The document describes several algorithms related to graph searching and optimization problems: 1) Binary search and depth-first search algorithms for searching arrays and graphs. 2) Breadth-first search and shortest path algorithms using BFS for graphs. 3) Topological sort and n-queens algorithms using backtracking to solve optimization problems on graphs and boards. 4) Hamiltonian cycle algorithm applying backtracking to find cycles in graphs. The algorithms are represented using pseudocode with common structures like adjacency lists, matrices, queues, and recursion to traverse and search data structures efficiently.

1. CHAPTER 4 Searching

2. Algorithm 4.1.1 Binary Search This algorithm searches for the value key in the nondecreasing array L [ i ], ... , L[ j ]. If key is found, the algorithm returns an index k such that L [ k ] equals key . If key is not found, the algorithm returns -1, which is assumed not to be a valid index. Input Parameters: L , i , j , key Output Parameters: None bsearch ( L , i , j , key ) { while ( i = j ) { k = ( i + j )/2 if ( key == L [ k ]) // found return k if ( key < L [ k ]) // search first part j = k - 1 else // search second part i = k + 1 } return -1 // not found }

3. Algorithm 4.2.2 Depth-First Search This algorithm executes a depth-first search beginning at vertex start in a graph with vertices 1, ... , n and outputs the vertices in the order in which they are visited. The graph is represented using adjacency lists; adj [ i ] is a reference to the first node in a linked list of nodes representing the vertices adjacent to vertex i . Each node has members ver , the vertex adjacent to i , and next , the next node in the linked list or null, for the last node in the linked list. To track visited vertices, the algorithm uses an array visit ; visit[i ] is set to true if vertex i has been visited or to false if vertex i has not been visited.

4. Input Parameters: adj , start Output Parameters: None dfs ( adj , start ) { n = adj . last for i = 1 to n visit [ i ] = false dfs_recurs ( adj , start ) } dfs_recurs ( adj , start ) { println ( start ) visit [ start ] = true trav = adj [ start ] while ( trav != null) { v = trav . ver if (! visit [ v ]) dfs_recurs ( adj , v ) trav = trav . next } }

5. Algorithm 4.3.2 Breadth-First Search This algorithm executes a breadth-first search beginning at vertex start in a graph with vertices 1, ... , n and outputs the vertices in the order in which they are visited. The graph is represented using adjacency lists; adj [ i ] is a reference to the first node in a linked list of nodes representing the vertices adjacent to vertex i . Each node has members ve r, the vertex adjacent to i , and next , a reference to the next node in the linked list or null, for the last node in the linked list. To track visited vertices, the algorithm uses an array visit ; visit [ i ] is set to true if vertex i has been visited or to false if vertex i has not been visited. The algorithm uses an initially empty queue q to store pending current vertices. The expression q . enqueue ( val ) adds val to q . The expression q . front () returns the value at the front of q but does not remove it. The expression q . dequeue () removes the item at the front of q . The expression q . empty () returns true if q is empty or false if q is not empty.

6. Input Parameters: adj , start Output Parameters: None bfs ( adj , start ) { n = adj . last for i = 1 to n visit [ i ] = false visit [ start ] = true println ( start ) q . enqueue ( start ) // q is an initially empty queue while (! q . empty ()) { current = q . front () q . dequeue () trav = adj [ current ] while ( trav != null) { v = trav . ver if (! visit [ v ]) { visit [ v ] = true println ( v ) q . enqueue ( v ) } trav = trav . next } } }

7. Algorithm 4.3.4 Finding Shortest Path Lengths Using Breadth-First Search This algorithm finds the length of a shortest path from the start vertex start to every other vertex in a graph with vertices 1, ... , n . The graph is represented using adjacency lists; adj [ i ] is a reference to the first node in a linked list of nodes representing the vertices adjacent to vertex i . Each node has members ve r, the vertex adjacent to i , and next , a reference to the next node in the linked list or null, for the last node in the linked list. In the array length , length [ i ] is set to the length of a shortest path from start to vertex i if this length has been computed or to ∞ if the length has not been computed. If there is no path from start to i , when the algorithm terminates, length [ i ] is ∞ .

8. Input Parameters: adj , start Output Parameters: length shortest_paths ( adj , start , length ) { n = adj . last for i = 1 to n length [ i ] = ∞ length [ start ] = 0 q . enqueue ( start ) // q is an initially empty queue while (! q . empty ()) { current = q . front () q . dequeue () trav = adj [ current ] while ( trav != null) { v = trav . ver if ( length [ v ] == ∞) { length [ v ] = 1 + length [ current ] q . enqueue ( v ) } trav = trav . next } } }

9. Algorithm 4.4.1 Topological Sort This algorithm computes a topological sort of a directed acyclic graph with vertices 1, ... , n . The vertices in the topological sort are stored in the array ts . The graph is represented using adjacency lists; adj [ i ] is a reference to the first node in a linked list of nodes representing the vertices adjacent to vertex i . Each node has members ver , the vertex adjacent to i , and next , the next node in the linked list or null, for the last node in the linked list. To track visited vertices, the algorithm uses an array visit ; visit[i ] is set to true if vertex i has been visited or to false if vertex i has not been visited.

10. Input Parameters: adj Output Parameters: ts top_sort ( adj , ts ) { n = adj . last // k is the index in ts where the next vertex is to be // stored in topological sort. k is assumed to be global. k = n for i = 1 to n visit [ i ] = false for i = 1 to n if (! visit [ v ]) top_sort_recurs ( adj , i , ts ) } top_sort_recurs ( adj , start , ts ) { visit [ start ] = true trav = adj [ start ] while ( trav != null) { v = trav . ver if (! visit [ v ]) top_sort_recurs ( adj , v , ts ) trav = trav . next } ts [ k ] = start k = k - 1 }

11. Algorithm n -Queens, Initial Version The n -queens problem is to place n queens on an n × n board so that no two queens are in the same row, column, or diagonal. Using backtracking, this algorithm outputs all solutions to this problem. We place queens successively in the columns beginning in the left column and working from top to bottom. When it is impossible to place a queen in a column, we return to the previous column and move its queen down.

12. n_queens ( n ) { rn_queens (1, n ) } rn_queens ( k , n ) { for row [ k ] = 1 to n if ( position_ok ( k , n )) if ( k == n ) { for i = 1 to n print ( row [ i ] + “ ”) println () } else rn_queens ( k + 1, n ) } position_ok ( k , n ) for i = 1 to k - 1 // abs is absolute value if ( row [ k ] == row [ i ] || abs ( row [ k ] - row [ i ]) == k - i ) return false return true }

13. Algorithm 4.5.2 Solving the n -Queens Problem Using Backtracking The n -queens problem is to place n queens on an n × n board so that no two queens are in the same row, column, or diagonal. Using backtracking, this algorithm outputs all solutions to this problem. We place queens successively in the columns beginning in the left column and working from top to bottom. When it is impossible to place a queen in a column, we return to the previous column and move its queen down.

14. The value of row [ k ] is the row where the queen in column k is placed. The algorithm begins when n_queens calls rn_queens (1, n ). When rn_queens (k, n) is called, queens have been properly placed in columns 1 through k - 1, and rn_queens ( k , n ) tries to place a queen in column k . If it is successful and k equals n , it prints a solution. If it is successful and k does not equal n , it calls rn_queens ( k + 1, n ). If it is not successful, it backtracks by returning to its caller rn_queens ( k - 1 , n ). The value of row_used [ r ] is true if a queen occupies row r and false otherwise. The value of ddiag_used [ d ] is true if a queen occupies ddiag diagonal d and false otherwise. According to the numbering system used, the queen in column k , row r , is in ddiag diagonal n - k + r . The value of udiag_used [ d ] is true if a queen occupies udiag diagonal d and false otherwise. According to the numbering system used, the queen in column k , row r , is in udiag k + r - 1. The function position_ok ( k , n ) assumes that queens have been placed in columns 1 through k - 1. It returns true if the queen in column k does not conflict with the queens in columns 1 through k - 1 or false if it does conflict.

15. Input Parameter: n Output Parameters: None n_queens ( n ) { for i = 1 to n row_used [ i ] = false for i = 1 to 2 * n - 1 ddiag_used [ i ] = udiag_used [ i ] = false rn_queens (1, n ) } ...

16. ... // When rn_queens (k, n ) is called, queens have been // properly placed in columns 1 through k - 1. rn_queens ( k , n ) { for row [ k ] = 1 to n if ( position_ok ( k , n )) row_used [ row [ k ]] = true ddiag_used [ n - k + row [ k ]] = true udiag_used [ k + row [ k ] - 1] = true if ( k == n ) { // Output a solution. Stop if only one // solution is desired. for i = 1 to n print ( row [ i ] + “ ”) println () } else rn_queens ( k + 1, n ) row_used [ row [ k ]] = false ddiag_used [ n - k + row [ k ]] = false udiag_used [ k + row [ k ] - 1] = false } ...

17. ... // position_ok ( k , n ) returns true if the queen in column k // does not conflict with the queens in columns 1 // through k - 1 or false if it does conflict. position_ok ( k , n ) return !( row_used [ row [ k ]] || ddiag_used [ n - k + row [ k ]] || udiag_used [ k + row [ k ] - 1 ]) }

18. Form of Backtracking Algorithm Suppose that we solve a problem using backtracking as in Algorithm 4.5.2 in which the solution is of the form x [1], ... , x [ n ]. Suppose also that the values of x [ i ] are in the set S (e.g., in Algorithm 4.5.2, S = {1, . . . , n }). We require a function bound ( k ) with the following property. Whenever x [1], ... , x [ k - 1] is a partial solution and x [ k ] has been assigned a value, then bound ( k ) has to return true if x [1], ... , x [ k ] is a partial solution and false otherwise. The key to writing a useful back- tracking algorithm is to write an efficient bound function that eliminates many potential nodes from the search tree.

19. The general form of a backtracking algorithm is backtrack(n) { rbacktrack(1,n) } rbacktrack(k,n) { for each x[k]  S if (bound(k)) if (k == n) { // Output a solution. Stop if only for i = 1 to n // one solution is desired. print(x[i] + “ ”) println() } else rbacktrack(k + 1, n) }

20. Algorithm 4.5.4 Searching for a Hamiltonian Cycle This algorithm inputs a graph with vertices 1, ... , n . The graph is represented as an adjacency matrix adj ; adj [ i ][ j ] is true if ( i , j ) is an edge or false if it is not an edge. If the graph has a Hamiltonian cycle, the algorithm computes one such cycle ( x [1], ... , x [ n ], x[1]). If the graph has no Hamiltonian cycle, the algorithm returns false and the contents of the array x are not specified. In the array used , used [ i ] is true if i has been selected as one of the vertices in a potential Hamiltonian cycle or false if i has not been selected. The function path_ok ( adj , k , x ) assumes that ( x [1], ... , x [ k - 1]) is a path from x [1] to x [ k - 1] and that the vertices x [1], ... , x [ k - 1] are distinct. It then checks whether x [ k ] is different from each of x [1], ... , x [ k - 1] and whether ( x [ k - 1], x [ k ]) is an edge. If k = n , path_ok also checks whether ( x [ n ], x [1]) is an edge.

21. Input Parameter: adj Output Parameter: x hamilton ( adj , x ) { n = adj . last x [1] = 1 used [1] = true for i = 2 to n used [ i ] = false rhamilton ( adj ,2, x ) } rhamilton ( adj , k , x ) { n = adj . last for x [ k ] = 2 to n if ( path_ok ( adj , k , x )) { used [ x [ k ]] = true if ( k == n || rhamilton ( adj , k + 1, x )) return true used [ x [ k ]] = false } return false } ...

22. ... path_ok ( adj , k , x ) { n = adj . last if ( used [ x [ k ]]) return false if ( k < n ) return adj [ x [ k - 1]][ x [ k ]] else return adj [ x [ n - 1]][ x [ n ]] && adj [ x [1]][ x [ n ]] }