Discrete Structures / Discrete Mathematics

UNIT - V : BACKTRACKING General Method – 8-Queens Problem – Sum Of Subsets – Graph Coloring – Hamiltonian Cycles General Method Backtracking is a problem-solving technique used in algorithms to find solutions by exploring all possible options. In backtracking, a state-space tree is a tree-like structure that represents all possible states (solutions or non-solutions) of a problem. Backtracking algorithms explore this tree using a depth-first search approach, systematically checking each path until a solution is found or all possibilities have been exhausted. Here’s how it works, step-by-step: Start with a choice : Make a decision and move forward. Check feasibility : If this choice leads to a solution or partial solution, continue. If not, stop exploring this path. Go back and try a different path : If you reach a dead end, go back to the last point where you made a choice. Try a different path from there. Repeat until you find a solution ...

Divide and Conquer Technique - Binary Search, Quick Sort, Merge Sort

November 04, 2024

UNIT II - DIVIDE AND CONQUER TECHNIQUE General Method – Binary Search – Merge Sort – Quick Sort. GENERAL METHOD The divide and conquer technique is a problem-solving approach that works in three main steps: 1. Divide : Break down a large problem into smaller, simpler sub-problems. 2. Conquer : Solve each of the sub-problems individually. This can often be done recursively if the sub-problems are still large. 3. Combine : Combine the solutions of the sub-problems to get the solution to the original, larger problem. This approach is commonly used in algorithms like binary search, merge sort and quick sort, where breaking down a problem makes it easier and faster to solve. BINARY SEARCH Binary search is an efficient algorithm for finding a target value within a sorted list. Here’s how it works: Start in the Middle : Look at the middle element of the list. Compare : ...

GRAPH THEORY

September 24, 2020

GRAPH THEORY DEFINITION A graph G = (V, E) consists of a non-empty set V, called the set of vertices (points, nodes) and a set of ordered or unordered pairs of elements of V, called the edges E set (links), such that there is a mapping from the set E to the set of ordered or unordered pairs of elements of V. Example In the first graph, V = { A , B , C , D } E = {(A,B) , (A,C) , (A,D), (B,C), (B,D) , (C,D)} In the second graph, V = { E, F, G } E = {(F,E) , (E,G) , (G,F)} If an edge e ∈ E is associated with an ordered pair (u,v) or an unordered pair (u,v), where u,v ∈ V, then e is said to connect or join the nodes u and v. The edge e that connects the nodes u and v is said to be incident on each of the nodes. The pair of each node that is connected by an edge is called adjacent nodes . Isolated Nodes If any node has no edges connected with any other node then its degree will be zero(0) and it will be called as isolated node. Adjacent Nodes Her...

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Deepa PhD CS