Problem 1: Let A = [a1, a2, . . . , an] be an array of different integers and suppose thatthere exists an (unknown) index k such that the subarray [a1, a2, . . . , ak] is sorted in strictlyincreasing order (that is, if 1 ≤ i < j ≤ k, then ai < aj ), and the subarray [ak, ak+1, . . . , an]is sorted in strictly decreasing order (that is, if k ≤ i < j ≤ n, then ai > aj ). We wouldlike to find an index k.(i) Show that any comparison-based algorithm that solves this problem needs at leastΩ(log n) comparisons.(ii) Describe an optimal algorithm that solves this problem.
Problem 1: Let A = [a1, a2, . . . , an] be an array of different integers and suppose thatthere exists an (unknown) index k such that the subarray [a1, a2, . . . , ak] is sorted in strictlyincreasing order (that is, if 1 ≤ i < j ≤ k, then ai < aj ), and the subarray [ak, ak+1, . . . , an]is sorted in strictly decreasing order (that is, if k ≤ i < j ≤ n, then ai > aj ). We wouldlike to find an index k.(i) Show that any comparison-based algorithm that solves this problem needs at leastΩ(log n) comparisons.(ii) Describe an optimal algorithm that solves this problem.
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Problem 1: Let A = [a1, a2, . . . , an] be an array of different integers and suppose that
there exists an (unknown) index k such that the subarray [a1, a2, . . . , ak] is sorted in strictly
increasing order (that is, if 1 ≤ i < j ≤ k, then ai < aj ), and the subarray [ak, ak+1, . . . , an]
is sorted in strictly decreasing order (that is, if k ≤ i < j ≤ n, then ai > aj ). We would
like to find an index k.
(i) Show that any comparison-based algorithm that solves this problem needs at least
Ω(log n) comparisons.
(ii) Describe an optimal algorithm that solves this problem.
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