Algorithms

1. The Project for the Establishing the Korea ㅡ Vietnam College of Technology in BacGiang Algorithms April-May 2013 Dr. Youn-Hee Han

2. Tree • Tree consists of • A finite set of nodes (vertices) • A finite set of branches (edges, arcs) that connects the nodes • Degree of a node: # of branches • In-degree: # of branch toward the node (# of upward branch) • Out-degree: # of branch away from the node (# of downward branch) • Every non-empty tree has a root node whose in-degree is zero. • In-degree of all other nodes except the root node is one.

3. Tree • Terminology • A node is a structure which may contain a value or condition • A node is child of its predecessor • E and F are child nodes of B • A node is parent of its successor nodes • B is the parent node of E and F • A is the parent node of B, C, and D • A descendant node ofany node A is any node below A in a tree, where "below" means "away from the root." • B, E, F, C, and D are descendant nodes of A • An ancestor node of any node A is any node above A in a tree, where "above" means "toward the root." • Band A are ancestor nodes of F • Path • a sequence of nodes in which each nodeis adjacent to the next one

4. Tree • Terminology • A sibling node of A is a node with the same parent as A • B, C, and D are sibling nodes each other • E and F are sibling nodes each other • A leaf (or terminal) node is a node who hos no child nodes • C, D, E, E are leaf nodes • Node with out-degree zero

5. Tree • Terminology • A subtree of a tree T is a tree consisting of a node in T and all of its descendants in T • each node corresponds to the subtree of itself and all its descendants • each node is the root node of the subtree it determines • the subtree corresponding to the root node is the entire tree Subtree B can be divided into two subtrees, C and D

6. Tree • Recursive Definitions of Tree • A tree is a set of nodes that either: • 1. is Empty, or • 2. has a designated node, called the root, from which hierarchically descend zero or more subtrees, which are also trees. • Binary tree is a tree in which no node can have more than two subtrees • the child nodes are called left and right • Left/right subtrees are also binary trees

7. 1 1 5 3 2 2 3 4 5 4 6 6 7 7 Tree Traversal • Tree Traversal • process each node once and only once in a predetermined sequence • Two general approach • Depth-first traversal (depth-first search: DFS) • Breadth-first traversal (breadth-first search: BFS, level-order) breadth-first traversal depth-first traversal

8. Tree Traversal • Depth-first traversal • proceeds along a path from the root to the most distant descendent of that first child before processing a second child.

9. Tree Traversal • 3 types of depth-first traversal in Binary Tree • Preorder traversal • A node  left subtree right subtree • Inorder traversal • Left subtree a node right subtree • Postorder traversal • Left subtreeright subtree a node

10. Tree Traversal • Preordered depth-first traversal • Root  all subtrees of the node • When the function is returned, “backtracking” start void depth_first_tree_search(node v){ node u; visit v; // or process v for (each child u of v) // from left to right depth_first_tree_search(u) }

11. Tree Traversal • Breadth-first traversal (=level-order traversal) • Begins at the root node and explores all the neighboring nodes • Then for each of those nearest nodes, it explores their unexplored neighbor nodes, and so on, until it finds the goal  Attempts to visit the node, not already visited, closest to the root

12. Backtracking • Maze • When you reached a dead end, you’d go back to a fork and pursue another path. • Think how much easier it would be, if there were a sign, positioned to the dead end, to told you that the path led to nothing but dead ends

13. Backtracking • “Backtracking” is used to… • Solve problems in which a sequence of objects is chosen from a specified set so that the sequence satisfies some criterion • Such problem is called constraint satisfaction problem • It usually utilizes • tree data structure • tree traversal • In particular, Depth First Search (DFS) • If a more traversal (search) to current direction is determined to be useless, just backtrack to the choice point (sign). • If there is no choice points, the search to solve the problem comes to fail.

14. n-Queens Problem • n-Queens Problem • The classic example of the use of backtraking • Goal • Position n queens on an n * n chessboard so that no two queens threaten each other • “no queens threaten each other” means that no two queens may be in the same row, column, or diagonal. • In backtracking algorithm… • A sequence of objects • The n positions where the n queens are placed • A specified set • n2 possible positions on the chessboard • Criterion • “no queens threaten each other”

15. n-Queens Problem • 4-Queens Problem • # of places in the first row= 4 • # of places in the second row = 4 • # of places in the third row = 4 • # of places in the fourth row = 4 4  4  4  4 = 256 # of candidate solutions

16. n-Queens Problem • Tree for candidate solutions (i,j)  queens’ position (row: i, column:j) The first queen’s possible position The second queen’s possible position The third queen’s possible position The fourth queen’s possible position 후보답: Candidate Solution

17. n-Queens Problem • Basic backtracking strategy • A candidate solution is a path from the root to a leaf node • Use “depth-first traversal” • Should we traverse all possible paths? • 256 paths in 4-Queens Problem • No! we can traverse only promising node. • We meet a non-promising node, we go back to the node’s parent and proceed with the search on the next child • Promising node • We call a node non-promising if when visiting the node we determine that it cannot possibly lead to a solution • Otherwise, we call it promising

18. n-Queens Problem • Backtracking strategy • Do “depth-first traversal” of candidate solution tree • Checking whether each node is promising • If it is non-promising, we do prunning the candidate solution tree, and backtracking to the node’s parent • Rough sketch of backtracking void checknode(node v){ node u; if (promising(v))// it depends on the problem to be solved if (there is a solution at v) write the solution; else for (each child u of v) checknode(u); }

19. n-Queens Problem • Example – 4-Queens Problem • # of possible traversed nodes= 256 • # of traversed nodes in backtracking algorithm = 27

20. n-Queens Problem • Example – 4-Queens Problem

21. n-Queens Problem • A Backtracking Algorithm for n-Queens Problem • In main() function, we will execute “queens(0)” void checknode(node v){ node u; if (promising(v)) if (there is a solution at v) write the solution; else for (each child u of v) checknode(u); } void queens(index i){ index j; if(promising(i)) if(i == n) System.out.print(col[1] through col[n]); else for(j=1 ; j < n ; j++) { col[i+1] = j; // select the next location queens(i+1); } }

22. n-Queens Problem • Promising function in n-Queens Problem • It should check whether two queens are in the same column • col[i] = col[k] • It should also check whether two queens are in the same diagonal • |col[i] – col[k]| = |i – k| • The queen in row 6 is beingthreatened in its left diagonal by the queen in row 3, and its right diagonal by the queen in row 2 • col(6) – col(2) = 4 – 8 = -4 = 2 – 6 • Col(6) – col(3) = 4 – 1 = 3 = 6 – 3

23. n-Queens Problem • Promising function in n-Queens Problem boolean promising(index i){ index k = 1; boolean switch = true; while(k < i && switch) { // k <= n if(col[i] == col[k] || abs(col[i]-col[k]) == i-k) switch = false; k++; } return switch; }

24. [Programming Practice 6] • N-QueensAlgorithm • Visit • http://link.koreatech.ac.kr/courses/2013_1/AP-KOICA/AP-KOICA20131.html • Download “Nqueens.java” and run it • Analyze the source codes • Complete the source codes while insert right codes within the two functions • public static void queens(inti) • public static booleanisPromising(int n)

25. [Programming Practice 5] • N-Queens Algorithm • The output you should get