hw_2_soln - CS 473: Algorithms, Fall 2010 HW 2 Solutions 1....

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Unformatted text preview: CS 473: Algorithms, Fall 2010 HW 2 Solutions 1. (30 pts) Suppose you are given a directed graph G = ( V , E ) with non-negative edge lengths; ‘ ( e ) is the length of e ∈ E . You are interested in the shortest path distance between two given locations / nodes s and t . It has been noticed that the existing shortest path distance between s and t in G is not satisfactory and there is a proposal to add exactly one edge to the graph to improve the situation. The candidate edges from which one has to be chosen is given by E = { e 1 , e 2 ,..., e k } and you can assume that E ∩ E = ; . The length of the e i is α i ≥ 0. Your goal is figure out which of these k edges will result in the most reduction in the shortest path distance from s to t . Describe an algorithm for this problem that runs in time O (( m + n ) log n + k ) where m = | E | and n = | V | . Note that one can easily solve this problem in O ( k ( m + n ) log n ) by running Dijkstra’s algorithm k times, one for each G i where G i is the graph obtained by adding e i to G . Solution: R EDUCE S HORTEST P ATH ( G , E , s , t ) : Create an array d storing shortest paths from v i to v j where v i ∈{ s , t } and v j ∈ V d ( s , s ) ← For each vertex v ∈ V \ s d ( s , v ) ←∞ Execute D IJKSTRA ( G , s ) to compute d ( s , v ) , ∀ v ∈ V Reverse the edges in the graph G to create the graph G d ( t , t ) ← For each vertex v ∈ V \ t d ( t , v ) ←∞ Execute D IJKSTRA ( G , t ) to compute d ( t , v ) , ∀ v ∈ V least ← d ( s , t ) bestindex ← NONE For each edge e i = ( u i , v i ) ∈ E dist ← d ( s , u i )+ weight ( u i , v i )+ d ( t , v i ) IF least > dist least ← dist bestindex ← i Output e bestindex Description The problem asks us to find exactly one edge from the set of edges E , such that the addition of that edge will result in the most reduction of the shortest path from s to t. Initially, we start off with implementing the Dijkstra’s algorithm with the vertex s as the source. As Dijkstra’s algorithm is a single source shortest path algorithm, after the implementation of the algorithm, the shortest distance from the source s to all nodes in V is now available. We now reverse the edges of the original graph G to create the graph G . We now implement a Dijkstra’s algorithm on the graph G with node t as the source. We now get shortest path from any node in the graph G to the node t , this is because we have the shortest distance from t to any node in graph G . Then we pick edges e i ∈ E one by one, and look at the shortest path from s to t containing the edge e i . In order for calculating the shortest path from s to t that is guaranteed to contain an edge ( u , v ) , we need to sum the shortest distance from s to u , the weight of the edge ( u , v ) ∈ E and the shortest 1 distance from the node v to t . We pick the edge e i whose addition to the graph G results in the minimum length path from s to t ....
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This note was uploaded on 04/18/2011 for the course CS 473 taught by Professor Chekuri,c during the Spring '08 term at University of Illinois, Urbana Champaign.

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hw_2_soln - CS 473: Algorithms, Fall 2010 HW 2 Solutions 1....

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