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PRACTICAL IMPROVEMENT: A-STAR, KD-TREE, FIBONACCI HEAP CS16: Introduction to Data Structures & Algorithms Thursday, April 10, 2014 1

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Outline A* Shortest-path Search KD-trees for Nearest Neighbor Fibonacci Heap 2 Thursday, April 10, 2014

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Road Trip Thursday, April 10, 2014 3 ATL SF LA CLE DC PHL NYC PVD PHX STL CHI 10 40 15 10 20 15 5 10 20 15 10 Start End 20

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Road Trip Thursday, April 10, 2014 4 ATL SF LA CLE DC PHL NYC PVD PHX STL CHI 10 40 15 10 20 15 5 10 20 15 10 Start End 20 0 10 20 15 20 30 35 45

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Optimistic Estimates Thursday, April 10, 2014 5 ATL SF LA CLE DC PHL NYC PVD PHX STL CHI 10 40 15 10 20 15 5 10 20 15 10 Start End 20 0 inf 30 10 32 35 42 18 20 0 36 8 5 Optimistic estimate to End Shortest distance to Start PQ: (CHI,30) (…,inf) inf

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Optimistic Estimates Thursday, April 10, 2014 6 ATL SF LA CLE DC PHL NYC PVD PHX STL CHI 10 40 15 10 20 15 5 10 20 15 10 Start End 20 0 10 20 inf 15 inf 40 30 10 32 35 42 18 20 0 36 8 5 Optimistic estimate to End Shortest distance to Start PQ: (STL,30) (SF,40) (CLE,46) (NYC,50) (…,inf) inf

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Optimistic Estimates Thursday, April 10, 2014 7 ATL SF LA CLE DC PHL NYC PVD PHX STL CHI 10 40 15 10 20 15 5 10 20 15 10 Start End 20 0 10 20 40 15 inf 30 35 30 10 32 35 42 18 20 0 36 8 5 Optimistic estimate to End Shortest distance to Start PQ: (SF,35) (CLE,46) (ATL,48) (NYC,50) (DC,55) (PHL,72) (…,inf) inf

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Optimistic? It’s so important for estimates to be optimistic, ie never say it’s farther than it is, so that the best path will always be taken first Important to be admissible: a heuristic that never overestimates Inadmissible heuristics don’t always return the best path Thursday, April 10, 2014 8

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A* (A Star) Algorithm The algorithm is as follows: Decorate all nodes with: Distance ∞, except start node, which has distance 0, Optimistic estimate of node-to-end-node distance. Add all nodes to a priority queue, with the sum of both decorations as the priority. While the priority queue isn’t empty and you haven’t expanded the destination: Remove the minimum priority node Update the decorations and priority to the removed node’s neighbors if the distances have decreased Thursday, April 10, 2014 9

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A* Algorithm Pseuodocode Thursday, April 10, 2014 10 function astar(G, s, d): //Input: Graph G with vertices V, start vertex s, and destination vertex d //Output: u.dist set to shortest distance from s to u and path saved for v in V:// Initialize vertices’ decorations v.dist = infinity // dist-from-start v.est = heuristic(v, d) // dist-from-destination estimates v.prev = null // set previous pointers to null s.dist= 0 PQ = PriorityQueue(V) // add all nodes with (v.dist + v.est) as priority while PQ not empty and PQ.getMin()!=d://haven’t reached destination curr = PQ.removeMin() for all edges (curr, v)://cost() returns edge weight if (curr.dist + cost(curr, v)) < v.dist: // if this path is shorter v.dist = curr.dist+cost(curr,v) //Update to curr’s path+weight v.prev = curr// Maintain pointers for path PQ.replaceKey(v, v.dist+v.est)

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Other Applications and Heuristics What other problems do we use search to solve? Games like Chess, Go, Tic Tac Toe Dijkstras functions as BFS since the cost of all transitions are equal Heuristics can be much funkier than distance: chess could use number of your pieces on the board, sum of the values of pieces, moves until checkmate Runtime is dependent on difficulty of calculating heuristic Thursday, April 10, 2014 11

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1D-Tree Example – Nearest Neighbor Thursday, April 10, 2014 12 2 8 10 16 22 30 40 42

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1D-Tree Example – Nearest Neighbor Thursday, April 10, 2014 13 22 10 8 2 40 42 3016

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2D-Tree Example Now lets sort something a little more complicated, points in the Cartesian plane. A(10, 12), B(11, 9), C(3,4), D(5,3), E(7,8), F(1, 5) Thursday, April 10, 2014 14

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2D-Tree Example Sort list in the 1st dimension(x) F, C, D, E, A, B Center element becomes new node. Make two new lists, everything in the left list is less than or equal to the center, everything in the right list is greater than the center. Left list: F, C, DRight list: A, B Thursday, April 10, 2014 15 E

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2D-Tree Example Thursday, April 10, 2014 16 We now sort each list in the next dimension(y). F, C, D -> D, C, F A, B -> B, A Again we take the center element to be the new node Make our separate lists again. DFB E CA

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2D-Tree Example Sort our lists; in the first dimension this time DFB Make our new nodes Thursday, April 10, 2014 17 E CA DFB

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2D-Tree Example: Nearest Neighbor Thursday, April 10, 2014 18 X E Y C Y A X D X F X B X Y A(10, 12) B(11, 9) C(3,4) D(5,3) E(7, 8) F(1,5) G(1, 3.5)

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2D-Tree Example: Nearest Neighbor Thursday, April 10, 2014 19 X E Y C Y A X D X F X B X Y A(10, 12) B(11, 9) C(3,4) D(5,3) E(7, 8) F(1,5) G(1, 3.5)

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2D-Tree Example: Nearest Neighbor Thursday, April 10, 2014 20 X E Y C Y A X D X F X B X Y A(10, 12) B(11, 9) C(3,4) D(5,3) E(7, 8) F(1,5) G(1, 3.5)

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NNS: Pseudocode Move down the tree recursively starting with the root. Go left or right depending on whether the point is less than or greater than the current node in the split dimension. Once you reach a leaf node, save it as the "current best" Unwind the recursion of the tree, performing the following steps at each node: o If the current node is closer than the current best, then it becomes the current best. o Check whether there could be any points on the other side of the splitting plane that are closer to the search point than the current best. This is done by intersecting the splitting hyperplane with a hypersphere around the search point that has a radius equal to the current nearest distance. Since the hyperplanes are all axis-aligned this is implemented as a comparison to see whether the difference between the splitting coordinate of the search point and current node is less than the distance from the search point to the current best. If the hypersphere crosses the plane, there could be nearer points on the other side of the plane, so we must move down the other branch of the tree from the current node looking for closer points, following the same recursive process. If the hypersphere doesn't intersect the splitting plane, then we continue walking up the tree, and the entire branch on the other side of that node is eliminated. Thursday, April 10, 2014 21 Wikipedia

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3D-Tree example and Applications Thursday, April 10, 2014 22 Wikipedia

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Fibonacci Heap OperationBinary HeapFibonacci Heap insertlog(n)1 (amortized) decreaseKeylog(n)1 (amortized) removeMinlog(n) Thursday, April 10, 2014 23

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Fibonacci Heap DS: list of trees Thursday, April 10, 2014 24

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Fibonacci Heap DS: list of trees Thursday, April 10, 2014 25 insert(k, p): Add a new tree with key and priority If (p < current min): point current min at new tree decreasePri(n, p): decrease n.p if (n.p < n.parent.p): cut n from parent and make a new tree go up tree cutting marked ancestors until root or unmarked ancestor is reached mark first unmarked ancestor // note that roots are never marked

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Fibonacci Heap DS: list of trees Thursday, April 10, 2014 26 removeMin(): use pointer to get minimum make each child a new root combineTrees() point current min at new tree

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Outline A* Shortest-path Search KD-trees for Nearest Neighbor Fibonacci Heap 27 Thursday, April 10, 2014

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