Succinct Data Structures for Permutations, Functions and Suffix Arrays Ian Munro University of Waterloo Joint work with F. Fich, M. He, J. Horton, A. López-Ortiz, S. Srinivasa Rao, Rajeev Raman, Venkatesh Raman How do we encode a permutation or generalization … function or specialization … suffix array in a small amount of space and still perform queries in constant time ???

Permutations: a Shortcut Notation Let P be a simple array giving π; P[i] = π[i] Also have B[i] be a pointer t positions back in (the cycle of) the permutation; B[i]= π-t[i] .. But only define B for every tth position in cycle. (t is a constant; ignore cycle length “round-off”) So array representation P = [8 4 12 5 13 x x 3 x 2 x 10 1] 1 2 3 4 5 6 7 8 9 10 11 12 13 2 4 5 13 1 8 3 12 10

Representing Shortcuts In a cycle there is a B every t positions … But these positions can be in arbitrary order Which i’s have a B, and how do we store it? Keep a vector of all positions 0 indicates no B 1 indicates a B Rank gives the position of B[“i”] in B array So: π(i) and π -1(i) in O(1) time & (1+ε)n lg n bits Theorem: Under a pointer machine model with space (1+ ε) n references, we need time 1/ε to answer π and π -1 queries; i.e. this is as good as it gets.

Getting n lg n Bits: an Aside This is the best we can do for O(1) operations But using Benes networks: 1-Benes network is a 2 input/2 output switch r+1-Benes network … join tops to tops 1 2 3 4 5 6 7 8 3 5 7 8 1 6 4 2 R-Benes Network R-Benes Network

A Benes Network Realizing the permutation (3 5 7 8 1 6 4 2) 1 2 3 4 5

What can we do with it? Divide into blocks of lg lg n gates … encode their actions in a word. Taking advantage of regularity of address mechanism and also Modify approach to avoid power of 2 issue Can trace a path in time O(lg n/(lg lg n) This is the best time we are able get for π and π-1 in minimum space. Observe: This method “violates” the pointer machine lower bound by using “micropointers”.

Back to the main track: Powers of π Consider the cycles of π ( 2 6 8)( 3 5 9 10)( 4 1 7) Keep a bit vector to indicate the start of each cycle ( 2 6 8 3 5 9 10 4 1 7) Ignoring parentheses, view as new permutation, ψ. Note: ψ-1(i) is position containing i … So we have ψ and ψ-1 as before Use ψ-1(i) to find i, then bit vector (rank, select) to find πk or π-k

Functions Now consider arbitrary functions [n]→[n] “A function is just a hairy permutation” All tree edges lead to a cycle

Challenges here Essentially write down the components in a convenient order and use the n lg n bits to describe the mapping (as per permutations) To get fk(i): Find the level ancestor (k levels up) in a tree Or Go up to root and apply f the remaining number of steps around a cycle

Level Ancestors There are several level ancestor techniques using O(1) time and O(n) WORDS. Adapt Bender & Farach-Colton to work in O(n) bits But going the other way …

f-k is a set Moving Down the tree requires care f-3( ) = ( ) The trick: Report all nodes on a given level of a tree in time proportional to the number of nodes, and Don’t waste time on trees with no answers

Final Function Result Given an arbitrary function f: [n]→[n] With an n lg n + O(n) bit representation we can compute fk(i) in O(1) time and f-k(i) in time O(1 + size of answer).

Back to Text … And Suffix Arrays Text T[1..n] over (a,b)*# (a<#<b) There are 2n-1 such texts, which of the n! suffix arrays are valid? 1 2 3 4 5 6 7 8 SA= 4 7 5 1 8 3 6 2 is a b b a a b a # SA-1= 4 8 6 1 3 7 2 5 M= 4 7 1 5 8 2 3 6 isn’t ..why?

Ascending to Max M is a permutation so M-1 is its inverse i.e. M-1[i] says where i is in M Ascending-to-Max:  1  i  n-2 M-1[i] < M-1[n] and M-1[i+1] < M-1[n]  M-1[i] < M-1[i+1] M-1[i] > M-1[n] and M-1[i+1] > M-1[n]  M-1[i] > M-1[i+1] 4 7 5 1 8 3 6 2 OK 4 7 1 5 8 2 3 6 NO

Non-Nesting Non-Nesting: 1  i,j  n-1 and M-1[i]<M-1[j] M-1[i] < M-1[i+1] and M-1[j] < M-1[j+1]  M-1[i+1] < M-1[j+1] M-1[i] > M-1[i+1] and M-1[j] > M-1[j+1]  M-1[i+1] < M-1[j+1] 4 7 5 1 8 3 6 2 OK 4 7 1 5 8 2 3 6 NO

Characterization Theorem for Suffix Arrays on Binary Texts Theorem: Ascending to Max & Non-nesting  Suffix Array Corollary: Clean method of breaking SA into segments Corollary: Linear time algorithm to check whether SA is valid

Cardinality Queries T= a b a a a b b a a a b a a b b # Remember lengths longest run of a’s and of b’s SA (broken by runs, but not stored explicitly) 8 3 | 9 4 12| 1 10 5 13|16 | 7 2 11 15|6 14 Ba,  bit vector .. If SA-1[i-1] in an “a” section store 1 in Ba,[SA-1[i]], else 0 Ba 0 0 1 1 0 0 1 1 1 0 0 1 1 0 1 1 Create rank structure on Ba, and similarly Bb, (Note these are reversed except at #) Algorithm Count(T,P) s ← 1; e ←n; i ← m; while i>0 and se do if P[i[=a then s← rank1(Ba,s-1)+1; e←rank1(Ba,e) else s← na + 2 + rank1(Bb,s-1); e←na + 1 +rank1(Bb,e) i ← i-1 Return max(e-s+1,0) Time: O(length of query)

Listing Queries Complex methods Key idea: for queries of length at least d, index every dth position .. For T and forT(reversed) So we have matches for T[i..n] and T[1,i-1] View these as points in 2 space (Ferragina & Manzini and Grossi & Vitter) Do a range query (Alstrup et al) Variety of results follow

General Conclusion Interesting, and useful, combinatorial objects can be: Stored succinctly … O(lower bound) +o() So that Natural queries are performed in O(1) time (or at least very close) This can make the difference between using them and not …