problem solving by search ii - cw.fel.cvut.cz€¦ · problem solving by search ii tom a s svoboda...
TRANSCRIPT
Problem solving by search II
Tomas Svoboda
Vision for Robots and Autonomous Systems, Center for Machine PerceptionDepartment of Cybernetics
Faculty of Electrical Engineering, Czech Technical University in Prague
June 18, 2019
1 / 27
Outline
I Graph search
I Heuristics (how to search faster)
I Greedy
I A∗. A-star search.
2 / 27
Recap: Search
3 / 27
A tree search recap
function tree search(problem) return a solution or failureinitialize the frontier the initial state of the problemloop
if the frontier is empty then return failureelse choose a node from frontier and remove from frontierend ifif the node contains a goal state then return the solutionend ifExpand the node and add the resulting nodes to frontier
end loopend function
4 / 27
A tree search recap
function tree search(problem) return a solution or failureinitialize the frontier the initial state of the problemloop
if the frontier is empty then return failureelse choose a node from frontier and remove from frontierend ifif the node contains a goal state then return the solutionend ifExpand the node and add the resulting nodes to frontier
end loopend function
4 / 27
A tree search recap
function tree search(problem) return a solution or failureinitialize the frontier the initial state of the problemloop
if the frontier is empty then return failureelse choose a node from frontier and remove from frontierend ifif the node contains a goal state then return the solutionend ifExpand the node and add the resulting nodes to frontier
end loopend function
4 / 27
A tree search recap
function tree search(problem) return a solution or failureinitialize the frontier the initial state of the problemloop
if the frontier is empty then return failureelse choose a node from frontier and remove from frontierend ifif the node contains a goal state then return the solutionend ifExpand the node and add the resulting nodes to frontier
end loopend function
4 / 27
A tree search recap
function tree search(problem) return a solution or failureinitialize the frontier the initial state of the problemloop
if the frontier is empty then return failureelse choose a node from frontier and remove from frontierend ifif the node contains a goal state then return the solutionend ifExpand the node and add the resulting nodes to frontier
end loopend function
4 / 27
A tree search recap
function tree search(problem) return a solution or failureinitialize the frontier the initial state of the problemloop
if the frontier is empty then return failureelse choose a node from frontier and remove from frontierend ifif the node contains a goal state then return the solutionend ifExpand the node and add the resulting nodes to frontier
end loopend function
4 / 27
A tree search recap
function tree search(problem) return a solution or failureinitialize the frontier the initial state of the problemloop
if the frontier is empty then return failureelse choose a node from frontier and remove from frontierend ifif the node contains a goal state then return the solutionend ifExpand the node and add the resulting nodes to frontier
end loopend function
4 / 27
A Maze, what could possibly go wrong?
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5 / 27
Analyze the demo run. What happened? Why did it take that long?
Tree search the maze
function tree search(env) return asolution or failure
initialize the frontierwhile frontier do
node = frontier.pop()if goal in node then
breakend ifnodes = env.expand(node.state)Add nodes to frontier
end whileend function
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6 / 27
Make a frontier and expand columns on a paper and follow the algo-
rigthm by putting and removing (scratching out) nodes from the list.
A graph searchfunction graph search(env) return a solution or failure
init frontier by the start stateinitialize the explored set to be emptywhile frontier do
node = frontier.pop()if goal in node then breakend ifnodes = env.expand(node.state)add node to exploredfor all nodes do
if node not in explored (or in frontier) thenadd nodes to frontier
end ifend for
end whileend function
Do not forget: node is not the same as state!7 / 27
Think about what is node and what state. What is main difference? Howare they connected? Where do they appear? What is node/state in themaze problem?
The main idea: Do not expand a state twice.
A graph searchfunction graph search(env) return a solution or failure
init frontier by the start stateinitialize the explored set to be emptywhile frontier do
node = frontier.pop()if goal in node then breakend ifnodes = env.expand(node.state)add node to exploredfor all nodes do
if node not in explored (or in frontier) thenadd nodes to frontier
end ifend for
end whileend function
Do not forget: node is not the same as state!7 / 27
Think about what is node and what state. What is main difference? Howare they connected? Where do they appear? What is node/state in themaze problem?
The main idea: Do not expand a state twice.
A graph searchfunction graph search(env) return a solution or failure
init frontier by the start stateinitialize the explored set to be emptywhile frontier do
node = frontier.pop()if goal in node then breakend ifnodes = env.expand(node.state)add node to exploredfor all nodes do
if node not in explored (or in frontier) thenadd nodes to frontier
end ifend for
end whileend function
Do not forget: node is not the same as state!7 / 27
Think about what is node and what state. What is main difference? Howare they connected? Where do they appear? What is node/state in themaze problem?
The main idea: Do not expand a state twice.
A graph searchfunction graph search(env) return a solution or failure
init frontier by the start stateinitialize the explored set to be emptywhile frontier do
node = frontier.pop()if goal in node then breakend ifnodes = env.expand(node.state)add node to exploredfor all nodes do
if node not in explored (or in frontier) thenadd nodes to frontier
end ifend for
end whileend function
Do not forget: node is not the same as state!7 / 27
Think about what is node and what state. What is main difference? Howare they connected? Where do they appear? What is node/state in themaze problem?
The main idea: Do not expand a state twice.
A graph searchfunction graph search(env) return a solution or failure
init frontier by the start stateinitialize the explored set to be emptywhile frontier do
node = frontier.pop()if goal in node then breakend ifnodes = env.expand(node.state)add node to exploredfor all nodes do
if node not in explored (or in frontier) thenadd nodes to frontier
end ifend for
end whileend function
Do not forget: node is not the same as state!7 / 27
Think about what is node and what state. What is main difference? Howare they connected? Where do they appear? What is node/state in themaze problem?
The main idea: Do not expand a state twice.
A graph searchfunction graph search(env) return a solution or failure
init frontier by the start stateinitialize the explored set to be emptywhile frontier do
node = frontier.pop()if goal in node then breakend ifnodes = env.expand(node.state)add node to exploredfor all nodes do
if node not in explored (or in frontier) thenadd nodes to frontier
end ifend for
end whileend function
Do not forget: node is not the same as state!7 / 27
Think about what is node and what state. What is main difference? Howare they connected? Where do they appear? What is node/state in themaze problem?
The main idea: Do not expand a state twice.
The BFS graph search
function BFS graph search(env) return a solution or failurenode ← env.observe()frontier ← FIFOqueue(node)explored ← set()while frontier not empty do
node ← frontier.pop()explored.add(node.state) . adding state not node!child nodes ← env.expand(node.state)for all child nodes do
if child node.state not in explored or in frontier thenif child node contains Goal then return child nodeend iffronter.insert(child node)
end ifend for
end whileend function
8 / 27
Why adding/checking state and note node in expanded data structure?
Can I do the simple presence check for all kind of graph search algorithms?
The BFS graph search
function BFS graph search(env) return a solution or failurenode ← env.observe()frontier ← FIFOqueue(node)explored ← set()while frontier not empty do
node ← frontier.pop()explored.add(node.state) . adding state not node!child nodes ← env.expand(node.state)for all child nodes do
if child node.state not in explored or in frontier thenif child node contains Goal then return child nodeend iffronter.insert(child node)
end ifend for
end whileend function
8 / 27
Why adding/checking state and note node in expanded data structure?
Can I do the simple presence check for all kind of graph search algorithms?
The BFS graph search
function BFS graph search(env) return a solution or failurenode ← env.observe()frontier ← FIFOqueue(node)explored ← set()while frontier not empty do
node ← frontier.pop()explored.add(node.state) . adding state not node!child nodes ← env.expand(node.state)for all child nodes do
if child node.state not in explored or in frontier thenif child node contains Goal then return child nodeend iffronter.insert(child node)
end ifend for
end whileend function
8 / 27
Why adding/checking state and note node in expanded data structure?
Can I do the simple presence check for all kind of graph search algorithms?
The BFS graph search
function BFS graph search(env) return a solution or failurenode ← env.observe()frontier ← FIFOqueue(node)explored ← set()while frontier not empty do
node ← frontier.pop()explored.add(node.state) . adding state not node!child nodes ← env.expand(node.state)for all child nodes do
if child node.state not in explored or in frontier thenif child node contains Goal then return child nodeend iffronter.insert(child node)
end ifend for
end whileend function
8 / 27
Why adding/checking state and note node in expanded data structure?
Can I do the simple presence check for all kind of graph search algorithms?
The BFS graph search
function BFS graph search(env) return a solution or failurenode ← env.observe()frontier ← FIFOqueue(node)explored ← set()while frontier not empty do
node ← frontier.pop()explored.add(node.state) . adding state not node!child nodes ← env.expand(node.state)for all child nodes do
if child node.state not in explored or in frontier thenif child node contains Goal then return child nodeend iffronter.insert(child node)
end ifend for
end whileend function
8 / 27
Why adding/checking state and note node in expanded data structure?
Can I do the simple presence check for all kind of graph search algorithms?
The BFS graph search
function BFS graph search(env) return a solution or failurenode ← env.observe()frontier ← FIFOqueue(node)explored ← set()while frontier not empty do
node ← frontier.pop()explored.add(node.state) . adding state not node!child nodes ← env.expand(node.state)for all child nodes do
if child node.state not in explored or in frontier thenif child node contains Goal then return child nodeend iffronter.insert(child node)
end ifend for
end whileend function
8 / 27
Why adding/checking state and note node in expanded data structure?
Can I do the simple presence check for all kind of graph search algorithms?
The BFS graph search
function BFS graph search(env) return a solution or failurenode ← env.observe()frontier ← FIFOqueue(node)explored ← set()while frontier not empty do
node ← frontier.pop()explored.add(node.state) . adding state not node!child nodes ← env.expand(node.state)for all child nodes do
if child node.state not in explored or in frontier thenif child node contains Goal then return child nodeend iffronter.insert(child node)
end ifend for
end whileend function
8 / 27
Why adding/checking state and note node in expanded data structure?
Can I do the simple presence check for all kind of graph search algorithms?
What about actual costs graph search?
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f,3
c,1.2
f,2.4
G,2.9
d,2.2
G,3.2
b,1.7
e,2.7 f,3.7
d,3
9 / 27
When following the algorithm (animation) use the paper list of frontier
and expanded
What about actual costs graph search?
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f,2.4
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d,2.2
G,3.2
b,1.7
e,2.7 f,3.7
d,3
9 / 27
When following the algorithm (animation) use the paper list of frontier
and expanded
What about actual costs graph search?
a
S
b
c
d
e
f
G
1
3
1.2
1
2
1
0.5
11.2
0.5
1
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1.5S,0
a,1
b,2
e,3 f,4
e,2.5
f,3
c,1.2
f,2.4
G,2.9
d,2.2
G,3.2
b,1.7
e,2.7 f,3.7
d,3
9 / 27
When following the algorithm (animation) use the paper list of frontier
and expanded
What about actual costs graph search?
a
S
b
c
d
e
f
G
1
3
1.2
1
2
1
0.5
11.2
0.5
1
1
1
1.5S,0
a,1
b,2
e,3 f,4
e,2.5
f,3
c,1.2
f,2.4
G,2.9
d,2.2
G,3.2
b,1.7
e,2.7 f,3.7
d,3
9 / 27
When following the algorithm (animation) use the paper list of frontier
and expanded
What about actual costs graph search?
a
S
b
c
d
e
f
G
1
3
1.2
1
2
1
0.5
11.2
0.5
1
1
1
1.5S,0
a,1
b,2
e,3 f,4
e,2.5
f,3
c,1.2
f,2.4
G,2.9
d,2.2
G,3.2
b,1.7
e,2.7 f,3.7
d,3
9 / 27
When following the algorithm (animation) use the paper list of frontier
and expanded
What about actual costs graph search?
a
S
b
c
d
e
f
G
1
3
1.2
1
2
1
0.5
11.2
0.5
1
1
1
1.5S,0
a,1
b,2
e,3 f,4
e,2.5
f,3
c,1.2
f,2.4
G,2.9
d,2.2
G,3.2
b,1.7
e,2.7 f,3.7
d,3
9 / 27
When following the algorithm (animation) use the paper list of frontier
and expanded
What about actual costs graph search?
a
S
b
c
d
e
f
G
1
3
1.2
1
2
1
0.5
11.2
0.5
1
1
1
1.5S,0
a,1
b,2
e,3 f,4
e,2.5
f,3
c,1.2
f,2.4
G,2.9
d,2.2
G,3.2
b,1.7
e,2.7 f,3.7
d,3
9 / 27
When following the algorithm (animation) use the paper list of frontier
and expanded
What about actual costs graph search?
a
S
b
c
d
e
f
G
1
3
1.2
1
2
1
0.5
11.2
0.5
1
1
1
1.5S,0
a,1
b,2
e,3 f,4
e,2.5
f,3
c,1.2
f,2.4
G,2.9
d,2.2
G,3.2
b,1.7
e,2.7 f,3.7
d,3
9 / 27
When following the algorithm (animation) use the paper list of frontier
and expanded
What about actual costs graph search?
a
S
b
c
d
e
f
G
1
3
1.2
1
2
1
0.5
11.2
0.5
1
1
1
1.5S,0
a,1
b,2
e,3 f,4
e,2.5
f,3
c,1.2
f,2.4
G,2.9
d,2.2
G,3.2
b,1.7
e,2.7 f,3.7
d,3
9 / 27
When following the algorithm (animation) use the paper list of frontier
and expanded
What about actual costs graph search?
a
S
b
c
d
e
f
G
1
3
1.2
1
2
1
0.5
11.2
0.5
1
1
1
1.5S,0
a,1
b,2
e,3 f,4
e,2.5
f,3
c,1.2
f,2.4
G,2.9
d,2.2
G,3.2
b,1.7
e,2.7 f,3.7
d,3
9 / 27
When following the algorithm (animation) use the paper list of frontier
and expanded
What about actual costs graph search?
a
S
b
c
d
e
f
G
1
3
1.2
1
2
1
0.5
11.2
0.5
1
1
1
1.5S,0
a,1
b,2
e,3 f,4
e,2.5
f,3
c,1.2
f,2.4
G,2.9
d,2.2
G,3.2
b,1.7
e,2.7 f,3.7
d,3
9 / 27
When following the algorithm (animation) use the paper list of frontier
and expanded
What about actual costs graph search?
a
S
b
c
d
e
f
G
1
3
1.2
1
2
1
0.5
11.2
0.5
1
1
1
1.5S,0
a,1
b,2
e,3 f,4
e,2.5
f,3
c,1.2
f,2.4
G,2.9
d,2.2
G,3.2
b,1.7
e,2.7 f,3.7
d,3
9 / 27
When following the algorithm (animation) use the paper list of frontier
and expanded
The UCS graph searchfunction UCS graph search(env) return a solution or failure
node ← env.observe()frontier ← priority queue(node) . path cost for orderingexplored ← set()while frontier not empty do
node ← frontier.pop()if node contains Goal then return node . check here!end ifexplored.add(node.state)child nodes ← env.expand(node.state)for all child nodes do
if child node.state not in explored or in frontier thenfronter.insert(child node)
else if child node.state in frontier with higher cost thenreplace that with the child node
end ifend for
end whileend function
10 / 27
Does the algorithm always find the best (cheapest) path? Are there any
requirements for the path optimality function?
The UCS graph searchfunction UCS graph search(env) return a solution or failure
node ← env.observe()frontier ← priority queue(node) . path cost for orderingexplored ← set()while frontier not empty do
node ← frontier.pop()if node contains Goal then return node . check here!end ifexplored.add(node.state)child nodes ← env.expand(node.state)for all child nodes do
if child node.state not in explored or in frontier thenfronter.insert(child node)
else if child node.state in frontier with higher cost thenreplace that with the child node
end ifend for
end whileend function
10 / 27
Does the algorithm always find the best (cheapest) path? Are there any
requirements for the path optimality function?
The UCS graph searchfunction UCS graph search(env) return a solution or failure
node ← env.observe()frontier ← priority queue(node) . path cost for orderingexplored ← set()while frontier not empty do
node ← frontier.pop()if node contains Goal then return node . check here!end ifexplored.add(node.state)child nodes ← env.expand(node.state)for all child nodes do
if child node.state not in explored or in frontier thenfronter.insert(child node)
else if child node.state in frontier with higher cost thenreplace that with the child node
end ifend for
end whileend function
10 / 27
Does the algorithm always find the best (cheapest) path? Are there any
requirements for the path optimality function?
The UCS graph searchfunction UCS graph search(env) return a solution or failure
node ← env.observe()frontier ← priority queue(node) . path cost for orderingexplored ← set()while frontier not empty do
node ← frontier.pop()if node contains Goal then return node . check here!end ifexplored.add(node.state)child nodes ← env.expand(node.state)for all child nodes do
if child node.state not in explored or in frontier thenfronter.insert(child node)
else if child node.state in frontier with higher cost thenreplace that with the child node
end ifend for
end whileend function
10 / 27
Does the algorithm always find the best (cheapest) path? Are there any
requirements for the path optimality function?
The UCS graph searchfunction UCS graph search(env) return a solution or failure
node ← env.observe()frontier ← priority queue(node) . path cost for orderingexplored ← set()while frontier not empty do
node ← frontier.pop()if node contains Goal then return node . check here!end ifexplored.add(node.state)child nodes ← env.expand(node.state)for all child nodes do
if child node.state not in explored or in frontier thenfronter.insert(child node)
else if child node.state in frontier with higher cost thenreplace that with the child node
end ifend for
end whileend function
10 / 27
Does the algorithm always find the best (cheapest) path? Are there any
requirements for the path optimality function?
The UCS graph searchfunction UCS graph search(env) return a solution or failure
node ← env.observe()frontier ← priority queue(node) . path cost for orderingexplored ← set()while frontier not empty do
node ← frontier.pop()if node contains Goal then return node . check here!end ifexplored.add(node.state)child nodes ← env.expand(node.state)for all child nodes do
if child node.state not in explored or in frontier thenfronter.insert(child node)
else if child node.state in frontier with higher cost thenreplace that with the child node
end ifend for
end whileend function
10 / 27
Does the algorithm always find the best (cheapest) path? Are there any
requirements for the path optimality function?
The UCS graph searchfunction UCS graph search(env) return a solution or failure
node ← env.observe()frontier ← priority queue(node) . path cost for orderingexplored ← set()while frontier not empty do
node ← frontier.pop()if node contains Goal then return node . check here!end ifexplored.add(node.state)child nodes ← env.expand(node.state)for all child nodes do
if child node.state not in explored or in frontier thenfronter.insert(child node)
else if child node.state in frontier with higher cost thenreplace that with the child node
end ifend for
end whileend function
10 / 27
Does the algorithm always find the best (cheapest) path? Are there any
requirements for the path optimality function?
The UCS graph searchfunction UCS graph search(env) return a solution or failure
node ← env.observe()frontier ← priority queue(node) . path cost for orderingexplored ← set()while frontier not empty do
node ← frontier.pop()if node contains Goal then return node . check here!end ifexplored.add(node.state)child nodes ← env.expand(node.state)for all child nodes do
if child node.state not in explored or in frontier thenfronter.insert(child node)
else if child node.state in frontier with higher cost thenreplace that with the child node
end ifend for
end whileend function
10 / 27
Does the algorithm always find the best (cheapest) path? Are there any
requirements for the path optimality function?
Few examples of search strategies so far0
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Run the demos.11 / 27
What is wrong with UCS and other strategies?0
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Run the demo.12 / 27
Node selection, take argmin f (n)
I DFS: f (n) = −n.depthI BFS: f (n) = n.depth
I UCS: f (n) = n.path cost
The good: frontier as a priority queue The bad: All the f (n) correspond tothe cost from n to the start - only backward cost.
13 / 27
Node selection, take argmin f (n)
I DFS: f (n) = −n.depthI BFS: f (n) = n.depth
I UCS: f (n) = n.path cost
The good: frontier as a priority queue The bad: All the f (n) correspond tothe cost from n to the start - only backward cost.
13 / 27
Node selection, take argmin f (n)
I DFS: f (n) = −n.depthI BFS: f (n) = n.depth
I UCS: f (n) = n.path cost
The good: frontier as a priority queue The bad: All the f (n) correspond tothe cost from n to the start - only backward cost.
13 / 27
Heuristics
I A function that estimates how close a state to the goal.
I Designed for a particular problem.
I Examples:
I We will use h(n) - heuristic value of node n
14 / 27
Example of heuristics
Giurgiu
UrziceniHirsova
Eforie
Neamt
Oradea
Zerind
Arad
Timisoara
Lugoj
Mehadia
DobretaCraiova
Sibiu Fagaras
Pitesti
Vaslui
Iasi
Rimnicu Vilcea
Bucharest
71
75
118
111
70
75
120
151
140
99
80
97
101
211
138
146 85
90
98
142
92
87
86
Urziceni
NeamtOradea
Zerind
Timisoara
Mehadia
Sibiu
PitestiRimnicu Vilcea
Vaslui
Bucharest
GiurgiuHirsova
Eforie
Arad
Lugoj
DrobetaCraiova
Fagaras
Iasi
0
160
242
161
77
151
366
244
226
176
241
253
329
80
199
380
234
374
100
193
15 / 27
Greedy, take the node argmin h(n)Greedy Strategy
11
Combining#UCS#and#Greedy#
! UniformKcost#orders#by#path#cost,#or#backward-cost--g(n)#! Greedy#orders#by#goal#proximity,#or#forward-cost--h(n)-
! A*#Search#orders#by#the#sum:#f(n)#=#g(n)#+#h(n)-
S a d
b
G h=5
h=6
h=2
1
8
1 1
2
h=6 h=0 c
h=7
3
e h=1 1
Example:#Teg#Grenager#
S
a
b
c
e d
d G
G
g = 0 h=6
g = 1 h=5
g = 2 h=6
g = 3 h=7
g = 4 h=2
g = 6 h=0
g = 9 h=1
g = 10 h=2
g = 12 h=0
1
What is wrong (and nice) with the Greedy?
1Graph example: Ted Grenager16 / 27
Greedy, take the node argmin h(n)Greedy Strategy
11
Combining#UCS#and#Greedy#
! UniformKcost#orders#by#path#cost,#or#backward-cost--g(n)#! Greedy#orders#by#goal#proximity,#or#forward-cost--h(n)-
! A*#Search#orders#by#the#sum:#f(n)#=#g(n)#+#h(n)-
S a d
b
G h=5
h=6
h=2
1
8
1 1
2
h=6 h=0 c
h=7
3
e h=1 1
Example:#Teg#Grenager#
S
a
b
c
e d
d G
G
g = 0 h=6
g = 1 h=5
g = 2 h=6
g = 3 h=7
g = 4 h=2
g = 6 h=0
g = 9 h=1
g = 10 h=2
g = 12 h=0
1
What is wrong (and nice) with the Greedy?
1Graph example: Ted Grenager16 / 27
A∗ combines UCS and GreedyGreedy Strategy
11
Combining#UCS#and#Greedy#
! UniformKcost#orders#by#path#cost,#or#backward-cost--g(n)#! Greedy#orders#by#goal#proximity,#or#forward-cost--h(n)-
! A*#Search#orders#by#the#sum:#f(n)#=#g(n)#+#h(n)-
S a d
b
G h=5
h=6
h=2
1
8
1 1
2
h=6 h=0 c
h=7
3
e h=1 1
Example:#Teg#Grenager#
S
a
b
c
e d
d G
G
g = 0 h=6
g = 1 h=5
g = 2 h=6
g = 3 h=7
g = 4 h=2
g = 6 h=0
g = 9 h=1
g = 10 h=2
g = 12 h=0
UCS orders by backward (path) cost g(n)Greedy uses heuristics (goal proximity) h(n)
A∗ orders nodes by: f (n) = g(n) + h(n)
17 / 27
A∗ combines UCS and GreedyGreedy Strategy
11
Combining#UCS#and#Greedy#
! UniformKcost#orders#by#path#cost,#or#backward-cost--g(n)#! Greedy#orders#by#goal#proximity,#or#forward-cost--h(n)-
! A*#Search#orders#by#the#sum:#f(n)#=#g(n)#+#h(n)-
S a d
b
G h=5
h=6
h=2
1
8
1 1
2
h=6 h=0 c
h=7
3
e h=1 1
Example:#Teg#Grenager#
S
a
b
c
e d
d G
G
g = 0 h=6
g = 1 h=5
g = 2 h=6
g = 3 h=7
g = 4 h=2
g = 6 h=0
g = 9 h=1
g = 10 h=2
g = 12 h=0
UCS orders by backward (path) cost g(n)Greedy uses heuristics (goal proximity) h(n)
A∗ orders nodes by: f (n) = g(n) + h(n)
17 / 27
When to stop A∗
When#should#A*#terminate?#
! Should#we#stop#when#we#enqueue#a#goal?#
! No:#only#stop#when#we#dequeue#a#goal#
S
B
A
G
2
3
2
2 h = 1
h = 2
h = 0 h = 3
2
2Graph example: Dan Klein and Pieter Abbeel18 / 27
Is A∗ optimal? Is#A*#Op)mal?#
! What#went#wrong?#! Actual#bad#goal#cost#<#es)mated#good#goal#cost#! We#need#es)mates#to#be#less#than#actual#costs!#
A
G S
1 3 h = 6
h = 0
5
h = 7
3
What is the problem?
3Graph example: Dan Klein and Pieter Abbeel19 / 27
Is A∗ optimal? Is#A*#Op)mal?#
! What#went#wrong?#! Actual#bad#goal#cost#<#es)mated#good#goal#cost#! We#need#es)mates#to#be#less#than#actual#costs!#
A
G S
1 3 h = 6
h = 0
5
h = 7
3
What is the problem?
3Graph example: Dan Klein and Pieter Abbeel19 / 27
Admissible heuristics
A heuristic function h is admissible if:
0 ≤ h(n) ≤ h∗(n)
where h∗(n) is the true cost of going from n to the nearest goal.
20 / 27
Optimality of A∗ tree search
21 / 27
Properties - does heuristic matter?0
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22 / 27
A* graph search
function graph search(env)frontier.insert(startnode)explored = set()while frontier do
node = frontier.pop()if goal in node then breakend ifnodes = env.expand(node.state)explored.add(node)for all nodes do
if node not in explored thenfrontier.insert(node)
end ifend for
end whileend function
A*#Graph#Search#Gone#Wrong?#
S
A
B
C
G
1
1
1
2 3
h=2
h=1
h=4 h=1
h=0
S (0+2)
A (1+4) B (1+1)
C (2+1)
G (5+0)
C (3+1)
G (6+0)
State#space#graph# Search#tree#
Graph example: Dan Klein and Pieter Abbeel
What went wrong?
23 / 27
A* graph search
function graph search(env)frontier.insert(startnode)explored = set()while frontier do
node = frontier.pop()if goal in node then breakend ifnodes = env.expand(node.state)explored.add(node)for all nodes do
if node not in explored thenfrontier.insert(node)
end ifend for
end whileend function
A*#Graph#Search#Gone#Wrong?#
S
A
B
C
G
1
1
1
2 3
h=2
h=1
h=4 h=1
h=0
S (0+2)
A (1+4) B (1+1)
C (2+1)
G (5+0)
C (3+1)
G (6+0)
State#space#graph# Search#tree#
Graph example: Dan Klein and Pieter Abbeel
What went wrong?
23 / 27
Consistent heuristicsA*#Graph#Search#Gone#Wrong?#
S
A
B
C
G
1
1
1
2 3
h=2
h=1
h=4 h=1
h=0
S (0+2)
A (1+4) B (1+1)
C (2+1)
G (5+0)
C (3+1)
G (6+0)
State#space#graph# Search#tree#
Admissible h:h(A) ≤ true cost A→ G
Consistent h:h(A)− h(C ) ≤ true cost A→ C
f along a path never decreases!
24 / 27
Consistent heuristicsA*#Graph#Search#Gone#Wrong?#
S
A
B
C
G
1
1
1
2 3
h=2
h=1
h=4 h=1
h=0
S (0+2)
A (1+4) B (1+1)
C (2+1)
G (5+0)
C (3+1)
G (6+0)
State#space#graph# Search#tree#
Admissible h:h(A) ≤ true cost A→ G
Consistent h:h(A)− h(C ) ≤ true cost A→ C
f along a path never decreases!
24 / 27
Consistent heuristicsA*#Graph#Search#Gone#Wrong?#
S
A
B
C
G
1
1
1
2 3
h=2
h=1
h=4 h=1
h=0
S (0+2)
A (1+4) B (1+1)
C (2+1)
G (5+0)
C (3+1)
G (6+0)
State#space#graph# Search#tree#
Admissible h:h(A) ≤ true cost A→ G
Consistent h:h(A)− h(C ) ≤ true cost A→ C
f along a path never decreases!
24 / 27
Consistent heuristicsA*#Graph#Search#Gone#Wrong?#
S
A
B
C
G
1
1
1
2 3
h=2
h=1
h=4 h=1
h=0
S (0+2)
A (1+4) B (1+1)
C (2+1)
G (5+0)
C (3+1)
G (6+0)
State#space#graph# Search#tree#
Admissible h:h(A) ≤ true cost A→ G
Consistent h:h(A)− h(C ) ≤ true cost A→ C
f along a path never decreases!
24 / 27
Optimality of A∗
I admissible h for tree search
I consistent h for graph search
I What about UCS?
I Are all consistent heuristics also admissible?h(A)− h(C ) ≤ cost(A→ C )
25 / 27
Optimality of A∗
I admissible h for tree search
I consistent h for graph search
I What about UCS?
I Are all consistent heuristics also admissible?h(A)− h(C ) ≤ cost(A→ C )
25 / 27
Optimality of A∗
I admissible h for tree search
I consistent h for graph search
I What about UCS?
I Are all consistent heuristics also admissible?h(A)− h(C ) ≤ cost(A→ C )
25 / 27
Optimality of A∗
I admissible h for tree search
I consistent h for graph search
I What about UCS?
I Are all consistent heuristics also admissible?h(A)− h(C ) ≤ cost(A→ C )
25 / 27
How to find a heuristics?
Image from: https://commons.wikimedia.org/wiki/File:15-puzzle-loyd.svg 26 / 27
Which heuristics is the best?
27 / 27