Hamilton Path & Dijkstra's Algorithm
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This document summarizes topics related to discrete structures, including Hamiltonian paths and circuits, matching theory, and the shortest path problem solved using Dijkstra's algorithm. It defines Hamiltonian paths and circuits, discusses maximal and perfect matchings in graphs, and outlines Dijkstra's algorithm for finding the shortest path between nodes in a weighted graph by iteratively updating distances and marking visited nodes.
Hamilton Path & Dijkstra's Algorithm
- 2. TOPICS Hamiltonian Path and Circuit Matching Theory Shortest Path Problem ( Dijkstra’s Algorithm)
- 3. #1. HAMILTONIAN PATH & CIRCUIT • Let the graph be G = (V,E); where |V|>=3 (vertices/nodes) and E denote the Edges(connector of endpoints) • A simple path in such graph G that passes through every vertex exactly once is called Hamiltonian path. Every edge mayn't be used here. • A circuit is that path where starting and endpoints are same. • For example: A B A B D C D C Path -> A –B –C –D Circuit -> A-B-C-D-A
- 4. More Examples: A B C A B C D D E E F G H Hamilton Path Exists Hamilton Path Does not Exists a b c d
- 5. #THEOROM: Let G be a graph of n>=3 vertices, then G has a Hamiltonian path if for only two non adjacent vertices u and v of G satisfy the following condition deg (u) + deg (v) >= n For example: a b c d Here, total vertices (n) = 4 Take vertex a and d deg(a) = 1 & deg(d) = 2 1+ 2 < 4 So, Hamiltonian circuit does not exists.
- 6. #2. MATCHING THEORY • A matching graph is a subgraph of a graph where there are no edges adjacent to each other. • Simply, there should not be any common vertex between any two edges • A vertex is said to be a matched if it is incident to an edge otherwise unmatched. • The vertices should have a degree of 1 or 0 • Notation M(G) • In a matching, deg(V) = 1, then (V) is said to be matched deg(V) = 0, then (V) is said to be unmatched
- 7. MAXIMAL MATCHING • A maximal matching is a matching M of a graph G with the property that if any edge not in M is added to M, it is no longer a matching • A matching M of a graph G is maximal if every edge in G has a non-empty intersection with at least one edge in M. • It is also known as maximum cardinality matching. • It is a matching that contains the largest possible number of edges. • The number of edges in the maximum matching of ‘G’ is called its matching number. MAXIMUM MATCHING
- 8. PERFECT MATCHING • A matching (M) of graph (G) is said to be a perfect match, if every vertex of graph g (G) is incident to exactly one edge of the matching (M), i.e., deg(V) = 1 ∀V • Every perfect matching of graph is also a maximum matching of graph, because there is no chance of adding one more edge in a perfect matching graph.
- 10. #3. SHORTEST PATH PROBLEM Weighted Graph(Labelled Graph): A weighted graph is a graph G, in which each edge e, is assigned a non- negative real number. The number is the weight of the e. The length of a path in a weighted graph is the sum of the weights of the edges of this path The shortest path is the minimum length of the path.
- 12. #3.DIJKSTRA’S ALGORITHM (/DEIK-STRAS/) • Formulated by Edsgar W. Dijkstra • Dijkstra’s algorithm can be used to determine the shortest path from one node in a graph to every other node within the same graph data structure, provided that the nodes are reachable from the starting node. • This algorithm will continue to run until all of the reachable vertices in a graph have been visited. • Application: Google Maps, Satellite Navigation, Finding Shortest Path etc.
- 13. #3.DIJKSTRA’S ALGORITHM (/DEIK-STRAS/) Steps: a) Assign every node a tentative distance b) Set initial node as current and mark all nodes as unvisited. c) For current node, consider all unvisited nodes and calculate distance d) Compare current and calculated distance and assign the smaller value. e) When all neighbors are considered, mark them f) If the destination node is marked, Stop
- 15. Qn: Find the shortest path between a and z
- 16. THANK YOU!!!