Lab Manual

Algorithms Design (Pr)

COT-311

Lab Instructions

Whether an experiment contains one or

several practicals /programs

One

practical / program

Several

practicals / programs

Whether practical has been verified

and signed by the lab teacher?

Students write experiments in practical files and get them signed by the lab teacher

Students make entries in the list of contents of the practical files and get them signed by

the lab teacher

All Students need to perform the practical/program

Lab Teacher forms groups of the students based on

Assign all practicals /programs among all groups

?

Teacher decides whether the completed practicals / programs can be appropriately described

using flow chart, algorithm, query statement, etc.

Teacher issues necessary instructions to the students for writing practicals / programs

accordingly

In case of an experiment containing several practicals, a lab teacher needs to think whether a practical performed by the students in one group needs to be repeated by the other groups in lab on the same day?

OR A practical performed by the students in one group needs to be repeated as assignments to be completed by the students of other groups in their hostels? Here, an assignment includes both executing a program on computer and also writing the same in practical file.

OR A practical performed by the students in one group needs to be repeated as assignments, only writing practicals in their practical files, for the students of other groups in their hostels?

Teacher issues necessary instructions to the students accordingly.

If a student has not completed a practical, he/she is expected to complete it at his/her

own with the help of his/her fellow students in his/her hostel

The student completes the practical file and submits it to the concerned teacher in

his/her office or mail box on next working day

?

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Lab Manual Algorithm Design (Pr)

COT- 311

L T P

- - 3

Practical exam: 40

Sessional: 60

Note: Implement the following using C/C++ programming language on LINUX or Windows like

platform.

Experiment 1 (Quick and Heap sort)

I. (a) Implement the ascending and desending order using Quick Sort.

(b) Finding kth

minimum and maximum element in Heap.

II. (a)

(b)

Implement Quick Sort using first/last/any random element as pivot.

Build Min heap, Max heap and sort the given elements.

III. (a)

(b)

Implement Quick Sort with duplicate numbers in the given array/elements.

Delete kth

indexed element in Min heap and Max heap.

Experiment 2 (Dynamic Programming)

I. (a) Implement the matrix chain multiplication problem.

(b) Find the Longest palindromic subsequence (Hint: Subsequence is obtained by deleting

some of the characters from a string without reordering the remaining characters, which

is also a palindrome).

II. (a) Find optimal ordering of matrix multiplication.

(b) Find the longest common substring (Input: Character string as in book or binary matrix

of 0 and 1. Output: longest sequence of 1's either row wise or column wise).

Experiment 3 (Greedy Algorithms)

I. (a)

(b)

Implement the file or code compression using Huffman‟s algorithm.

You are given n events where each takes one unit of time. Event i will provide a profit

of gi dollars (gi > 0) if started at or before time ti where ti is an arbitrary real number.

(Note: If an event is not started by ti then there is no benefit in scheduling it at all. All

events can start as early as time 0).

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(c) Implement the activity-selection problem ( You are given n activities with their start and

finish times. Select the maximum number of activities that can be performed by a single

person, assuming that a person can only work on a single activity at a time.

Example: Consider the following 6 activities.

start[] = {1, 3, 0, 5, 8, 5}; finish[] = {2, 4, 6, 7, 9, 9};

The maximum set of activities that can be executed by a single person is {0, 1, 3, 4} ).

II. (a)

(b)

Implement the file or code compression using Huffman‟s algorithm.

Consider the following scheduling problem. You are given n jobs. Job i is specified by

an earliest start time si, and a processing time pi. We consider a preemptive version of

the problem where a job's execution can be suspended at any time and then completed

later. For example if n = 2 and the input is s1 = 2, p1 = 5 and s2 = 0, p2 = 3, then a legal

preemptive schedule is one in which job 2 runs from time 0 to 2 and is then suspended.

Then job 1 runs from time 2 to 7 and secondly, job 2 is completed from time 7 to 8. The

goal is to output a schedule that minimizes ∑Cj = 1, where Cj is the time when job j is

completed and j runs from 1 to n. In the example schedule given above, C1 =7 and C2

=8.

(c) Implement the activity-selection problem ( You are given n activities with their start and

finish times. Select the maximum number of activities that can be performed by a single

person, assuming that a person can only work on a single activity at a time.

Example: Consider the following 6 activities.

start[] = {1, 3, 0, 5, 8, 5}; finish[] = {2, 4, 6, 7, 9, 9};

The maximum set of activities that can be executed by a single person is {0, 1, 3, 4} ).

Experiment 4 (Binary Search Tree)

I. (a)

(b)

Finding the maximum and minimum element in BST.

Finding Kth

smallest/largest element in BST.

II. Write a program for converting the array to BST (Input: Unsorted/Sorted array).

III. Write a program for printing all the elements of BST in the range K1 and K2 (Input:

BST and two numbers K1 and K2).

IV. Write a program for removing the duplicates in array using BST.

Experiment 5 (Binary Tree)

I. Write a program to build a binary tree for the given elements (i.e., integers) and give

traversal functions: inorder, preorder, postorder.

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II. Write a program to transform given a tree into a binary tree.

III. Write a program to represent an arithmetic expression in binary tree format.

Experiment 6 (Graph Algorithms)

I. (a)

(b)

Find Minimum Cost Spanning Tree of a given undirected graph using Kruskal‟s

algorithm.

Implement the algorithm for Hash Tables to check both arrays have same set of

numbers/characters (Input: given two arrays of unordered numbers/characters).

II. (a)

Find Minimum Cost Spanning Tree of a given undirected graph using Prim‟s algorithm.

(b) Implement the algorithm for Hash Tables to removing the duplicate characters/ numbers

(Input as an array of characters/numbers).

III. (a)

(b)

Find Minimum Cost Spanning Tree of a given undirected graph using Kruskal‟s

algorithm.

Give an algorithm for finding non repeated character in a string/number using Hash

Tables. Example: the first non repeated character in the string “abbcc” is „a‟.

Experiment 7 (Shortest Path Finding in Graph)

I. (a) From a given vertex in a weighted connected graph, find shortest paths to other vertices

using Dijkstra's algorithm.

(b) Draw simple, connected weighted graph with 10 vertices and 18 edges, such that graph

contains minimum weight cycle with at least 4 edges. Show that the Bellman-Ford

algorithm find this cycle.

II. (a) Draw simple, connected weighted graph with 8 vertices and 16 edges, each with unique

edge weight. Identify one vertex as a start vertex and obtain shortest path using

Dijkstra's algorithm.

(b) From a given vertex in a weighted connected graph, find shortest paths to other vertices

using Bellman-Ford algorithm.

III. (a) Find the kth

shortest path between two given nodes of a graph.

(b) Give an algorithm to determine whether a directed graph with positive and negative

cost edges has negative cost cycle.

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Experiment 8 (All Pair’s Shortest Paths)

I. (a) Implement all pair‟s Shortest Paths Problem using Floyd‟s-Warshall algorithm.

(b) Implement all pair‟s Shortest Paths Problem using Johnson‟s algorithm.

Experiment 9 (BFS and DFS problems)

I. (a) Write an algorithm to print all the nodes reachable from a given starting node in a

digraph using BFS method.

(b) Obtain the Topological ordering of vertices in a given digraph.

II. (a) Check whether a given graph is connected or not using DFS method.

(b) Find the strongly connected components in a digraph.

Experiment 10 (Flow and Sorting Networks)

I. (a)

(b)

(c)

Implement the Ford-Fulkerson method for maximum flow network.

Implement the Maximum bipartite matching problem in the graph.

Implement the Bitonic sorting algorithm.

Note: Additional Programs are for those students who wish to do advance problems

Additional Problem 1 (Data Structures)

I. Write an algorithm for finding counting inversions in an array. Inversion is a pair such

that for an array A = {a1, a2, a3,...., an}, and ai > aj and i < j

II. Given an array. Let us assume that there can be duplicates in the list. Write an algorithm

to search for an element in the list in such a way that we get the highest index if there

are duplicates.

III. Write an algorithm for finding i and j in an array A for any key such that

A[j]2

+ A[i]2 == key

IV. Given key in a sorted array A with distinct values. Find i, j, k such that

A[i] + A[j] + A[k] == key

V. Suppose an array A has n distinct integers. Increasing sequence is given as

A[1].........A[k] and decreasing sequence is given as A[k+1]........A[n]. Write an

algorithm for finding k.

VI. You have 1 to N­1 array and 1 to N numbers and one number is missing. Write an

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algorithm for finding that missing number.

Additional Problem 2 (Dynamic Programming & Greedy Algorithms)

I. Implement the job sequencing with deadlines problem using the fixed tuple size

formulation.

II. (a)

(b)

Implement the algorithm to compute roots of optimal sub-trees.

Implement the optimal binary search tree problem.

III. (a) Write an algorithm for partitioning problem. Partition problem is to determine whether a

given set can be partitioned into two subsets such that the sum of elements in both

subsets is same.

(b) Find a subset of a given set S = {S1, S2,..…., Sm} of n positive integers whose sum is

equal to a given positive integer d. For example, if S = {1, 2, 5, 6, 8} and d = 9 there are

two solutions {1, 2, 6} and {1, 8}. A suitable message is to be displayed if the given

problem instance doesn't have a solution.

(c) Given a value N, if we want to make change for N cents, and we have infinite supply of

each of S = {S1, S2,..…., Sm} valued coins, how many ways can we make the change?

The order of coins doesn‟t matter. For example, for N = 4 and S = {1, 2, 3}, there are

four solutions: {1, 1, 1, 1}, {1, 1, 2}, {2, 2}, {1, 3}. So output should be 4. For N = 10

and S = {2, 5, 3, 6}, there are five solutions: {2, 2, 2, 2, 2}, {2, 2, 3, 3}, {2, 2, 6}, {2, 3,

5} and {5, 5}. So the output should be 5.

Additional Problem 3 (Advanced Data Structures)

I. Perform the following operations on B-Tree:

a) Creation

b) Insertion.

c) Deletion.

d) Searching.

II. Implement the conversion of the BST to B-tree.

III. Perform the following operations on Binomial heaps:

a) Creation

b) Insertion.

c) Deletion.

d) Searching.

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e) Uniting two Heaps

f) Extracting the minimum node

IV. Perform the following operations on Fibonacci heaps:

a) Creation

b) Insertion.

c) Deletion.

d) Searching.

e) Uniting two Heaps

f) Extracting the minimum node

Additional Problem 4 (Graph Algorithms)

I. Perform the following operations on Disjoint set:

a) MAKE-SET

b) UNION

c) FIND-SET

d) INSERT

e) DELETE

f) SPLIT

g) Returns the minimum object in SET

II. Perform the following Union-by-Weight operations on a universe of 10 elements (0-9,

each initially in their own set). Draw the forest of trees that result. U(1,5); U(3,7);

U(1,4);U(5,7); U(0,8); U(6,9); U(3,9). If the sets have equal weight, use the root with

the smaller value as the root of the new set.

III. Let G be a connected graph of order n. Write an algorithm for finding maximum number

of Cut­vertices that G can contain. Cut­vertex is a vertex which when removed from the

graph splits it into two disconnected components.

IV. An Euler circuit for an undirected graph is a path that starts and ends at the same vertex

and uses each edge exactly once. A connected undirected graph G has an Euler circuit.

If and only If energy vertex is of even degree. Give an algorithm and implement to find

the Euler Circuit in a graph with e edges provided one exists.

Additional Problem 5 (Other Techniques)

I. Implement N Queen's problem.

II. Implement Sudoku problem.

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III. Design an efficient solution for tower of Hanoi problem.

IV. Implement 0/1 Knapsack problem.

V Implement the travelling salesman problem (TSP).

VI. Given a rod of length n inches and an array of prices that contains prices of all pieces of

size smaller than n. Determine the maximum value obtainable by cutting up the rod and

selling the pieces. For example, if length of the rod is 8 and the values of different

pieces are given as following, then the maximum obtainable value is 22 (by cutting in

two pieces of lengths 2 and 6)

Length | 1 2 3 4 5 6 7 8

-----------------------------------------------------

Price | 1 5 8 9 10 17 17 20

VII Implement the Coin change problem.

VIII. Design the Graph coloring problem