The off-line minimum problem maintains a dynamic set T of elements from the domain {1, 2,...,n} under the operations INSERT and EXTRACT-MIN. A sequence S of n INSERT and m EXTRACT- MIN calls are given, where each key in {1, 2,...,n} is inserted exactly once. Let a sequence S be represented by I1 , E, I2, E, ... , E, Im+1 , where each Ij stands for a subsequence (possibly empty) of INSERT and each E stands for a single EXTRACT-MIN. Let Kj be the set of keys initially obtained from insertions in Ij. The algorithm to build an array extracted[1..m], where for i = 1, 2, ..., m, extracted[i] is the key returned by the ith EXTRACT-MIN call is given below: Off-Line-Minimum(m, n) for i = 1 to n determine j such that i ∈ Kj if j ≠ m + 1 extracted[j] = i let L be the smallest value greater than j for which KL exists KL = KL U Kj, Kj return extracted (1) Given the operation sequence 9, 4, E, 6, 2, E, E, 5, 8, E, 1, 7, E, E, 3; where each number stands for its insertion. Draw a table showing the building process of extracted[1..6]. (2) Argue the correctness of the array extracted[1..m] returned by the above algorithm, i.e. every time the current minimum is extracted. (3) Describe how to implement Off-Line-Minimum efficiently with a disjoint-set data structure (i.e. identify and replace the code in specific lines with proper disjoint-set operations) Analyze the worst-case running time of your implementation.

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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The off-line minimum problem maintains a dynamic set T of elements from the domain {1, 2,...,n}
under the operations INSERT and EXTRACT-MIN. A sequence S of n INSERT and m EXTRACT-
MIN calls are given, where each key in {1, 2,...,n} is inserted exactly once. Let a sequence S be
represented by I₁ , E, I₂_, E, ... , E, I_m+1 , where each I_j stands for a subsequence (possibly empty) of
INSERT and each E stands for a single EXTRACT-MIN. Let K_j be the set of keys initially obtained
from insertions in I_j. The algorithm to build an array extracted[1..m], where for i = 1, 2, ..., m,
extracted[i] is the key returned by the ith EXTRACT-MIN call is given below:

Off-Line-Minimum(m, n)
for i = 1 to n
determine j such that i ∈ K_j
if j ≠ m + 1
extracted[j] = i

let L be the smallest value greater than j for which K_L exists
K_L = K_L U K_j_, K_j
return extracted

(1) Given the operation sequence 9, 4, E, 6, 2, E, E, 5, 8, E, 1, 7, E, E, 3; where each
number stands for its insertion. Draw a table showing the building process of extracted[1..6].

(2) Argue the correctness of the array extracted[1..m] returned by the above algorithm,
i.e. every time the current minimum is extracted.

(3) Describe how to implement Off-Line-Minimum efficiently with a disjoint-set data
structure (i.e. identify and replace the code in specific lines with proper disjoint-set operations)
Analyze the worst-case running time of your implementation.

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