Quick Sort (Also known as “partition-exchange sort”)

By -

Quick Sort

(Also known as “partition-exchange sort”)

Definition:

Quick sort is a well –known sorting algorithm, based on divide & conquer approach. The

steps are:

1. Pick an element called pivot from the list

2. Reorder the list so that all elements which are less than the pivot come before the pivot and all elements greater than pivot come after it. After this partitioning, the pivot is in its final position. This is called the partition operation

3. Recursively sort the sub-list of lesser elements and sub-list of greater elements.

Features:

• Developed by C.A.R. Hoare

• Efficient algorithm

• NOT stable sort

• Significantly faster in practice, than other algorithms

Algorithm

ALGORITHM Quicksort (A[ l …r ])

//sorts by quick sort

//i/p: A sub-array A[l..r] of A[0..n-1],defined by its left and right indices l and r

//o/p: The sub-array A[l..r], sorted in ascending order

image

ALGORITHM Partition (A[l ..r])

//Partitions a sub-array by using its first element as a pivot

//i/p: A sub-array A[l..r] of A[0..n-1], defined by its left and right indices l and r (l < r)

//o/p: A partition of A[l..r], with the split position returned as this function’s value

image

Analysis:

• Input size: Array size, n

• Basic operation: key comparison

• Best, worst, average case exists:

Best case: when partition happens in the middle of the array each time.

Worst case: When input is already sorted. During key comparison, one half is empty, while remaining n-1 elements are on the other partition.

• Let C(n) denotes the number of times basic operation is executed in worst case: Then C(n) = C(n-1) + (n+1) for n > 1 (2 sub-problems of size 0 and n-1 respectively) C(1) = 1

Best case:

C(n) = 2C(n/2) + Θ(n)                 (2 sub-problems of size n/2 each)

• Solving the recurrence equation using backward substitution/ master theorem, we have:

C(n) = C(n-1) + (n+1) for n > 1; C(1) = 1

C(n) = Θ (n2)

C(n) = 2C(n/2) + Θ(n).

= Θ (n1log n)

= Θ (n log n)

NOTE:

The quick sort efficiency in average case is Θ( n log n) on random input.

Binary Search

Description:

Binary tree is a dichotomic divide and conquer search algorithm. Ti inspects the middle element of the sorted list. If equal to the sought value, then the position has been found. Otherwise, if the key is less than the middle element, do a binary search on the first half, else on the second half.

Algorithm:

Algorithm can be implemented as recursive or non-recursive algorithm.

ALGORITHM BinSrch ( A[0 … n-1], key)

//implements non-recursive binary search

//i/p: Array A in ascending order, key k

//o/p: Returns position of the key matched else –1

image

Analysis:

• Input size: Array size, n

• Basic operation: key comparison

• Depend on

Best – key matched with mid element

Worst – key not found or key sometimes in the list

• Let C(n) denotes the number of times basic operation is executed. Then Cworst(n) = Worst case efficiency. Since after each comparison the algorithm divides the problem into half the size, we have

image

• Solving the recurrence equation using master theorem, to give the number of times the search key is compared with an element in the array, we have:

C(n) = C(n/2) + 1

a = 1

image

Applications of binary search:

• Number guessing game

• Word lists/search dictionary etc

Advantages:

• Efficient on very big list

• Can be implemented iteratively/recursively

Limitations:

• Interacts poorly with the memory hierarchy

• Requires given list to be sorted

• Due to random access of list element, needs arrays instead of linked list.

Comments

Post a Comment

Popular posts from this blog

Data Structure Visualization:Introduction and Value of Data Structure Rendering

By -
Introduction Important advances in programming languages have occurred since the early days of assembly languages and op codes, and modern programming languages have significantly simplified the task of writing computer programs. Unfortunately, tools for un d e rstanding programs have not achieved the accompanying levels of improvement. Software developers still face difficulties in understanding, analyzing, debugging, and improving code. As an example of the difficulties still evident, consider a developer who has just implemented a new algorithm and is now ready to examine the program’s behavior. After issuing a command to begin program execution, the developer examines output data and/or inter- active behavior, attempting to ascertain if the program functioned correctly, and if not, the reasons for the program’s failure. This task can be laborious and usually requires the use of a debugger or perhaps even the addition of explicit checking code, usually through output statements. Furt
Read more

Binary Space Partitioning Trees:BSP Tree as a Hierarchy of Regions.

By -
Image
BSP Tree as a Hierarchy of Regions Another way to look at BSP Trees is to focus on the hierarchy of regions created by the recursive partitioning, instead of focusing on the hyperplanes themselves. This view helps us to see more easily how intersections are efficiently computed. The key idea is to think of a BSP Tree region as serving as a bounding volume: each node v corresponds to a convex volume that completely contains all the geometry represented by the sub tree rooted at v . Therefore, if some other geometric entity, such as a point, ray, object, etc., is found to not intersect the bounding volume, then no intersection computations need be performed with any geometry within that volume. Consider as an example a situation in which we are given some test point and we want to find which object if any this point lies in. Initially, we know only that the point lies somewhere space ( Figure 20.12). By comparing the location of the point with respect to the first partitioning hyper pl
Read more

Drawing Trees:HV-Layout

By -
Image
HV-Layout An hv -layout of a binary tree is an upward (but not strictly upward) straight line orthogonal drawing with edges drawn as rightward horizontal or downward vertical segments. The ”hv” stands for horizontal-vertical. For any vertex v in a binary tree T we either have: 1. A child of v is either • aligned horizontally with v and to the right of v or • vertically aligned below v . 2. The smallest rectangles that cover the area of the subtrees of the children of v in the layout do not intersect. Figure 45.12 shows an example of a hv -layout A hv -layout is generated by applying a dived and conquer approach. The divide step constructs the hv -layout of the left and the right subtrees, while the conquer step either performs a ho rizontal or a ver t i c a l combination . Figure 45.13 shows the two types of com- bination. If the left subtree a node v is placed to the left in a horizontal combination and below in a vertical combination, the layout preserves
Read more