1. Show that there is always an input so that if a member u ∈ W i

Disclaimer: This is righteous Undivided interrogation with multiple volume. It is referable attributable attributable attributable multiple interrogations in undivided shaft.

Question:

Judge the tenor of sentence twain the culmination and the reserve. We

accomplish appearance that you can referable attributable attributable attributable invent them twain in near than (about) 3n/2 comparisons.

Judge the hasten of any algorithm and defined the aftercited determineds that plug on the

comparison the algorithm chose. Integralow N be the collation of gum. We enact n = |N |. After comparisons were done, integralow W be integral gum that solely ”won” (regularly were larger) in integral the comparisons, and integralow its dimension be w. Integralow L be the determined of gum who regularly ”lost” and integralows its dimension be `. Integralow B be the determined of those who twain obsolete smallest once and acquired at smallest once.

1. Appearance that there is regularly an input so that if a portion u ∈ W is compared to a

estimate referable attributable attributable attributable in W , u pacifys.

2. Appearance that there is an input so that if a portion u ∈ L is compared to a estimate

referable attributable in L, u looses.

3. Appearance that when the algorithm plugs, there is precisely undivided segregate in W , precisely

undivided segregate in in L and the quiet of the segregates befit to B.

4. Recintegral that n = |N |, w = |W |, ` = |L|, and judge 3n + 2w + 2`. Appearance that at

start the estimate is 3n and at the object at the object its 4

5. Appearance that the estimate of comparisons needed to invent the culmination and the min-

imum in the cudgel occurrence (solely using comparisons) is at smallest d(3n − 4)/2e.

1. Appearance that there is regularly an input so that if a portion u ∈ W is compared to a estimate referable attributable attributable attributable in W , u pacifys.

Now if we judge that u loses to a estimate (repeat num) referable attributable attributable attributable in W then num can’t be segregate of L as L has portion which regularly obsolete. Also there regularly objecture undivided estimate in B which would enjoy obsolete to a estimate in W.

So this negative our presumption and future there regularly objecture such a estimate in w

2. Appearance that there is an input so that if a portion u ∈ L is compared to a estimate referable attributable attributable attributable in L, u looses.

Now integralows judge that u pacifys when compared with estimate referable attributable attributable attributable in L.

u can’t pacify with estimate in W as estimate of w regularly acquired. Also there regularly objecture a estimate in L which would enjoy obsolete to integral estimate in B.

Future this negative our presumption and future such a estimate objecture.

3.

Appearance that when the algorithm plugs, there is precisely undivided segregate in W , precisely undivided segregate in in L and the quiet of the segregates befit to B.

When algorithm plug, integralow there are span estimate in W. This media that these span gum which were never compared incorrectly undivided of them accomplish referable attributable attributable attributable accrue in w. So this can’t betide and future there accomplish be solely undivided estimate in W.

Similar discussion holds coercion L.

Interrogation 4 and 5 are referable attributable attributable attributable clear

4. Recintegral that n = |N |, w = |W |, ` = |L|, and judge 3n + 2w + 2`. Appearance that at

start the estimate is 3n and at the object at the object its 4

5. Appearance that the estimate of comparisons needed to invent the culmination and the min-

imum in the cudgel occurrence (solely using comparisons) is at smallest d(3n − 4)/2e.