Maps Problem 1 Geography and CS Where am I? “Localization” Problem 2 How do I get there? Philip Chan “Navigation” Localization--Where am I? Cell phone Localization GPS—Global Positioning System Problem 1 Localization--Where am I? Localization--Where am I? Cell phone Cell phone Reference points: Reference points: cell towers GPS—Global Positioning System GPS—Global Positioning System Reference points: Reference points: satellites 1
Localization--Where am I? Localization--Where am I? Cell phone Cell phone Reference points: cell towers Reference points: cell towers Need 3 reference points GPS—Global Positioning System GPS—Global Positioning System Reference points: satellites Reference points: satellites How many reference points are needed to fix How many reference points are needed to fix the location? the location? Localization [2D] Localization--Where am I? (Problem Formulation) Cell phone Given (input) Reference points: cell towers Coordinates of the reference points Need 3 reference points GPS—Global Positioning System Distances from the reference points Reference points: satellites Need 4 reference points, Find (output) but 3 are ok if I know that I’m not floating in Coordinates of the location space above the satellites How many reference points are needed to fix the location? Localization [2D] Algorithm (Problem Formulation) Given (input) What is the mathematical relationship among the variables? Coordinates of the reference points ( x 1 , y 1 ), ( x 2 , y 2 ), ( x 3 , y 3 ) Distances from the reference points d 1 , d 2 , d 3 Find (output) Coordinates of the location ( x , y ) 2
Algorithm What is the mathematical relationship among Navigation the variables? Hint: given two points [two pairs of (x,y) coordinates], what is the distance between them? Problem 2 Navigation Navigation [Problem understanding] [Problem Formulation] Finding a route from the origin to the Given (input) destination Map Address of the origin “Static” directions Address of the destination Mapquest, Google maps Find (output) Turn-by-turn directions “Dynamic” on-board directions GPS navigation Simplification if the car deviates from the route, it finds a new route In the same city, all two-way streets, all left and right turns are allowed, no overpass/tunnels… Navigation Navigation [Problem Formulation Graph Problem] [Problem Formulation Graph Problem] Given (input) Given (input) Map ? Map edge=street, vertex=intersection, weight=length Address of the origin ? Address of the origin vertex Address of the destination ? Address of the destination vertex Find (output) Find (output) Turn-by-turn directions ? Turn-by-turn directions ? Simplification Simplification In the same city, all two-way streets, all left and In the same city, all two-way streets, all left and right turns are allowed, no overpass/tunnels… right turns are allowed, no overpass/tunnels… 3
Navigation Map/Street Data (input) [Problem Formulation Graph Problem] Given (input) Need more thoughts: Map edge=street, vertex=intersection, weight=length What do we need to know about the streets? Address of the origin vertex How could they be represented? Address of the destination vertex Find (output) Turn-by-turn directions shortest path Simplification In the same city, all two-way streets, all left and right turns are allowed, no overpass/tunnels … Map/Street Data (input) Map/Street Data (input) Tessellation or Vector? Tessellation or Vector? Tessellation: Tessellation: “image” of the streets Vector: Vector: “description” of the streets Map/Street data (input) Map/Street data (input) Vector Vector Name Name Two end points in x,y coordinates Two end points in x,y coordinates Range of house numbers Range of house numbers What if the street is curvy (not straight)? What if the street is curvy (not straight)? “Polyline” Additional intermediate x,y coordinates and house numbers Street name, ( x 1 , y 1 , h 1 ), ( x 2 , y 2 , h 3 ), … 4
Map/Street data (input) Map/Street data (input) What if a straight street has multiple What if a straight street has multiple intersections? intersections? Polyline (like curvy street) Additional x,y coordinates and house numbers Algorithm Overview Vertices in the graph 1. Preprocessing What should be a vertex? Convert the map, origin & destination into a Intersections graph How about intermediate points in the polyline 2. Main algorithm of a curvy street? Dijkstra’s shortest path algorithm 3. Postprocessing Convert shortest path to turn-by-turn directions Vertices in the graph Curvy streets vs intersections What should be a vertex? An intermediate point of a polyline could be: intersection a vertex Intersections part of a curvy street not a vertex How about intermediate points in the polyline of a curvy street? Vector representation: No, fewer vertices, but need to sum segment distances Street name, ( x 1 , y 1 , h 1 ), ( x 2 , y 2 , h 3 ), … (Yes, make program simpler) How could we tell the difference? Each vertex corresponds to a pair of x,y coordinates What is the weight of an edge? 5
Curvy Streets vs Intersections Curvy Streets vs Intersections Additional info in vector representation Additional info in vector representation intersection: Pointer to the cross street s [assuming intersection: Pointer to the cross street s [assuming only one cross street; a list otherwise] only one cross street; a list otherwise] curvy street: no pointer curvy street: no pointer Street name, ( x 1 , y 1 , h 1, s 1 ), ( x 2 , y 2 , h 2, s 2 ), … Street name, ( x 1 , y 1 , h 1, s 1 ), ( x 2 , y 2 , h 2, s 2 ), … No additional info in vector representation Intersection: Two streets with the same vertex ID A convenient vertex ID would be? Curvy Streets vs Intersections Converting Address to Vertex Additional info in vector representation For the origin and destination intersection: Pointer to the cross street s [assuming Given street name and house number only one cross street; a list otherwise] Create: curvy street: no pointer One temporary vertex (unless at an intersection) Street name, ( x 1 , y 1 , h 1, s 1 ), ( x 2 , y 2 , h 2, s 2 ), … Two temporary edges, why? No additional info in vector representation Intersection: Two streets with the same vertex ID A convenient vertex ID would be? (concatenation of) x, y coordinates Time-space tradeoffs? Converting Address to Vertex Converting Address to Vertex For the origin and destination Tradeoffs between: Given street name and house number Replace original edge with temporary vertex 1. Create: & edges [then reverse the process later] One temporary vertex (unless at an intersection) Add temporary vertex & edges [then reverse the 2. Two temporary edges, why? process later] What are the x,y coordinates of the new temporary vertex? What are the weights of the two new temporary edges? 6
Main Algorithm Main Algorithm—Greedy Algorithm If you do not know about Dijkstra’s algorithm Greedy algorithm Pick the closest vertex (shortest edge) 1. How would you solve the shortest path Go to the vertex 2. problem? Repeat until the destination vertex is reached 3. Does this always find the shortest path? If not, what could be a counter example? Main Algorithm-- Dijkstra’s shortest Main Algorithm-- Dijkstra’s shortest path algorithm path algorithm What are the key ideas? What are the key ideas? Similar to BFS: pick a leaf and expand its children Different in which leaf to pick, how? Main Algorithm-- Dijkstra’s shortest Main Algorithm-- Dijkstra’s shortest path algorithm path algorithm What are the key ideas? What are the key ideas? Similar to BFS: Similar to BFS: pick a leaf and expand its children pick a leaf and expand its children Different in which leaf to pick, how? Different in which leaf to pick, how? the shortest length so far the shortest length so far instead of the fewest # of levels in BFS instead of the fewest # of levels in BFS BFS is a special case of Dijkstra’s, why? 7
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