Pathfinding. Prof. Dr. Taoufik Nouri

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1 Pathfinding Prof. Dr. Taoufik Nouri

2 Pathfinding Depth First Search Breadth First Search Dijkstra A*

3 Pathfinding Pathfinding

4 AI-Game Engine Navigation 1:Soccer

5 Navigation 2: Raven

6 Pathfinding 1 Pathfinding is the path between two nodes of a graph. A graph is a set of points called nodes connected to each other through connections. There is an infinite number of paths from one node to another. Usually pathfinding is the shortest way between two nodes

7 Pathfinding 2 Graphs can have a cost associated to the connections like distances etc. This kind of graphs is called weighted graph. There is also, directional graphs: paths between nodes is considered one way. Graphs can be more complicated: Weighted directional graphs, you can not go back. A graph can be of any dimensions and any shapes

8 Pathfinding Very important for real world problems: 1. The airport system is a graph. What is the best flight from one city to another? 2. Traffic flow can be modeled with a graph. What are the shortest routes? 3. Traveling Salesman Problem: What is the best order to visit a list of cities in a graph?

9 Graph Representation v5 v4 v3 v6 We can use an adjacency matrix representation. v2 v1 v7 For each edge (u,v) we set A[u][v] to true; else it is false. If there are weights associated with the edges, insert those instead. v1 v2 v3 v4 v5 v6 v7 v v v3 v4 v5 v6 v7

10 Adjacency List v5 v4 v3 v6 We can use an adjacency list representation. v2 v1 v7 For each vertex we keep a list of adjacent vertices. If there are weights associated with the edges, that information must be stored as well. v1 v2 v4 v3 v2 v4 v5 v3 v6 v4 v6 v7 v3 v5 v4 v7 v6 v7 v6

11 Graph GraphNodes are indexed nodes see class GraphNode GraphNodeTypes.h class GraphEdge {protected: //An edge connects two nodes. Valid node indices are always positive. int m_ifrom; int m_ito; //the cost of traversing the edge double m_dcost; public: //ctors GraphEdge(int from, int to, double cost):m_dcost(cost), m_ifrom(from), m_ito(to) } see GraphEdgeTypes.h The graph can be a matrix or a lists

Pathfinding 3 This is a weighted graph Cities are nodes connected through roadsdistances-relief-terrain kante(saintgall, zurich, 80, Terrain etc). kante(zurich, basel, 85). kante(basel, bern, 100).

12 Pathfinding 3 This is a weighted graph Cities are nodes connected through roadsdistances-relief-terrain kante(saintgall, zurich, 80, Terrain etc). kante(zurich, basel, 85). kante(basel, bern, 100). kante(lausanne,fribourg,31). kante(lausanne, geneve, 100). kante(zurich, bern, 120). kante(bern, fribourg, 30). kante(bern, neuchatel, 48). kante(neuchatel, lausanne, 78). kante(zurich, solothurn, 99). kante(solothurn, biel, 27). kante(biel, neuchatel, 29).

13 Pathfinding 4 Usually, we define a graph as regularly spaced grid where we can travel to: N, NW, NE, S, SW, SE etc. Green is start position, Red is Target Note: There is no obstacle in our grid

14 Pathfinding 5 Here is a graph with obstacle Graph is a powerfull representation tool. But this power can be realized when they are operated upon by algorithms designed to explore them, either to find a specific node or to find a path berween nodes. this is what we will do next

15 General search algorithm Start with "frontier" = { (v,v) } Until frontier is empty remove an edge (n,m) from the frontier set mark n as parent of m mark m as visited if m = w, return otherwise for each edge <i,j> from m add (i, j) to the frontier» if j not previously visited

16 Questions Answer the following questions: What is pathfinding? Why is pathfinding useful or necessary what is a graph What is a weighted graph What are the traditional types of pathfinding methods

17 Pathfinding Depth First Search Breadth First Search Dijkstra A*

18 Depth First Search DFS Stack Parent >? 2->? 3->? 4->? 5->? 6->? Firt In Last Out

19 DFS Node 5 is marked Visited Stack Parent >? 2->? 3->? 4->? 5->5 6->? The edges leading from node 5 are placed on the Stack

20 DFS Node 2 is marked Visited Stack Parent >? 2->5 3->? 4->? 5->5 6->? Node 2 is pushed on the Stack, Node 5 is parent of 2

21 DFS Node 1 is marked Visited Stack Parent >2 2->5 3->? 4->? 5->5 6->? Node 1 is pushed on the stack, 2 is parent of 1

22 DFS Node 3 is marked Visited, OHH Node 3 is the Target, the algorithm exit Stack Parent >2 2->5 3->1 4->? 5->5 6->? Node 1 is parent of 3, Edge 1-3 is poped from the stack The path from S to T is stored in a vector m-route in the table parent. A method getpathtotarget will extract this information and return a vector What about the path 5-4-3

23 DFS Depth limitation is the major problem of DFS The path is not allways the shortest It is easy to implement using recursion Some situation DFS will make spagethi if not restricted in depth see GraphAlgorithms.h implementation

24 DFS in Action

25 Breadth First Search BFS The BFS fans out from the source node and examines each of the nodes ist edges lead to before fanning out from those nodes and examining all the edges they connect to and so on. look at that!!!

26 BFS Queue Parent 1->? 2->? 3->? 4->? 5->? 6->? 5-5 First In First Out

27 BFS Node 6, 4 and 2 are marked Visited Queue Parent 1->? 2->? 3->? 4->? 5->5 6->? The edges faning from node 5 are placed on the Queue

28 BFS Node 6 is marked Visited Queue Parent 1->? 2->? 3->? 4->5 5->5 6-> Node 6 pointed to node 4(visited), Node 6 is not placed on the Queue

29 BFS Node 4 is marked Visited Queue Parent 1->? 2->5 3->? 4->5 5->5 6-> The edges leading from node 4 to 3 are placed on the Queue

30 Node 3 is marked Visited BFS Queue Parent 1->? 2->5 3->4 4->5 5->5 6-> Edge 4-3 is next. Node 4 is noted to be node 3's parent. Node 3 is Target. BFS exits. m_routes Vector work back through the parents from T to S and we get

31 BFS in Action BFS returns allways the shortest path if exist. Simple to implement Very Slow on large graphe. Doesn't consider path cost! see GraphAlgorithms.h implementation

32 DFS-BFS Summary Depth first search begins by diving down as quickly as possible to the leaf nodes of the tree (or graph). Traversal can be done by: visiting the node first, then its children (pre-order traversal): a b d h e i j c f k g visiting the children first, then the node (post-order traversal): h d i j e b k f g c a visiting some of the children, then the node, then the other children (in-order traversal) h d b i e j a f k c g Breadth-first search traverses the tree or graph level-by-level, visiting the nodes of the tree above in the order a b c d e f g h i j k.

33 Exercices How are the traditional types of pathfinding methods implemented Implement in a pseudo-code the following pathfinding techniques:bfs, DFS

34 Shortest Path Trees SPT Dijkstra's Algorithm This is a weighted graph. Weight is distance, cost etc. How to find the shortest path from a source node to a Target node?

35 Dijkstra's Algorithm T T SPT 5 S Starting from node 5, source.

36 Dijkstra's Algorithm T T SPT 5 2 S The algorithm take edge 2 and 6. Added the shortest 2 (1.9) to the SPT

37 Dijkstra's Algorithm T T SPT 5 2 S 6 The algorithm examines edge 2-3, the cost to node 3 is 5( ). The cost to from node 5 to 6 is 3.0. Node 6 is added to SPT

38 Dijkstra's Algorithm T T SPT 5 2 S 6 The process is repeated once more. Cost to node 4 is less( contra to node 3. Node 4 is added to SPT

39 Dijkstra's Algorithm T T SPT 5 2 S The shortest path is: with the cost The process is repeated once more. Cost to node 3 is more( contra to node 3. Node 3 is added to SPT through 5-2-3

40 Djikstra's Algorithm Use a priority queue a data structure in which the item with the smallest "value" is always first items can be added in any order Use the "value" of an edge as the total cost of the path through that edge always expand the node with the least cost so far If an edge leads to a previously expanded node compare costs if greater, ignore edge if lesser, replace path and estimate at node with new value "Greedy" algorithm

41 Dijskra's Algoritm in Action In this case the cost is the distance, from source to target see GraphAlgorithms.h implementation

42 Best First Search A* Dijkstra's algorithm searches by minimizing the cost of the path in a directed weighet graph. In some situations, we have partial knowledge of the structure of the search space that can be applied to guide search. It can be improved by estimating the cost to the target. This estimate is usually referred to as a heuristic and the algorithm is called Best-first search (A*). We can inspect all the currently-available transitions, and rank them on the basis of our partial knowledge. Here high rank means that the transition looks promising in relation to the goal.

43 Best First Search A* If good Heuristics are used,a* is guaranteed to give the optimal paths. Heuristics: any cost estimate to the goal, ex. Euclidean distance, Block distance, it can be coupled with penalties or bonuses for traveling on a specific type of terrain(jungle, desert, forest,plain ). Performance of A* is based on the value of Heuristics Poor heuristics provides bad performance with A*

44 Best First Search A* Heuristic estimates Max(dx, dy) Manhatten or Blocks (dx+dy) Euclidean sqrt(dx*dx + dy*dy)

45 Best First Search A* You are at A going to Target T. f(t) = g(a) + h(a). f(t) total cost, g(a) cost until A, h(a) estimate cost from A to T. We continue our search with the lowest f(t).

46 A* in Action see GraphAlgorithms.h implementation

47 Comparison No walls

48 Comparison with walls

49 Questions about A* What is A*? What are the advantages of A*? What are the disadvantages of A*? How can heuristics be used to help A*?

50 Exercices Implement in a pseudo-code the following pathfinding techniques: DJ,A*

51 Summary We introduced the concepts of : node, edge and graph. Pathfinding: BFS, BFS, DJ and best-first search A* Their Advantages/Desadvantages Demo and comparison For source code/exe files s. [1]

52 Exercices 1. Solve with hand some path finder using these four algorithms. 2. Implement the DFS Method 3. Implement the A* search Method 4. Create a weighted Graph and walk through it step by step with Dijkstra's 5. Try Dijkstra's on graph with negative weights 6. Experiment with pathfinder chapt. 5 [1]

53 Exercices 7. Implement or read the implementations of DFS, BFS, Dijkstra and A* 8. Read the solution of exercice 7 in chap. 5 [1]

54 Other Implementations In the next slide, you have the best search first using prolog. It is compact and efficient. Demo-- Vortrag

55 Exercice: Best search using prolog pfad_mit_breitensuche(anfp, EndP, Route) :- weg([[anfp]], EndP, R), reverse(r, Route ). /* to use this programme, write: go(saintgall,geneve,route). moeglicher_knoten( AnfP, Kantenzug, ZP) :- The answer look like this: Route = [saintgall,zurich,bern,fribourg,lausanne,geneve] */ kante(saintgall, zurich, 80). kante(zurich, basel, 85). kante(basel, bern, 100). kante(lausanne,fribourg,31). kante(lausanne, geneve, 100). kante(zurich, bern, 120). kante(bern, fribourg, 30). kante(bern, neuchatel, 48). kante(neuchatel, lausanne, 178). kante(zurich, solothurn, 99). kante(solothurn, biel, 27). kante(biel, neuchatel, 29). % Best first search kurz_pfad(routes, Dest, Route) :- shortest(routes, Shortest, RestRoutes), proceed(shortest, Dest, RestRoutes, Route). % proceed finds all the legal extensions to the path and adds them to the list pfad_ohne_zyklus(anfp, EndP):- weg(anfp, EndP, [AnfP]). (kante(anfp, ZP, Dist); kante(zp, AnfP, Dist)), not(element(zp, Kantenzug)), weg(zp, EndP, [ZP Kantenzug]). element(e, [E Rest]). element(e, [Kopf Rest]):- element(e, Rest). pfad_mit_knotenliste(anfp, EndP, Pfad):- weg(anfp, EndP, [], P), reverse(p, Pfad). weg(x, X, Kantenzug, [X Kantenzug]). weg(anfp, EndP, T, Kantenzug) :- collect_found(s, L) :-getnext(x),!,collect_found([x S], L). collect_found(l, L). getnext(x) :- retract(found(x)),!, X \== mark. (kante(anfp, ZP,Dist); kante(zp, AnfP,Dist)), not(element(zp, Kantenzug)). append([], Liste, Liste). append([kopf Rest], Liste, [Kopf Ergebnis]):- append( Rest, Liste, Ergebnis). pfad(x, X). pfad(anfp, EndP):- (kante(anfp, ZP,Dist); kante(zp, AnfP,Dist) ), pfad(zp, EndP). weg(x, X, Kantenzug). weg(anfp, EndP, Kantenzug):- append([], Liste, Liste). append([kopf Rest], Liste, [Kopf Ergebnis]) :-append( Rest, Liste, Ergebnis). moeglicher_knoten(anfp, T, ZP), weg(zp, EndP, [AnfP T], Kantenzug). moeglicher_knoten(anfp, Kantenzug, ZP) :- (kante(anfp, ZP,Dist); kante(zp, AnfP,Dist)), not(element(zp, Kantenzug)). reverse([], []). reverse([k Rest], Neueliste):-reverse( Rest, TL),append(TL, [K], Neueliste). proceed(r(dist, Route), Dest,_, Route):- Route =[Dest _]. % r(m, P) M is the total length of the path and P is the list of places visited proceed(r(dist, [Last Trail]), Dest, Routes, Route):findall(r(D1, [Z, Last Trail]), legalnode(last, Trail, Z, Dist, D1), List), % write(route),write(dist),nl, % write(routes),nl, % write([list]),nl, % write([d1,z,dist,d1,last,trail]),nl, append(list, Routes, NewRoutes), kurz_pfad(newroutes, Dest, Route). % shortest returns the shortest path on the list shortest([route Routes], Shortest, [Route Rest]):shortest(Routes, Shortest, Rest), shorter(shortest, Route),!. shortest([route Rest], Route, Rest). shorter(r(m1,_), r(m2,_)):- M1 < M2. % legal node adds the distance to the next town legalnode(x, Trail, Y, Dist, NewDist) :- (kante(x, Y, Z); kante(y, X, Z)), legal(y, Trail), NewDist is Dist + Z. legal(x, []). % legal ist gleich not member or not element legal(x, [H T]) :- X \== H, legal(x, T). % To start write: go(zurich,geneve,j). go(start, Dest, Route) :- kurz_pfad([r(0, [Start])], Dest, R), reverse(r, Route),!.

56 Pseudo-code of DFS

57 Pseudo-code of BFS

58 Pseudo-code of DJ

59 Pseudo-code of A*

60 Pseudo-code of Trace

9/17/2015 7:56 AM. CSCE 625 Programing Assignment #1 due: Tues, Sep 22 (by start of class) Objective

9/17/2015 7:56 AM. CSCE 625 Programing Assignment #1 due: Tues, Sep 22 (by start of class) Objective CSCE 625 Programing Assignment #1 due: Tues, Sep 22 (by start of class) Objective The goal of this assignment is to implement and compare the performance of Breadth-first search (BFS), Depth-First Search