1. the bike mapa look into a practical application of graph theory
2. demo! the bike map
3. how was it done? by creating a graph - a set of nodes and edges in computer science, graph theory provides algorithms for doingall sorts of useful things with graphs - routing, coloring, network ﬂow, etc. the bike map
4. starting simple 1st and Folsom (twilio) 1st and Howard (peet’s)http://maps.googleapis.com/maps/api/geocode/json?address=1st St and Folsom St,+San+Francisco,+CA&sensor=false{ "results" : [ { "address_components" : [ ... ], "formatted_address" : "Folsom Street & 1st Street, San Francisco, CA 94105, USA", "geometry" : { "location" : { "lat" : 37.78733430, "lng" : -122.39453480 }, ... }, "types" : [ "intersection" ] } ], "status" : "OK"} the bike map
5. starting simple 1st and Folsom (twilio) 1st and Howard (peet’s)http://maps.googleapis.com/maps/api/elevation/json?locations=37.78733430,-122.39453480&sensor=false{ "results" : [ { "elevation" : 12.22284126281738, "location" : { "lat" : 37.78733430, "lng" : -122.39453480 }, "resolution" : 0.5964969992637634 } ], "status" : "OK"} the bike map
6. voila! new google.maps.Polyline({ path: [<latlng for 1st & folsom>, <latlng for 1st & howard>], strokeColor: <color based on elevation difference divided by distance>, ... map: BikeMap.MAP_OBJECT, })...now just repeat for all intersections in the city... the bike map
7. that was too easy... where did those intersections come from? how did you know they existed? how did you know they were next to each other? the bike map
8. a city as a graph or, how to generate a lot of nodesgoal: get a list of all street intersections in a city - these are our nodes i’ll just write them all out by hand! i’m not above data entry! charlie the bike map
9. omg san francisco is huge
10. a city as a graph or, how to generate a lot of nodesgoal: get a list of all street intersections in a city - these are our nodes i’ll just write them all out by hand! i’m not above data entry! maybe I can use nextbus / proximobus data... charlie the bike map
11. proximobus proximobus.appspot.comhttp://proximobus.appspot.com/agencies/sf-muni/routes/F/stops.json{"items": [ {"latitude": 37.7875299, "display_name": "Market St & 3rd St", "id": "15640", "longitude":-122.40352}, {"latitude": 37.76979, "display_name": "Market St & Buchanan St", "id": "15659", "longitude":-122.4261499}, {"latitude": 37.7805699, "display_name": "Market St & 7th St North", "id": "15656","longitude": -122.41244}, {"latitude": 37.7911099, "display_name": "Market St & Battery St", "id": "15657", "longitude":-122.39907}, {"latitude": 37.7840799, "display_name": "Market St & 5th St North", "id": "15655","longitude": -122.40799}, {"latitude": 37.7774099, "display_name": "Market St & 9th St", "id": "15652", "longitude":-122.41634}, ...]} great! i’ll just map all the transit lines. charlie the bike map
12. bike on geary, market, and van ness, get run over
13. a city as a graph or, how to generate a lot of nodesgoal: get a list of all street intersections in a city - these are our nodes i’ll just write them all out by hand! i’m not above data entry! maybe I can use nextbus / proximobus data... write out all the streets by hand... and then compute intersections? charlie the bike map
14. a city as a graph or, how to generate a lot of nodes goal: get a list of all street intersections in a city - these are our nodesidea: write out all the streets by hand... and then compute intersections? • Bucket streets into non-intersecting groups • Take all possible combinations of streets that may intersect • Verify with Google that the intersection exists • Fairly good hit rate with non-intersecting groups • Data entry sucks, but this is exponentially less than before the bike map
15. a city as a graph or, how to generate a lot of nodes goal: get a list of all street intersections in a city - these are our nodes even better idea: use a reverse geocoding service to ﬁnd nearby streets for every point in a polygonal area, cluster streets based on latlng slopehttp://api.geonames.org/findNearbyStreetsJSON?lat=37.770443&lng=-122.448172&username=demo{ "streetSegment" : [ {"line":"-122.447123 37.770935,-122.448768 37.770731", "distance":"0.06", "name":"Page St"}, {"line":"-122.448768 37.770731,-122.448579 37.769798", "distance":"0.06", "name":"ClaytonSt"}, {"line":"-122.446934 37.770008,-122.448579 37.769798", "distance":"0.08", "name":"Haight St"}, ],} the bike map
16. a city as a graph or, how to generate a lot of nodes goal: get a list of all street intersections in a city - these are our nodeseven better idea: use a reverse geocoding service to ﬁnd nearby streets for every point in a polygonal area, cluster streets based on latlng slope • Create a polygon of lat/lng coordinates for your region • Query every few hundred feet within that polygon for nearby streets (see ray casting point in polygon algorithm) • Keep track of streets and which way they tend to point • Use clustering algorithm (or just guesstimate) to make non-intersecting buckets the bike map
17. a city as a graph or, how to generate a lot of nodes goal: get a list of all street intersections in a city - these are our nodeseven better idea: use a reverse geocoding service to ﬁnd nearby streets for every point in a polygonal area, cluster streets based on latlng slope the bike map
18. a city as a graph part 2: the edges! remember: a graph is a set of nodes and edges thankfully, generating the edges isn’t too complicated• We sort the intersections for each unique street• Each edge is just a straight line from one intersection to the next the bike map
19. a city as a graph part 2: the edges!there are, however, some interesting edge cases... ha ha • I can’t simply ride straight through Alamo Square Park from Grove / Scott to Grove / Steiner. • We have to manually deﬁne “breaks” along a street. the bike map
20. a city as a graph part 2: the edges!there are, however, some interesting edge cases... ha ha • To save time and space, we’ve assumed all intersection-to-intersection paths are STRAIGHT. • To render a curved street, we have to conﬁgure it so, and then our scraper script will pick up Google Directions for that path, which we save. the bike map
21. graphs at work optimal path searchnow that we have our graph, we can harness a general optimal path search algorithm called A* search this will let us suggest biking routes for users! the bike map
22. what A* needs to work• a cost function: how much does it cost to move from point A to B? (more on the bike map’s cost function later)• a heuristic function: approximate how much it will cost me to get to the end goal (example: distance from point A to B) the bike map’s heuristic function ground_distance2 + min(0, elevation diff)2 = heuristic2 heuristic = sqrt(104) = ~10.2 hill = 2m ground distance = 10m Note: downhill is considered no cost, hence, the min(0) the bike map
23. how A* worksStrategy:1. Start with a queuecontaining the start node.2. Dequeue the node withthe lowest sum [f-score] ofheuristic cost (estimatedfuture cost) and alreadysunk cost (cost to get tocurrent node).3. Get all the neighbors ofsaid node, compute their f-scores, and add them toyour queue.4. Repeat. End when youdequeue the goal. the bike map
24. A* properties• A* search is greedy - by using the heuristic, it will keep trying to go downwhat it thinks is the best path in the immediate future• A* search is like uniform cost search - by keeping track of sunk cost, it willmake sure not to go further down already expensive roads unless needed• A* search is optimal - it will always* return you the perfect path (proofsonline), assuming your cost function is correct, and...• A* search (for graphs) relies on consistent heuristics - a consistentheuristic is one that satisﬁes this condition: h(x) <= cost(x, y) + h(y), or, inEnglish, one where the cost of moving from x to y is never less than theheuristic difference from x to y. the bike map
25. tbm’s cost function recall tbm’s heuristic function ground_distance2 + min(0, elevation diff)2 = heuristic2 and the heuristic consistency criteria h(x) - h(y) <= cost(x, y) the bike map’s cost function(scalar penalty > 1) *sqrt(ground_distance2 + min(0, elevation diff)2) = cost the cost function is the heuristic multiplied by a scalar penalty > 1. the heuristic is never negative. thus, it must be consistent. the bike map
26. and what’s the scalar penalty?• Grade steeper than 2% = 10% penalty• Grade steeper than 6% = 100% penalty• Grade steeper than 10% = 200% penalty(we punish steep hills almost exponentially - wouldn’t you?)• Riding on a bike path = no penalty• Riding on a bike route = 5% penalty• Riding on a regular road = 15% penalty the bike map
Be the first to comment