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COMP3411/9814 Artificial Intelligence
创建日期：2024-06-13 16:35:25
标签： COMP3411
Artificial Intelligence

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COMP3411/9814 Artificial Intelligence

Assignment 2 – Heuristics and Search
Marks: 12% of final assessment
In this assignment you will be examining search strategies for the 15-puzzle,
admissible heuristics for a trajectory planning problem, and pruning in alpha-
beta search trees. You should submit a report in .pdf format with answers
to the questions below.
Question 1: Search Strategies for the 15-Puzzle (2 marks)
For this question you will construct a table showing the number of states
expanded when the 15-puzzle is solved, from various start positions, using
four different search strategies:
(i) Breadth First Search
(ii) Iterative Deepening Search
(iii) Greedy Search (using the Manhattan Distance heuristic)
(iv) A* Search (using the Manhattan Distance heuristic)
Unzip the file and change directory to path search
unzip path_search.zip
cd path_search
Run the code by typing:
python3 search.py --start 2634-5178-AB0C-9DEF --s bfs
The --start argument specifies the start position, which in this case is:
2 6 3 4
5 1 7 8
A B C
9 D E F
Start State
1 2 3 4
5 6 7 8
9 A B C
D E F
Goal State
The Goal State is shown on the right. The --s argument specifies the search
strategy (bfs for Breadth First Search).
The code should print out the number of expanded nodes (by thousands)
as it searches. It should then print a path from the Start State to the Goal
State, followed by the number of nodes Generated and Expanded, and the
Length and Cost of the path (which are both equal to 12 in this case).
1
(a) Draw up a table in this format:
Start State BFS IDS Greedy A*
start1
start2
start3
Run each of the four search strategies from three specified start positions,
using the following combinations of command-line arguments:
Start Positions:
start1: --start 2634-5178-AB0C-9DEF
start2: --start 1034-728B-5D6A-E9FC
start3: --start 5247-61C0-9A83-DEBF
Search Strategies:
BFS: --s bfs
IDS: --s dfs --id
Greedy: --s greedy
A*S earch: --s astar
In each case, record in your table the length of the path, and the number
of nodes Expanded during the search. Include the completed table in
your report.
(b) Briefly discuss the efficiency of these four search strategies, with regard
to the number of nodes Expanded, and the length of the resulting path.
Question 2: Heuristic Path Search for 15-Puzzle (2 marks)
In this question you will be exploring a search strategy known as Heuristic
Path Search, which is a best-first search using the objective function:
fw(n) = (2− w)g(n) + wh(n),
where w is a number between 0 and 2. Heuristic Path Search is equivalent
to Uniform Cost Search when w = 0, to A* Search when w = 1, and Greedy
Search when w = 2. It is Complete for all w between 0 and 2.
(a) Prove that Heuristic Path Search is optimal when 0 ≤ w ≤ 1, assuming
h() is admissible. (You may also assume that A* Search is optimal.)
Hint: show that minimizing fw(n) = (2 − w)g(n) + wh(n) is the same
as minimizing f ′
w
(n) = g(n) + h′(n) for some function h′(n) with the
property that h′(n) ≤ h(n) for all n.
2
(b) Draw up a table in this format (the top row has been filled in for you):
start4 start5 start6
IDA* Search 45 545120 50 4178819 56 169367641
58 13770561HPS, w = 1.1
HPS, w = 1.2
HPS, w = 1.3
HPS, w = 1.4
Run search.py on each of the three start states shown below, using the
Iterative Deepening version of Heuristic Path Search, with w = 1.1, 1.2, 1.3
and 1.4 .
Start Positions:
start4: --start A974-3256-FD8B-EC01
start5: --start 153E-A02C-9FBD-8476
start6: --start 418E-7AD0-9C52-3FB6
Search Strategies:
HPS, w = 1.1: --s heuristic --w 1.1 --id
HPS, w = 1.2: --s heuristic --w 1.2 --id
HPS, w = 1.3: --s heuristic --w 1.3 --id
HPS, w = 1.4: --s heuristic --w 1.4 --id
In each case, record in your table the length of the path, and the number
of nodes Expanded during the search. Include the completed table in
your report.
(c) Briefly discuss how the path Length and the number of Expanded nodes
change as the value of w varies between 1.0 (equivalent to IDA*) and 1.4.
Question 3: Graph Paper Grand Prix (5 marks)
We saw in lectures how the Straight-Line-Distance heuristic can be used to
find the shortest-distance path between two points. However, in many robotic
applications we wish to minimize not the distance but rather the time taken
to traverse the path, by speeding up on the straight bits and avoiding sharp
corners.
In this question you will be exploring a simplified version of this problem in
the form of a game known as Graph Paper Grand Prix (GPGP).
3
To play GPGP, you first need to draw the outline of a racing track on a sheet
of graph paper, and choose a start location S = (rS, cS) as well as a Goal
location G = (rG, cG) where r and c are the row and column. The agent
begins at location S, with velocity (0,0). A “state” for the agent consists of a
position (r, c) and a velocity (u, v), where r, c, u, v are (positive or negative)
integers.
G
S
At each time step, the agent has the opportunity to increase or decrease each
component of its velocity by one unit, or to keep it the same. In other words,
the agent must choose an acceleration vector (a, b) with a, b ∈ {−1, 0,+1}. It
then updates its velocity from (u, v) to (u′, v′) = (u+ a, v + b), and updates
its position – using the new velocity – from (r, c) to (r + u′, c + v′). The
aim of the game is to travel as fast as possible, but without crashing into
any obstacles or running off the edge of the track, and eventually stop at the
Goal with velocity (0, 0).
4
Gk0 n
S
We first consider a 1-dimensional version of GPGP where the vehicle moves
through integer locations on a number line, with no obstacles. Assume the
Goal is at location n, and that the agent starts at location 0, with velocity k.
We will use M(n, k) to denote the minimum number of time steps required
to arrive and stop at the Goal. Clearly M(−n,−k) = M(n, k) so we only
need to compute M(n, k) for k ≥ 0.
(a) Starting with the special case k = 0, compute M(n, 0) for 1 ≤ n ≤ 21 by
writing down the optimal sequence of actions for all n between 1 and 21. For
example, if n = 7 then the optimal sequence is [+ + ◦ − ◦−] so M(7, 0) = 6.
(When multiple solutions exist, you should pick the one which goes “fast
early” i.e. with all the +’s at the beginning.)
(b) Assume n ≥ 0. By extrapolating patterns in the sequences from part (a),
explain why the general formula for M(n, 0) is
M(n, 0) =
⌈
2
√
n
⌉
,
where ⌈z⌉ denotes z rounded up to the nearest integer.
Hint: Do not try to use recurrence relations. You should instead use this
identity:
⌈
2
√
n
⌉
=


2s+ 1, if n = s2 + k, 1 ≤ k ≤ s
2s+ 2, if n = s2 + s+ k, 1 ≤ k ≤ s
2s+ 2, if n = (s+ 1)2
(c) Assuming the result from part (b), show that if k ≥ 0 and n ≥ 1
2
k(k−1)
then
M(n, k) =
⌈
2
√
n + 1
2
k(k+1)
⌉
− k
Hint: Consider the path of the agent as part of a larger path.
(d) Derive a formula for M(n, k) in the case where k ≥ 0 and n < 1
2
k(k−1).
(e) Write down an admissible heuristic (that approximates the true number
of moves as closely as possible) for the original 2-dimensional GPGP game in
terms of the function M() derived above.
Hint: Consider the horizontal and vertical motion separately, keeping in mind
that, unlike the 15-puzzle, the agent can move in the horizontal and vertical
direction simultaneously. Your heuristic should be of this form:
h (r, c, u, v, rG, cG) = max(M(.., ..),M(.., ..))
5
Question 4: Game Trees and Pruning (3 marks)
(a) Consider a game tree of depth 4, where each internal node has exactly
two children (shown below). Fill in the leaves of this game tree with all
of the values from 0 to 15, in such a way that the alpha-beta algorithm
prunes as many nodes as possible. Hint: make sure that, at each branch
of the tree, all the leaves in the left subtree are preferable to all the leaves
in the right subtree (for the player whose turn it is to move).
MIN
MAX
MAX
MIN
(b) Trace through the alpha-beta search algorithm on your tree, clearly show-
ing which of the original 16 leaves are evaluated.
(c) Now consider another game tree of depth 4, but where each internal node
has exactly three children. Assume that the leaves have been assigned
in such a way that the alpha-beta algorithm prunes as many nodes as
possible. Draw the shape of the pruned tree. How many of the original
81 leaves will be evaluated?
Hint: If you look closely at the pruned tree from part (b) you will see
a pattern. Some nodes explore all of their children; other nodes explore
only their leftmost child and prune the other children. The path down
the extreme left side of the tree is called the line of best play or Principal
Variation (PV). Nodes along this path are called PV-nodes. PV-nodes
explore all of their children. If we follow a path starting from a PV-node
but proceeding through non-PV nodes, we see an alternation between
nodes which explore all of their children, and those which explore only
one child. By reproducing this pattern for the tree in part (c), you should
be able to draw the shape of the pruned tree (without actually assigning
values to the leaves or tracing through the alpha-beta algorithm).
(d) What is the time complexity of alpha-beta search, if the best move is
always examined first (at every branch of the tree)? Explain why.
6
Submission
This assignment must be submitted electronically – either through WebCMS,
or from the command line, using:
give cs3411 hw2 hw2.pdf
The submission deadline is Tuesday 2nd April, 2pm. Late submissions will
incur a penalty of 5% per day, up to a maximum of 5 days.
Group submissions will not be allowed. By all means, discuss the assignment
with your fellow students. But you must write (or type) your answers indi-
vidually. Do not copy anyone else’s assignment, or send your assignment to
any other student.
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MAS316/414 ACCT2522 MSIN0181 ATMS 201 COMP3425 MATH7861 SOLA2060 MA 325 7CCEMCTH ECON 432 INFT2051 SOLA 2060 ECOS2002 Stats141XP BIOA02H3 S2022 MA4601 K5 COMP5329 EG 284 IB3A70 WECO1020 COMP3027 Energy ACS61011 MAST20026 STAT 533 MATA31 ENVS257 BEM2031 EC 486 COMP0172 CS3214 EECE5644 IB2B20 COSC2536 COSC1147 COMP6214 5CCM242A COMP0130 AMME2000 MTMG50: EG238 ACF6006 QBUS2810 COSC2391 ECON7560 MATH377 STA305 CS 486 COMP1117B EL648 ELEC3909 ECON 241 COMP 4901W BUSI4428 ENEN20002 ENGN4524 ELEC3115 CSC 413 PSYC 100B ECON1002 MATH256 ANSC20005 MTHS2008 MATH40082 COSC 2406 FIT3174 STA 238 CMT202 Differential Equations MTHM017 GARCH 101 MATH2069 CSC336S ACSC12 ETF2121 CEME 2001 DB B474F ARHT1001 EECS 3311 3220 STATS 786 CS 484 COMP20008 MATH4091 ECE 455 CSCI 204 BUS B200F BA 875 BU5562 COSC 1147 X11 GEOG 101 IMM250F CSE 584A MATH38172 LAB 4 ELEC5882M FINS5523 EE 430 COMP 465 6CCS3PRE 7CCSMNSE FINANCE ACMB02H3 ENGG2112 ELEC5616 SCM B371 CHEM3120 CS22 ECMT2130 MAST30011 ADSPBF706 CENG0013 ACCOUNTING BUSS1040 CHEM 181 MAT136H1 GY 6183 COMP3170 AGRI90089 PHYSICS 1002 ECON10003 IEOR E4004 ACS61012 MATH 40550 MKTG2113 TELE4653 ENSC1004 ECE 232E MK726 MTRX2700 SHEET 3 CSCA08 COMP30027 Math 2XX3 statistic MMAN3200 BISM7213 ECMM109 ECON 2022 FNCE 20005 GR5067 CSCB09 ME 3017 CS 5100 PHYS 2903 COMS 4771 COMP222 COMP 1046 CSC148H1 ISYS2038 FINS2615 OT5206 COMP2048 A6 MECH5770M MA826/20 CMPT 726 STA457 INFS6071 INFS588 integral ECON 216 IRDR 0017 MA323 PA 15213 PSYC2001 ENV222 Statistics ACCT5930 ACCT12 LAND2121 ME 5405 COMM1180 TELE3113 A7 ECE 568 ACCT2019 INFS1200 ENGG1340 CS 1400 ASST3 FINM7409 SWEN30006 INFO2222 FIN20150 SOSS1000 CSSE7231 FINS5513 COMS3200 CECN 702 ECON4060H MN3515 HW2 PSYA02H3 ECON 2112 COMP90051 CS 1652 MA30059 AF1605 MKTG7501 ECO-6003B COM4521 CTM ECON3350 COMP 3804 ELEC9762 BISM 7255 ST22 MATH20014 MNE3122 COMP343 MATH5165 ELE7029 ENGR30002 SEEM3500 ECON7350 RISK5002 FINM7405 COMP90059 MOS 3320B STAT 431 BISM7221 MATH3014 MTH783P EBA 35302 ACS133 BUS2206 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