Today More about Trees. Introduction to Computers and Programming. Spanning trees. Generic search algorithm. Prim s algorithm Kruskal s algorithm

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1 Introduction to omputers and Programming Prof. I. K. Lundqvist Lecture 8 pril Today More about Trees panning trees Prim s algorithm Kruskal s algorithm eneric search algorithm epth-first search example Handling cycles readth-first search example

2 Trees panning Trees tree is a connected graph without cycles panning tree of a graph, is a tree that includes all the vertices from. connected graph is a tree iff it has N vertices and N- edges irline Routes FO oston The The resulting spanning tree tree is is not not unique Wash graph is a tree iff there is one and only one path joining any two of its vertices L allas FO oston Wash L allas Minimum panning Tree Prim Initialization Prim s lgorithm Finds a subset of the edges (that form a tree) including every vertex and the total weight of all the edges in tree is minimized ody hoose starting vertex Initialization reate the Fringe et Loop until the MT contains all the vertices in the graph Remove edge with minimum weight from Fringe et dd the edge to MT Update the Fringe et Pick any vertex x as the starting vertex Place x in the Minimum panning Tree (MT) For each vertex y in the graph that is adjacent to x dd y to the Fringe et For each vertex y in the Fringe et et weight of y to weight of the edge connecting y to x et x to be parent of y

3 Prim ody 8 OR O JFK While number of vertices in MT < vertices in the graph Find vertex y with minimum weight in the Fringe et dd vertex and the edge x,y to the MT Remove y from the Fringe et For all vertices z adjacent to y If z is not in the Fringe et dd z to the Fringe et et parent to y et weight of z to weight of the edge connecting z to y lse If Weight(y,z) < Weight(z) then et parent to y et weight of z to weight of the edge connecting z to y FO 9 LX 97 N 8 FO 9 LX 98 N TL 9 MI OR 7 TL 9 O 9 JFK Minimum spanning tree Prim MI Minimum panning Tree 8 OR O JFK Kruskal s lgorithm Finds a minimum spanning tree for a connected weighted graph FO 9 LX 97 8 N 98 7 TL 9 9 reate a set of trees, where each vertex in the graph is a separate tree reate set containing all edges in the graph While not empty Remove edge with minimum weight from if that edge connects two different trees, then add it to the forest, combining two trees into a single tree Otherwise discard that edge FO 9 LX 8 N 98 MI OR 7 TL 9 O 9 JFK Minimum spanning tree Kruskal MI

4 More about Trees panning trees Prim s algorithm Kruskal s algorithm eneric search algorithm epth-first search example Handling cycles readth-first search example epth First earch (F) Idea: xplore descendants before siblings xplore siblings left to right Where do we place the children on the queue? ssume we pick first element of dd path extensions to? of imple earch lgorithm Let be a list of partial paths, Let be the start node and Let be the oal node. epth-first Pick first element of ; dd path extensions to front of. Initialize with partial path (). If is empty, fail. lse, pick a partial path N from. If head(n) =, return N (goal reached!). lse: a) Remove N from b) Find all children of head(n) and create all the one-step extensions of N to each child. () c) dd all extended paths to d) o to step.

5 imple earch lgorithm Let be a list of partial paths, Let be the start node and Let be the oal node. epth-first Pick first element of ; dd path extensions to front of. Initialize with partial path (). If is empty, fail. lse, pick a partial path N from. If head(n) =, return N (goal reached!). lse: a) Remove N from b) Find all children of head(n) and create all the one-step extensions of N to each child. () c) dd all extended paths to d) o to step. epth-first epth-first Pick first element of ; dd path extensions to front of Pick first element of ; dd path extensions to front of () ( ) () ( ) ( ) dded paths in blue dded paths in blue

6 imple earch lgorithm Let be a list of partial paths, Let be the start node and Let be the oal node. epth-first Pick first element of ; dd path extensions to front of. Initialize with partial path (). If is empty, fail. lse, pick a partial path N from. If head(n) =, return N (goal reached!). lse: a) Remove N from b) Find all children of head(n) and create all the one-step extensions of N to each child. c) dd all extended paths to () ( ) ( ) dded paths in blue d) o to step. epth-first epth-first Pick first element of ; dd path extensions to front of Pick first element of ; dd path extensions to front of () ( ) ( ) ( ) ( ) ( ) () ( ) ( ) ( ) ( ) ( ) dded paths in blue dded paths in blue

7 epth-first epth-first Pick first element of ; dd path extensions to front of Pick first element of ; dd path extensions to front of () ( ) ( ) ( ) ( ) ( ) () ( ) ( ) ( ) ( ) ( ) ( ) ( ) dded paths in blue dded paths in blue epth-first epth-first Pick first element of ; dd path extensions to front of Pick first element of ; dd path extensions to front of () ( ) ( ) ( ) ( ) ( ) () ( ) ( ) ( ) ( ) ( ) ( ) ( ) dded paths in blue ( ) ( ) ( )( ) ( )

8 imple earch lgorithm Let be a list of partial paths, Let be the start node and Let be the oal node.. Initialize with partial path (). If is empty, fail. lse, pick a partial path N from. If head(n) =, return N (goal reached!). lse: a) Remove N from b) Find all children of head(n) and create all the one-step extensions of N to each child. c) dd all extended paths to d) o to step. () ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) epth-first Pick first element of ; dd path extensions to front of epth-first epth-first Pick first element of ; dd path extensions to front of Pick first element of ; dd path extensions to front of () ( ) ( ) ( ) ( ) ( ) () ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( )( ) ( )( )

9 More about Trees panning trees Prim s algorithm Kruskal s algorithm Issue: tarting at and moving top to bottom, will depth-first search ever reach? eneric search algorithm epth-first search example Handling cycles readth-first search example () ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )( ) epth-first ffort can be wasted in more mild cases visited multiple times Multiple paths to, & How much wasted effort can be incurred in the worst case? Idea: How o We void Repeat Visits? Keep track of nodes already visited. o not place visited nodes on. oes this maintain correctness? ny goal reachable from a node that was visited a second time would be reachable from that node the first time. oes it always improve efficiency? uarantees each node appears at most once at the head of a path in.

10 imple earch lgorithm Let be a list of partial paths, Let be the start node and Let be the oal node.. Initialize with partial path () as only entry; set = ( ). If is empty, fail. lse, pick some partial path N from. If head(n) =, return N (goal reached!) More about Trees panning trees Prim s algorithm Kruskal s algorithm. lse a) Remove N from b) Find all children of head(n) not in and create all the one-step extensions of N to each child. c) dd to all the extended paths; eneric search algorithm epth-first search example Handling cycles readth-first search example d) dd children of head(n) to e) o to step. readth First earch (F) Idea: xplore relatives at same level before their children xplore relatives left to right 7 readth-first Pick first element of ; dd path extensions to end of () 8 9 Where do we place the children on the queue? ssume we pick first element of dd path extensions to? of

11 readth-first Pick first element of ; dd path extensions to end of readth-first Pick first element of ; dd path extensions to end of () () ( ) ( ),, readth-first Pick first element of ; dd path extensions to end of readth-first Pick first element of ; dd path extensions to end of () ( ) ( ),, () ( ) ( ),, ( ) ( ) ( ),,,,

12 readth-first Pick first element of ; dd path extensions to end of readth-first Pick first element of ; dd path extensions to end of () ( ) ( ) ( ) ( ) ( ),,,,,, () ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )*,,,,,,,,,,, ( ) ( ) ( ),,,,,,,,,, epth First earch (F) epth-first: dd path extensions to front of Pick first element of readth First earch (F) readth-first: dd path extensions to back of Pick first element of ummary Most problem solving tasks may be formulated as state space search. Mathematical representations for search are graphs and search trees. epth-first and breadth-first search may be framed, among others, as instances of a generic search strategy. ycle detection is required to achieve efficiency and completeness. Test_ordered_binary.adb

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Problem Solving as State Space Search. Assignments

Problem Solving as State Space Search. Assignments Problem olving as tate pace earch rian. Williams 6.0- ept th, 00 lides adapted from: 6.0 Tomas Lozano Perez, Russell and Norvig IM rian Williams, Fall 0 ssignments Remember: Problem et #: Java warm up