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Question 4: For each of the following, find the dominant term(s) having the sharpest increase in n and give the time complexi

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Answer #1

Find the dominant term and Time complexity:

Expression 2n2log4n +30m2log100m 7(n+5y*+nlogn20 30n logn+3n log1On (2n3(10m))+)100 Dominant Term 2n log4n 7(n+5) 30nPlogn (n

Description:

(1) 2n2log4n + 30m2log100m (consider n > m)

Simplifying above expression to find time complexity:

2n2log4n + 30m2log100m

= 2n2(log104 + log10n) + 30m2 (log10100+ log10m)

= 2n2log104 + 2n2log10n +30m2log10100+30m2log10m

Since n>m Dominant term for expression 2n2log4n + 30m2log100m is 2n2​​​​​​​log4n

Time complexity: O(n2logn)

(2) 7(n+5)4 + nlogn20

Simplifying above expression to find time complexity:

7(n+5)4 + nlogn20

= 7(n4 +20n3 + 150n2 + 500n +625) + 20nlogn

= (7n4 +140n3 + 1050n2 + 3500n +4375) + 20nlogn

Dominant term for expression 7(n+5)4 + nlogn20  is 7(n+5)4

Time complexity: O(n4)

(3) 30n5logn + 3n log10n

Simplifying above expression to find time complexity:

30n5logn + 3n log10n

= 30n5logn + 3n log10n10

= 30n5logn + 30nlogn

Dominant term for expression 30n5logn + 3n log10n is 30n5logn

Time complexity: O(n5logn)

(4) (2n3(10m3)) + (n/2(n2))2 /100 (consider n > m)

Simplifying above expression to find time complexity:

(2n3(10m3)) + (n/2(n2))2 /100

= (2n3(10m3)) + ( (n/2)2 (n2)2 )/ 100

= 20n3 m3+ ((n2 * n4 ) / 200

= 20n3 m3 + n6 / 200

Since n>m Dominant term for expression (2n3(10m3)) + (n/2(n2))2 /100 is (n/2(n2))2 /100

Time complexity: O(n6)

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