Algorithms Assignment
(a) CLRS 15.3-2 Draw the recursion tree for the Merge-Sort procedure from Section 2.3.1 on
an array of 16 elements. Explain why memorization fails to speed up a good divide-and-conquer
algorithm such as Merge-Sort.
(b) CLRS 15.3-5 Suppose that in the rod-cutting problem of Section 15.1, we also had limit li
on the number of pieces of length i that we are allowed to produce, for i = 1, 2, …, n. Show that
the optimum-substructure property described in Section 15.1 no longer holds.
(c) Using the longest common sequence, show in c[i, j] what “spanking” and “amputation” have
in common.
(d) CLRS 15.4.2 Give pseudocode to reconstruct an LCS from the complete c table and the
original sequence X =< x1, x2, …, xm > and Y =< y1, y2, …, yn > in O(m, n) time, without
using the b table.
(e) Given the sequence K =< k1, k2, …, k5 >=< 1, 2, 3, 4, 5 > with associated probabilities
pi =< 0.25, 0.2, 0.05, 0.2, 0.3 > what is the estimated search cost, E[search cost] for the tree in
Figure ???
2
2. Problem 2 Chapter 29 Note this is a Collaborative Problem
50 Points Total 12.5 Each
In the course content, it is explained how to solve two-player zero-sum games using linear programming.
One of the games described is “Rock-Paper-Scissors.” In this problem the game is
examined closer. Assume the the following is a “loss” matrix for Player 1, e.g., the matrix shows
how much Player 1 has lost rather than gained so the sign is reversed:
A =
0 1 −1
−1 0 1
1 −1 0
(a) What is the expected loss for Player 1 when Player 1 plays a mixed strategy x = (x1, x2, x3)
and Player 2 plays a mixed strategy y = (y1, y2, y3)?
(b) Show that Player 1 can achieve a negative expected loss (i.e., an expected gain) if Player 2
plays any strategy other than y = (y1, y2, y3) = ( 1
3
,
1
3
,
1
3
).
(c) Show that x = ( 1
3
,
1
3
,
1
3
) and y = ( 1
3
,
1
3
,
1
3
) from Nash equilibrium.
(d) Let x = ( 1
3
,
1
3
,
1
3
) as in part (c). Is it possible for (x, y) to be a Nash equilibrium for some
mixed strategy y06= ( 1
3
,
1
3
,
1
3
)? Explain.
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