Analysis of random splitters

Question 1

What is the goal of the median-finding problem?

Answer 1

To find the middle element of a sorted set of numbers.

Question 2

What is the Select algorithm used for?

Answer 2

Select(S, k) finds the kth largest element in a set S of n numbers.

Question 3

What is the base case for the Select algorithm?

Answer 3

If |S| = 1, then the single element in S is returned.

Question 4

Why is a ‘splitter’ chosen in the Select algorithm?

Answer 4

A splitter divides the set S into subsets S- and S+, allowing the search for the kth largest element to focus on a smaller subset.

Question 5

What is the expected time complexity of the Select algorithm?

Answer 5

The expected time complexity is O(n).

Question 6

How does the choice of a splitter affect the Select algorithm’s performance?

Answer 6

A well-centered splitter significantly reduces the size of the subset considered in recursive calls, improving performance.

Question 7

What happens if the splitter in Select is the minimum element?

Answer 7

The algorithm performs poorly, as the recursive call reduces the subset size by only one element.

Question 8

Why is randomization used to choose the splitter in Select?

Answer 8

Randomization ensures a high probability of choosing a well-centered splitter, avoiding worst-case scenarios.

Question 9

What is the probability of selecting a ‘central’ splitter in the Select algorithm?

Answer 9

The probability is 1/2, as half of the elements in the set are central.

Question 10

What is the recurrence for Select when random splitters are used?

Answer 10

T(n) ≤ T((3/4)n) + cn, which resolves to O(n) using geometric series analysis.

Question 11

What is the Quicksort algorithm?

Answer 11

Quicksort is a sorting algorithm that uses a random splitter to divide a set into two subsets, recursively sorting each.

Question 12

What is the worst-case running time of Quicksort?

Answer 12

The worst-case running time is O(n^2), occurring when splitters are poorly chosen (e.g., smallest elements).

Question 13

What is the expected running time of Quicksort?

Answer 13

The expected running time is O(n log n), achieved through randomization of splitters.

Question 14

What is a central splitter in Quicksort?

Answer 14

A central splitter divides the set so that each subset contains at least a quarter of the elements.

Question 15

What is the role of randomization in Quicksort?

Answer 15

Randomization increases the likelihood of selecting central splitters, ensuring efficient sorting.

Question 16

What is Modified Quicksort?

Answer 16

Modified Quicksort finds a central splitter before making recursive calls, ensuring balanced division of subsets.

Question 17

What is the expected time spent in one phase of Modified Quicksort?

Answer 17

The expected time spent in one phase, excluding recursive calls, is O(|S|), where S is the set size.

Question 18

How many iterations are expected to find a central splitter in Modified Quicksort?

Answer 18

The expected number of iterations is 2, as half of the elements are central.

Question 19

What is the recurrence for Modified Quicksort’s time complexity?

Answer 19

T(n) ≤ 2T((3/4)n) + cn, leading to an expected running time of O(n log n).

Question 20

How does the number of subproblems relate to the phase type in Modified Quicksort?

Answer 20

There are at most (4/3)^j subproblems of type j, where j is the phase type.

Question 21

What is the difference between Quicksort and Modified Quicksort?

Answer 21

Quicksort accepts all splitters, while Modified Quicksort retries until finding a central splitter.

Question 22

What is the purpose of splitting the input set in Quicksort?

Answer 22

Splitting organizes elements smaller than the splitter into one subset and larger elements into another for efficient sorting.

Question 23

How does recursion in Quicksort handle subsets?

Answer 23

Quicksort recursively sorts subsets created by splitting, then combines them with the splitter to form the sorted output.

Question 24

What is the intuition behind the efficiency of Quicksort?

Answer 24

Randomly chosen splitters are likely to balance subsets, leading to efficient sorting on average.

Question 25

What is the convergence property of Quicksort’s recurrence?

Answer 25

The recurrence forms a geometric series, ensuring O(n log n) expected time for sorting.

Question 26

What is the geometric series’ role in analyzing Quicksort?

Answer 26

The geometric series bounds the total work across recursive calls, showing the algorithm’s efficiency.

Question 27

How does Modified Quicksort guarantee balanced recursive calls?

Answer 27

By ensuring splitters are central, Modified Quicksort reduces each subset by at least 25%.

Question 28

What is the relationship between randomization and performance in Quicksort?

Answer 28

Randomization reduces the likelihood of worst-case scenarios, maintaining expected O(n log n) performance.

Question 29

What happens when Modified Quicksort updates the splitter?

Answer 29

It retries splitting with a new random splitter until a central splitter is found.

Question 30

How is the running time of Modified Quicksort bounded?

Answer 30

The running time is bounded by summing the expected costs of processing subsets of decreasing size, yielding O(n log n).