Lecture-05-transitive-closure
Page 1: Introduction
Course Information
Course Name: CSC 226 Algorithms and Data Structures: II
Instructor: Navid Mahdian
Contact: navidmahdian@uvic.ca
Location: ECS 626
Page 2: Transitive Closure
Definition
Given a directed graph (digraph) G, the transitive closure of G, denoted as G*, contains the same vertices as G.
Condition: If there is a directed path from vertex u to vertex v (where u ≠ v) in G, then G* has a directed edge from u to v.
Purpose of Transitive Closure
Provides reachability information within a digraph.
Page 3 & 4: Air Canada Routes
Air Canada Map Options
Choices include: World, Canada, United States, Europe, Asia/Pacific, Latin America.
Key Locations
Cities featured: Flin Flon, Prince Rupert, Grande Prairie, Calgary, Vancouver, Winnipeg, Seattle, St. John's, Montreal, Toronto, Halifax, and many U.S. cities.
Note
Some routes may be seasonal and subject to change based on demand.
Page 5: Transitive Closure Algorithms
Graph Construction Algorithms
Transitive closure G* can be constructed using:
Depth First Search (DFS) or Breadth First Search (BFS) from each vertex.
Identifying Strongly Connected Components (SCC) and closure on Directed Acyclic Graphs (DAG).
Floyd-Warshall's Algorithm.
Matrix Multiplication.
Complexity
The time complexity is generally O(n³).
Page 6: Floyd-Warshall’s Algorithm for Transitive Closure
Overview
Floy-Warshall’s algorithm uses dynamic programming to construct transitive closures.
Similar to Floyd's algorithm for all-pairs shortest paths.
Utilizes overlapping paths between vertices to minimize recalculations.
Page 7: Dynamic Programming Technique
Usage
Primarily addresses optimization problems where the "best" solution needs to be identified.
Comparison to Brute-Force
The brute-force approach can be exponential in nature, defining all potential solutions.
Structural Exploitation
Problems with a specific structure can be simplified to obtain polynomial-time algorithms.
Page 8: Dynamic Programming Characteristics
Key Elements of Problems
Simple Subproblems: Problems can be broken down into smaller subproblems with similar structure.
Subproblem Optimality: The optimal global solution is a combination of optimal subproblems.
Subproblem Overlap: Efficiency improves as many subproblems may overlap, allowing for memoization to be used.
Page 9: Classic Dynamic Programming Problems
Historical Context
Introduced by Richard Bellman (1940-53).
Examples of Problems
0-1 Knapsack Problem: Maximize value within weight limits.
Computing binomials, Longest Common Subsequence, Floyd-Warshall Algorithm, and others.
Page 10: Floyd-Warshall Transitive Closure Steps
Step 1: Number vertices as v1, v2,..., vn.
Step 2: Analyze paths using vertices v1,..., vk as intermediates for various edge additions.
Page 11: The Floyd-Warshall Algorithm
Initial Conditions
Given a digraph G with n vertices and m edges, number vertices as v1, v2,..., vn.
Iterative Process
In the k-th iteration, Gk constructed by adding edges based on paths found in Gk-1.
Page 12: Algorithm Structure of Floyd-Warshall
Algorithm Steps
Input: Digraph G with n vertices.
Create an arbitrary numbering for vertices.
Iterate through k from 1 to n, modifying Gk in each step based on edges and paths identified from Gk-1.
Page 13: Example with Floyd-Warshall Algorithm
Visual representation of vertex connections.
Page 14 - 19: Iterative Steps of Floyd-Warshall
Iterations 1-6: Illustrate how directed paths are expanded for intermediate vertices through consecutive improvements of Gk.
Page 20: Conclusion of Floyd-Warshall's Iterations
Completion of all iterations leading to G* denoting the transitive closure of G.
Page 21: Adjacency Matrix Representation
Proposal to utilize a 0-1 matrix representation for graph manipulation within Floyd-Warshall's framework.
Page 22 - 23: Example Adjacency Matrices
Illustrations of adjacency matrices indicating vertex connections from the example.
Page 24: Additional Steps in Matrix Representation
Detailed processing of each vertex pair indication and addition of directed edges based on existing intermediate vertices.
Pages 25 - 130: Continued Illustration of Iteration Progress
Showcasing the iterative addition of directed edges through k=1 to k=7 and how each round expands possible connections in G.
Page 131 - 142: Final Steps for Transitive Closure
Finalization of edges added for all directed paths extending through multiple intermediate vertices.
Page 143 - 160: Runtime Implementation of Floyd-Warshall
Complete algorithmic flow for implementing the Floyd-Warshall procedure with end conditions to return the transitive closure G*.