State Transition Graph of 8-Puzzle E W W E W W 3 2 E

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1 rrays. Lists. inary search trees. ash tables. tacks. Queues. tate Transition raph of 8-Puzzle oal: Understand data structures by solving the puzzle problem lementary tructures Trees earch tructures equence tructures Priority queues raphs ata tructures Motivating pplication

2 raph: a very general data structure V: set of nodes : set of edges (pairs of nodes) Path: a sequence of edges in which destination node of an edge in V = {,,,,,,,,, = {(,), (,),(,),... dge (,): is source of edge is destination of edge sequence is source node of next edge in sequence (,),(,),(,) (ii) (,),(,) xamples:(i) ource of a path: source of rst edge on path of path: destination of last edge on path estination Reachability: node n is said to be reachable from node m if there is a path from m to n. raphs can represent state transitions, road maps, mazes... n some graphs, edges may have additional information. - puzzle graph: edges annotated with /// n some graphs, certain nodes may be special - puzzle graph: initial and final (sorted) nodes are special raph in example is a RT graph Undirected graph: no arrows on edges analogy: -way street vs -way street There may be many paths from one node to another. (,) and (,),(,),(,),(,),(,) xample: imple path: a path in which every node is the source and of at most two edges on the path destination (not a simple path) (,),(,),(,) xample: ycle: a simple path whose source and destination nodes are the same. xample: (i) (,),(,) (ii) (,),(,),(,) 8 6 raph: set of odes and dges between nodes V = {,,,,,,,,, = {(,), (,),(,),... ome terminology edges: connection between two nodes Out-edges of a node n: set of edges whose source is node n (eg. of are f(,),(,)g) out-edges Out-degree of a node n: number of out-edges of node n n-edges of a node n: set of edges whose destination is node n (eg. in-edges of are f(,),(,)g) n-degree of a node n: number of in-edges of node n egree of a node n in an undirected graph: number of edges to n in that graph attached djancency: node n is said to be adjacent to node m if (m,n) is an nodes 4 5 edge in the graph ntuitively, we can get from m to n by traversing one edge. 5 7

3 you get turned on by this stu, become a major and take f 8/48 etc. 40, will have time only to study some simple problems like e with the goal of understanding modern data reachability, Path problems X Y Z These kinds of problems are studied in graph theory. Length of a path: number of edges in path Minimal path problems: ind the shortest path from node to node. ind the shortest path from every node to. or puzzle problem, this corresponds to path with fewest moves. structures. ometimes, edges have distances. Length of path = sum of lengths of distances on path. This is more appropriate for path problems in graphs representing maps. 0 Path problems X Y Z Many interesting problems can be phrased as path problems in graphs. Path problems Y X Z () s there a way to reach the sorted state, starting from any scrambled position of tiles? RP R (similar to search in array) Reachability problem: s there a path from a given node to the node representing sorted position? or puzzle problem, answer is no for some nodes! am Loyd: cannot reach sorted state from ycle: Path that starts and ends at same node. Travelling salesman s problem: ind the smallest length cycle that passes through all nodes. asy to come up with slow algorithms for this problem. o one knows if there is an efficient algorithm for this problem. (P/P-complete problems) 9

4 should not get stuck in cycles (correctness). should be exhaustive: if we terminate without reaching sorted. write a program to determine if sorted state is reachable oal: scrambled state for puzzle problem from tart in the scrambled state. enerate states adjacent to scrambled state.... top if you either generate sorted state or you have generated raph search is similar to linear search except that we are Think: for something in a graph rather than in an array. searching sorted state must be unreachable from scrambled state state, (correctness) should not repeatedly examine states adjacent to a. state(eciency) before adding w to too set, check if it is already Modication: in the done set. there it has already been explored. if isadvantage: if node has not been explored, we will look it up too : set of nodes whose adjancies might need to be examined. done : set of nodes whose adjacencies have been examined. Requirements: dvantage: we do not put it into too set and get it out again in done set twice. ode: see next slide. 6 4 Key dea: Keep two sets of nodes dea: Pseudocode for raph earch algorithm: initialize too set with scrambled configuration while (too set is not empty) enerate states adjacent to those states. {Remove a node v from too set if (v is in done set) continue //we reach here if we have never explored v before for each node w adjacent to v do //there is edge (v -> w) all states reachable from scrambled state. {f w is the goal node, declare victory Otherwise, add w to too set add v to done set 5

5 (v ; > v) is an edge, we would add v to too set when f v. exploring lgorithm for olving Puzzle : some possible initial steps Modication: handling self-loops more eciently This is not necessary, so let us x code. initialize too set with scrambled configuration while (too set is not empty) {Remove a node v from too set if (v is in done set) continue //we reach here if we have never explored v before add v to done set //this optimizes self-loops for each node w adjacent to v do //there is edge (v -> w) f (w is not in done set) { f (w is the goal node) declare victory : configuration in too set : configuration in one set add w to too set ote: we are not keeping track of moves that brought us to a given configuration 0 8 Let us try to execute a few steps of the algorithm for this problem. initialize too set with scrambled configuration (too set is not empty) while a node v from too set {Remove (v is in done set) continue if reach here if we have never explored v before //we each node w adjacent to v do //there is edge (v -> w) for (w is not in done set) { f f (w w to too set add v to done set add is the goal node) declare victory orted tate crambled tate 7 9

6 Order in which we explore nodes (order in which they are from too set) is very important, and can make a big removed can we write code so that it works for any sequence ow nswer: use subtyping structure? do we write code so that it works for any search ow nswer: use subtyping structure? raph search algorithm works for any graph, not just puzzle transition graph (all we need to some way to determine state can we write code so that it works for any graph? Use ow to return all adjacent vertices of a node. terators QU TRUTUR: n what order should we get nodes () the too set? hat data structure can we design to give us from nodes in that order? ow can we accommodate the fact that the too set grows and shrinks? the R TRUTUR: ow do we organize the one set so () we can search it eciently? that riting generic code: ode for imulating Puzzle class searcher{ dierence in how quickly we nd solution. public static void main(tring[] args) { Puzzle p0 = new rraypuzzle() p0.move() Most time-consuming part: searching if node is in one set. p0.move() earchtructure s = new T() eqtructure q = new QsList() graphearch(p0,q,s) what nodes are adjacent to a given node). 4 To make this a program, we need to answer the following questions: Two key interfaces: eqtructure: all sequence structures implement this interface earchtructure: all search structures implement this interface nswer: stacks, queues, priority queues or search structure, fast search is important! or sequence structure, fast lookup is not important! nswer: binary search trees, hash tables

7 code we have written will work for any search structure and The structure that implement the interfaces dened before. sequence puzzle state transition graph is hardwired into the code. e The use it to perform a graph search in a general graph. e will cannot data structure grows and shrinks. ow do we implement a too sequence structure? good if (done.search(p)) continue //if not, lets explore this node done.insert(p) //determine adjacent nodes and process them tring Moves = "" for (int i = 0 i < Moves.length() i++) { ubtyping is wonderful! Puzzle np = p.duplicate() char dir = Moves.chart(i) //try to make the move boolean OK = np.move(dir) x this later. if (OK) { //move succeeded, so we have a legitimate node if (! done.search(np)) { if (np.isorted()) { ystem.out.println("urrah") ow do we implement a good search structure? return too.put(np) 8 6 //specialized to puzzle problem //will work for any search structure and sequence structure public static void graphearch(puzzle p0, eqtructure too, earchtructure done){ if (p0.isorted()) { ystem.out.println("lready sorted") return //no more too nodes //initialize work-list ystem.out.println("ould not reach sorted state") too.put(p0) //while there are too nodes while (! too.ismpty()) { //get a too node Puzzle p = (Puzzle)too.get() //have we explored this node already? 5 7