1. Owen Astrachan ola@cs.duke.edu November 9, 2018
Compsci 201 Backtracking
Owen Astrachan
November 9, 2018
11/9/2018
Compsci 201, Fall 2018, Backtracking

2. Compsci 201, Fall 2018, Backtracking
S is for …
Stack
Last in, First Out, source of overflow!
Software
Joys and sorrows, eating the world
11/9/2018
Compsci 201, Fall 2018, Backtracking

3. Compsci 201, Fall 2018, Backtracking
Plan for Today
Backtracking: try, retry, try again
Paradigm for searching and much more
Demonstration of recursion
Lead-in to Percolation assignment and APTs
Community hygiene
How we can all succeed, thrive, move forward
11/9/2018
Compsci 201, Fall 2018, Backtracking

4. Compsci 201, Fall 2018, Backtracking
Flood Fill, Blob Count
11/9/2018
Compsci 201, Fall 2018, Backtracking

5. Compsci 201, Fall 2018, Backtracking
Applications
Using draw/paint program, fill a region
Redeye removal
Give image partially transparent background
Finding edges or blobs in image processing
11/9/2018
Compsci 201, Fall 2018, Backtracking

6. Compsci 201, Fall 2018, Backtracking
Blob Counting Ideas
How do I count “my region” or “my blob” size?
Ask my neighbors their size
Add +1 to their result
Avoid double-counting!
Colors indicate calls
White calls green
Green calls tan and gray
Gray calls mustard, …
11/9/2018
Compsci 201, Fall 2018, Backtracking

7. Compsci 201, Fall 2018, Backtracking
Bookkeeping Details
Blobs marked with ‘*’ – aka asterisk
We fill or mark cells when counting
Filling avoids counting cells twice
Filling allows for visualization as well
BlobModel.blobFill method: (row,col) for start
lookFor character, e.g., ‘*’
fillWith character, e.g., ‘1’ or ‘7’ or …
If blob not big enough? Erase: lookFor = fillWith
11/9/2018
Compsci 201, Fall 2018, Backtracking

8. Compsci 201, Fall 2018, Backtracking
Two Stage Analysis
Before looking for blobs? Make a copy of data
Data stored in two-dimensional array int[][]
char values can be stored as int values
Look at every possible blob starting location
If it's a blob? Mark with number
If it's not a blob? Erase any markings
There is a minimal size, erasing cleans up
11/9/2018
Compsci 201, Fall 2018, Backtracking

9. Lookfor for Blobs Everywhere
Two parameters in each call to blobFill
lookFor
fillWith
11/9/2018
Compsci 201, Fall 2018, Backtracking

10. Blobs starting at (row,col)
Ask each of four neighbors for their blob-size
Add me, +1, to what they report back
11/9/2018
Compsci 201, Fall 2018, Backtracking

11. Compsci 201, Fall 2018, Backtracking
When does process stop?
If (row,col) is not in bounds? Do nothing
If grid[row][col] != lookFor? Do nothing
Otherwise (what do we know here?)
In bounds AND looking for ‘*’ for example
Look at horizontal and vertical neighbors
Use results of recursive calls to create return
11/9/2018
Compsci 201, Fall 2018, Backtracking

12. Compsci 201, Fall 2018, Backtracking
blobFill(row,col)
Returns a value, make sure value used
Both in original calls
And in recursive calls
How do we know this will terminate?
Each recursive call “marks” a cell
myGrid[row][col] = fillWith
Unless fillWith == lookFor, not infinite!
11/9/2018
Compsci 201, Fall 2018, Backtracking

13. Compsci 201, Fall 2018, Backtracking
WOTO for Blobs
11/9/2018
Compsci 201, Fall 2018, Backtracking

14. Compsci 201, Fall 2018, Backtracking
Sir Tim Berners-Lee
Invented the Web, http
Not the Internet
Turing award 2016
One way to think about the magnitude of the changes to come is to think about how you went about your business before powerful Web search engines.
You probably wouldn't have imagined that a world of answers would be available to you in under a second. The next set of advances will have a different effect, but similar in magnitude.
11/9/2018
Compsci 201, Fall 2018, Backtracking

15. Level-Order aka Breadth-First
We saw level-order tree traversal using Queue
From top, then all children (one-away)
Then their children (two-away), then …
11/9/2018
Compsci 201, Fall 2018, Backtracking

16. Use BFS instead of recursion
Details: queue, parallel grid [][], rowDelta
Easier
Similar
11/9/2018
Compsci 201, Fall 2018, Backtracking

17. How to search 8 neighbors?
We need to make 8 recursive calls
W, NW, N, NE, E, SE, S, SW
See coding “trick” below
int[] rd = {0,-1,-1,-1,0,1,1,1};
int[] cd = {-1,-1,0,1,1,1,0,-1};
for(int d = 0; d < rd.length; d+= 1){
int nr = row + rd[d];
int nc = col + cd[d];
size += blobFill(nr,nc, …) }
11/9/2018
Compsci 201, Fall 2018, Backtracking

18. Backtracking and Blob Fill
Explore a move (blob fill) if it works? Fabulous!
If it does not work? Undo the move, try again
Similar to Sudoku solving? Crossword puzzles?
Tentatively try number of word, follow through
May need to undo and try alternative
11/9/2018
Compsci 201, Fall 2018, Backtracking

19. Compsci 201, Fall 2018, Backtracking
Exhaustive Search
Can explore every possible move in tic-tac-toe
Cannot explore every possible move in chess
Brute-force doesn't work
Be smart, try move? Then undo, try another
Backtracking in search tree
Smart pruning
11/9/2018
Compsci 201, Fall 2018, Backtracking

20. N-Queens: Know History
Can we place N queens on NxN board so no queen attacks another
4x4 or 8x8 or …
11/9/2018
Compsci 201, Fall 2018, Backtracking

21. Compsci 201, Fall 2018, Backtracking
Nqueen Concepts
For each column c in [0..N)
Try to place queen in each row of column c
grid[r][c] ok? Place queen, try c+1
If not ok? Or Doesn't work? Try next row, r+1
When have all queens been placed?
If c == N and success? Done!
Can't do column c? return false, c-1 continues
11/9/2018
Compsci 201, Fall 2018, Backtracking

22. Code for Nqueens Backtracking
Done when c == N
Place queens, recurse, unplace and try again
Return true
All placed
Recursive
Use myBoard
Track moves
11/9/2018
Compsci 201, Fall 2018, Backtracking

23. Compsci 201, Fall 2018, Backtracking
Backtracking Summary
Enumerate all possible moves/choices
Nqueen? Each column and each row in column
Blob-fill? Each neighbor: fill, and unfill
Board often two-dimensional array/grid
Record move, recurse, undo if not done
11/9/2018
Compsci 201, Fall 2018, Backtracking

24. Compsci 201, Fall 2018, Backtracking
Backtracking APTs
Often use grid[][] to store state/moves
In Java this is actually an array of arrays
int[][] a = new int[4][4] for example
What is a[0]? What is a[0][0]?
Often move must be explicitly undone
Sometimes just try everything
11/9/2018
Compsci 201, Fall 2018, Backtracking

25. Collaborative APT Solving
Shall we play a game?
Each player plays perfectly
If I go here, will you win?
11/9/2018
Compsci 201, Fall 2018, Backtracking

26. Compsci 201, Fall 2018, Backtracking
GridGame Ideas
Have boolean method winWithBoard(board)
For each legal move
Place on board – board now changed
Call winWithBoard(board) -- if true? I lose
Undo move --- take off board and continue
If you could see ahead, you'd know if the move was good. winWithBoard is an oracle
11/9/2018
Compsci 201, Fall 2018, Backtracking

27. What can we do with a board?
".X.X"
"..X."
".X.."
"...."
Can you determine if [r][c] is legal?
[1][0] is legal, why?
[3][1] is NOT legal, why?
Suppose there are no legal moves? Answer: Zero/0
Suppose I place an 'X' and then ask
How many ways to win does opponent have?
If answer is Zero/0, what does placing 'X' do?
This leads to backtracking, believe the code!!!
11/9/2018
Compsci 201, Fall 2018, Backtracking

28. GridGame backtracking, count wins
private int countWinners(char[][] board) {
int wins = 0;
for(int r=0; r < 4; r++){
for(int c=0; c < 4; c++){
if (canMove(board,r,c)){
board[r][c] = 'X';
int opponentWins = countWinners(board);
if (opponentWins == 0){
wins += 1; }
board[r][c] = '.';
return wins;
".X.X"
"..X."
".X.."
"...."
".X.X"
"X.X."
".X.."
"...."
".X.X"
"X.X."
"...."
11/9/2018
Compsci 201, Fall 2018, Backtracking

29. Red to try each open space and …
".X.X"
"..X."
".X.."
"...."
".X.X"
"X.X."
".X.."
"...."
".X.X"
"X.X."
"...."
".X.X"
"X.X."
"X..."
Red loses
".X.X"
"..X."
".X.."
"...."
".X.X"
"..X."
".X.."
"...X"
".X.X"
"X.X."
".X.."
"...X"
Red wins!
Not all moves shown, but result is correct, symmetry!
11/9/2018
Compsci 201, Fall 2018, Backtracking

30. GridGame Backtracking
Create grid/board, create legal move method
Return winCount(original board)
11/9/2018
Compsci 201, Fall 2018, Backtracking

31. Collaborative APT Solving
Take a step toward the cheese
If that works? Add +1 to total
If that doesn't work? Try another stop
Grid representing "arena"
Methods?
Think verbs
11/9/2018
Compsci 201, Fall 2018, Backtracking

32. Transform input to grid
["X..X.X.", "XX.C.X.", " ",
"..X.X..", " ", "R.XX..." ]
Input: String[], transform to char[][]
[0][0] is upper left, 0th row/column
Start at rat and ...
Try each step closer to cheese
X..X.X.
XX.C.X.
..X.X..
R.XX...
11/9/2018
Compsci 201, Fall 2018, Backtracking

33. Transform, Initialize, Solve
State and behavior
11/9/2018
Compsci 201, Fall 2018, Backtracking

34. Base cases for cheese-finding
Off the grid? No paths to cheese
On an 'X'? No paths to cheese
On the cheese? One path to cheese
11/9/2018
Compsci 201, Fall 2018, Backtracking

35. Try every possible step that …
Closer to the cheese only, see line 50
What do we return? Recursive help
11/9/2018
Compsci 201, Fall 2018, Backtracking

36. Compsci 201, Fall 2018, Backtracking
WOTO
11/9/2018
Compsci 201, Fall 2018, Backtracking

37. Don't Know Much about History
Usenet, Chess, Checkers, …
Alan Biermann, Tom Truscott: Internet Pioneer
11/9/2018
Compsci 201, Fall 2018, Backtracking