計概。黃智沂主講

1. CSBB LAB 新生訓練 基礎計算機科學 Speaker: 黃智沂 2 nd Year Student of Ph.D. Program

16. 上限 & 下限 演算法最短的執行時間 慢 快 這是某個演算法 因此，這問題的解法 最慢不過如此了 ( 上限 ) 所有的演算法 都不可能更快了 ( 下限 )

21. 空間複雜度 c c c c c O(1) 100K 100,000 10K 10,000 1K 1,000 100 100 10 10 O(n) 輸入量 n

31. Multi-tape Turing Machines: Informal Description … We add a finite number of tapes Control … a 1 a 2  Tape 1 head 1 … a 1 a 2  Tape 2 head 2

34. a b  a b  Tape 1 a b a Tape 2 State in M2: s Solve by 2-tape Turing Machine M2 : a b  a b  Tape 1 a b a Tape 2 State in M2: s’

35. Using States to “Remember” Information Equivalent configuration in a Turing Machine M : a b  a b a b # a b

39. Definition: A Non-Deterministic TM is a 7-tuple T = (Q, Σ , Γ ,  , q 0 , q accept , q reject ), where: Q is a finite set of states Γ is the tape alphabet, where   Γ and Σ  Γ q 0  Q is the start state Σ is the input alphabet, where   Σ  : Q  Γ -> Pow( Q  Γ  {L,R}) q accept  Q is the accept state q reject  Q is the reject state, and q reject  q accept

41. Non-Deterministic TM is a Parallel Universe

42. the set of languages decided by a O(t(n))-time non-deterministic Turing machine. Definition: NTIME(t(n)) is TIME(t(n))  NTIME(t(n))

44. Deterministic Polynomial Time P = TIME(n k )  k  N

45. Non-deterministic Polynomial Time NP = NTIME(n k )  k  N

48. A language is in NP if and only if there exist polynomial-length certificates for membership to the language. SAT is in NP because a satisfying assignment is a polynomial-length certificate that a formula is satisfiable.

50. Karp’s Complete Problems

51. The World by Karp P 2-SAT, Shortest-Path, Minimum-Cut, Arc-Cover ? NP-Hard NP-Complete SAT, Clique, Hamiltonian-Circuit, Chromatic Number . . . Equivalence of Regular Expression, Equivalence of ND Finite Automata, Context Sensitive Recognition Linear-Inequalities Graph-Isomorphism, Non-Primes NP ? ? In NPC In P

52. NP vs. P 這代表什麼？

53. How to use NPC? NP = The set of all the problems for which you can verify an alleged solution in polynomial time.

54. 最好的狀況當然是證明沒有好的方法存在。例如知名的 Sorting Problem 的 Lower Bound 是 O(n lg n).

55. 可是這通常比找出演算法更難

78. 最佳排比演算法

80. 最佳排比演算法

81. 最佳排比演算法 ( 續 )

83. 最佳區域排比的演算法

88. A matrix of similarity scores PAM 120

89. A maximum-scoring segment

90. BLOSUM62 versus PAM250 (For Protein)

92. BLAST Step 1: Build the hash table for Sequence A. (3-tuple example) For DNA sequences: Seq. A = AGATCGAT 12345678 AAA AAC .. AGA 1 .. ATC 3 .. CGA 5 .. GAT 2 6 .. TCG 4 .. TTT For protein sequences: Seq. A = ELVIS Add xyz to the hash table if Score(xyz, ELV) ≧ T; Add xyz to the hash table if Score(xyz, LVI) ≧ T; Add xyz to the hash table if Score(xyz, VIS) ≧ T;

93. BLAST Step2: Scan sequence B for hits.

94. BLAST Step2: Scan sequence B for hits. Step 3: Extend hits. hit Terminate if the score of the sxtension fades away. (That is, when we reach a segment pair whose score falls a certain distance below the best score found for shorter extensions.) BLAST 2.0 saves the time spent in extension, and considers gapped alignments.

95. Gapped BLAST (I) The two-hit method

96. Gapped BLAST (II) Confining the dynamic-programming

97. BLAT

104. Bio-Map protein-gene interactions protein-protein interactions PROTEOME GENOME Citrate Cycle METABOLISM Bio-chemical reactions

105. Boehring-Mennheim

106. An Introduction to Graph Theory Definitions and Examples Undirected graph Directed graph isolated vertex adjacent loop multiple edges simple graph : an undirected graph without loop or multiple edges degree of a vertex: number of edges connected (indegree, outdegree) G =( V , E )

107. x y path : no vertex can be repeated a-b-c-d-e trail : no edge can be repeat a-b-c-d-e-b-d walk : no restriction a-b-d-a-b-c closed if x = y closed trail: circuit (a-b-c-d-b-e-d-a, one draw without lifting pen) closed path: cycle (a-b-c-d-a) a b c d e length : number of edges in this (path,trail,walk)

108. a x b remove any cycle on the repeated vertices Def 11.4 Let G =( V , E ) be an undirected graph. We call G connected if there is a path between any two distinct vertices of G . a b c d e a b c d e disconnected with two components

110. Def. 11.6 multigraph of multiplicity 3 multigraphs

111. Subgraphs, Complements, and Graph Isomorphism a b c d e a b c d e b c d e a c d spanning subgraph V 1 = V induced subgraph include all edges of E in V 1

112. Subgraphs, Complements, and Graph Isomorphism Def. 11.11 complete graph: K n a b c d e K 5 Def. 11.12 complement of a graph G G a b c d e a b c d e

113. Subgraphs, Complements, and Graph Isomorphism Graph Isomorphism 1 2 3 4 a b c d w x y z

114. Subgraphs, Complements, and Graph Isomorphism Ex. 11.8 q r w z x y u t v a b c d e f g h i j a-q c-u e-r g-x i-z b-v d-y f-w h-t j-s, isomorphic Ex. 11.9 degree 2 vertices=2 degree 2 vertices=3 Can you think of an algorithm for testing isomorphism?

115. Module

116. Module

117. Network Motif

118. Graph Alignment: NetworkBLAST/PathBLAST

123. Weighted Bipartite Matching

124. Weighted Bipartite Matching Given a weighted bipartite graph, find a matching with maximum total weight. Not necessarily a maximum size matching. A B

144. Example: A C B D G E F Karger’s Min-Cut Algorithm

145. Example: A C B D G E F contract

146. Example: A C B D G E F contract A C B D E FG

147. Example: A C B D G E F contract A C B D E FG contract

148. Example: A C B D G E F contract A C B D E FG contract A C B E FGD

156. Illustrating Classification Task

157. Decision Tree

158. Example of a Decision Tree Refund MarSt TaxInc YES NO NO NO Yes No Married Single, Divorced < 80K > 80K Splitting Attributes Training Data Model: Decision Tree categorical categorical continuous class

159. Another Example of Decision Tree categorical categorical continuous class MarSt Refund TaxInc YES NO NO Yes No Married Single, Divorced < 80K > 80K There could be more than one tree that fits the same data! NO

160. Decision Tree Classification Task Decision Tree

161. Apply Model to Test Data Test Data Start from the root of tree. Refund MarSt TaxInc YES NO NO NO Yes No Married Single, Divorced < 80K > 80K

162. Apply Model to Test Data Test Data Refund MarSt TaxInc YES NO NO NO Yes No Married Single, Divorced < 80K > 80K

163. Apply Model to Test Data Refund MarSt TaxInc YES NO NO NO Yes No Married Single, Divorced < 80K > 80K Test Data

164. Apply Model to Test Data Refund MarSt TaxInc YES NO NO NO Yes No Married Single, Divorced < 80K > 80K Test Data

165. Apply Model to Test Data Refund MarSt TaxInc YES NO NO NO Yes No Married Single, Divorced < 80K > 80K Test Data

166. Apply Model to Test Data Refund MarSt TaxInc YES NO NO NO Yes No Married Single, Divorced < 80K > 80K Test Data Assign Cheat to “No”

167. Support Vector Machine

168. S upport Vector Machines Which hyperplane? y=1 y=-1

169. S upport Vector Machines Margin Margin = |d + |+|d - | y=1 y=-1 d + d - d -

170. S upport Vector Machines Maximum Margin d + d + d - d - y=1 y=-1 Support vectors

171. S upport Vector Machines Margin d b i1 b i2 y=1 y=-1 d x 1 x 2 x 1 -x 2 Page:261 5.32 5.33 5.34

173. Artificial Neural Network

178. Sigmoid Unit x 1 x 2 x n w 1 w 2 w n  x 0 = 1 w 0

184. S = M A L A Y A L A M $ 1 2 3 4 5 6 7 8 9 10 $ YALAM$ M $ ALAYALAM$ $M YALAM$ $M YALAM$ $M YALAM$ A AL LA 6 2 8 4 7 3 1 9 5 10 Suffix Trees Paths from root to leaves represent all suffixes of S

185. M A L A Y A L A M $ 1 2 3 4 5 6 7 8 9 10 $ YALAM$ M $ ALAYALAM$ $M YALAM$ $M YALAM$ $M YALAM$ A AL LA 6 2 8 4 7 3 1 9 5 10 Suffix Tree

188. Finding a Pattern in a String Find “ALA” $ YALAM$ M $ ALAYALAM$ M$ YALAM$ M$ YALAM$ M$ YALAM$ A AL LA 6 2 8 4 7 3 1 9 5 10 Two matches - at 6 and 2

189. (10, 10) (5, 10) (1, 1) (10, 10) (2, 10) (3, 4) (5, 10) (9, 10) (2, 2) (5, 10) (9, 10) (3, 4) (9, 10) (5, 10) 6 2 8 4 7 3 1 9 5 10 Edge Encoding S = M A L A Y A L A M $ 1 2 3 4 5 6 7 8 9 10

190. N äive Suffix Tree Construction Before starting: Why exactly do we need this $ , which is not part of the alphabet? $ 10 M$ 9 AM$ 8 LAM$ 7 ALAM$ 6 YALAM$ 5 AYALAM$ 4 LAYALAM$ 3 ALAYALAM$ 2 MALAYALAM$ 1

191. N äive Suffix Tree Construction $MALAYALAM LAYALAM$ 1 2 LAYALAM$ 3 A 2 3 4 4 YALAM$ etc. $ 10 M$ 9 AM$ 8 LAM$ 7 ALAM$ 6 YALAM$ 5 AYALAM$ 4 LAYALAM$ 3 ALAYALAM$ 2 MALAYALAM$ 1

195. Suffix Array – Reducing Space M A L A Y A L A M $ 1 2 3 4 5 6 7 8 9 10 Suffix Array : Lexicographic ordering of suffixes Derive Longest Common Prefix array Suffix 6 and 2 share “ALA” Suffix 2,8 share just “A”. lcp achieved for successive pairs . $ 10 YALAM$ 5 M$ 9 MALAYALAM$ 1 LAYALAM$ 3 LAM$ 7 AYALAM$ 4 AM$ 8 ALAYALAM$ 2 ALAM$ 6 10 5 9 1 3 7 4 8 2 6 - 0 0 1 0 2 0 1 1 3

196. Example Text Position Suffix Array 3 1 1 0 2 0 1 0 0 lcp Array M M A L A Y A L A $ 1 2 3 4 5 6 7 8 9 10 3 7 4 10 5 8 9 1 2 6 $ 10 YALAM$ 5 M$ 9 MALAYALAM$ 1 LAYALAM$ 3 LAM$ 7 AYALAM$ 4 AM$ 8 ALAYALAM$ 2 ALAM$ 6

202. 0 0 0 0 0 0 0 0 0 0 0 0 Initial with all 0 1 1 1 1 1 x 1 x 2 Each element of S is hashed k times Each hash location set to 1 1 1 1 1 1 y To check if y is in S, check the k hash location. If a 0 appears , y is not in S 1 1 1 1 1 y If only 1s appear, conclude that y is in S This may yield false positive

224. Range trees The query time : Querying a 1D-tree requires O(log n+k) time. How many 1D trees (associated structures) do we need to query? At most 2  height of T = 2 log n Each 1D query requires O(log n+k’) time.  Query time = O(log 2 n + k) Answer to query = Union of answers to subqueries: k = ∑k’ . Query: [x,x’] x x’