Problem Hashing (a) Consider a hash table of size M = 5. If items with keys k = 17, 3, 22 are inserted in this order, draw the resulting hash table(s) if we resolve collisions using: 1. Chaining, with h(k) = k mod 5 %3D 2. Linear Probing, with h(k) = k mod 5 %3D 3. Cuckoo Hashing, with ho(k) = k mod 5, h1(k) = [k/5]. Note: both tables have M = 5, so there are a total of 10 cells in the data structure. %3D 4. Double Hashing, with ho(k) 1+y5 = k mod 5, h1(k) = 1 + [4(kø – [kø])], where (b) Which of h1(k) = k mod 5, h2(k) = [k/5], and h3(k) = 5 – (k mod 5) are suitable hash functions for a hash table T = [0, . ,4] of size 5, with keys k E N? %3D .... (c) (i) Suppose we are using an open-addressed hash table of size M to store n items, where n < M/2, with an ideal random hash function. For any i, let X; denote the number of probes required for the ith insertion into the table. Prove that Pr[X; > k] < 1/2* for all i and k. (ii) Prove that Pr[X; > 2 log n] < 1/n². (iii) Let X 2 log n] < 1/n. = max; X; be the length of the longest probe sequence. Prove that Pr[X >
Problem Hashing (a) Consider a hash table of size M = 5. If items with keys k = 17, 3, 22 are inserted in this order, draw the resulting hash table(s) if we resolve collisions using: 1. Chaining, with h(k) = k mod 5 %3D 2. Linear Probing, with h(k) = k mod 5 %3D 3. Cuckoo Hashing, with ho(k) = k mod 5, h1(k) = [k/5]. Note: both tables have M = 5, so there are a total of 10 cells in the data structure. %3D 4. Double Hashing, with ho(k) 1+y5 = k mod 5, h1(k) = 1 + [4(kø – [kø])], where (b) Which of h1(k) = k mod 5, h2(k) = [k/5], and h3(k) = 5 – (k mod 5) are suitable hash functions for a hash table T = [0, . ,4] of size 5, with keys k E N? %3D .... (c) (i) Suppose we are using an open-addressed hash table of size M to store n items, where n < M/2, with an ideal random hash function. For any i, let X; denote the number of probes required for the ith insertion into the table. Prove that Pr[X; > k] < 1/2* for all i and k. (ii) Prove that Pr[X; > 2 log n] < 1/n². (iii) Let X 2 log n] < 1/n. = max; X; be the length of the longest probe sequence. Prove that Pr[X >
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![Problem
Hashing
(a) Consider a hash table of size M = 5. If items with keys k = 17,3, 22 are inserted in
this order, draw the resulting hash table(s) if we resolve collisions using:
1. Chaining, with h(k) = k mod 5
2. Linear Probing, with h(k) = k mod 5
3. Cuckoo Hashing, with ho(k) = k mod 5, h1(k) = [k/5]. Note: both tables have
M = 5, so there are a total of 10 cells in the data structure.
4. Double Hashing, with ho(k)
1+y5
k mod 5, h1(k) = 1+ [4(kø – [kø])], where
= k mod 5, h2(k)
(b) Which of h1(k)
hash functions for a hash table T = [0, . , 4] of size 5, with keys k E N?
[k/5], and h3(k) = 5 – (k mod 5) are suitable
%3D
(c) (i) Suppose we are using an open-addressed hash table of size M to store n items,
where n < M/2, with an ideal random hash function. For any i, let X; denote
the number of probes required for the ith insertion into the table. Prove that
Pr[X; > k] < 1/2k for all i and k.
(ii) Prove that Pr[X; > 2 log n] < 1/n².
(iii) Let X
2 log n] < 1/n.
max; X; be the length of the longest probe sequence. Prove that Pr[X >](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6bd173dc-b595-47dd-9ed3-7eeb4e63db70%2F9803ee75-1a6a-4f1c-989d-6d72f30a8404%2F636y4p7_processed.png&w=3840&q=75)
Transcribed Image Text:Problem
Hashing
(a) Consider a hash table of size M = 5. If items with keys k = 17,3, 22 are inserted in
this order, draw the resulting hash table(s) if we resolve collisions using:
1. Chaining, with h(k) = k mod 5
2. Linear Probing, with h(k) = k mod 5
3. Cuckoo Hashing, with ho(k) = k mod 5, h1(k) = [k/5]. Note: both tables have
M = 5, so there are a total of 10 cells in the data structure.
4. Double Hashing, with ho(k)
1+y5
k mod 5, h1(k) = 1+ [4(kø – [kø])], where
= k mod 5, h2(k)
(b) Which of h1(k)
hash functions for a hash table T = [0, . , 4] of size 5, with keys k E N?
[k/5], and h3(k) = 5 – (k mod 5) are suitable
%3D
(c) (i) Suppose we are using an open-addressed hash table of size M to store n items,
where n < M/2, with an ideal random hash function. For any i, let X; denote
the number of probes required for the ith insertion into the table. Prove that
Pr[X; > k] < 1/2k for all i and k.
(ii) Prove that Pr[X; > 2 log n] < 1/n².
(iii) Let X
2 log n] < 1/n.
max; X; be the length of the longest probe sequence. Prove that Pr[X >
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