Algorithms

CSE374
Han Li
02/28/2021
Problem 1
Demonstrate the insertion of keys 5, 28, 19, 15, 20, 33, 12, 17, 10 into a hash table with collisions resolved by chaining. Let the table have 9 slots and let the hash function h(k) =k mod 9.
The hash table insertion of all 9 keys
i
T[i]
0
[]
1
[10,19,28]
2
[20]
3
[12]
4
[]
5
[5]
6
[33,15]
7
[]
8
[17]
Problem 2
Professor Marley hypothesizes that substantial performance gains can be obtained if we modify the chaining scheme so that each list is kept in sorted order. How does the professor’s modification affect the running time for the following.
Whatever Successful or Unsuccessful Searches will take time O(length).
Thus, if the factor is constant, the average time complexity of O(1) will not be changed.
Insertions: The insertion operation earlier took O (1), after that, it will take time O(length).
Deletions: For doubly linked list will take O(1), for single linked list will take time O(length).
Problem 3
Consider a hash table of size m = 1000 and a corresponding hash function h(k) = [m(kAmod1)] For A= (√5−1)/2. Compute the locations to which the keys 61, 62, 63, 64, and 65 are mapped.
The hash function is ℎ(?)=(?∗(?∗? ??? 1))
A= (√5−1)/2 is equal to 0.618, so we have following
ℎ(61)=(?∗(?∗? ??? 1)) =(1000∗(61∗0.618 ??? 1))=698 ℎ(62)=(?∗(?∗? ??? 1)) =(1000∗(62∗0.618 ??? 1))=316 ℎ(63)=(?∗(?∗? ??? 1))
=(1000∗(63∗0.618 ??? 1))=934 ℎ(64)=(?∗(?∗? ??? 1)) =(1000∗(64∗0.618 ??? 1))=552
ℎ(65)=(?∗(?∗? ??? 1)) =(1000∗(61∗0.618 ??? 1))=170
Problem 4
Suppose that n women check their coats at a concert. However, at the end of the night, the attendant has lost the claim checks and does not know which coat belongs to whom. All of the women came dressed in black coats that were nearly identical, but of different sizes. The attendant can have a woman try a coat, and find out whether the coat fits (meaning it belongs to that woman), or the coat is too big, or the coatis too small. However, the attendant cannot compare the sizes of two coats directly, or compare the sizes of two women directly. Describe how the attendant can determine which coat belongs to which woman unexpected O(n log n) time. Give a brief analysis of the running time of your algorithm.
We can use Quicksort to solve such problems. For ladies and coats, we only need to choose a lady, use her as the pivot, and then find out who is bigger and smaller than her to compare. Then continue to compare between the two separate categories, and finally find the result. ?(?)=?(?2)+∅(1) ?(?)=??(??)+∅(???????)
We have a=1,b=2,k=0,p=1 ?(?)=∅(????)
For n number of coats, we have ?(?)=∅(?????)

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