Algorithm Design and Complexity Analysis

Posted on Dec 20, 2024 in Computers

1. Defining an Algorithm and Its Characteristics

An algorithm is a finite set of well-defined instructions to solve a problem in a step-by-step manner.

Characteristics of an Algorithm:

• Finiteness: Must terminate after a finite number of steps.
• Definiteness: Each step must be clear and unambiguous.
• Input: Accepts zero or more inputs.
• Output: Produces at least one output.
• Effectiveness: Instructions must be basic and executable.

2. Time and Space Complexity of an Algorithm

• Time Complexity: Measures the amount of time an algorithm takes to run as a function of the input size (n). Example: O(n), O(n²).
• Space Complexity: Measures the memory space required by the algorithm during execution, including inputs, outputs, and auxiliary variables.

3. Divide and Conquer vs. Dynamic Programming

4. Understanding Big-O Notation

Big-O Notation: Describes the upper bound of an algorithm’s time complexity in the worst-case scenario.

Example: If an algorithm’s time complexity is 5n² + 3n + 2, Big-O is O(n²).

Importance:

• Compares the efficiency of algorithms.
• Provides scalability insights.

5. Bubble Sort Algorithm Explained

Algorithm:

1. Start from the first element.
2. Compare adjacent elements and swap if needed.
3. Repeat for all elements until the array is sorted.

Pseudocode:

for i = 0 to n-1
    for j = 0 to n-i-2
        if arr[j] > arr[j+1]
            swap(arr[j], arr[j+1])

Time Complexity:

• Best Case: O(n).
• Worst Case: O(n²).

6. Greedy Algorithm Technique

Definition: Greedy algorithms solve problems by choosing the locally optimal solution at each step, aiming for a global optimum.

Example: Huffman Encoding

Used for data compression. At each step, the algorithm merges the two smallest weights/nodes.

7. Merge Sort Algorithm Steps

Merge Sort uses Divide and Conquer to sort an array.

Steps:

1. Divide the array into two halves.
2. Recursively sort both halves.
3. Merge the two sorted halves.

Time Complexity: O(n log n) for all cases.

8. Backtracking Explained with an Example

Backtracking: A technique for solving problems recursively by trying to build a solution incrementally and backtracking when a solution fails.

Example: N-Queens Problem

• Place queens one by one in columns.
• If placing in a column leads to failure, backtrack to the previous column.

9. Understanding NP-Completeness

• P (Polynomial): Problems solved in polynomial time.
• NP (Non-deterministic Polynomial): Problems where solutions are verified in polynomial time.
• NP-complete: Problems that are both in NP and as hard as any NP problem.

Example: Traveling Salesman Problem (TSP).

10. Prim’s vs. Kruskal’s Algorithms

11. Recursive Algorithm for GCD

Algorithm (Euclid’s Method):

1. Base Case: If b = 0, return a.
2. Recursive Case: GCD(a, b) = GCD(b, a mod b).

Pseudocode:

GCD(a, b):
    if b == 0:
        return a
    return GCD(b, a % b)

12. The Knapsack Problem and Its Variations

Knapsack Problem: Optimize the total value of items placed in a knapsack without exceeding its capacity.

Types:

1. 0/1 Knapsack: Items are either included or not.
2. Fractional Knapsack: Items can be divided into fractions.

Dynamic Programming solves 0/1 Knapsack, and Greedy solves Fractional Knapsack.

13. Dijkstra’s Algorithm Explained

Dijkstra’s algorithm finds the shortest path from a source node to all other nodes in a weighted graph.

Steps:

1. Initialize distances to infinity; set source distance to 0.
2. Select the node with the smallest distance.
3. Update distances of adjacent nodes.
4. Repeat until all nodes are processed.

Time Complexity: O(V²) or O((V+E) log V) using a priority queue.

Additional Problems and Solutions

1. (a) Prove: 2n² + 5n + 2 = θ(n²)

To prove, compare lower and upper bounds:

• Lower Bound: 2n² ≤ 2n² + 5n + 2 for n > 0.
• Upper Bound: 2n² + 5n + 2 ≤ 3n² (as 5n + 2 is dominated by n² for large n).

Hence, 2n² + 5n + 2 is θ(n²).

(b) Solve the recurrence relation: T(n) = 3T(n/2) + n

Using Master’s Theorem:

a = 3, b = 2, f(n) = n. log_b(a) = log₂(3) ≈ 1.58.

Since f(n) = n and log_b(a) > 1, T(n) = θ(n^log₂(3)).

(c) Binary Search Algorithm:

Algorithm:

1. Set low = 0, high = n-1.
2. While low ≤ high:

• Compute mid = (low + high) // 2.
• If arr[mid] == key, return mid.
• If arr[mid] < key, set low = mid + 1.
• Else, set high = mid – 1.

Complexity: Best Case: O(1), Worst Case: O(log n).

(d) Adjacency List Representation:

For a graph with vertices A, B, C, D:

A -> B, C
B -> A, D
C -> A
D -> B

(e) Quick Sort Steps:

List: 8, 29, 30, 6, 25, 7, 8, 9

1. Pivot = 8. Partition: [6, 7, 8] | [29, 30, 25, 9].
2. Recursively sort left and right partitions.

Final Sorted List: 6, 7, 8, 8, 9, 25, 29, 30.

2. (a) Definitions:

• Connected Graph: A graph where a path exists between every pair of vertices.
• Cycle in Undirected Graph: A path that starts and ends at the same vertex without repeating edges.

(b) Euclid’s Algorithm for GCD:

Algorithm:

1. If b == 0, return a.
2. Else, GCD(a, b) = GCD(b, a % b).

Example: GCD(48, 18) = GCD(18, 12) = GCD(12, 6) = 6.

3. (a) Spanning Tree and Applications:

A spanning tree is a subgraph that connects all vertices with the minimum number of edges.

Applications: Network design, circuit design.

(b) Dijkstra’s Algorithm:

1. Initialize distances to infinity; source = 0.
2. Select the vertex with the smallest distance.
3. Update distances of adjacent vertices.
4. Repeat until all vertices are visited.

Complexity: O(V²) or O((V+E) log V) using a priority queue.

4. (a) Greedy Technique:

Greedy algorithms solve problems by choosing the best option at each step, aiming for a global optimum.

Example: Huffman Encoding.

(b) DFS Traversal (Starting from A):

Traversal: A -> B -> C -> D (adjust according to the graph structure).

5. (a) Mathematical Induction Proof:

Prove: 1 + 2 + 3 + … + n = n(n+1)/2

• Base Case: For n=1, LHS=1, RHS=1. True.
• Inductive Step: Assume true for n=k, prove for n=k+1.
• LHS: 1 + 2 + … + k + (k+1) = k(k+1)/2 + (k+1) = (k+1)(k+2)/2.

(b) Prim’s Algorithm for MST:

1. Start with any vertex.
2. Add the smallest edge connecting visited and unvisited vertices.
3. Repeat until all vertices are included.

Complexity: O(V²) or O(E log V) with a heap.