Md. Abul Kashem, Chowdhury Sharif Hasan, and Anupam Bhattacharjee

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Presentation transcript:

Md. Abul Kashem, Chowdhury Sharif Hasan, and Anupam Bhattacharjee An Algorithm for Solving the Minimum Vertex-Ranking Spanning Tree Problem on Series-Parallel Graphs ICECE 2006 Md. Abul Kashem, Chowdhury Sharif Hasan, and Anupam Bhattacharjee

Graphs, Cycles and Trees root parent nodes child A Graph A cycle A tree Anupam Bhattacharjee, CSE, BUET

Spanning Trees A connected graph with no cycles is a spanning tree In a connected cyclic graph, if we delete edges to remove cycles. If there remains no cycle, we call it a spanning tree of the graph. Anupam Bhattacharjee, CSE, BUET

SP graphs: Series Connection Anupam Bhattacharjee, CSE, BUET

SP Graphs: Parallel Connection Anupam Bhattacharjee, CSE, BUET

Vertex-Ranking Valid Ranking Invalid Ranking A labeling (ranking) of the vertices of G with positive integers such that every path in G with end vertices of the same label i contains an internal vertex with label j > i. Valid Ranking Invalid Ranking Anupam Bhattacharjee, CSE, BUET

Minimum Vertex-Ranking A Vertex-Ranking is minimum if least number of ranks are needed to rank the graph. A minimum vertex-ranking A non-optimal Vertex-Ranking Anupam Bhattacharjee, CSE, BUET

Minimum Vertex-Ranking Spanning Tree The problem is to find a spanning tree of a graph whose vertex-ranking needs least number of ranks. Input: A graph Output: A tree with minimum vertex-ranking Anupam Bhattacharjee, CSE, BUET

Binary Decomposition Tree Anupam Bhattacharjee, CSE, BUET

Solution types Two types of partial solutions: A one-tree type solution: a spanning tree is kept A two-tree type solution: a spanning forest having exactly two components (trees) with terminal vertices in different trees is kept. Anupam Bhattacharjee, CSE, BUET

Steps of the algorithm An SP graph is given: ac ab bc s Steps of the algorithm An SP graph is given: Step#1: Binary decomposition tree Step#2: Equivalence class computation for each leaf node Anupam Bhattacharjee, CSE, BUET

For s-node Solution Computation ab bc Anupam Bhattacharjee, CSE, BUET

For p-node Solution Computation ac s DESIRED Anupam Bhattacharjee, CSE, BUET

Points to note Total running time of the algorithm is O(n5 log4 n). Some open problems still: Develop a polylog-time parallel algorithm for solving the minimum vertex- ranking spanning tree problem on series-parallel graphs. Develop a polynomial-time sequential algorithm for solving the minimum edge-ranking spanning tree problem on series-parallel graphs Anupam Bhattacharjee, CSE, BUET

Thank you Anupam Bhattacharjee, CSE, BUET