Back tracking and branch and bound class 20

1. Backtracking & Branch and Bound

2. 2 Backtracking Suppose you have to make a series of decisions, among various choices, where You don’t have enough information to know what to choose Each decision leads to a new set of choices Some sequence of choices (possibly more than one) may be a solution to your problem Backtracking is a methodical way of trying out various sequences of decisions, until you find one that “works”

3. 3 Backtracking Algorithm Based on depth-first recursive search Approach Tests whether solution has been found If found solution, return it Else for each choice that can be made Make that choice Recur If recursion returns a solution, return it If no choices remain, return failure Some times called “search tree”

4. Backtracking Performs a depth-first traversal of a tree Continues until it reaches a node that is non-viable or non-promising Prunes the sub tree rooted at this node and continues the depth-first traversal of the tree

5. 5 Backtracking Algorithm – Example Find path through maze Start at beginning of maze If at exit, return true Else for each step from current location Recursively find path Return with first successful step Return false if all steps fail

6. Backtracking Backtracking is a technique used to solve problems with a large search space, by systematically trying and eliminating possibilities. A standard example of backtracking would be going through a maze. At some point in a maze, you might have two options of which direction to go: Junction Portion A Portion B

7. Backtracking Junction Portion B Portion A One strategy would be to try going through Portion A of the maze. If you get stuck before you find your way out, then you "backtrack" to the junction. At this point in time you know that Portion A will NOT lead you out of the maze, so you then start searching in Portion B

8. Backtracking Clearly, at a single junction you could have even more than 2 choices. The backtracking strategy says to try each choice, one after the other, if you ever get stuck, "backtrack" to the junction and try the next choice. If you try all choices and never found a way out, then there IS no solution to the maze. Junction B C A

9. Backtracking – Eight Queens Problem Find an arrangement of 8 queens on a single chess board such that no two queens are attacking one another. In chess, queens can move all the way down any row, column or diagonal (so long as no pieces are in the way). Due to the first two restrictions, it's clear that each row and column of the board will have exactly one queen.

10. The backtracking strategy is as follows: Place a queen on the first available square in row 1 . Move onto the next row, placing a queen on the first available square there (that doesn't conflict with the previously placed queens). Continue in this fashion until either: you have solved the problem, or you get stuck. When you get stuck, remove the queens that got you there, until you get to a row where there is another valid square to try. Backtracking – Eight Queens Problem Animated Example: http://www.hbmeyer.de/backtrack/achtdamen/eight.htm#up Q Q Q Q Q Q Continue…

11. Backtracking – Eight Queens Problem When we carry out backtracking, an easy way to visualize what is going on is a tree that shows all the different possibilities that have been tried. On the board we will show a visual representation of solving the 4 Queens problem (placing 4 queens on a 4x4 board where no two attack one another).

12. Backtracking – Eight Queens Problem The neat thing about coding up backtracking, is that it can be done recursively, without having to do all the bookkeeping at once. Instead, the stack or recursive calls does most of the bookkeeping (ie, keeping track of which queens we've placed, and which combinations we've tried so far, etc.)

13. void solveItRec ( int perm[], int location, struct onesquare usedList []) { if (location == SIZE) { printSol (perm); } for ( int i =0; i <SIZE; i ++) { if ( usedList [ i ] == false) { if (!conflict(perm, location, i )) { perm[location] = i ; usedList [ i ] = true; solveItRec (perm, location+1, usedList ); usedList [ i ] = false; } } } } perm[] - stores a valid permutation of queens from index 0 to location-1. location – the column we are placing the next queen usedList[] – keeps track of the rows in which the queens have already been placed. Found a solution to the problem, so print it! Loop through possible rows to place this queen. Only try this row if it hasn’t been used Check if this position conflicts with any previous queens on the diagonal mark the queen in this row mark the row as used solve the next column location recursively un-mark the row as used, so we can get ALL possible valid solutions.

14. Backtracking – 8 queens problem - Analysis Another possible brute-force algorithm is generate the permutations of the numbers 1 through 8 (of which there are 8! = 40,320), and uses the elements of each permutation as indices to place a queen on each row. Then it rejects those boards with diagonal attacking positions. The backtracking algorithm, is a slight improvement on the permutation method, constructs the search tree by considering one row of the board at a time, eliminating most non-solution board positions at a very early stage in their construction. Because it rejects row and diagonal attacks even on incomplete boards, it examines only 15,720 possible queen placements. A further improvement which examines only 5,508 possible queen placements is to combine the permutation based method with the early pruning method: The permutations are generated depth-first, and the search space is pruned if the partial permutation produces a diagonal attack

15. Sudoku and Backtracking Another common puzzle that can be solved by backtracking is a Sudoku puzzle. The basic idea behind the solution is as follows: Scan the board to look for an empty square that could take on the fewest possible values based on the simple game constraints. If you find a square that can only be one possible value, fill it in with that one value and continue the algorithm. If no such square exists, place one of the possible numbers for that square in the number and repeat the process. If you ever get stuck, erase the last number placed and see if there are other possible choices for that slot and try those next.

16. Mazes and Backtracking A final example of something that can be solved using backtracking is a maze. From your start point, you will iterate through each possible starting move. From there, you recursively move forward. If you ever get stuck, the recursion takes you back to where you were, and you try the next possible move. In dealing with a maze, to make sure you don't try too many possibilities, one should mark which locations in the maze have been visited already so that no location in the maze gets visited twice. (If a place has already been visited, there is no point in trying to reach the end of the maze from there again.

17. Example: N-Queens Problem Given an N x N sized chess board Objective: Place N queens on the board so that no queens are in danger

18. One option would be to generate a tree of every possible board layout This would be an expensive way to find a solution

19. Backtracking Backtracking prunes entire sub trees if their root node is not a viable solution The algorithm will “backtrack” up the tree to search for other possible solutions

20. Efficiency of Backtracking This given a significant advantage over an exhaustive search of the tree for the average problem Worst case: Algorithm tries every path, traversing the entire search space as in an exhaustive search

21. Branch and Bound Where backtracking uses a depth-first search with pruning, the branch and bound algorithm uses a breadth-first search with pruning Branch and bound uses a queue as an auxiliary data structure

22. The Branch and Bound Algorithm Starting by considering the root node and applying a lower-bounding and upper-bounding procedure to it If the bounds match, then an optimal solution has been found and the algorithm is finished If they do not match, then algorithm runs on the child nodes

23. Example: The Traveling Salesman Problem Branch and bound can be used to solve the TSP using a priority queue as an auxiliary data structure An example is the problem with a directed graph given by this adjacency matrix:

24. Traveling Salesman Problem The problem starts at vertex 1 The initial bound for the minimum tour is the sum of the minimum outgoing edges from each vertex Vertex 1           min (14, 4, 10, 20)       =  4 Vertex 2           min (14, 7, 8, 7)          =  7 Vertex 3           min (4, 5, 7, 16)           =  4 Vertex 4           min (11, 7, 9, 2)           =  2 Vertex 5           min (18, 7, 17, 4)        =  4 Bound                                                 = 21

25. Traveling Salesman Problem Next, the bound for the node for the partial tour from 1 to 2 is calculated using the formula: Bound = Length from 1 to 2 + sum of min outgoing edges for vertices 2 to 5 = 14 + (7 + 4 + 2 + 4) = 31

26. Traveling Salesman Problem The node is added to the priority queue The node with the lowest bound is then removed This calculation for the bound for the node of the partial tours is repeated on this node The process ends when the priority queue is empty

27. Traveling Salesman Problem The final results of this example are in this tree: The accompanying number for each node is the order it was removed in

28. Efficiency of Branch and Bound In many types of problems, branch and bound is faster than branching, due to the use of a breadth-first search instead of a depth-first search The worst case scenario is the same, as it will still visit every node in the tree