1 Suppose you’ve been sent back in time and have arrived at the sce

1. Suppose you’ve been sent back in time and have arrived
at the scene of an ancient Roman battle. Moreover,
suppose you have just learned that it is your job to assign
n spears to n Roman soldiers, so that each man has a
spear. You observe that the men and spears are of various
heights, and you have been told (in Latin) that the army is
at its best if you can minimize the total difference in
heights between each man and his spear. That is, if the ith
man has height mi and his spear has height si, the n you
want to minimize the sum, i=1 to n |mi si|. Consider a
greedy strategy of repeatedly matching the man and spear
that minimizes the difference in heights between these
two. Prove or disprove that this greedy strategy results in
the optimal assignment of spears to men.
2. Consider again the time-travel problem of the previous
exercise,but now consider a greedy algorithm that sorts
the men by increasing heights and sorts the spears by
increasing heights, and then assigns the ith spear in the
ordered list of spears to the ith man in the ordered list of
Roman soldiers. Prove or disprove that this greedy
strategy results in the optimal assignment of spears to
men.
Solution
1. Given: Army- N men and N spears are of various
heighThe Army is at its best if you can minimize the total
difference in heights between each man and his spear.
That is, if the ith man has height mi and his spear has
height si, then you want to minimize the sum, i=1 to n |mi
si|
Greedy Approach says always select the local optimum
value.Greedy Approach will do the optimal assignment of
spears to men.
Proof: – let we given n spear of height s1,s2, ——, sn and
n army men of height a1,a2—–an.
using greedy approach, we will select army men of
minimum height and corresponding spear of minimum
height. so the difference between ai -si will be minimum.
Now we left with n-1 spear and person. Again and again,
we will select spear and army men of miminum height from
remaining local poll untill all men are assigned a spear.So
in this way overall differnece will be minimum and greedy
will give optimal solution.
2.Given:-greedy algorithm that sorts the men by increasing
heights and sorts the spears by increasing heights, and
then assigns the ith spear in the ordered list of spears to
the ith man in the ordered list of Roman soldiers.
Yes,this greedy strategy will also results in the optimal
assignment of spears to men.
Proof:-Beacause greedy always follow local optimal
approach. As all spear and men are sorted according to
their heights. so assignment of ith spear to ith men is best
one. and we get solution.
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