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Controlled Randomized Rounding Benjamin Doerr joint work with Tobias Friedrich, Christian Klein, Ralf Osbild

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ADFOCS Benjamin DoerrControlled Randomized Rounding August 21 - August 25, 2006, Saarbrücken, Germany Advanced Course on the Foundations of Computer Science Tamal Dey Joel SpencerIngo Wegener Surface Reconstruction and Meshing: Algorithms with Mathematical Analysis Erdős Magic, Erdős-Rényi Phase Transition Randomized Search Heuristics: Concept and Analysis

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Overview Introduction Two Applications Problem: Matrix rounding with small errors in row and columns Previous and new results The Algorithms Alternating cycle trick. Two speed-ups. Summary Benjamin DoerrControlled Randomized Rounding

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Introduction Benjamin DoerrControlled Randomized Rounding Application 1: Increase Readability SPDCDUDie GrünenFDPDie Linkesum Saarbrücken Saarlouis St. Wendel Homburg sum Bundestagswahl 2005, Zweitstimmen in den vier Wahlkreisen im Saarland

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Introduction Benjamin DoerrControlled Randomized Rounding Application 1: Increase Readability SPDCDUDie GrünenFDPDie Linkesum Saarbrücken Saarlouis St. Wendel Homburg sum SPDCDUDie GrünenFDPDie Linkesum Saarbrücken Saarlouis St. Wendel Homburg sum Bundestagswahl 2005, Zweitstimmen in den vier Wahlkreisen im Saarland

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Introduction Benjamin DoerrControlled Randomized Rounding Application 2: Confidentiality Protection k-100k100k-1M1M k-100k100k-1M1M Kleinkleckersdorf income statistics 2000 Kleinkleckersdorf income statistics 2001 Dr. Hein Blød, born 1941, has a 1M+ annual income! Solution: Round all number to multiples of 5.

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Introduction Benjamin DoerrControlled Randomized Rounding Round a [0,1] matrix to a {0,1} matrix s.t. rounding errors in row totals are less than one; rounding errors in column totals are less than one; rounding error in grand total is less than one. Controlled Rounding Classical result: All matrices have controlled roundings. Bacharach 66, Cox&Ernst 82: Statistics. Baranyai 75: Hypergraph coloring. Basic Problem

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Introduction Benjamin DoerrControlled Randomized Rounding Unbiased = Randomized: Pr(y ij = 1) = x ij, Pr(y ij = 0) = 1 – x ij. Result: Unbiased controlled roundings exist. Cox 87. Follows also from GKPS (FOCS 02). Problem: Extension 1

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Introduction Benjamin DoerrControlled Randomized Rounding Small errors in initial intervals of row/columns: Strongly controlled roundings Observation: Errors less than two in arbitrary intervals. Allows reliable range queries. # of year olds with income 10k-100k? Problem: Extension 2 8 i 8 b : ¯ ¯ ¯ ¯ b P j = 1 ( x ij ¡ y ij ) ¯ ¯ ¯ ¯ < 1 8 j 8 b : ¯ ¯ ¯ ¯ b P i = 1 ( x ij ¡ y ij ) ¯ ¯ ¯ ¯ < 1 8 i 8 a 8 b : ¯ ¯ ¯ ¯ b P j = a ( x ij ¡ y ij ) ¯ ¯ ¯ ¯ < 2 8 j 8 a 8 b : ¯ ¯ ¯ ¯ b P i = a ( x ij ¡ y ij ) ¯ ¯ ¯ ¯ < 2

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Introduction Benjamin DoerrControlled Randomized Rounding Unbiased strongly controlled roundings exist. Can be generated in time O((mn) 2 ); O(mn ), if numbers have binary length at most ; O(mn b 2 ), if numbers are multiples of 1/b; [Confidentiality protection: rounding integers to multiples of b] Our Result:

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The Algorithms Benjamin DoerrControlled Randomized Rounding Simplifying assumptions: Row/columns sums integral Only aim at low errors in whole rows/columns (all intervals: more technical). Alternating Cycle Trick: Choose an alternating cycle (of non-zeroes). 2.Compute possible modifications: ε min = -0.1, ε max = (a) Non-randomized: Modify with any ε [here: ε = ε max ]. (b) Unbiased: Suitable random choice. Result: At least one entry becomes 0 or 1. Time complexity: One iteration O(mn), total O((mn) 2 ). +ε+ε +ε+ε -ε-ε -ε-ε

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The Algorithms Benjamin DoerrControlled Randomized Rounding Assumptions: All number have finite binary expansion. Simplifications as before. Fast Alternating Cycle Trick: Choose an alternating cycle (with 1s in last digit). 2.Allow only modifications ε 1 = and ε 2 = (a) Non-randomized: Modify with either value. (b) Unbiased: Pick each value with 50% chance [here: ε = ε 2 ]. Result: Bit-length in whole cycle reduces. Time complexity: Amortized O(1) to reduce bit-length of one entry. Total: O(mn bit_length). +ε+ε +ε+ε +ε+ε +ε+ε -ε-ε -ε-ε -ε-ε -ε-ε

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The Algorithms Benjamin DoerrControlled Randomized Rounding Assumptions: All numbers multiples of 1/b. Simplifications as before. Multiples of 1/b (here b=5): 1/53/54/52/50 3/54/503/50 2/5 4/51/54/53/5 1.Choose an alternating cycle (of non-zeroes). 2.Allow only modifications. ε 1 = -1/b and ε 2 = +1/b. 3.(a) Non-randomized: Be clever (derandomize!). (b) Unbiased: Pick each value with 50% chance [here: ε = ε 2 ]. Time complexity: Amortized O(b 2 ) to fully round one entry. Total: O(mn b 2 ). Proof: Entries perform random walk in {0, 1/5, 2/5, 3/5, 4/5, 1}. -ε-ε -ε-ε +ε+ε +ε+ε 2/5 1

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Benjamin DoerrControlled Randomized Rounding Unbiased strongly controlled roundings: randomized roundings rounding errors in initial intervals of rows/column: < 1. Result: Can be generated in time O((mn) 2 ); O(mn ), if numbers have binary length at most ; O(mn b 2 ), if numbers are multiples of 1/b; Summary: Thank you!

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