Download COMP 261 Lecture 14

COMP 261 Lecture 17
Test Revision
2
Outline
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•
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Data Structures for Graphs
Tries
Dijkstra’s Algorithm
A* Search
Articulation Points
Minimum Spanning Tree
3D Graphics
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Data Structures for Graphs
• Traditional Adjacency Matrix
• Traditional Adjacency List
• Object-based structure (Adjacency list-like)
4
Adjacency Matrix
• Complexity of actions
– Find all nodes
• O(numNodes)
– Find all edges
• O(numNodes2)
– Outgoing edges of a node
• O(numNodes)
– Incoming edges of a node
• O(numNodes)
– Outgoing neighbours of a node
• O(numNodes)
– Incoming neighbours of a node
• O(numNodes)
– Edge from vertex v to u?
• O(1)
A B C D E F G H
A
B
7
7
C
5
6
1
1
E
4
F
10
I
15 3
J
16
6
17
19
19
9
2 13
17
2
13
16
H
3
9
4
G
J
10 15
5
D
I
18
11 8 14
11
12
8
20
18 14 12 20
Cannot handle
multi-graph
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Adjacency List
• Complexity of actions
– Find all nodes
• O(numNodes)
– Find all edges
• O(numEdges)
– Outgoing edges of a node
• O(numEdges)
– Incoming edges of a node
• O(numEdges)
– Outgoing neighbours of a node
• O(numOutNeighbours)
– Incoming neighbours of a node
• O(numEdges)
– Edge from vertex v to u?
• O(numOutNeighbours)
A
B
C
D
E
F
G
H
I
J
B
A
B
C
D
E
B
A
A
H
C
D
E
F
G
F
G
B
I
G
I
J
I
J
H
J
C
I
J
E
G
J
C
D
F
G
H
I
I
J
Cannot handle
multi-graph either
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Object-based Data Structure
1
25
2
key
A
B
5
C
C
B
7
A
7
5
6
10
24
1
9
2
3
1
D
9
I
4
11
3
J
4
E
2
J
H
10
20
11
18
14
13
23
17
16
13
F
G
6
25
20
8
14
23
25
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Tries
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
had
has
hat
hats
have
he
heal
health
heat
ice
iced
in
ink
irk
iron
jab
jabs
job
a
e
d s t
s
j
i
c
v
a
e
l
t
h
e
t
s
n
r
a
k
k
b
s
o
b
Tries
• 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
children
nodes
z _
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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?
Dijkstra’s Algorithm and A* Search
• Path finding
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
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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
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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?
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Articulation Points
• What is an articulation point?
• How to find articulations points in a graph?
B
A
C
D
G
F
E
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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?
A 1/1
B 2/1
• Min count number each node can reach
back to?
D
4/1
• Articulation point if a subtree node has
reach back no smaller than its count
5/4
E
number
F
• Root node
6/4
C 3/1
G 7/7
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Minimum Spanning Tree
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•
•
•
What is a MST?
Prim’s Algorithm
Kruskal’s Algorithm
Union-Find Data Structure
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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
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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
Union–Find Data Structure
• MakeSet(x), Find(x), Union(x,y)
• How to never merge longer tree into shorter tree?
Z0
A4
T0
C3
G0
E2
R0
X3
J2
D2
S0
U1
K0
R1
Y1
Q1
W0
H0
N0
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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
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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
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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
y
(x2,y2,z2)
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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
EdgeList
while y <= round(b.y) [increase]
Right
Left
xleft(y) ← x, zleft(y) ← z,
z
x
z
x
x ← x + slopex , z ← z + slopez
ymin
y++
ymin+1
else
// going up, visit right
while y >= round(b.y) [decrease]
xright(y) ← x, zright(y) ← z,
x ← x – slopex , z ← z – slopez
y--
ymax
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Use EdgeList
• Two interpolations
– 1: get x and z bounds from y
– 2, get z from x and bounds
z
(xright(y), zright(y))
(x, z)
(xleft(y), zleft(y))
x
(xleft(y), y, zleft(y))
(xright(y), y, zright(y))