4-2=2 and 2-1=1 and the minimum 1 is stored in diff [4].

Compressed Child Table [1..8] [1..6][7..8] [1..4] [5..6] [2..4] [2..3] [1] [2] [3] [8] [7] [4] [5] [6] child table difference child table diff sign Difference child table  diff array  sign array In a diff array, instead of storing child (i,j), we store min{j- child (i,j), child (i,j)-i}.  For an interval [1..4] whose child(1,4) = 2, we compute 4-2=2 and 2-1=1 and the minimum 1 is stored in diff [4]. Whether diff [i] stores j- child (i,j) or child (i,j)-i is indicated by sign [i]. It stores 0 if j- child (i,j) is stored in diff [i] and 1 if child (i,j)-i is stored.  Since diff [4] stores child(1,4)-1, sign [4] stores 1. child table  difference child table  compressed child table

Compressed Child Table Compressed child table  Compressed diff array  sign array child table  difference child table  compressed child table

Compressed Child Table diff Compressed child table  Compressed diff array  C array: a concatenated bit string of the integers in the diff array  D array: a bit string of the same length as C array where most bits are 0 except the starting bit of each integer in the diff array  Data structures for rank and select for D array to find the ith leftmost 1 in the D array  sign array child table  difference child table  compressed child table C array D array

Compressed Child Table Space consumption of a compressed child table  Compressed child table requires 5n + o(n) bits.  C array: 2n bits  D array: 2n bits  Data structures for rank and select: o(n) bits  sign array: n bits

Construction Algorithm We construct the compressed child table directly from the lcp array without building a suffix tree or an lcp-interval tree as intermediate data structures.  The child table can be constructed directly from the lcp array in O(n) time due to Kim et al [SPIRE2004].  They first construct the extended the lcp array and then compute the child table. We modify their construction algorithm so that it constructs the compressed child table directly from the compressed lcp array.

Construction Algorithm The construction algorithm consists of two procedures EXTLCP and CHILD.  Procedure EXTLCP constructs the compressed extended lcp array from the compressed lcp array.  Procedure CHILD constructs the compressed child table which are the C, D, and sign arrays from the compressed extended lcp array.

Construction Algorithm Pseudo-code for EXTLCP

Construction Algorithm To optain the O(n log|Σ|)-bit working space, the size of temporary data structures should be O(n log|Σ|).

Construction Algorithm To optain the O(n log|Σ|)-bit working space, the size of temporary data structures should be O(n log|Σ|). Arrays ranking an numchild is of size O(n log|Σ|) because a node may have |Σ| childrens and each entry of the array consumes log|Σ| bits

Construction Algorithm To optain the O(n log|Σ|)-bit working space, the size of temporary data structures should be O(n log|Σ|). The size of the stack is O(n log|Σ|) because it can be encoded by δ-code.

Construction Algorithm Pseudo-code for CHILD We also developed some techniques to reduce the working space.

Conclusion We presented a new compressed suffix tree supporting O(n log ε n log|Σ|)-time pattern search that consumes 5n + o(n) bit-space. We also presented a construction algorithm for our compressed suffix tree running in O(n log|Σ|)-bit working space and O(n log ε n) time.

Compressed Child Table Space consumption of a compressed child table  Compressed child table requires 5n + o(n) bits.  C array: 2n bits  S(n) = max {k=1..n/2} {S(k)+S(n-k)+log(k+1)}  D array: 2n bits  Data structures for rank and select: o(n) bits  sign array: n bits