Decrease-and-Conquer. Dr. Yingwu Zhu

Transcription

1 Decrease-and-Conquer Dr. Yingwu Zhu

2 Decrease-and-Conquer 1. Reduce problem instance to smaller instance of the same problem 2. Solve smaller instance 3. Extend solution of smaller instance to obtain solution to original instance

3 Fake-Coin Puzzle There are n identically looking coins one of which is fake. There is a balance scale but there are no weights; the scale can tell whether two sets of coins weigh the same and, if not, which of the two sets is heavier (but not by how much). Design an efficient algorithm for detecting the fake coin. Assume that the fake coin is known to be lighter than the genuine ones. How to solve this problem by applying decrease-and-conquer?

4 Fake-Coin Puzzle There are n identically looking coins one of which is fake. There is a balance scale but there are no weights; the scale can tell whether two sets of coins weigh the same and, if not, which of the two sets is heavier (but not by how much). Design an efficient algorithm for detecting the fake coin. Assume that the fake coin is known to be lighter than the genuine ones. Decrease by factor 2 algorithm Decrease by factor 3 algorithm

5 Binary search Another Example

6 Problems to Discuss Insertion sorting Graph traversal Generating Permutations & Subsets Selection problem BST

7 Insertion Sort To sort array A[0..n-1], sort A[0..n-2] recursively and then insert A[n-1] in its proper place among the sorted A[0..n-2] Example: Sort 6, 4, 1, 8,

8 Pseudocode of Insertion Sort

9 A Different Look at Insertion Sort Apply decrease-and-conquer!

10 Graph Traversal Many problems require processing all graph vertices (and edges) in systematic fashion Data structures to represent a graph Adjacency matrix Adjacency list Graph traversal algorithms: Depth-first search (DFS) Breadth-first search (BFS)

11 Depth-First Search (DFS) Visits graph s vertices by always moving away from last visited vertex to unvisited one, backtracks if no adjacent unvisited vertex is available. Uses a stack a vertex is pushed onto the stack when it s reached for the first time a vertex is popped off the stack when it becomes a dead end, i.e., when there is no adjacent unvisited vertex Redraws graph in tree-like fashion (with tree edges and back edges for undirected graph)

12 Edge Types 1. Tree edge: parent-child 2. Forward edge: node to its descendant 3. Back edge: node to its ancestor 4. Cross edge: others

13 Pseudocode of DFS

14 Recursive DFS How? T(n) =?

15 Example: DFS traversal of undirected graph a b c d e f g h DFS traversal stack: DFS tree:

16 Notes on DFS DFS can be implemented with graphs represented as: adjacency matrices: Θ(V 2 ) adjacency lists: Θ( V + E ) Time efficiency is proportional to the size of data structure. Yields two distinct ordering of vertices: order in which vertices are first encountered (pushed onto stack) order in which vertices become dead-ends (popped off stack) Applications: checking connectivity, finding connected components checking acyclicity (back edge in DFS trees)

17 Breadth-first search (BFS) Visits graph vertices by moving across to all the neighbors of last visited vertex Instead of a stack, BFS uses a queue Similar to level-by-level tree traversal Redraws graph in tree-like fashion (with tree edges and cross edges for undirected graph)

18 Pseudocode of BFS

19 Example of BFS traversal of undirected graph a b c d e f g h BFS traversal queue: BFS tree:

20 Notes on BFS BFS has same efficiency as DFS and can be implemented with graphs represented as: adjacency matrices: Θ(V 2 ) adjacency lists: Θ( V + E ) Yields single ordering of vertices (order added/deleted from queue is the same) Applications: same as DFS, but can also find paths from a vertex to all other vertices with the smallest number of edges

21 Dags and Topological Sorting A dag: a directed acyclic graph, i.e. a directed graph with no (directed) cycles a b a b a dag not a dag c d c d Arise in modeling many problems that involve prerequisite constraints (construction projects, document version control) Vertices of a dag can be linearly ordered so that for every edge its starting vertex is listed before its ending vertex (topological sorting). Being a dag is also a necessary condition for topological sorting be possible.

22 Topological Sorting Example Order the following items in a food chain tiger human fish sheep shrimp plankton wheat

23 Topological Sorting How to apply decrease-and-conquer here? Discussion.

24 Source Removal Algorithm Source removal algorithm (decrease by one and - conquer) Repeatedly identify and remove a source (a vertex with no incoming edges) and all the edges incident to it until either no vertex is left (problem is solved) or there is no source among remaining vertices (not a dag) Example: a b c d e f g h Efficiency: same as efficiency of the DFS-based algorithm

25 DFS-based Algorithm DFS-based algorithm for topological sorting Perform DFS traversal, noting the order vertices are popped off the traversal stack Reverse order solves topological sorting problem Back edges encountered? NOT a dag! Example: a b c d e f g h Efficiency:

26 Permutations and Subsets How to apply decrease-and-conquer to: Generate permutations Generate subsets

27 Generating Permutations Problem: permutation of n elements: 1,,n n! permutations

28 Generating Permutations Problem: permutation of n elements: 1,,n n! permutations Solution 1: decrease by one and conquer 1,,n-1 permutations Insert n into n positions for each of the permutation Minimal-change: insert n into a permutation from R-2-L and then switch direction every time a new permutation of [1,..,n-1] needs to be processed. Example: 123

29 Generating Permutations Solution 2: Johnson-Trotter Algorithm, p179 Each element is associate an arrow The element k is mobile if its arrow points to a smaller number adjacent to it: Alg: initialize 1 2 n while the last permutation has a mobile element do find its largest mobile element k swap k and the adjacent integer k s arrow points to reverse the direction of all the elements that are larger than k add the new permutation to the list

30 Generating Subsets Problem: generating all 2^n subsets of set A={a1,, an} Discussion: how?

31 Generating Subsets Problem: generating all 2^n subsets of set A={a1,, an} Solution 1: decrease by one and conquer Generating all 2^(n-1) subsets for {a1,, an-1} Add an to each subset

32 Generating Subsets Problem: generating all 2^n subsets of set A={a1,, an} Solution 1: decrease by one and conquer Generating all 2^(n-1) subsets for {a1,, an-1} Add an to each subset Solution 2: Using a bit string of length n Example: 000, 001,., 111

33 Selection Problem Find the k-th smallest element in a list of n numbers k = 1 or k = n median: k = n/2 Example: 4, 1, 10, 9, 7, 12, 8, 2, 15 median =? More general: find k-th smallest element? Brute force approach? Discussion!

34 Algorithms for the Selection Problem Problem: select a k-th smallest element in an array More efficient solutions?

35 Algorithms for the Selection Problem Brute force: The sorting-based algorithm: Sort and return the k-th element Efficiency (if sorted by mergesort): Θ(nlog n) Better solution: How to apply decrease-and-conquer?

36 Algorithms for the Selection Problem all are A[s] all are A[s] s Better solution: A faster algorithm is based on using the quicksort-like partition of the list. Let s be a split position obtained by a partition: Assuming that the list is indexed from 1 to n: If s = k, the problem is solved; if s > k, look for the k-th smallest elem. in the left part; if s < k, look for the (k-s)-th smallest elem. in the right part. Note: The algorithm can simply continue until s = k.

37 Tracing the Median / Selection Algorithm Example: Here: n = 9, k = 9/2 = 5

38 Binary Search Tree Algorithms BST Binary tree k Order property: For each node, all the nodes on its left subtree have data less than the node s data; all the nodes on its right subtree larger than the node s data Recursive data structure Empty tree Or, a root node, left subtree is a BST, and right subtree is a BST <k >k

39 Binary Search Tree Algorithms Several algorithms on BST requires recursive processing of just one of its subtrees, e.g., Searching Insertion of a new key Finding the smallest (or the largest) key k <k >k

40 Searching in Binary Search Tree Algorithm BTS(x, v) //Searches for node with key equal to v in BST rooted at node x if x = NIL return -1 else if v = K(x) return x else if v < K(x) return BTS(left(x), v) else return BTS(right(x), v) Efficiency worst case: T(n) = n average case: T(n) 2ln n 1.39log 2 n