Lesson 6
Branch and Bound (or Uniform Cost) Search

Branch and Bound – Paths are developed in terms of increasing cost
(more accurately, in terms of non-decreasing cost).

Once a path to the goal is found, it is likely that this path is optimal.

To change “likely” to “guaranteed”, one must continue generating partial
paths until each has a cost greater than or equal to the path found to the
goal.
Branch and Bound Search
To conduct a branch-and-bound search,

Form a one-element queue consisting of a zero-length path that contains
only the root node.

Until the first path is the queue terminates at the goal node or the queue is
empty,
o Remove the first path from the queue; create new paths by
extending the first path to all the neighbors of the terminal node.
o Reject all new paths with loops.
o Add the remaining new paths, if any, to the queue.
o Sort the entire queue by path length with least-cost paths in front.

If the goal node is found, announce success; otherwise announce failure.
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Branch and Bound (Uniform Cost) Search – cont.
A
A
4
Start with
root node
A.
Note: Heurisitic
estimates are not
used in this search!
11
B
Paths from root
are generated.
0
A
C
4
15
13
D
12
10
I
18
4
E
F
16
6
J G1
B
H
3
K L
1
M
7
N
Since B has the least cost,
we expand it.
G2
A
11
C
11
C
B
13
D 19
11
4
B
15
15
E 17
4
13
D 19
Of our 3 choices, C
has the least cost so
we’ll expand it.
E 17
H
15
Node H has the least cost thus far, so we expand it.
4
C
B
C
13
11
4
11
B
E 17
14
A
4
D 19
3
F
A
15
C
11
4
A
4
11
3
15
3
F
15
We have a goal, G2 but
1
need to expand other
branches to see if there is
15 N
another goal with less
distance.
19
H
13
4
E 17
F
6
7
G2
21
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3
L
21
3
1
M N
18 15
Note: Both
nodes F and N
have a cost of
15, we chose to
expand the
leftmost node
first. We
continue
expanding until
all remaining
H
paths are
greater
than 21,
7
the cost of G2
G2
21
Our B & B Example – continued
A
4
11
B
C
15
19 D
13
4
17 E
F
16
6
10
J
29
Step 4
3
H
3
1
7
M N
G2
K L
33 21
18 15
21
Step 2
Step 3 Step 1
All partial paths must be extended until their costs ≥ shortest path to goal.

Step 1:
The path to node N cannot be extended.

Step 2:
The next shortest path, A -> B -> E is extended Its cost now
exceeds 21.

Step 3:
The path to node M, much like to node N, cannot be
extended.

Step 4:
The last partial path with a cost ≤ 21 is extended. Its cost,
29, now exceeds the start -> goal path.
The shortest path to a goal is A -> C -> H -> G2 with a cost of 21.
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Branch and Bound with Estimates
Now, we will augment Branch and Bound Search with underestimates of
remaining distance.
A:18
11
4
C:4
B:14
15
13
D:9
12
E:12
10
I:21
4
F:1
18
J:31
3
16
G1:0
6
K:34
H:3
3
L:21
1
M:18
7
N:8
G2:0
Hence, we generate paths in terms of their estimated overall length.
A:18
A:18
A:18
11
A is not a
goal; we
continue
11
C:4
11+4=15
C:4
15
4
Observe, that in this search, we
went to C first instead of B as we did
in plain B&B.
F:1
A:18
11
C:4
4
F:1
15
3
16
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H:3
(11+3)+3=17
(11+4)+1=16
Branch and Bound with Estimates
A:18
11
4
B:14
C:4
4+14=18
15
3
4
F:1
16
H:3
17
A:18
11
4
18
3
4
15
Step 3
15
C:4
B:14
16
D:9
H:3
F:1
Step 1
1
3
M:18
N:8
7
17
Step 2
G2:0
(11+3+7)+ 0 = 21
Once again, we are not done until all paths with an estimated cost ≤ 21 are
extended.

Step 1:
The path A -> C -> F is exended to M. We are over 21.

Step 2:
A -> C -> H is extended to N. Over 21.

Step 3:
A -> B is extended to D. Over 21.
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Branch & Bound with Underestimates
To conduct a branch-and-bound search with a lower-bound estimate,

Form a one-element queue consisting of a zero-length path that
contains only the root node.

Until the first path in the queue terminates at the goal node or the
queue is empty,
o Remove the first path from the queue; create new paths by
extending the first path to all the neighbors of the terminal node.
o Reject all new paths with loops.
o Add the remaining new paths, if any, to the queue.
o Sort the entire queue by the sum of the path length and a lowerbound estimate of the cost remaining, with least-cost paths in
front.

If the goal node is found, announce success; otherwise announce
failure.
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Branch and Bound with Dynamic Programming
An Example (Artifical Intelligence by: Patrick Henry Winston, 3rd Edition, AddisonWesley):
A
B
4
C
4
3
5
5
S
G
3
4
D
2
E
4
F
A basic search problem. A path is to be found from the start node, S, to the goal
node, G. Search procedures explore nets such as these, learning about
connections and distance as they go.
An illustration of the dynamic-programming
S
principle. The numbers beside the nodes are
accumulated distances. There is no point in
A
expanding the instance of node D at the end
Expanded
next ->
of S-A-D, because getting to the goal via the
instance of D at the end of S-D is obviously
B
D
7
8
D
4
<- Never
expanded
more efficient.

Assume cost 1 > cost 2
o Why would one wish to get to the goal by first taking a more
expensive path to I?
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Branch and Bound with Dynamic Programming – continued

The Principle of Optimality – optimal paths are
constructed from optimal subpaths. i.e.
S
Cost 1
Cost 2
I
I
an optimal subpath from S to G that
passes through some intermediate node
I is composed of an optimal S -> I path, followed by
G
an optimal I -> G path.
To conduct a branch-and-bound search with dynamic programming.

Form a one-element queue consisting of a zero-length path that
contains only the root node.

Until the first path in the queue terminates at the goal node or the
queue is empty,
o Remove the first path from the queue; create new paths by
extending the first path to all the neighbors of the terminal node.
o Reject all new paths with loops.
o Add the remaining new paths, if any, to the queue.
o If two or more paths reach a common node, delete all those
paths except the one that reaches the common node with the
minimum cost.
o Sort the entire queue by path length with least-cost paths in
front.

If the goal node is found, announce success; otherwise announce
failure.
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Branch and Bound with Dynamic Programming – continued
Branch-and-bound search, augmented by dynamic programming, determines
that path S-D-E-F-G is optimal. The numbers beside the nodes are accumulated
path distances. Many paths, those shown terminated with underbars, are found
to be redundant. Thus, dynamic programming reduces the number of nodes
expanded.
S
S
S
A
A
D
3
4
3
D
A
4
4
B
D
7
8
B
D
A
E
7
8
9
6
S
3
D
S
A
4
D
B
D
A
7
8
9
E
3
7
6
B
F
11
10
A
B
D
A
8
9
7
6
E
B
F
11
12
11
10
A
B
E
C
S
3
4
D
4
D
D
A
8
9
E
C
E
B
11
12
11
6
F
G
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10
A* Search
A* Search

Branch and Bound, with

Estimate of remaining distance, and

Dynamic Programming principle employed
To conduct A* Search

Form a one-element queue consisting of a zero-length path that
contains only the root node.

Until the first path in the queue terminates at the goal node or the
queue is empty,
o Remove the first path from the queue; create new paths by
extending the first path to all the neighbors of the terminal node.
o Reject all new paths with loops.
o If two or more paths reach a common node, delete all those
paths except the one that reaches the common node with the
minimum cost.
o Sort the entire queue by the sum of the path length and a lowerbound estimate of the cost remaining, with least-cost paths in
front.

If the goal node is found, announce success; otherwise announce
failure.
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Branch and Bound (Uniform Cost) Search
Complete:
If one guarantees that the cost of every step is some small positive
constant.
Optimal:
With above caveat.
Time and Space Complexity:
Recall b is branching factor and d is the depth
of the shallowest goal node. Can be much greater than bd … this is because this
search may explore large trees of small steps before exploring paths involving
large and perhaps fruitful steps.

A* Search is optimal whenever the estimate of remaining distance
employed, h(n) is an admissible heuristic, i.e. h(n) never overestimates
the remaining distance.

A* is also complete.
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Traversing a Maze
Artificial Intelligence Illuminated by Ben Coppin, Jones & Bartlett
F
L
C
B
J
G
D
I
K
E
IN
A
N
M
OUT
A:
entrance
M:
exit
C, E, F, G, H, J, I: dead ends
B, D, I, K:
choices exist
Imagine an ant crawling along the left wall…. the path A, B, E, F, C, D, G, H, I, J,
K, L, M would be taken. A depth first search would follow this same path
A
B
E
C
F
D
G
H
I
J
K
L
M
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N
Maze Example – A Search Approach
Now, imagine that one had a heuristic measure of remaining distance (how was it
obtained…??).
Ten times straight-line distance to goal … in inches.
L
H
J
C
G
B
19.5
25
D
E
30
23.5
17
10
24
N
A
21
K
23.5
27.5
11
30
M
We employ hillclimbing.
1) We first go to D as it has the
lowest estimated remaining distance
to the goal.
A
30 B
25 C
24 D
2) I is selected next.
3) We proceed to K.
I 17
23.5 G
H
23.5
4) And finally M is selected.
K
10
19.5 J
21
L
M
0
11
N
Note: Had we arrived at a dead end
e.g. C, E, F, G, H, J, L, or N
hillclimbing would be stuck.
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3 Puzzle Revisited
14 of 14