1. Breadth-First Search Graph Algorithm Type #3

2. Depth-First Search

3. More Orderly Technique?

4. Breadth First Search (BFS) int main () { Node *sourceNode = getNodebyID (sourceID); bool found = bfsPath (sourceNode, destID);... } bool bfsPath (Node* sourceNode, int destID) {... } source dest start – 0 hops 1 hop away 2 hops away 3 hops away

5. Breadth First Search bool bfsPath (Node* sourceNode, int destID) { list wavefront; wavefront.push_back (sourceNode); while (wavefront not empty) { Node *currNode = wavefront.front (); wavefront.pop_front(); // Remove node from wavefront if (currNode->id == destID) return (true); for each (outEdge of currNode) { Node *toNode = outEdge.toNode; wavefront.push_back (toNode); } } return (false); // No path exits! } source dest 0 1 2 3 4 5 wavefront = {0} {1, 2} {2, 3} {3,4,3} {4,3,5} {5} wavefront = {} {2} {3} {4,3} {3,5} {5,5} {5}

6. BFS: How Do I Print Out the Path? source dest bool bfsPath (Node* sourceNode, int destID) { list wavefront; wavefront.push_back (sourceNode); while (wavefront not empty) { Node *currNode = wavefront.front (); wavefront.pop_front(); // Remove node from wavefront if (currNode->id == destID) return (true); for each (outEdge of currNode) { Node *toNode = outEdge.toNode; wavefront.push_back (toNode); } return (false); // No path exits! } 0 1 2 3 4 5 wavefront = {0} {1, 2} {2, 3} {3,4,3} {4,3,5} {5} wavefront = {} {2} {3} {4,3} {3,5} What do I know here? {5,5} {5}

7. BFS: How Do I Print Out the Path? Need more information! struct waveElem { Node *node; int edgeID; // ID of edge used to reach this node waveElem (Node *n, int id) {node = n; edgeID = id;} }; #define NO_EDGE -1 // Illegal edge ID  no edge Why did I use an int instead of an Edge?

8. BFS: How Do I Print Out the Path? source dest bool bfsPath (Node* sourceNode, int destID) { list wavefront; wavefront.push_back (waveElem (sourceNode, NO_EDGE)); while (wavefront not empty) { waveElem curr = wavefront.front (); wavefront.pop_front(); // Remove node from wavefront if (curr.node->id == destID) return (true); for each (outEdge of curr.node) { Node *toNode = outEdge.toNode; wavefront.push_back ( waveElem(toNode, outEdge.id); } return (false); // No path exits! } 0 1 2 3 4 5 {4/c,5/f} f c {5/f} Know I got here from edge f. Good enough? a b de

9. BFS: How Do I Print Out the Path? bool bfsPath (Node* sourceNode, int destID) { list wavefront; wavefront.push_back (waveElem (sourceNode, NO_EDGE)); while (wavefront not empty) { waveElem curr = wavefront.front (); wavefront.pop_front(); // Remove node from wavefront curr.node->reachingEdge = curr.edgeID; if (curr.node->id == destID) return (true); for each (outEdge of curr.node) { Node *toNode = outEdge.toNode; wavefront.push_back ( waveElem(toNode, outEdge.id); } } return (false); // No path exits! } source dest f c a b de NO_EDGE a b ec f d

10. BFS: How Do I Print Out the Path? int main () { Node *sourceNode = getNodebyID (sourceID); bool found = bfsPath (sourceNode, destID); if (found) list path = bfsTraceBack (destID); } list bfsTraceBack (int destID) { list path; Node *currNode = getNodebyID (destID); prevEdge = currNode->reachingEdge; while (prevEdge != NO_EDGE) { path.push_front (prevEdge); currNode = prevEdge.fromNode; prevEdge = currNode->reachingEdge; } return (path); } source dest f c a b de NO_EDGE a b c f d path = {}path = {f} path = {f,d}path = {f,d,b}

11. reachingEdge = areachingEdge = c Min. Path Issues source dest c a b d wavefront = 0 1 2 3 {0}{1,2}{2,3}{3,1} 4 {1,4}{4,3} b d e e What happened? Node 1 was re-expanded Overwrote the reachingEdge Messed up the path traceback

12. d source reachingEdge = a Solution? dest c a b d wavefront = 0 1 2 3 {0}{1,2}{2,3}{3,1} 4 {1,4} {4} b e e Mark nodes as visited when you reach them Never allow them to be visited again

13. d source reachingEdge = a Solution? dest c a b d wavefront = 0 1 2 3 {0}{1,2}{2,3}{3,1} 4 {1,4} {4} b e e Another test case With travel times t = 5 s t = 20 s

14. e d source reachingEdge = a Visited Flag: Problem dest c a b d 0 1 2 3 4 b e t = 5 s t = 20 s Shortest path Path we obtain

15. a, 20 sc, 10 s Solution #2 source dest c a b d wavefront = 0 1 2 3 {0}{1,2}{2,3}{3,1} 4 {1,4}{4,3} b, 5 s d, 26 s e, 29 s e 1.Store shortest travel time from source at each node 2.Only process node if shortest travel time to it has dropped t = 5 s t = 20 s t = 6 s t = 3 s Half fixed. Got right path, but wrong travel time at dest!

16. a, 20 s Solution #2 source dest c a b d wavefront = 0 1 2 3 {0}{1,2}{2,3}{3,1} b, 5 s d, 26 s Another Test Case t = 5 s t = 20 s t = 6 s This path is wrong !

17. a, 20 sc, 10 s Solution #2++ source dest c a b d wavefront = 0 1 2 3 {0/0s}{1/20s,2/5s}{2/5s,3/26s}{3/26s,1/10s} 4 {1/10s,4/29s}{4/29s,3/16s} b, 5 s d, 26 s e, 29 s e 1.Store shortest travel time from source at each node 2.Only update reachingEdge if shortest travel time drops 3.Don’t stop BFS on first reach of destination Continue BFS until all entries in wavefront have higher travel time than best path found so far t = 5 s t = 20 s t = 6 s t = 3 s d, 16 s e, 19 s {3/16s}{4/19s}