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LEVEL II, TERM II CSE – 243 MD. MONJUR-UL-HASAN LECTURER DEPT OF CSE, CUET EMAIL: MAIL@MONJUR-UL-HASAN.INFO Backtracking Algorithm

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An Example Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET 2 Find out all 3-bit binary numbers for which the sum of the 1's is greater than or equal to 2. The only way to solve this problem is to check all the possibilities: (000, 001, 010,....,111) The 8 possibilities are called the search space of the problem. They can be organized into a tree.

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An Example (cont…) Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET 3 0 0 _ 0 0 00 0 1 0 1 _ 0 1 00 1 1 1 _ _0 _ _ 1 0 _ 1 0 01 0 1 1 1 _ 1 1 01 1 1

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Backtracking: Definition 4 Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET Start Success! Failure For some problems, the only way to solve is to check all possibilities. Backtracking is a systematic way to go through all the possible configurations of a search space. Problem space consists of states (nodes) and actions (paths that lead to new states). When in a node can only see paths to connected nodes If a node only leads to failure go back to its "parent“ node. Try other alternatives. If these all lead to failure then more backtracking may be necessary.

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Goals of Backtracking Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET 5 Possible goals Find a path to success Find all paths to success Find the best path to success Not all problems are exactly alike, and finding one success node may not be the end of the search Start Success!

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Backtracking and Recursive Backtracking is easily implemented with recursion because: 1. The run-time stack takes care of keeping track of the choices that got us to a given point. 2. Upon failure we can get to the previous choice simply by returning a failure code from the recursive call. 6 Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET

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Improving Backtracking: Search Pruning Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET 7 Search pruning will help us to reduce the search space and hence get a solution faster. The idea is to a void those paths that may not lead to a solutions as early as possible by finding contradictions so that we can backtrack immediately without the need to build a hopeless solution vector.

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A More Concrete Example: Suduku Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET 8 8 9 by 9 matrix with some numbers filled in all numbers must be between 1 and 9 Goal: Each row, each column, and each mini matrix must contain the numbers between 1 and 9 once each no duplicates in rows, columns, or mini matrices

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Suduku with Brut force Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET 9 A brute force algorithm is a simple but general approach Try all combinations until you find one that works This approach isn’t clever, but computers are fast Then try and improve on the brute force resuts

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Sodoku (Cont…) Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET 10 After placing a number in a cell is the remaining problem very similar to the original problem? Brute force Sudoku Soluton if not open cells, solved scan cells from left to right, top to bottom for first open cell When an open cell is found start cycling through digits 1 to 9. When a digit is placed check that the set up is legal now solve the board 1

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Solving Sudoku – Later Steps Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET 11 112124 1248 12489 uh oh!

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Solving Sudoku – Dead End Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET 12 We have reached a dead end in our search With the current set up none of the nine digits work in the top right corner 12489

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Solving Suduku - Backing Up Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET 13 When the search reaches a dead end in backs up to the previous cell it was trying to fill and goes onto to the next digit We would back up to the cell with a 9 and that turns out to be a dead end as well so we back up again so the algorithm needs to remember what digit to try next Now in the cell with the 8. We try and 9 and move forward again. 12489 1249

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Problem with Suduku Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET 14 Brute force algorithms are slow The don't employ a lot of logic For example we know a 6 can't go in the last 3 columns of the first row, but the brute force algorithm will plow ahead any way But, brute force algorithms are fairly easy to implement as a first pass solution backtracking is a form of a brute force algorithm

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Solving Suduku-Search pruning Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET 15 After trying placing a digit in a cell we want to solve the new sudoku board Isn't that a smaller (or simpler version) of the same problem we started with?!?!?!? After placing a number in a cell the we need to remember the next number to try in case things don't work out. We need to know if things worked out (found a solution) or they didn't, and if they didn't try the next number If we try all numbers and none of them work in our cell we need to report back that things didn't work

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Solving Suduku-Recursive Backtracking Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET 16 Problems such as Suduko can be solved using recursive backtracking recursive because later versions of the problem are just slightly simpler versions of the original backtracking because we may have to try different alternatives

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General Back Tracking Algorithm Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET 17 void checknode (node v) { if (promising ( v )) if (aSolutionAt( v )) write the solution else //expand the node for ( each child u of v ) checknode ( u ) }

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General Back Tracking Algorithm (Cont…) Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET 18 Checknode uses the functions: promising(v) which checks that the partial solution represented by v can lead to the required solution aSolutionAt(v) which checks whether the partial solution represented by node v solves the problem.

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Eight Queen Problem: A Classic Puzzle Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET 19 Attempts to place 8 queens on a chessboard in such a way that no queen can attack any other. A queen can attack another queen if it exists in the same row, columm or diagonal as the queen. This problem can be solved by trying to place the first queen, then the second queen so that it cannot attack the first, and then the third so that it is not conflicting with previously placed queens.

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N Queen Problem (N=8) Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET 20 Place N Queens on an N by N chessboard so that none of them can attack each other Number of possible placements? In 8 x 8 64 * 63 * 62 * 61 * 60 * 58 * 59 * 57 = 4,426,165,368 roughly four and a half billion possible set ups n choose k How many ways can you choose k things from a set of n items? In this case there are 64 squares and we want to choose 8 of them to put queens on

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Solution: N Queen Puzzle Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET 21 We know that for queens: 1. each row will have exactly one queen 2. each column will have exactly one queen 3. each diagonal will have at most one queen The previous calculation includes set ups like following: Includes lots of set ups with multiple queens in the same column Number of set ups 8 * 8 * 8 * 8 * 8 * 8 * 8 * 8 = 16,777,216 We have reduced search space by two orders of magnitude by applying some logic

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N Queen Solution: Chess board Store Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET 22 The previous calculation help us to model the chessboard not as a 2-D array, but as a set of rows, columns and diagonals. To simplify the presentation, we will study for smaller chessboard, 4 by 4 First: we need to define an array to store the location of so far placed queens PositionInRow 1 3 0 2

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N Queen Solution: Chess board Store (Cont..) Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET 23 We need an array to keep track of the availability status of the column when we assign queens. FTFT Suppose that we have placed two queens

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N Queen Solution: Chess board Store (Cont..) Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET 24 We have 7 left diagonals, we want to keep track of available diagonals after the so far allocated queens T T F T F T T

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N Queen Solution: Chess board Store (Cont..) Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET 25 We have 7 left diagonals, we want to keep track of available diagonals after the so far allocated queens T F T T F T T

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N Queen Solution: Chess board Store (Cont..) Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET 26 Now Run a loop from 1 to 4 with counter i and place a Queen to safe position of the i th column. Now try to make a solution for N Queen. Try to find all possible solution. [4,3] [3,1] [2,4] [1,2]

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N Queen Puzzle: Algorithm Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET 27 putQueen(int row) { for col=1 to N if (column[col]==available && leftDiagonal[row+col]==available && rightDiagonal[row-col+norm]== available) { positionInRow[row]=col; column[col]=!available; leftDiagonal[row+col]=!available; rightDiagonal[row-col+norm]=!available; if (row< squares-1) putQueen(row+1); else PRINT solution found column[col]=available; leftDiagonal[row+col]=available; rightDiagonal[row-col+norm]= available; } Base Update Forward Recursi. Back Track

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Backtracking Algorithm for optimization problems Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET 28 Procedure checknode (node v ) { node u ; if ( value(v) is better than best ) best = value(v); if (promising (v) ) for (each child u of v) checknode (u ); } best is the best value so far and is initialized to a value that is equal or worse than any possible solution. value(v) is the value of the solution at the node.

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Backtracking for optimization problems (Cont…) Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET 29 To deal with optimization we compute: best - value of best solution achieved so far value(v) - the value of the solution at node v Modify promising(v) Best is initialized to a value that is equal to a candidate solution or worse than any possible solution. Best is updated to value(v) if the solution at v is “better” By “better” we mean: larger in the case of maximization and smaller in the case of minimization

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Modifying promising Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET 30 A node is promising when it is feasible and can lead to a feasible solution and “there is a chance that a better solution than best can be achieved by expanding it” Otherwise it is nonpromising A bound on the best solution that can be achieved by expanding the node is computed and compared to best If the bound > best for maximization, (< best for minimization) the node is promising

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Modifying promising for Maximization Problems Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET 31 For a maximization problem the bound is an upper bound, the largest possible solution that can be achieved by expanding the node is less or equal to the upper bound If upper bound > best so far, a better solution may be found by expanding the node and the feasible node is promising

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Modifying promising for Minimization Problems Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET 32 For minimization the bound is a lower bound, the smallest possible solution that can be achieved by expanding the node is less or equal to the lower bound If lower bound < best a better solution may be found and the feasible node is promising

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Optimization Example:0-1 Knapsack Problem The 0-1 knapsack problem: N items which cost p 1,p 2,p 3 ….p n and have weight w 1, w 2,w 3 ….w n. (where the i-th item is worth p i dollars and weight w i pounds.) v i and w i are integers. A thief can carry at most C (integer) pounds. How to take as valuable a load as possible. An item cannot be divided into pieces. The fractional knapsack problem: The same setting, but the thief can take fractions of items. 33 Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET

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Knapsack Problem: In Mathematical Term Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET 34 Input description: A set of items 1 n, where item i has size w i and value p i, and a knapsack capacity C Problem description: Find a subset S that maximizes given that i.e. all the items fit in a knapsack of size C

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0-1 Knapsack - Example Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET 35 50 E.g.1: 10 20 30 50 Item 1 Item 2 Item 3 $60$100$120 10 20 $60 $100 + $160 50 20 $100 $120 + $220 30 $6/pound$5/pound$4/pound W E.g.2: Item: 1 2 3 4 5 6 7 Benefit:5 8 3 2 7 9 4 Weight:7 8 4 10 4 6 4 Knapsack holds a maximum of 22 pounds Fill it to get the maximum benefit

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Brute force tree Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET 36 A B C D EE D EE C D EE D EE B C D EE D EE C D EE D EE In Out Weight = 15.12 Value = $315 Weight = 8.98 Value = $235 Weight = 9.98 Value = $225

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0-1 Knapsack Problem: Brute force Algorithm Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET 37 Let’s first solve this problem with a straightforward algorithm Since there are n items, there are 2 n possible combinations of items. We go through all combinations and find the one with the most total value and with total weight less or equal to W Running time will be O(2 n ) In the tree representation, we can think of the previous algorithm doing a DFS in the tree If we reach a point where a solution no longer is feasible, there is no need to continue exploring We can “backtrack” from this point and potentially cut off much of the tree and many of the solutions We can backtrack if we know the best possible solution in current subtree is worse than current best solution obtained so far.

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0-1 Knapsack: Algorithm Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET 38 Inputs: Positive integers n and W; arrays w and p, each indexed from 1 to n, and each containing positive integers sorted in nonincreasing order according to the values of p[i]/w[i]. Outputs: an array bestset indexed from 1 to n, where the values of bestset[i] is "yes" if the ith item is included in the optimal set and is "no" otherwise; an integer maxprofit that is the maximum profit. void knapsack (index i, int profit, int weight) { if (weight maxprofit){ // This set is best maxprofit = profit; // so far. numbest = i; // Set numbest to bestset = include; // number of items } // considered. Set // bestset to this // solution. if (promising(i)){ include [i + 1] = "yes"; // Include w[i + 1]. knapsack(i + 1, profit + p[i + 1], weight + w[i + 1]); include [i + 1] = "no"; // Do not include knapsack (i + 1, profit, weight); // w[i + 1]. }

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0-1 Knapsack: Algorithm (Cont…) Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET 39 bool promising (index i) { index j, k; int totweight; float bound; if (weight >= W) // Node is promising only return false; // if we should expand to else { // its children. There must j = i + 1; // be some capacity left for bound = profit; // the children. totweight = weight;

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0-1 Knapsack: Algorithm (Cont…) Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET 40 while (j <= n && totweight + w[j] < = W){ // Grab as many items as totweight = totweight + w[j]; // possible. bound = bound + p[j]; j++; } k = j; // Use k for consistency if (k <=n) // with formula in text. bound = bound + (W - totweight) * p[k]/w[k]; // Grab fraction of kth return bound > maxprofit; // item. }

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0-1 Knapsack: Implementation (Cont…) Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET 41 numbest = 0; maxprofit = 0; knapsack(0, 0, 0); print maxprofit; // Write the maximum profit. for (j = 1; j <= numbest; j++) // Show an optimal set of items. print bestset[i];

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Subset finding problem We say that running time of an alg. is O(f(n)) If there exists c such that for large enough n: RunningTime(n) < c·f(n) Translation: From some point c·f(n) is an upper bound on the running time. Why only for “large enough n”? 42 Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET

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Other Problems Do not assume that because we analyze running time asymptotically we no longer care about the constants. If you have two algorithms with different asymptotic limits – easy to choose. (may be misleading if constants are really large). Usually we are happy when something runs in polynomial time and not exponential time. O(n 2 ) is much much better than O(2 n ). 43 Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET

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