Search, Trees, Games, Backtracking

Trees help with search





Set, map: binary search tree, balanced and otherwise
Quadtree and more, scenes and objects in games
3-4 Trees, KD-trees, search for files or points in “space”
Gametrees, search for game-playing
Intelligent search



How do we find the “best move” in a game?
How do we find optimal needle in haystack?
When is brute-force, exhaustive search (im)possible?
CPS 100, Spring 2009
8.1
Search, Backtracking,Heuristics

How do you find a needle in a haystack?


How does a computer play chess?
Why would you write that program?


How does Mapquest/Googlemap find routes from
one place to another?



Shortest path algorithms
Longest path algorithms
Optimal algorithms and heuristic algorithms


When is close good enough? How do measure “closeness”?
When is optimality important, how much does it cost?
CPS 100, Spring 2009
8.2
Exhaustive Search/Heuristics

We use binary search trees to organize data, in
searching we don’t need to examine all the data to
find what we’re looking for



What do we do when the search space is huge?



Where is the smallest item in a search tree? Largest?
How are trees balanced?
How many chess boards are there?
How many routes are there between my house and yours?
Exhaustive search: look at everything!
CPS 100, Spring 2009
8.3
Classic problem: N queens

Can queens be placed on a
chess board so that no queens
attack each other?



Make the board NxN, this is
the N queens problem



Place one queen/column
Horiz/Vert/Diag attacks
Backtracking




Easily place two queens
What about 8 queens?
Tentative placement
Recurse, if ok done!
If fail, undo tentative, retry
wikipedia-n-queens
CPS 100, Spring 2009
8.4
Backtracking idea with N queens

Try to place a queen in each column in turn


Try first row in column C, if ok, move onto next column
If solved, great, otherwise try next row in column C, place queen,
move onto the next column
• Must unplace the placed queen to keep going

What happens when we start in a column, where to start?


If we fail, move back to previous column (which remembers
where it is/failed)
When starting in a column anew, start at beginning
• When backing up, try next location, not beginning

Backtracking in general, record an attempt go forward

If going forward fails, undo the record and backup
CPS 100, Spring 2009
8.5
N queens backtracking: Queens.java
public boolean solve(int col){
if (col == mySize) return true;
// try each row until all are tried
for(int r=0; r < mySize; r++){
if (myBoard.safeToPlace(r,col)){
myBoard.setQueen(r,col,true);
if (solve(col+1)){
return true;
}
myBoard.setQueen(r,col,false);
}
}
return false;
}
CPS 100, Spring 2009
8.6
Basic ideas in backtracking search

We need to be able to enumerate all possible choices/moves


We try these choices in order, committing to a choice
If the choice doesn’t pan out we must undo the choice
• This is the backtracking step, choices must be undoable

Process is inherently recursive, so we need to know when the
search finishes



When all columns tried in N queens
When we have found the exit in a maze
When every possible moved tried in Tic-tac-toe or chess?
• Is there a difference between these games?

Summary: enumerate choices, try a choice, undo a choice, this
is brute force search: try everything
CPS 100, Spring 2009
8.7
Pruning vs. Exhaustive Search

If we consider every possible placement of 4 queens
on a 4x4 board, how many are there? (N queens)



What about if we avoid diagonal attacks?


4x4x4x4 if we don’t pay attention to any attacks
4x3x2x1 if we avoid attacks in same row
Pruning search space makes more search possible, still
could be lots of searching to do!
Estimate how long to calculate # solutions to the Nqueens problem with our Java code….
CPS 100, Spring 2009
8.8
Queens Details

How do we know when it’s safe to place a queen?




No queen in same row, or diagonal
For each column, store the row that a queen is in
See QBoard.java for details
For GUI version, we use a decorator



The QBoardGUI is an IQueenState class and it has an
IQueenState object in it
Appears as an IQueenState to client, but uses an existing
one to help do its work
One of many object oriented design patterns, seen in Huff
in the BitInputStream class
CPS 100, Spring 2009
8.9
Daphne Koller


2004, Macarthur
2008, first ACM/Infosys
“The world is noisy and messy
…You need to deal with the noise
and uncertainty.”
“I find it distressing that the view of

the field is that you sit in your office
by yourself surrounded by old pizza
boxes and cans of Coke, hacking
away at the bowels of the Windows
operating system,” she said. “I spend
most of my time thinking about
things like how does a cell work or
how do we understand images in the
http://tinyurl.com/3tdlug
world around us?”
CPS 100, Spring 2009
8.10
Computer v. Human in Games

Computers can explore a large
search space of moves quickly


Computers cannot explore every
move (why) so must use heuristics



Rules of thumb about position,
strategy, board evaluation
Try a move, undo it and try
another, track the best move
What do humans do well in these
games? What about computers?

CPS 100, Spring 2009
How many moves possible in
chess, for example?
What about at Duke?
8.11
Games at Duke

Alan Biermann




Tom Truscott



CPS 100, Spring 2009
Natural language processing
Compsci 1: Great Ideas
Duchess, checkers, chess
Duke undergraduate working
with/for Biermann
Usenet: online community
Second EFF Pioneer Award (with
Vint Cerf!)
8.12
Heuristics

A heuristic is a rule of thumb, doesn’t always work, isn’t
guaranteed to work, but useful in many/most cases


What heuristic is good for Sudoku?




Search problems that are “big” often can be approximated or
solved with the right heuristics
Is there always a no-reasoning move, e.g., 5 goes here?
What about “if I put a 5 here, then…”
Do something else?
http://en.wikipedia.org/wiki/Algorithmics_of_sudoku
What other optimizations/improvements can we make?

For chess, checkers: good heuristics, good data structures
CPS 100, Spring 2009
8.13
Boggle Program
CPS 100, Spring 2009
8.14
Boggle Search for Word

Starting at board location (row,col) to find a string s



How do we know when we’re done?



We want to keep track of where we are in the string
We want to keep track of what board locations we’ve used
Base case of recursive, backtracking call
Where we are in the string?
How do we keep track of used locations?


Store in array list: tentatively use current one, recurse
If we don’t succeed, take off the last one stored!
CPS 100, Spring 2009
8.15
Backtracking, minimax, game search

We’ll use tic-tac-toe to illustrate the idea, but it’s a silly game
to show the power of the method


Minimax idea: two players, one maximizes score, the other
minimizes score, search complete/partial game tree for best
possible move



What games might be better? Problems?
In tic-tac-toe we can search until the end-of-the game, but this
isn’t possible in general, why not?
Use static board evaluation functions instead of searching all the
way until the game ends
Minimax leads to alpha-beta search, then to other rules and
heuristics
CPS 100, Spring 2009
8.16
Minimax, see TicTac.java

Players alternate, one might be
computer, one human (or two
computer players)

Simple rules: win scores +10,
loss scores –10, tie is zero



What happens otherwise?
As game tree is explored is
there redundant search?

X
O
What can we do about this?
CPS 100, Spring 2009
X X
O
X
X
O
X maximizes, O minimizes
Assume opponent plays smart

X
X
O
O
X X
X
O
O
X
X XO
O
X
X
O
X X
O
O
8.17
Interlude for trees


Joyce Kilmer
Balanced Trees




Splay
Red-Black
AVL
B-tree
CPS 100, Spring 2009
8.18
Tree functions (repeat, review)

Compute height of a tree, what is complexity?
int height(Tree root){
if (root == null) return 0;
else {
return 1 + Math.max(height(root.left),
height(root.right) );
}
}

Modify function to compute number of nodes in a
tree, does complexity change?

What about computing number of leaf nodes?
CPS 100, Spring 2009
8.19
Balanced Trees and Complexity

A tree is height-balanced if
 Left and right subtrees are height-balanced
 Left and right heights differ by at most one
boolean isBalanced(Tree root){
if (root == null) return true;
return
isBalanced(root.left) && isBalanced(root.right) &&
Math.abs(height(root.left) – height(root.right)) <= 1;
}
}
CPS 100, Spring 2009
8.20
Rotations and balanced trees


Height-balanced trees
 For every node, left and
right subtree heights differ
by at most 1
 After insertion/deletion
need to rebalance
 Every operation leaves tree
in a balanced state: invariant
property of tree
Find deepest node that’s
unbalanced then make sure:
 On path from root to
inserted/deleted node
 Rebalance at this
unbalanced point only
CPS 100, Spring 2009
Are these trees heightbalanced?
8.21
What is complexity?

Assume trees are “balanced” in analyzing
complexity



How to develop recurrence relation?



Roughly half the nodes in each subtree
Leads to easier analysis
What is T(n)?
What other work is done?
How to solve recurrence relation


Plug, expand, plug, expand, find pattern
A real proof requires induction to verify correctness
CPS 100, Spring 2009
8.22
Balanced trees we won't study

B-trees are used when data is both in memory and
on disk



File systems, really large data sets
Rebalancing guarantees good performance both
asymptotically and in practice. Differences between cache,
memory, disk are important
Splay trees rebalance during insertion and during
search, nodes accessed often more closer to root

Other nodes can move further from root, consequences?
• Performance for some nodes gets better, for others …

No guarantee running time for a single operation, but
guaranteed good performance for a sequence of
operations, this is good amortized cost (ArrayList.add)
CPS 100, Spring 2009
8.23
Balanced trees we will study


Both kinds have worst-case O(log n) time for tree
operations
AVL (Adel’son-Velskii and Landis), 1962




Nodes are “height-balanced”, subtree heights differ by 1
Rebalancing requires per-node bookkeeping of height
http://webpages.ull.es/users/jriera/Docencia/AVL/AVL tree
applet.htm
Red-black tree uses same rotations, but can
rebalance in one pass, contrast to AVL tree



In AVL case, insert, calculate balance factors, rebalance
In Red-black tree can rebalance on the way down, code is
more complex, but doable
Standard java.util.TreeMap/TreeSet use red-black
CPS 100, Spring 2009
8.24
Rotation doLeft (see AVLSet.java)

N
N
C
A
B
A
B
C

Why is this called doLeft?
 N will no longer be root,
new value in left.left
subtree
 Left child becomes new
root
Unbalanced by two (not one!)
 If left, left (or right, right)
• doLeft (doRight)
Node doLeft(Node root)
 Otherwise need two
{
• doLeft/doRight
Node newRoot = root.left;  First to get to left, left
root.left = newRoot.right;
• Or to right, right
newRoot.right = root;
return newRoot;
}
CPS 100, Spring 2009
8.25
Rotation to rebalance
?????
N
N
C
A
Suppose we add a new node in
right subtree of left child of root
 Single rotation can’t fix
 Need to rotate twice

C First stage is shown at bottom
 Rotate blue node right

A
B
B
• (its right child takes its place)

C
A
B1
B2
CPS 100, Spring 2009
B1
A
B2
This is left child of unbalanced
Node doRight(Node root)
{
Node newRoot = root.right;
root.right = newRoot.left;
newRoot.left = root;
return newRoot;
}
8.26
Double rotation complete

Calculate where to rotate and what
case, do the rotations
Node doRight(Node root)
{
Node newRoot = root.right;
root.right = newRoot.left;
newRoot.left = root;
return newRoot;
}
C
A
B1
B1
B2
C
A
Node doLeft(Node root)
{
Node newRoot = root.left;
root.left = newRoot.right;
newRoot.right = root;
return newRoot;
}
B1
A
CPS 100, Spring 2009
B2
B2
C
8.27
AVL tree practice
Insert into AVL tree:
 18 10 16 12 6 3 8 13 14
 After adding 16: doLeftRight

18
10
10
doLeft
18
18
10
doRight
16
16
16
10
6
16
6
10
18
18
12
After 3, doLeft on 16
10
10
10
18
12
18
16
10

16
6
6
3
16
16
12
18
3
8
12
18
3
CPS 100, Spring 2009
8.28
AVL practice: continued, and finished


After adding 13, ok
After adding14, not ok
 doRight at 12
10
6
3
10
6
3
16
8
12
16
8
12
10
18
6
13
14
3
16
8
13
12
CPS 100, Spring 2009
18
18
14
8.29
Dame Wendy Hall
Multimedia, “web before the
web”, President of ACM,
cofounder of WSRI (web science
research institute)

"....We are living in very
unpredictable times....Every
country, whether developed or the
developing, need people with IT
skills....What we need are people
who can innovate. I think there is a
unique opportunity to say these
are the types of skills we need
young people to have going
forward...."
CPS 100, Spring 2009
8.30