CSS 343 Data Structures, Algorithms, and Discrete Math II. Graphs II. Yusuf Pisan

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1 CSS 343 Data Structures, Algorithms, and Discrete Math II Graphs II Yusuf Pisan

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4 Shortest Path: Dijkstra's Algorithm Shortest path from given vertex to all other vertices Initial weight is first row of adjacency matrix Maintain a vertexset, initially has vertex-0 in it for step = 2 to n 1. Find the vertex with the lowest weight that is not in the vertexset (vertex-4 in this case) 2. Add vertex-4 to vertexset 3. For vertices that are adjacent to vertex-4 AND not in vertexset, check whether it is shorter to go via vertex-4 and update weight (can get to vertex-2, weight[2] goes from INF to 5) 4

5 Shortest Path: Dijkstra's Algorithm 5

6 Shortest Path: Dijkstra's Algorithm To keep track of path, keep an array, previous[u] = v, every time we update the weight At start, previous[1] previous[2] previous[3] previous[4] = = = = 0 UNDEFINED 0 0 After adding vertex 4 previous[2] = 4 6

7 Shortest Path: Dijkstra's Algorithm Implementation details If using Adjacency Matrix You can use INT_MAX for weight when not connected You have to check for INT_MAX explicitly. Adding to it will lead to overflow and negative numbers If using Adjacency List Unconnected vertices are not represented 7

8 Shortest Path Calculate shortest path from 'a' to all vertices using Dijkstra 8

9 Step v vertexset a b c d e f g h i z 1 - a 0 4 INF INF 15 INF 6 INF INF INF 2 b a,b 3 g a,b,g 4 c a,b,g,c 5 e a,b,g,c,e 6 f a,b,g,c,e,f h a,b,g,c,e,f,h 8 d a,b,g,c,e,f,h,d 9 i a,b,g,c,e,f,h,d,i z a,b,g,c,e,f,h,d,i,z indicates update considered, but current path is shorter 9

10 Shortest Path Calculate shortest path from 'a' to 'f' using Dijkstra Since we do not need all the distances, how do we change the algorithm? 10

11 Shortest Path: Dijkstra's Algorithm Assumes distance of node back to itself is 0 Need to change priorities! 11

12 Wolf, Goat & Cabbage -- (0, 0, 0) 12

13 Wolf, Goat & Cabbage -- (0, 0, 0) 13

14 Wolf, Goat & Cabbage -- (0, 0, 0) Solution: Finding a path from (0,0,0) to (1,1,1) 14

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19 Best-First Search Generalizing breadth-first search to find a specific vertex Instead of queue us priorityqueue Cost function f(n) is the priority Add initial vertex to PQ Mark v as visited while PQ is not empty w = top of queue if w is targetvertex, return for each unvisited vertex u adjacent to w Mark u as visited push u to PQ with cost f(u) If f(n) is depth(n) or all edges are of length 1, then we have breadth-first search if f(n) is sum of edge costs from start to n, we have Dijkstra 19

20 Uniform-Cost Search Best-First Search with f(n) as sum of edge costs from start to n, we have Dijkstra -- also known as Uniform Cost Search Start from S Add (A 1), (G 12) Get (A 1) from PQ Add (C 2), (B 4) Get (C 2) from PQ // yes add G again, shorter Add (D 3), (G 4) Get (D 3) from PQ Add (G 6) Get (B 4) from PQ Add (D 7) to PQ Get (G 4) from PQ Found goal G, return path to G Explores options in every direction. No information about goal location 20

heuristic should help us make better choices Instead

21 Heuristic An estimate of how close a state is to a goal Designed for a particular search problem A good heuristic should help us make better choices Instead of exploring all states, try to go closer to goal 21

22 Greedy Search Best-first with f(n) = heuristic estimate of distance to goal Expands the node that appears to be closest to goal 22

23 Was this the best route? 451km Arad - Sibiu 140 Sibiu - Fagaras 99 Fagaras - Bucharest

24 A* Search Best first search with f(n) = g(n) + h(n) g(n) = sum of costs from start to n h(n) = estimate of lowest cost path n goal Can view as cross-breed: g(n) ~ uniform cost search h(n) ~ greedy search 24

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29 A* Pros Produces optimal cost solution! Does so quite quickly (focused) Cons Maintains priority queue Which can get exponentially big 29

30 Review Graph terminology Travelling Salesman Problem Implementation Options: Matrix vs List Basic Search Depth-First Breadth-First Topological Sort Minimum Spanning Tree - Prim's Algorithm Shortest Path - Dijkstra's Algorithm Puzzles that can be solved using graphs Best-first search - BFS or Dijkstra depending on f(n) Uniform Cost Search - Best-first search variation becomes Dijkstra Informed Search Greedy Search A* Search 30

Route planning / Search Movement Group behavior Decision making

Game AI Where is the AI Route planning / Search Movement Group behavior Decision making General Search Algorithm Design Keep a pair of set of states: One, the set of states to explore, called the open