Linear Probing - Coursera Algorithms

Linear probing is used to resolve collisions in hashing. It takes less space and has better cache performance.

An array of size M > number of elements N is used such that:

1. Hash - map key to i between 0 and M-1.
2. Insert - at table index i, if not free - try i+1, i+2 etc

3. Search - search table index i, if occupied and no match - try i+1, i+2 etc

Following is the implementation :

Separate Chaining - Coursera Algorithms

To deal with collisions in hashing, separate chaining implementation can be used, where M linked lists(<number of elements N) are used such that :

1. Hash - map key to i between 0 and M-1.
2. Insert - at the front of ith chain.

3. Search - in ith chain.

Easier to implement and clustering less sensitive to poorly designed hash function.

Following is the implementation:

Hash Functions - Coursera Algorithms

We can do better than the sequential/binary search or search through BST/red-black BST, by storing the data in key-indexed table. Hash functions compute the array index from key. We need to handle the hash implementation, equality test and collision resolution when two keys hash to the same index.

There are several hashing implementations, some of them being :
1. 31*initial_hash_value+ current element value or current object hashcode.
2. Modular - Math.abs(key.hashCode() & 0x7fffffff)%M returning a hash between 0 and M-1

Following code experiments with what happens when we override equals/hashcode:

Heap Sort - Coursera Algorithms

Heap Sort is an in place sort that builds a max binary heap and then repeatedly deletes the maximum element. It's not stable and it's inner loop takes longer than Quick Sort. It has poor cache usage. To build a heap, it takes O(N) and for sort, it takes O(NlogN).

Following is the program:

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Binary Heap

Binary Heap - Coursera Algorithms

A binary tree is either empty or points to a root node with links to left and right binary trees. A tree is complete if it is balanced perfectly, excluding the bottom level. A binary heap is an array representation of a heap ordered complete binary tree.

Max node is root at a[1]. For a node k, it's parent is at k/2 and it's children are at 2k and 2k+1.
Swim: When a child node is greater than parent, keep exchanging them till they are ordered.
Sink: When a parent node is lesser than child, keep exchanging them till they are ordered.
In insert operation, add the child at end and swim it. --> O(logN)
In delete operation, exchange root with node at end and sink the top node. -->O(logN)

Following is the program:

Three Way Quick Sort - Coursera Algorithms

Three Way Quick Sort uses multiple partition elements, such that the there are no larger entries to their left and no smaller entries to their right. It is in place, but not a stable sort. This is used to deal with duplicate elements.

Time complexity: O(NlgN)


Following is the program, partition elements are from lt to gt:

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Quick Sort

Insertion Sort 

Quick Select - Coursera Algorithms

Quick select finds k^th smallest element by partitioning. It takes linear time and O(N²) in worst case.

Following is the program:

Quick Sort - Coursera Algorithms

Quick Sort uses a partition element, such that the there are no larger entries to it's left and no smaller entries to it's right. It is in place, but not a stable sort.

Time complexity: O(NlgN)
Worst case : O(N²)

Following is the program:

Merge Sort - Coursera Algorithms

Merge sort uses divide and conquer approach to sort the elements. It takes at most NlgN compares and 6NlgN array accesses with total running time of NlgN.

Following is an optimized merge sort - it uses insertion sort for less number of elements. Please see Insertion Sort for the insertion sort algorithm. It also has a method for bottom up merge sort which is not as efficient as the recursive merge sort.

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Insertion Sort

Shell Sort - Coursera Algorithms

Shell sort moves entries by more than one position at a time. It is basically insertion sort with a stride length h - it considers h interleaved sequences and sorts them using insertion sort. When the increment(h) is big, the sub-arrays are smaller and when the increment gets shorter in subsequent iterations, the array is already partially sorted. In both cases, insertion sort works great and hence, it's the best option here.

The optimal value for h is provided by Knuth as 3X+1, but no accurate model has been found for this algorithm yet.

It takes O(N3/2) compares in worst case. It's fast unless array size is large and it has less footprint for code. These make it a perfect choice for hardware sort prototype, linux/kernel, embedded systems etc.

Following is the program:

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Insertion Sort

Insertion Sort - Coursera Algorithms

Insertion Sort swaps an element with a larger entry to it's left.

It take ~1/4 N2 compares and ~1/4 N2 exchanges.
Best case(already sorted) : N-1 compares and zero exchanges.
Worst case(reverse sorted):  ~1/2 N2 compares and ~1/2 N2 exchanges.
Partially sorted(when number of inversions <= cN): linear time as number of exchanges equal the number of inversions.

Following is the program, it also uses shuffle to make the order as random as possible. This is done by swapping each element with a random index r such that r is between 0 and iteration i. This takes linear time.

Selection Sort - Coursera Algorithms

Selection sort finds the minimum element and swaps it with the first element, till the complete array is sorted. It takes N²/2 compares and N exchanges.

Following is the program:

Decimal To Binary Conversion Using Stack

Stack can be used to convert a number into it's binary form.

1. Push the remainder into stack after dividing the number by 2.
2. Keep doing 1, till number>0.
3. Pop the binary remainders 1/0 to get the binary form.

 Following is the program:

 

Arithmetic Expression Evaluation Using Stack

Arithmetic expressions can be evaluated using two stacks - one for operands and other for numeric values.

Algorithm:

1. If element is numeric, push to numeric stack.
2. If element is operand, push to operand stack.
3. If element is "(", ignore. 
4. If element is ")", pop operand from operand stack and pop two numerics from numeric stack - perform the operation and push the result back into numeric stack. 

 Following is the program based on the generic stack implementation using arrays.

Generic Queue Using Resized Array And Iterator - Coursera Algorithms

To make the stack generic and to hide the internal representation of a queue from client, we can use Generics and Iterator.

Following is the program:

Generic Queue Using Linked List And Iterator - Coursera Algorithms

To make the stack generic and to hide the internal representation of a queue from client, we can use Generics and Iterator.

Following is the program:

Generic Stack Using Resized Array And Iterator - Coursera Algorithms

To make the stack generic and to hide the internal representation of a stack from client, we can use Generics and Iterator.

Following is the program:

Generic Stack Using Linked List And Iterator - Coursera Algorithms

To make the stack generic and to hide the internal representation of a stack from client, we can use Generics and Iterator.

Following is the program:

Queue Using Array Resizing - Coursera Algorithms

Queues follow FIFO standard. To implement, we need to maintain two indices to first and last elements of the array, for enqueue and dequeue operations.

Following is the program:

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Queue Using Linked List - Coursera Algorithms

Queue Using Linked List - Coursera Algorithms

Queues follow FIFO standard. To implement, we need to maintain two pointers to first and last nodes of linked list, as we insert at the end of the list and removed the first element of the list for enqueue and dequeue operations.

Following is the program: