1. COMP 261 Lecture 17
Test Revision

2. Outline Data Structures for Graphs Tries Dijkstra’s Algorithm
A* Search
Articulation Points
Minimum Spanning Tree
3D Graphics

3. Data Structures for Graphs
Traditional Adjacency Matrix
Traditional Adjacency List
Object-based structure (Adjacency list-like)

4. Adjacency Matrix Complexity of actions Cannot handle multi-graph
Find all nodes
O(numNodes)
Find all edges
O(numNodes2)
Outgoing edges of a node
Incoming edges of a node
Outgoing neighbours of a node
Incoming neighbours of a node
Edge from vertex v to u?
O(1) B 7 5 6 C 1 D 4 E 16 F 19 G 11 H 10 A 3 9 17 8 15 2 13 18 14 12 I J 20
Cannot handle multi-graph

5. Adjacency List Complexity of actions Cannot handle multi-graph either
Find all nodes
O(numNodes)
Find all edges
O(numEdges)
Outgoing edges of a node
Incoming edges of a node
Outgoing neighbours of a node
O(numOutNeighbours)
Incoming neighbours of a node
Edge from vertex v to u? A B C D E F G H I J
Cannot handle multi-graph either

6. Object-based Data Structure1 2 25
key A A B C D E F G H I J 7 10 24 5 6 3 1 4 11 23 17 25 18 20 13 8 14 9 2 16 B C J

7. Tries

8. Tries Tree: set of strings/keys (if map string → value )
children indexed by elements of the key
nodes corresponding to complete key are marked ( contain a value)
tree under a node = set of all keys with the prefix so far. h j i
had
ice
has
iced
hat in
hats
ink
have
irk he
iron
heal
jab
health
jabs
heat
job a e c n r a o a e k k b b d s t v s e l t s s t h

9. Tries nodes Each node contains
marker (if set) or associated value (if hashmap)
a map: char → node
value a b c d e f g h i j k z _
children
nodes

10. Tries How to add a word into the trie?
How to get a given word from the trie?
How to get all words that match a given prefix from the trie?

11. Dijkstra’s Algorithm and A* Search
Path finding

12. A General Graph Search Strategy
Start a “fringe" with one node
Repeat until fringe is empty or other criteria is met:
Choose a fringe node to visit
Add its neighbours to fringe
Used by both Dijkstra’s Algorithm and A* Search

13. Dijkstra’s Algorithm
Find the shortest paths from a start node to all other nodes
Early stop if the goal node is given
Use a priority queue
Priority: costFromStart
Always select the unvisited node with minimal costFromStart

14. A* Search Find the shortest path from a start node to a goal node
Use a priority queue
Priority: estimated total cost = costFromStart + estCostToGoal
Use heuristic to estimate estCostToGoal
Admissible and Consistent heuristic
Always select the unvisited node with minimal estTotalCost
What if the heuristic is not admissible/not consistent?

15. Articulation Points What is an articulation point?
How to find articulations points in a graph? B A C D F G E

16. Find Articulation Points
DFS of the graph, set node count number
For each node, split other nodes into
Subtree nodes
Non-subtree nodes
Whether there is an alternative path (edges in the graph but not in the tree) from each subtree node to the non-subtree nodes?
Min count number each node can reach back to?
Articulation point if a subtree node has reach back no smaller than its count number
Root node A
1/1 B
2/1 C
3/1 D
4/1 G
7/7
5/4 E F
6/4

17. Minimum Spanning Tree What is a MST? Prim’s Algorithm
Kruskal’s Algorithm
Union-Find Data Structure

18. Prim’s Algorithm Grow from a start node
At each step, the min-weight edge that connects a node in the tree and a node not in the tree
Stop when all the nodes in the tree

19. Kruskal’s Algorithm Start from a lot of single-node trees (forest)
At each step, merge two trees into one tree by adding a min-weight edge
Stop when there is a single tree

20. Union–Find Data Structure
MakeSet(x), Find(x), Union(x,y)
How to never merge longer tree into shorter tree? Z0 X3 J2 A4 T0 D2 C3 G0 U1 E2 S0 K0 R1 Y1 R0 Q1 N0 W0 H0

21. 3D Graphics Right-hand rule Thumb pointing to z axis
Curl of other fingers from x to y (anti-clockwise viewing from z axis)
Left-handed
Right-handed
wikipedia

22. Object Transformation
A unified [matrix * vector] transformation operator
Convert 3D vectors to 4D vectors
Use 1 for the value in the 4th dimension
Translation
Scaling
Rotation

23. Rendering a Polygon Calculate EdgeList x (x3,y3,z3) (x1,y1,z1)
Need to keep only the closest pixels (small z) ⇒
work out z value of each pixel
(x2,y2,z2) y

24. EdgeList with Z-buffer
Anti-clockwise
for each edge (a, b) in {(v1,v2), (v2, v3), (v3, v1)}
slopex ← (b.x – a.x) / (b.y – a.y), slopez ← (b.z – a.z) / (b.y – a.y)
x ← a.x, z ← a.z, y ← round(a.y)
if a.y < b.y then // going down, visit left
while y <= round(b.y) [increase]
xleft(y) ← x, zleft(y) ← z,
x ← x + slopex , z ← z + slopez
y++
else // going up, visit right
while y >= round(b.y) [decrease]
xright(y) ← x, zright(y) ← z,
x ← x – slopex , z ← z – slopez
y--
EdgeList
Left
Right x z x z
ymin
ymin+1 …
ymax

25. Use EdgeList Two interpolations 1: get x and z bounds from y
2, get z from x and bounds
(xleft(y), zleft(y))
(xright(y), zright(y))
(x, z) x z
(xleft(y), y, zleft(y))
(xright(y), y, zright(y))