CAD Algorithms. Shortest Path

Transcription

1 lgorithms Shortest Path lgorithms Mohammad Tehranipoor epartment September 00 Shortest Path Problem: ind the best way of getting from s to t where s and t are vertices in a graph. est: Min (sum of the edge lengths of a path) On a weighted graph: Shortest path = Min ( f(p)), where p Є P (P is the set of all paths) xample: Traveling Salesman Problem (TSP) astest path or shortest path are the same. The length of a path can be actual mileage or time to drive it. September 00

2 ssumption ll edges have lengths (weights) that are positive numbers. It makes the algorithm a lot easier. It can be applied to both directed and undirected graph There are algorithms that handle negative weights/lengths/costs. ellman-ord lgorithm September 00 Path from istance Objective: ind the shortest path from s to t s x t Path: d(s,t) Therefore: d(s,t) = d(s,x) + d(x,t) d(s,x) < d(s,t) September 00

3 ijkstra s lgorithm Given a connected weighted graph G=(V,), a weight d: R + and a fixed vertex s in V, find the shortest path from s to each vertex v in V. ijkstra s algorithm works almost the same as Prim s algorithm. The only difference is that it chooses an edge once at a time instead of vertex to find the shortest path or minimize d(s,x) + length(x,y). September 00 ijkstra s lgorithm : Let T be a single vertex s : While (T has fewer than n vertices) // n: total number of vertices : { : ind edge (x,y) : with x in T and y not in T Minimizing d(s,x)+length(x,y) : dd (x,y) to T : d(s,y) = d(s,x) + length (x,y) : } September 00

4 xample Problem: ind the shortest paths from to all vertices 0 September 00 ont. T = {} 0 September 00

5 ont. T = {} 0 September 00 ont. T = {, } 0 September 00 0

6 ont. T = {, } 0 September 00 ont. T = {,,, } 0 September 00

7 ont. T = {,,,, } 0 0 September 00 ont. T = {,,,,, } September 00

8 Prim s lgorithm { T = ; U = {first vertex}; while (U V) { let (u, v) be the lowest cost edge such that u U and v V - U; T = T {(u, v)} U = U {v} } } September 00 Prim s lgorithm Problem: ind the shortest paths from to all vertices U = {} September 00

9 September 00 ont. U = {} September 00 ont. U = {, }

10 0 September 00 ont. U = {,, } September 00 0 ont. T = {,,, }

11 September 00 ont. T = {,,,, } September 00 ont. T = {,,,,, } This algorithm is more suitable for MST problem.

12 lgorithms Minimum Spanning Tree lgorithms Mohammad Tehranipoor epartment September 00 MST MST is a well-solved combinatorial optimization problem. efinition: tree is a connected graph without cycles. Properties of Trees: graph is a tree if and only if there is only and only one path joining any two of its vertices. connected graph is a tree if and only if it has N vertices and N- edges. September 00

13 MST (ont.) efinition: subgraph that spans (reaches out to) all vertices of a graph is called a spanning subgraph. subgraph that is a tree and that spans (reaches out to) all vertices of the original graph is called a spanning tree. mong all the spanning trees of a weighted and connected graph, the one (possibly more) with the least total weight is called a minimum spanning tree (MST). September 00 xample simple example: graph may have many spanning trees. This graph has spanning trees. September 00

14 spanning trees: xample: Trees Tree #: MST September 00 Problem Suppose the edges of the graph have weights. The weights of a tree is sum of weights of its edges. ifferent trees may have different weights. Problem: How to find minimum spanning tree? Solutions: Heuristic Greedy xhaustive search September 00

15 Why Minimum Spanning Tree? xample: Phone Network esign: One has a business with several offices and wants to lease phone lines to connect them up with each other; the phone company charges differently to connect different pair of cities. How he can find a set of lines that connects all the offices with a minimum total cost. Traveling Salesman Problem (TSP): nother convenient way to solve this problem is to find the shortest path that visits each point at least once. VLSI Problem: Global Routing MST is used in global routing Steiner Tree is used in detailed (channel) routing September 00 MST/TSP In general the MST weight is less than the TSP. In MST, we are not limited only to paths. What is path: Path Path Not a path Not a TSP solution September 00 0

16 How to ind MST? n exhaustive search: List all spanning trees and find the minimum one. Not applicable in practice especially in VLSI problems. Heuristic lgorithms Greedy lgorithms Prim s lgorithm September 00 Kruskal s lgorithm It is easy to understand and the best for solving problems by hands. Kruskal s lgorithm: Step : Sort the edges of graph G in increasing order by weight Step : Keep a subgraph S of G, initially empty Step : or each edge e in sorted order If the endpoints of e are disconnected in S dd e to S Step : Return S September 00

17 xample dge Weight S={} September 00 ont. dge Weight dges:,,,,,,,, S={} September 00

18 ont. dge Weight dges:,,,,,,,, S={, } September 00 ont. dge Weight dges:,,,,,,,, S={,, } September 00

19 ont. dge Weight dges:,,,,,,,, S={,,, } September 00 ont. dge Weight dges:,,,,,,,, S={,,, } September 00

20 ont. dge Weight dges:,,,,,,,, S={,,, } September 00 ont. dge Weight dges:,,,,,,,, S={,,,, } Total ost = September

21 xample graph may have more than one MST. Total ost = Total ost = This algorithm is known as a greedy algorithm because it chooses the cheapest edge at each step and adds to S. September 00 Prim s lgorithm Step : Pick any vertex as a starting vertex. all it s. Step : ind the nearest neighbor of s (call it p). hoose sp and p such that sp is the cheapest edge in the graph that does not close the selected edges. dd sp to S. Step : Return subgraph S. September 00

22 xample Prim s lgorithm: 0 0 September 00 ont. 0 0 September 00

23 ont. 0 0 September 00 ont. 0 0 September 00

24 ont. 0 0 September 00 ont. 0 0 September 00

25 ont. 0 0 September 00 ont. 0 0 September 00 0

26 ont. 0 0 September 00 ont. 0 0 September 00

27 ont. 0 0 September 00 ont. 0 0 September 00

28 ont. 0 0 September 00 ont. 0 0 September 00

29 More lgorithms oruvka s lgorithm: The idea is to do steps like Prim s algorithm in parallel all over the graph at the same time. Hybrid lgorithm: ombine two of the classical algorithms and do better than either one alone. September 00