Data Structures for Range Minimum Queries in Multidimensional Arrays
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1 Data Structures for Range Minimum Queries in Multidimensional Arrays Hao Yuan Mikhail J. Atallah Purdue University January 17, 2010 Yuan & Atallah (PURDUE) Multidimensional RMQ January 17, / 28
2 Outline Introduction Results Overview Details Step 1: Comparison-Efficient Data Structures Step 2: Random Access Machine Implementation Future Work Yuan & Atallah (PURDUE) Multidimensional RMQ January 17, / 28
3 Definitions Given a d-dimensional array A with N entries, a Range Minimum Query (RMQ) asks the minimum element in the query range q = [a 1, b 1 ] [a 2, b 2 ] [a d, b d ], i.e., RMQ(A, q) = min A[q] = min A[k 1,..., k d ]. (k 1,...,k d ) q Yuan & Atallah (PURDUE) Multidimensional RMQ January 17, / 28
4 Applications String Pattern Matching: 1D RMQ and its related Least Common Ancestor (LCA) problems are fundamental building blocks in suffix trees/arrays Computational Biology: Finding min/max number in an alignment tableau (genome sequence analysis) Image Processing: Finding the lightest/darkest point in a range (Dilate/Erode Filter) Databases: Range Min/Max Query in OLAP Data Cube Example Select the highest paid employee whose age is between 30 and 40 and joined the company during the period between 1995 and 2005 Yuan & Atallah (PURDUE) Multidimensional RMQ January 17, / 28
5 Previous Work - 1D RMQ 1D Range Minimum Query Linear Reduction to Least Common Ancestor (LCA) Problem [Gabow, Bentley and Tarjan 1984] LCA: O(N) Preprocessing, O(1) Querying [Harel and Tarjan 1984] RMQ & LCA: Much Studied (Parallelization, Simplification, Distributed Algorithms, etc) [Schieber and Vishkin 1988] [Bender and Farach-Colton 2000] [Alstrup et al. 2002] Yuan & Atallah (PURDUE) Multidimensional RMQ January 17, / 28
6 Previous Work - Semigroup Model Related to the semi-group sum problem (MIN is a semi-group operator) Data Structures: O(M) preprocessing time and space (M N), O(α(M, N)) querying time One Dimensional: [Yao 1982], [Alon and Schieber 1987] Multidimensional (fixed d): [Chazelle and Rosenberg 1989] Yuan & Atallah (PURDUE) Multidimensional RMQ January 17, / 28
7 Multidimensional RMQ Unit-Cost RAM Model: O(1) cost for: Read/Write Memory, +,,, /, <<, >> Comparison-Based: Array entries can only be compared Table: Results for d-dimensional RMQ (d is fixed). The O( ) is omitted. Preprocess Time Space Querying Time Gabow et al N log d 1 N N log d 1 N log d 1 N Chazelle and Rosenberg 1989 M M α d (M, N) Poon 2003 N(log N) d N 1 Amir et al (d = 2) N log [k+1] N kn 1 Our result N N 1 Yuan & Atallah (PURDUE) Multidimensional RMQ January 17, / 28
8 Overview General Approach Design Comparison-Efficient Algorithm: Only comparisons between input array entries are counted Implement the Algorithm in RAM: All the computations are counted Example: Minimum Spanning Tree Verification [Komlós 1984] [Dixon, Rauch and Tarjan 1992] Yuan & Atallah (PURDUE) Multidimensional RMQ January 17, / 28
9 Overview Following the general approach: Comparison-Efficient Data Structures New 1D RMQ Preliminary: O(N log N)-comparison preprocessing and 1-comparison querying Speedup the preprocessing to O(N) comparisons New data structure generalizes to two or higher dimensional cases Preprocessing: O(N) comparisons Querying: O(1) comparisons RAM Implementations Micro blocks of size ɛ log N Solve big size query by well-known algorithms Solve small size query by table lookup Yuan & Atallah (PURDUE) Multidimensional RMQ January 17, / 28
10 Overview Following the general approach: Comparison-Efficient Data Structures New 1D RMQ Preliminary: O(N log N)-comparison preprocessing and 1-comparison querying Speedup the preprocessing to O(N) comparisons New data structure generalizes to two or higher dimensional cases Preprocessing: O(N) comparisons Querying: O(1) comparisons RAM Implementations Micro blocks of size ɛ log N Solve big size query by well-known algorithms Solve small size query by table lookup Yuan & Atallah (PURDUE) Multidimensional RMQ January 17, / 28
11 Overview Following the general approach: Comparison-Efficient Data Structures New 1D RMQ Preliminary: O(N log N)-comparison preprocessing and 1-comparison querying Speedup the preprocessing to O(N) comparisons New data structure generalizes to two or higher dimensional cases Preprocessing: O(N) comparisons Querying: O(1) comparisons RAM Implementations Micro blocks of size ɛ log N Solve big size query by well-known algorithms Solve small size query by table lookup Yuan & Atallah (PURDUE) Multidimensional RMQ January 17, / 28
12 Overview Following the general approach: Comparison-Efficient Data Structures New 1D RMQ Preliminary: O(N log N)-comparison preprocessing and 1-comparison querying Speedup the preprocessing to O(N) comparisons New data structure generalizes to two or higher dimensional cases Preprocessing: O(N) comparisons Querying: O(1) comparisons RAM Implementations Micro blocks of size ɛ log N Solve big size query by well-known algorithms Solve small size query by table lookup Yuan & Atallah (PURDUE) Multidimensional RMQ January 17, / 28
13 Overview Following the general approach: Comparison-Efficient Data Structures New 1D RMQ Preliminary: O(N log N)-comparison preprocessing and 1-comparison querying Speedup the preprocessing to O(N) comparisons New data structure generalizes to two or higher dimensional cases Preprocessing: O(N) comparisons Querying: O(1) comparisons RAM Implementations Micro blocks of size ɛ log N Solve big size query by well-known algorithms Solve small size query by table lookup Yuan & Atallah (PURDUE) Multidimensional RMQ January 17, / 28
14 Overview Following the general approach: Comparison-Efficient Data Structures New 1D RMQ Preliminary: O(N log N)-comparison preprocessing and 1-comparison querying Speedup the preprocessing to O(N) comparisons New data structure generalizes to two or higher dimensional cases Preprocessing: O(N) comparisons Querying: O(1) comparisons RAM Implementations Micro blocks of size ɛ log N Solve big size query by well-known algorithms Solve small size query by table lookup Yuan & Atallah (PURDUE) Multidimensional RMQ January 17, / 28
15 Overview Following the general approach: Comparison-Efficient Data Structures New 1D RMQ Preliminary: O(N log N)-comparison preprocessing and 1-comparison querying Speedup the preprocessing to O(N) comparisons New data structure generalizes to two or higher dimensional cases Preprocessing: O(N) comparisons Querying: O(1) comparisons RAM Implementations Micro blocks of size ɛ log N Solve big size query by well-known algorithms Solve small size query by table lookup Yuan & Atallah (PURDUE) Multidimensional RMQ January 17, / 28
16 Overview Following the general approach: Comparison-Efficient Data Structures New 1D RMQ Preliminary: O(N log N)-comparison preprocessing and 1-comparison querying Speedup the preprocessing to O(N) comparisons New data structure generalizes to two or higher dimensional cases Preprocessing: O(N) comparisons Querying: O(1) comparisons RAM Implementations Micro blocks of size ɛ log N Solve big size query by well-known algorithms Solve small size query by table lookup Yuan & Atallah (PURDUE) Multidimensional RMQ January 17, / 28
17 Overview Following the general approach: Comparison-Efficient Data Structures New 1D RMQ Preliminary: O(N log N)-comparison preprocessing and 1-comparison querying Speedup the preprocessing to O(N) comparisons New data structure generalizes to two or higher dimensional cases Preprocessing: O(N) comparisons Querying: O(1) comparisons RAM Implementations Micro blocks of size ɛ log N Solve big size query by well-known algorithms Solve small size query by table lookup Yuan & Atallah (PURDUE) Multidimensional RMQ January 17, / 28
18 Overview Following the general approach: Comparison-Efficient Data Structures New 1D RMQ Preliminary: O(N log N)-comparison preprocessing and 1-comparison querying Speedup the preprocessing to O(N) comparisons New data structure generalizes to two or higher dimensional cases Preprocessing: O(N) comparisons Querying: O(1) comparisons RAM Implementations Micro blocks of size ɛ log N Solve big size query by well-known algorithms Solve small size query by table lookup Yuan & Atallah (PURDUE) Multidimensional RMQ January 17, / 28
19 Overview Following the general approach: Comparison-Efficient Data Structures New 1D RMQ Preliminary: O(N log N)-comparison preprocessing and 1-comparison querying Speedup the preprocessing to O(N) comparisons New data structure generalizes to two or higher dimensional cases Preprocessing: O(N) comparisons Querying: O(1) comparisons RAM Implementations Micro blocks of size ɛ log N Solve big size query by well-known algorithms Solve small size query by table lookup Yuan & Atallah (PURDUE) Multidimensional RMQ January 17, / 28
20 Multidimensional RMQ If only count comparisons: 2D RMQ Lower Bound: [Demaine, Landau and Weimann, 2009] If NO COMPARISON is allowed at the query stage, then Ω(N log N) comparisons preprocessing is required Our Result: O(2.89 d (d + 1)!N) comparisons preprocessing, 2 d 1 comparisons querying Yuan & Atallah (PURDUE) Multidimensional RMQ January 17, / 28
21 Canonical Ranges x CR(x) = [5, 8] Yuan & Atallah (PURDUE) Multidimensional RMQ January 17, / 28
22 Pre-Computations For each p CR(x), define LeftMin(x, p) = RightMin(x, p) = x min A[k] k CR(x) and k p min A[k] k CR(x) and k p p LeftMin(x, 7) = min{ A[5], A[6], A[7] } RightMin(x, 7) = min{ A[7], A[8] } Yuan & Atallah (PURDUE) Multidimensional RMQ January 17, / 28
23 Query min A[6..14] LCA(6, 14) x y RightMin(x, 6) LeftMin(y, 14) Yuan & Atallah (PURDUE) Multidimensional RMQ January 17, / 28
24 Pre-Computations Naïve Algorithm: O(N log N) Comparisons Faster Approach Sort the canonical ranges by their lengths Compute the LeftMin and RightMin entries for canonical ranges in the sorted order For length-one canonical range CR(w), LeftMin(w, p) = RightMin(w, p) = A[p] (p CR(w)) For a canonical range CR(w) covering more than one position, compute the LeftMin and RightMin arrays in O(log CR(w) ) time (instead of O( CR(w) )) Yuan & Atallah (PURDUE) Multidimensional RMQ January 17, / 28
25 Pre-Computations Case 1: p CR(x) w x p y LeftMin(w, p) = LeftMin(x, p) Yuan & Atallah (PURDUE) Multidimensional RMQ January 17, / 28
26 Pre-Computations Case 2: p CR(y) w x p y LeftMin(w, p) = min { min CR(x), LeftMin(y,p) } Yuan & Atallah (PURDUE) Multidimensional RMQ January 17, / 28
27 Pre-Computations Case 2: p CR(y) w x p y LeftMin(w, p) = min { min CR(x), LeftMin(y,p) } Monotonicity (Non-Increasing): LeftMin(y, p) LeftMin(y, p + 1) Binary Search! Yuan & Atallah (PURDUE) Multidimensional RMQ January 17, / 28
28 Binary Search Example w x p y min CR(x) = 40 LeftMin(y,...) = 90, 70, 50, 20, 10 LeftMin(w,...) = 40, 40, 40, 20, 10 Yuan & Atallah (PURDUE) Multidimensional RMQ January 17, / 28
29 Comparison Complexity T (n): the number of comparisons to compute the LeftMin and RightMin entries for canonical ranges whose size is at most n T (1) = 0 T (n) = 2T ( n 2 ) + O(log n) for n 2 We have the preprocessing comparison complexity T (n) = O(n), and need to do 1 comparison at the query stage. Yuan & Atallah (PURDUE) Multidimensional RMQ January 17, / 28
30 2D Case 2D Canonical Range: Cartesian Product of Two 1D Canonical Ranges x y D Canonical Range Yuan & Atallah (PURDUE) Multidimensional RMQ January 17, / 28
31 2D Pre-Computations For each 2D canonical range r and a point p r, compute the 4 Dominance Min array entries TopLeftMin(r, p) TopRightMin(r, p) p BotLeftMin(r, p) BotRightMin(r, p) Yuan & Atallah (PURDUE) Multidimensional RMQ January 17, / 28
32 2D Query For any query range q, we can always divide it into 4 parts, which are all pre-computed BotRightMin(r1, p1) BotLeftMin(r2, p2) TopRightMin(r3, p3) TopLeftMin(r4, p4) Yuan & Atallah (PURDUE) Multidimensional RMQ January 17, / 28
33 Efficient Pre-Computations For any canonical range r, cut the middle of its longer side to obtain two smaller canonical ranges r 1 and r 2 Do binary search row by row (or column by column) O(N) comparisons for 2D preprocessing Generalize to any fixed dimension d: Preprocess: O(2.89 d (d + 1)!N) comparisons Query: 2 d 1 comparisons Binary Search r1 r2 p BotLeftMin(r, p) Yuan & Atallah (PURDUE) Multidimensional RMQ January 17, / 28
34 Efficient Pre-Computations For any canonical range r, cut the middle of its longer side to obtain two smaller canonical ranges r 1 and r 2 Do binary search row by row (or column by column) O(N) comparisons for 2D preprocessing Generalize to any fixed dimension d: Preprocess: O(2.89 d (d + 1)!N) comparisons Query: 2 d 1 comparisons Binary Search r1 r2 p BotLeftMin(r, p) Yuan & Atallah (PURDUE) Multidimensional RMQ January 17, / 28
35 Efficient Pre-Computations For any canonical range r, cut the middle of its longer side to obtain two smaller canonical ranges r 1 and r 2 Do binary search row by row (or column by column) O(N) comparisons for 2D preprocessing Generalize to any fixed dimension d: Preprocess: O(2.89 d (d + 1)!N) comparisons Query: 2 d 1 comparisons Binary Search r1 r2 p BotLeftMin(r, p) Yuan & Atallah (PURDUE) Multidimensional RMQ January 17, / 28
36 Efficient Pre-Computations For any canonical range r, cut the middle of its longer side to obtain two smaller canonical ranges r 1 and r 2 Do binary search row by row (or column by column) O(N) comparisons for 2D preprocessing Generalize to any fixed dimension d: Preprocess: O(2.89 d (d + 1)!N) comparisons Query: 2 d 1 comparisons Binary Search r1 r2 p BotLeftMin(r, p) Yuan & Atallah (PURDUE) Multidimensional RMQ January 17, / 28
37 Overview of RAM Implementations Divide the array into micro blocks of size ɛ log N Each block is a d-dimensional cube, with side length (ɛ log N) 1 d For example in 2D, make each block ɛ log N by ɛ log N q1 q2 q3 Yuan & Atallah (PURDUE) Multidimensional RMQ January 17, / 28
38 Overview of RAM Implementations For query that crosses the border of any micro block: there exists O(N)-time preprocessing and constant-time querying data structures to solve it, using dimension reductions and the help of the data structures in [Yao 1982] [Chazelle and Rosenberg 1989] q1 q2 q3 Yuan & Atallah (PURDUE) Multidimensional RMQ January 17, / 28
39 Overview of RAM Implementations For query that is complete within a micro block, use table lookup technique (Four Russian s Trick) to get the locations of at most 2 d candidates to compare at the querying stage Based on our linear-comparison preprocessing data structure q1 q2 q3 Yuan & Atallah (PURDUE) Multidimensional RMQ January 17, / 28
40 Micro Block Key Idea: If two micro blocks have the same type, then they should share the same data structures Type of a micro block: Comparison results (true/false sequence) of the linear-comparison preprocessing algorithm Assume cɛ log N comparisons to preprocess a block: at most 2 cɛ log N = N cɛ possible types Choose ɛ < 1 c, then there are only a sublinear number of types: N cɛ polylog(ɛ log N) = o(n) Recognizing the types for all micro blocks in linear time: Build a linear-depth decision tree according to the linear-comparison preprocessing algorithm Yuan & Atallah (PURDUE) Multidimensional RMQ January 17, / 28
41 Summary of RAM Our unit-cost RAM data structure Preprocess in O(2.89 d (d + 1)!N) time and (2 d d!n) space Query in O(3 d ) time Yuan & Atallah (PURDUE) Multidimensional RMQ January 17, / 28
42 Future Work Future Work: Extend the lower bound of [Demaine, Landau and Weimann, 2009] If at most t comparisons are allowed at the querying stage, find the lower bound for the number of comparisons required to preprocess the input Dynamic Updates [Poon, 2003] Extend our results to the external memory model Yuan & Atallah (PURDUE) Multidimensional RMQ January 17, / 28
43 Future Work Future Work: Extend the lower bound of [Demaine, Landau and Weimann, 2009] If at most t comparisons are allowed at the querying stage, find the lower bound for the number of comparisons required to preprocess the input Dynamic Updates [Poon, 2003] Extend our results to the external memory model Yuan & Atallah (PURDUE) Multidimensional RMQ January 17, / 28
44 Future Work Future Work: Extend the lower bound of [Demaine, Landau and Weimann, 2009] If at most t comparisons are allowed at the querying stage, find the lower bound for the number of comparisons required to preprocess the input Dynamic Updates [Poon, 2003] Extend our results to the external memory model Yuan & Atallah (PURDUE) Multidimensional RMQ January 17, / 28
45 Future Work Future Work: Extend the lower bound of [Demaine, Landau and Weimann, 2009] If at most t comparisons are allowed at the querying stage, find the lower bound for the number of comparisons required to preprocess the input Dynamic Updates [Poon, 2003] Extend our results to the external memory model Yuan & Atallah (PURDUE) Multidimensional RMQ January 17, / 28
46 END Thank You! Yuan & Atallah (PURDUE) Multidimensional RMQ January 17, / 28
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